Frontiers in Quantum Computing 2020039648, 2020039649, 9781536185157, 9781536186574

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PHYSICS RESEARCH AND TECHNOLOGY

FRONTIERS IN QUANTUM COMPUTING

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PHYSICS RESEARCH AND TECHNOLOGY

FRONTIERS IN QUANTUM COMPUTING

LUIGI MAXMILIAN CALIGIURI EDITOR

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Library of Congress Cataloging-in-Publication Data Names: Caligiuri, Luigi Maxmilian, 1972- editor. Title: Frontiers in quantum computing / Luigi Maxmilian Caligiuri. Description: New York : Nova Science Publishers, [2020] | Series: Physics research and technology | Includes bibliographical references and index. | Identifiers: LCCN 2020039648 (print) | LCCN 2020039649 (ebook) | ISBN 9781536185157 (hardcover) | ISBN 9781536186574 (adobe pdf) Subjects: LCSH: Quantum computing. Classification: LCC QA76.889 .F76 2020 (print) | LCC QA76.889 (ebook) | DDC 006.3/843--dc23 LC record available at https://lccn.loc.gov/2020039648 LC ebook record available at https://lccn.loc.gov/2020039649

Published by Nova Science Publishers, Inc. † New York

To Fernanda Sophie, Soraya and Domenica, wih love.

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

ix Quantum (Hyper)Computation by Means of Water Coherent Domains – Part I: The Phsyical Level Luigi Maxmilian Caligiuri The Quantum Phase Operator and Its Role in Quantum Computing Luigi Maxmilian Caligiuri Quantum (Hyper)Computation by Means of Water Coherent Domains – Part II: The Computational Level Luigi Maxmilian Caligiuri Computing Hyperincursive Discrete Relativistic Quantum Majorana and Dirac Equations and Quantum Computation Daniel M. Dubois Quantum Computing and the Quantum Mind: A New Approach to Quantum Gravity Paola Zizzi and Massimo Pregnolato

1

39

57

103

153

viii Chapter 6

Contents Can Instantaneous Quantum Algorithms Be Developed? Richard L. Amoroso

Chapter 7

Sentient Androids Richard L. Amoroso

Chapter 8

Imminent Advent of Universal Quantum Computing (UQC) Richard L. Amoroso

Chapter 9

Brain - Quantum Hypercomputing System Takaaki Musha

Chapter 10

Parametric Resonance, Particle Stochastic Interactions with a Periodic Medium, and Quantum Simulations Mario J. Pinheiro

177 223

269 331

389

About the Editor

401

Index

403

PREFACE Quantum Computing is surely one of the most interesting and promising field belonging to the so-called Quantum Information Science, born by the synergy between Quantum Mechanics and Information Theory. Quantum Mechanics has radically changed our vision and understanding of the physical reality and has had also an enormous technological and societal impact. The developing of Information Theory which includes the foundations of both computer science and communications theory, on the other hand, has begun the information “revolution” which has determined, among other things, the deep impact the computers have on our everyday life. We can also say that it was just quantum mechanics to make possible such information revolution by allowing the invention of the transistor which is based on purely quantum concepts like, for example, the FermiDirac statistics, the idea of “hole” and so on. It is interesting to note that success of the information revolution is based on two interconnect facets: the increase of computational power of machines and the progressive miniaturization of electronic circuits as described by the so-called Moore’s Law, according to which the number of transistors that could form a single integrated circuit would approximatively double every 18-24 months. The Moore’s Law (which, we recall, is not a physical law) so would set an ultimate limit to the dimension of the smallest integrated circuit even

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Luigi Maxmilian Caligiuri

feasible, just coinciding with that at which quantum effects will become unavoidable dominant. Nevertheless, instead of representing a weakness point, this could allow the development of a totally new idea of computation, that considers the principles and consequences of quantum mechanics to “manipulate” and transmit information. Among these we cite, to limit ourselves to just some of them, the principle of superposition of states, the entanglement, the quantum teleporting, etc. The implementation of quantum computing also implies a radically new mode of representing the fundamental piece of information that we name qubit (quantum bit), a continuous superimposition of the zero- and one-logic state, and then a shift from the Boolean logic to a quantum logic able to process the information and perform logic operations on such quantum bits by using the law of quantum mechanics. Such possibility to manipulate information at a quantum level is also founded on another and relatively recent conceptual acquisition, namely that the information itself is rooted into the physical world so that, ultimately, it possesses a quantum nature. At the physical layer, quantum computation needs a suitable system able to actually realize the qubits used for storing and processing information. Being the qubit a linear superposition of the two logical states zero and one, the most suitable physical systems able to represent quantum information are the two-levels systems in which the two logical states can be respectively associated to a couple of well-defined energetic states of the system (as, for example, a spin ½ particle, two levels of an atom, the vertical and horizontal polarizations of a photons, etc.). A quantum computer is then a system in which both the physical and computational layers are suitably implemented, namely a system of many qubits, satisfying a defined set of conditions (Di Vincenzo’s criteria), whose time-evolution can be controlled to realize the quantum transformations (unitary operations) required to perform the quantum algorithm on multiple qubits states.

Preface

xi

As in Boolean logic, even in quantum logic it is possible to find a set of universal transformations, namely a set of quantum gates, able to realize, in principle, any type of quantum computation. The possibility to codify, in principle, an infinite amount of information (through the continuous degrees of freedoms caracterizing it) in a single qubit and the consideration of multiple qubits systems have suggested, since its first proposals, the possibility that a quantum computer could be intrinsically very more powerful than its classical counterpart. It is now currently believed that in general this is the case, although, in specific contexts, this possibility actually depends both on the particularly algorithm to be implemented and on the physical realization of the computing quantum system. Another very interesting feature characterizing the quantum computing it its reversibility (reflecting the unitarity of the mathematical operators representing quantum transformations and the corresponding logical gates) that implies, from an energetic standpoint, its non-dissipative nature. Despite the general theoretical basis of quantum computing are sufficiently easy to understand, the actual realization of a useful and really scalable quantum computer has posed great difficulties so far. One of the most important and well-known question is related to the phenomenon of quantum decoherence, according to which the interaction between a quantum system and its surrounding environment could destroy the integrity of the quantum state, unless the dynamical evolution of the system realizing the quantum computing process happens in time less than the decoherence time. This also poses further questions, as per se important, related to the computational speed and scalability that can be achieved trough a real quantum computer. In chapter 1, the results of theory of QED coherence in condensed matter applied to liquid water will be discussed with particular reference to the formation of a spectrum of excited energy levels of such coherent domain in the form of “cold” vortex of quasi-free electrons. A consequence of the existence of such excited levels, a distinctive feature of liquid water, is the interaction between water coherent domains through the exchange of virtual photons by tunnel effect. This interaction determines the transition between

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two excited energy levels of the interacting coupled coherent domains that is described by a “Rabi”-like equation. This result opens the door, as it will be discussed in chapter 3, to the possibility to use water coherent domains to perform quantum (hyper)computation. Chapter 2 introduces the very important definition of quantum phase operator, discussing some of its most important properties, and showing how to use it to describe the evolution of the quantum macroscopic coherent domains of water. Chapter 3 shows how to use the quantum tunneling interaction between water coherent domains described in chapter 1, to perform, in principle, any kind of quantum computation. More specifically, it has been proposed that enclosing coherent domains of water inside waveguides made of metamaterials could allow for the implementation of a superfast network of interacting coherent domains that could be used as a novel architecture for a kind of quantum supercomputer based on the coherent dynamics of liquid water that would avoid the issues of quantum decoherence and scalability. Contrary to the currently (but not entirely correct) accepted picture of quantum mechanics, according to which it would describe physical systems at a microscopic scale only, and to the idea that the increase in computational power is an effect of miniaturization, these results instead show the existence of macroscopic quantum states describing the coherent domains of water and how they could be used to realize any type of quantum computation at a computational speed very close to the ultimate computational limit imposed so far by quantum mechanics. Even more interestingly, the possibility to break this barrier by suitably parallelizing the interaction between water coherent domains is also discussed in chapter 3, arguing this would represent the basis to implement the so-called quantum “hypercomputation”, namely the fastest type of computational even possible. Chapter 4 discusses the computing of “hyperincursive” discrete relativistic quantum Majorana and Dirac equations, and their importance for quantum computation. In chapter 5 the features of quantum computers which are related to quantum gravity (via entanglement) and to the quantum side of the mind (via

Preface

xiii

quantum computational logic and quantum meta-language) are presented. It is argued that Quantum computing exploits the inner quantumcomputational side of quantum gravity. The subject of chapter 6 is the study of the possibility to develop “Instantaneous Quantum Computing Algorithms (IQCA)”, so going beyond the current limits of quantum theory imposed by the Standard Model (SM) of particle physics as currently described by the Copenhagen Interpretation due to the limitations imposed by the Heisenberg uncertainty relation and Pauli exclusion principle. Chapter 7 focuses on the possibility to use the results of the so-called “Unified Field Mechanics” (UFM), together with a class of quantum computer modeled with physical parameters of mind-body interaction, to the construction of sentient androids. In chapter 8, the issue of decoherence in the realization of a Universal Quantum Computer is faced by considering a version of Einstein’s long sought Unified Field Theory based on modification to string/M-theory. Chapter 9 describes, starting from standpoint of a model of the brain based on superluminal tunneling photons and considering brain’s microtubules, a brain-like computer that would be more powerful than Turing machines and so would allow non-Turing computation, that may hold the key to the origin of human consciousness itself. According to this model biological brains would achieve large quantum bit computation characterized by higher performances than silicon processors. Finally, in chapter 10, a non-Markovian stochastic model is discussed to show the emergence of structures in a medium, where the nonlinear coupling between the medium modes of oscillation and the characteristic frequencies of the medium could be related to the operational mode for quantum information processing. The topics considered in this book show that the synthesis between quantum mechanics and information science, leading to quantum computation offers completely new opportunities and perspectives of development in both fundamental science and advanced technological applications.

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This volume doesn’t mean to represent a complete guide to quantum computing but to discuss some of the most interesting and fascinating developments of such field in different frontier areas. Many of the questions here discussed are still opened and a vivid research activity is currently in progress worldwide to give an answer to them while it is more and more clear that quantum information and quantum computation could offer to us revolutionary opportunities and perspectives of development in both fundamental science and advanced technological applications.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 1

QUANTUM (HYPER)COMPUTATION BY MEANS OF WATER COHERENT DOMAINS – PART I: THE PHSYICAL LEVEL Luigi Maxmilian Caligiuri* Foundation of Physics Research Center (FoPRC), Cosenza, Italy

ABSTRACT According to the theory of QED coherence in condensed matter, liquid water must be considered as a “condensed vapor” in which an initially independent ensemble of molecules is spontaneously driven towards a coherent dynamical state, in which all of them oscillate in tune among each other and with an auto-generated and self-trapped electromagnetic field within macroscopic spatial domains called “coherent domains.” A peculiar feature of such QED coherent dynamics is to produce an evanescent e.m. field spreading across the boundaries of coherent domains and the possibility to have a wide spectrum of excited energy levels of the coherent domain in the form of “cold” vortex of quasi-free electrons. In this chapter we’ll show how these features allows coherent domains to interact each other through the exchange of virtual photons (belonging to their respective *

Corresponding Author’s Email: [email protected].

2

Luigi Maxmilian Caligiuri evanescent e.m. field modes) by tunnel effect. We then study some main features of such so far unrecognized type of “evanescent-tunnelingcoupling” interaction occurring in liquid water finally getting a “Rabi” equation describing the transition between two excited coherent states. This result opens the door to the fascinating perspective of using water coherent domains to perform quantum computation.

Keywords: QED coherence in matter, water coherent domains, supercoherence, evanescent fields, quantum tunneling, tunnelingcoupling interaction

AN OVERVIEW OF QED COHERENCE IN MATTER AND THE DYNAMICS OF COHERENT LIQUID WATER In QFT the quantization of the e.m. field is realized by replacing the complex amplitudes and of the Fourier decomposition of radiation fields with the photon annihilation and creation operators and respectively. For many decades, condensed matter has been considered as a collection of microscopic components, atoms or molecules, keep together by a set of intermolecular short-range forces when temperature is below a defined value and kinetic energy of corpuscles is lower than potential energy between them. In this framework, representing the commonly accepted one, the acting forces, as the Van Der Waals or Lennard-Jones forces, are supposed to have a range of action no longer than two or three molecular diameters so it is an unsolved dynamical issue the emergence of long - range order in macroscopic condensed matter systems, as for example, in crystals and living organisms. Yet in 1916 physicist W. Nernst proposed a very different point of view according to which, the quantum fluctuations of microscopic components of matter, predicted by Heisenberg uncertainty principle, could synchronize establishing a phased common oscillation over extended space-time regions that completely characterizes the condensed system as a whole. This

Quantum (Hyper)Computation by Means …

3

fundamental principle of macroscopic coherence has been reinterpreted within the more correct framework of QFT by Umezawa [1]. The possibility to admit, for a given system, the contemporary existence of matter fluctuations on one side and a non-fluctuating and well-defining evolution on the other is based on the required invariance of the system’s Lagrangian under local phase transformation of matter field, due to the action of the gauge field

that, at atomic and molecular scale, is just

the electromagnetic field. This gauge invariance allows the quantum fluctuations of matter and e.m. fields to tune together and, if some conditions are fulfilled, to giving rise to a common phased oscillation of the matter system spanning a macroscopic spatial region covered by e.m. field. This process is, in some respect, conceptually similar to that determining the Lamb-shift of the energy levels of the hydrogen atom due to the coupling between the quantum fluctuations of vacuum (QV), the so-called Zero-Point (ZP) fluctuations, and the electron orbital density current through the interaction term . The above dynamics arises from a fundamental principle, namely that any quantum system (either composed by particles or fields) is always subject to fluctuations as well as the vacuum. The occurrence of the Lamb shift actually shows a fluctuating field is always present as the result of the quantum fluctuations of vacuum oscillators and that we should always consider that atoms and molecules are coupled to a quantized e.m. field. If we consider a matter system composed of electrical charged particles (electrons and nuclei) and characterized by a discrete energy spectrum of values

, a QV (or thermal bath) fluctuation, able to excite one of its

levels from the fundamental state (with energy ), must have a wavelength

) to the state where

(with . In the

case of water molecules the typical energy difference between two levels can be estimated as

corresponding to a wavelength

that is much greater than the typical size of water molecule . This means that an e.m. vacuum fluctuation, able to excite a

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Luigi Maxmilian Caligiuri

molecule, is composed by a photon “extending” over a length more than one thousand greater than the dimension a single molecule itself and includes, at the usual density, in its volume If

, more than 20.000 water molecules.

is the Lamb -shit type probability that a photon of the vacuum

fluctuations excites the level (quantified by the “oscillator strength” for the transition ), for an atom or a molecule with electrons, we have (1)

and the probability that a photon, due to its subsequent decay, could in turn excite one of the

molecules contained in

is given by

(2)

where If

is the water density. the emitted photon will probably come back to QV or fly

away, but when

, it will surely excite another molecule in the

volume and the process will repeat; other photons from quantum fluctuations then will undergo the same phenomenon and, if

, after a

suitable time interval, a considerable e.m. field will develop in the region of extension by attracting more and more molecules able to resonate with the growing e.m. As we’ll see in the following, this attractive process will go on, increasing the system density, until it reaches a defined value when the autogenerated e.m. field exponentially increases, so spontaneously driving the system towards a more stable state (because characterized by a lower energy and so strongly favored), where the further density increase is counterbalanced by the repulsive intermolecular forces and in which the above quantum fluctuations become strongly amplified and phase correlated.

Quantum (Hyper)Computation by Means …

5

These coherent oscillations are then confined within defined spatial regions, called “Coherence Domains” (CDs), associated to the wavelength of the tuning electromagnetic field, whose extension is of the order of

(3) where

(from now on in this chapter, unless otherwise specified, we’ll

adopt the natural units system ) is the energy associated to the transition between a given couple of levels of energy of the quantized matter system driving the coherent evolution of the whole system. Without entering into a detailed mathematical analysis of the coherent dynamics of a general two-levels system, already detailed discussed in other works [2, 3], we’ll limit ourselves to its application to the water molecules. Furthermore, since we are interested in the coherent evolution of a system oscillating between two discrete energy levels, we’ll consider a twocomponent matter field, respectively associated with the space-time distribution of the molecules among the considered energy levels. In the case of multilevel atoms or molecules the contribution to the described dynamics of the other energetic levels, different than the two considered, takes place through second order radiative corrections that are suppressed by the dominant dynamics that drives the system towards the coherent state. The mathematical description of dynamics evolution of matter + e.m. field interacting system is achieved by considering its Lagrangian function and the corresponding Eulero-Lagrange equations. Such equations contain, as dynamical variables, the matter field associated to the space-time distribution of matter components (water molecules) corresponding to the different energetic configurations and the e.m. field, described in terms of its potential vector . The interesting physical evolution over long time is that considering only the e.m. modes that couple with the matter oscillations, namely the modes defined by the condition

6

Luigi Maxmilian Caligiuri (4) Among all the possible quantum transitions

of the matter fields,

it will be considered the transition between the ground state particular excited state

such as

and

and a

.

Furthermore, by limiting our description to a limited region whose size is the of order of the wavelength of the coupling mode (corresponding to the CD “extension”),

, we can neglect the spatial dependence of both

the matter and e.m. fields inside the CD (whose volume is

). The

dynamic equations, describing the time-evolution of the electromagnetic field + matter interacting ensemble, can be written as

(5)

where

is a two-components matter field describing the space-time

distribution of water molecules, and

the space - averaged e.m, vector potential

, with

(6)

the term

is the electron plasma frequency (

is the mass of the electron) (7)

Quantum (Hyper)Computation by Means … and

7

represents the so-called photon “mass” term that can be written as the

sum of two terms: (8) in which

(9)

includes the contributions of the transitions from ground state to excited discrete states and

(10)

where

is the ionization threshold and

the oscillator strength for

unit of frequency, is the contribution of the transitions from ground state to excited states of the continuum spectrum (namely those for which the excitation energy is larger than ionization threshold) and strength for the electronic transition from the ground state state

is the oscillator to the excited

.

The short-time behavior of system can be studied by differentiating the third of (5) and substituting it into the second one, so obtaining the differential equation

(11)

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Luigi Maxmilian Caligiuri

to be solved assuming the “initial” conditions that define the “perturbative ground state” (the PGS) in which all the fields just perform their zero-point oscillations and the electromagnetic field is too low to ensure the phased collective behavior of the matter + e.m. system. The transition towards the CGS would then require the exponential increase of

able to overcome its nearly zero initial value and create the

coherent tuning field. It can be shown [2] that this occurs, for a given

,

when (12) with

(13)

Since the value of

depends on the density

, the value of

will then determine the corresponding critical value of density such as the (12) holds. As long as

the system

remains in the “vapor” phase with all the molecules in PGS. But when (and

as well), the system will undergo a “phase transition”

from the PGS, in which the electromagnetic and matter fields perform ZP very weak uncoupled fluctuations only, towards the CGS, the “liquid” state, in which almost every ZP fluctuation couple with the oscillations of atoms or molecules in the ensemble and then a strong electromagnetic field arises from QV tuning all the matter constituents to oscillate in phase with it and among themselves inside any CD. This results in the emergence of a macroscopic quantum state in which atoms and molecules lose their

Quantum (Hyper)Computation by Means …

9

individuality and become part of a whole electromagnetic field + matter entangled system. This condensation process happens within a time interval less than the typical period of oscillations of the coherent state, namely

(14)

and goes on until the short-range electrostatic repulsion acts against the indefinitely increase of system density so determining the equilibrium state. The stationary solution, corresponding to CGS, can be found by writing the coherent dynamical equations (10) in the phase representation by posing

(15)

with

. In the coherent equations (24) the field

molecules in the ground state excited state

while the field

whereas the field

describes the

the molecules in the

represent the photon absorption and the

field the photon emission. The second term in (11) has a special meaning (as we’ll see in the following) since it describes the “mass” got by the e. m. field when it is coupled with the water molecules and continuously exchanges photons with them. The properties of water CGS, deriving from the equations (5), have already been studied in a series of previous papers [2, 4-9]. We note the parameters

,

and

, depending on the physical quantities defining the

system, assume renormalized values due to the quantum transition

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Luigi Maxmilian Caligiuri

(condensation) of each molecule from the non-coherent state one

to coherent

. The fields amplitudes defining such coherent

state are the complete solution of the system (15) given by [2, 5, 9]

(16)

where

and

respectively indicate the re-normalized common values

of e.m. and matter field oscillation

and of the mass term

the energy of the matter system in the coherent state

while

is

. The solution

(16) also implies the condition namely a “phase-locking” constraint between the fields. It can be shown that the following condition holds [2, 5, 9]: (17) that implies, due to the smallness of common frequency of oscillation

for water [2, 5, 9], the of electromagnetic field and

matter inside a CD is lower than the value characterizing the perturbative state of the incoherent phase

, namely

Quantum (Hyper)Computation by Means …

11 (18)

where

is the phase factor ruling the behavior of the vector potential . The frequency re-normalization

has a very

important consequence on the photon mass-term, whose value, as we’ll see in the following, will be

, showing that after the runaway water

dynamics is completely changed with respect the non-coherent state. The presence of a negative term in

also allows for

to assume

negative values, meaning the coherent state to have an energy lower than the non-coherent one, so resulting more stable and representing the true vacuum state of the system. The quantum phase transition from gas-like non - coherent state to coherent liquid state takes place spontaneously when the water density exceeds the smallest critical density corresponding to the various possible electronic transitions and, once the systems has “selected” the corresponding , all the other transitions are highly suppressed and don’t have

energy

effect on the dynamical evolution of the matter + e. m. field system. The level driving the transition towards the coherent state is just that characterized by the shortest time

required by the coherent oscillation

to build up and consolidate which, in the case of water, is of the order of . According to [2, 5, 9], the smallest value of critical density in water is associated to the electronic transition between the ground state and the 5d excited stated of the molecule corresponding to the level at about . The selection of this level as the excited partner of water coherent dynamics could also explain why the intermolecular distance in liquid water is larger than twice the molecular radius (respectively equal to and ). Finally, the thermodynamics properties of liquid water, such as density and specific heat, obtained in this case also show a good agreement with experimental data.

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Luigi Maxmilian Caligiuri

The numerical solution of system (16), so far obtained for the coherent state, are shown in Table 1 [5, 9]. Table 1. Calculated value of the parameters characterizing the coherent state of liquid water corresponding to oscillator strengths respectively used in: (a) [9] and (b) [2, 5]. Energy units in eV, density units in

(a) (b)

The negativity of

states, as we have anticipated, the coherent state

is more stable than the non - coherent gas-like state and represents the true ground state of the system (being energetically favored) for this reason named “coherent ground state” (CGS) towards which the non - coherent state spontaneously runs away when

.

The energy gap per molecule, characterizing the coherent state with respect the non-coherent one, that ensures the stability of the former is then equal to

(19)

The frequency renormalization has an even more impressive consequence since it determines the spontaneous creation of a resonating cavity whose e.m. mode coincides with the common coherent frequency of oscillation of matter and e.m. field inside it. As we’ll discuss in more detail in the following, according to the principles of Quantum Optics, this condition implies the self-generated coherent e.m. field is trapped within a well-defined macroscopic region coinciding with the CD, being unable to be irradiated outwards, and whose dynamics is fundamental for our discussion.

Quantum (Hyper)Computation by Means …

13

QUANTUM DYNAMICS OF COHERENT DOMAINS IN LIQUID WATER Spatial Behavior of the Coherent Electromagnetic Field The spatial behavior of the coherent field amplitude

is one of the

most interesting and meaningful features of the coherent state. Assuming spherical symmetry, we can define the radius

as the radial dimension of

the smallest CD in equilibrium with its own coherent e.m. which depends on the quantum “condensation” process. For a water CD we have [2, 3], due to the condition

,

(20)

so that, for

,

.

The radial profile of the coherent e. m. field then admits, for

, an

evanescent-like solution given by

(21)

where

is the vector potential at the CD’s center (

) and whose

spatial behavior is shown in Figure 1. The origin of such evanescent e.m. field is the notable energy shift

due to the coherent

dynamics. For two closely packed CDs, the overall spatial profile of by the function

, with

(

is given ) and

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Luigi Maxmilian Caligiuri

(22)

The energy gap per molecule behavior as

given by (19) has the same spatial

.

Figure 1. Spatial profile of coherent e.m. field

of a CD.

This can be easily view by noting that, within a CD, then, since

, we have

. Furthermore

, and is practically

constant since it represents a sort of “average photon mass” so we can write (where

is a constant positive value within a CD)

(23)

where we have set

and

The energy gap has then, as spatial profile, the function in Figure 2.

. shown

Quantum (Hyper)Computation by Means …

15

Figure 2. Spatial profile of energy gap in a coherence domain for two isolated CDs (dotted curve) and for two closely packed “interacting” CDs (dashed line).

The coherent e.m. field generated inside a CD extends its influence outside the CD itself by means of its own evanescent tail so allowing two CDs to interact each other through the overlapping of the evanescent tails belonging to their respective coherent e.m. fields. It is easy to see that, for a pair of closely packed coherent domains (whose intercentre distance is of order

), the energy gap curve

associated to the superposition of the coherent e.m. fields generated by them is lower than the energy gap curve associated to each CD field when isolated. This overlapping determines an energy gaining up to four times larger than in the isolated CDs at the point

. At this distance the difference

between the two curves can be considered as a “binding” energy of the two CDs, as shown in Figure 2. This extremely important result shows that two or more nearby CDs interact each other through their respective coherent e.m. fields and, in particular, by means of their evanescent tails. Although the large frequency rescaling

keep the e.m.

radiation field trapped inside every CD preventing it to escape to the vacuum (where

), a coherent e.m. field, in the form of non-radiating

evanescent wave, leaks out far beyond the CD’s borders.

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Luigi Maxmilian Caligiuri

As we’ll shown in the following, it is just this evanescent e.m. field, acting as a “propagation channel” of virtual photons between water CDs, that would allow the realization of quantum computing in liquid water.

Thermodynamics of “Interfacial” Water The above formulation of coherent dynamics doesn’t consider the effect of thermal collision of water molecules on the stability of liquid coherent state and then it formally holds for . We now briefly discuss this effect and review its consequences on the equilibrium between the incoherent (vapor-like) state and the coherent liquid state of water. The thermodynamics of coherent liquid water has been already examined [2, 5]; here we limit the discussion to just some features that will be relevant for our model. The coherent ground state of liquid water is characterized by an energy gap per molecule (whose value for

, is given by

) that

prevents each of them to “escape” from the coherent phase. If the thermal fluctuations are able to transfer to a given CD an amount of energy , they put some water molecules out of phase with the coherent oscillations, pulling them out from the coherent domain. These molecules then start to behave as a dense gas (a Van der Waals gas in which strong short-range attraction forces act), occupying the interstitial spaces between the CDs, whose sizes increase with the temperature. On the other hand, the energy minimization requirement keeps neighbouring CDs closely packed so that the non - coherent gas is trapped inside the spaces between the CDs. The pressure of this gas of non - coherent molecules increases with the temperature until it is able to escape from CDs array, vaporizing to the open. As soon as the thermal fluctuations will be able to transfer a sufficiently high number of molecules to the non - coherent phase, a more and more increasing number of CDs will be broken and all the molecules will pass to the gaseous non - coherent state.

Quantum (Hyper)Computation by Means …

17

The thermodynamics of coherent domains can be then described by considering the coherent and non - coherent fractions of water

and

, respectively indicating the number of molecules belonging, at each instant, to the coherent and non - coherent phases. In a given system we then must have, at a given temperature, the following constraint: (24)

in the case of water, the function

, assuming no rotational motion due

to the coherent alignment of electric molecular dipoles, has been calculated as [2, 5]

(25)

where

is the space-dependence non - coherent fraction

(26)

and

is the total energy gap between the CGS and the

non - coherent state (

being is the term due to short-range electrostatic

force between molecules),

is water molecular mass and

“average” momentum belonging to the “excitation curve”

is the of the

single-particle levels of liquid water in the normal phase. It can be calculated, through (25) that, for water at room temperature ( ), the coherent fraction is about

.

18

Luigi Maxmilian Caligiuri

From a statistical viewpoint, for every , the system oscillates between two states: the one in which some molecules are in the coherent state and the one in which they are in non - coherent one, according to whether the coherent electrodynamic attraction is overtaken or not by the thermal collisions. On average a molecule will then spend equal time between the two fractions and in order to reveal this two-phases structure, the observation time would require a resolution shorter than the oscillation time between the two fractions. Nevertheless, the situation appears to be radically different if we consider the “interfacial water,” namely the water closer than a fraction of micron to a surface. The special properties of interfacial water have been studied for a long time [10-12] especially in connection with the so called “EZ water” (i.e., “exclusion-zone” water). In particular, as studied by the Pollack group [13-15], water near a surface or a wall would be subjected to a new type of strongly attractive interaction able to counteract the effect of thermal collision. As a consequence, such interfacial water behaves as overcooled water (namely water at subzero temperature) that is characterized by glass-like structural and dynamical properties as the temperature decreases. In this “glassy” state, water can be considered as a liquid possessing an enormously high viscosity in which perturbing just one molecule would influence all of them. According to the above thermodynamics, this is just what happens in coherent water when we consider a sufficiently low temperature. In particular, when

, we have

) and to each decrease of about

( corresponds an

increase of one order of magnitude of water density [16, 17]. Thus, the “glassy” state characterizing interfacial water can be considered as a fully coherent state, whose enormous viscosity and rigid dynamics are due to strong phase-correlation between molecules, for it is impossible to move one molecule without moving the whole CD. From a dynamical point of view, the arising of such fully coherent behavior can be explained by considering other coherent processes, occurring in liquid water close to a surface, in addition to that described

Quantum (Hyper)Computation by Means …

19

above and concerning electron clouds [18]. The layer of fully coherent water close to a surface can then reach an extension up to about [18] able to include some thousands of CDs as occurs in EZ water.

Excited Spectrum of Coherent Domains in Water A very important consequence of the coherent dynamics of liquid water is the possibility to have “excited” coherent oscillations, namely excited states of CGS. This is a specific feature of liquid water due to the particular couple of levels involved in the coherent oscillation, namely the ground state and the excited level at that lies just below the ionization threshold of

. For the water coherent state

we have,

from Table 1, , meaning than more than 0.10 of electron per water molecule can be considered as quasi-free particles. A water CD can include, at the density of liquid water, a number of molecules of the order of , corresponding to a number of quasifree electrons of about

, meaning that each water CD acts as an

enormous reservoir of quasi-free electrons. These electrons can be easily excited by external, even weak, energy source so originating metastable coherent excited levels in the form of “cold electron vortices” [19] as far as this energy supply is lower than the energy gap per molecule characterizing the water CD. This energy, because of the coherence, is absorbed by the CD as a whole. Very interestingly, such vortices cannot be excited nor decay thermally since they are coherent and have a quantized magnetic moment that allows them to align to an external static magnetic field. For this reason, they are characterized by a very long lifetime. The energy of an excited coherent vortex, associated to a single CD, is given by [19]

(27)

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Luigi Maxmilian Caligiuri

where

is the angular momentum,

the momentum of inertia,

the gyromagnetic ratio and the external magnetic field. It is easy to show that the moment of inertia is given by:

(28)

showing the excited energy level depends on the size of the CD and on the number density of water molecules. The probability to excite each component of the spectrum

is uniform i.e., it is independent of the

energy level to be excited. The excited levels so obtained are associated to higher values of angular momentum

of the coherent vortex and to a shift in frequency

of the

tuned oscillations. It is important to observe that, being the spectrum

a collective

property of the entire CD, it has practically no upper limit since, although every molecule in the coherent state cannot absorb an energy higher than , all the

resonant molecules can account for a total energy of

excitation of the order of

a very high quantity (including frequency

up to visible and HV spectrum) since, at room temperature,

.

As we have seen, coherent states are characterized by a well-defined value of the common phase of oscillation of matter and e.m. field as follows from the phase-lock constraint relating the matter and e.m. field phases in the coherent state. From a QFT standpoint this means the wavefunctions describing coherent domains are eigenstates of a suitable quantum phase operator whose features will be studied in details in the chapter 2 of this book. The existence of coherent excited levels in liquid water opens the possibility to considering the onset of a coherence among CDs, namely the so-called “supercoherence” [9, 19-22].

Quantum (Hyper)Computation by Means …

21

The idea of super-coherence is conceptually similar to that of coherence between electron clouds and e.m. field oscillations just described above in the sense we now consider the possibility that a certain, generally high, number of CDs become able to oscillate in phase with each other and the resulting coherent oscillation could cover a much more extended region. In general, coherence among CDs doesn’t imply the achievement of a critical density to occur, as for the formation of single a CD, but is just limited to the possibility for every single isolated CD to oscillate in phase with any other (despite, as we’ll prove in the following, super-coherence, within our model, can be also determined by a real quantum-type interaction between CDs), allowing them to be much less closely spaced. Anyway, for the supercoherence to occur is needed a CD could discharge their excitation energy outwards in order to give the required oscillations. As a consequence of energy discharge (or absorption) the CD changes its own frequency of oscillation. Supercoherence also further increases the stability of the involved CDs against the damaging effect of thermal fluctuations so further stabilizing the value of

.

In some previous papers [8, 19, 20], a possible chemo-electrodynamical scheme for the energy discharging of a CD has been proposed requiring the presence of suitable “guest” macromolecules able to “resonate” with the involved CDs. According to this proposal, the energy “discharge” of a CD would happen when the energy accumulated in the excited state (including the few guest molecules), coming from the environment, reaches the threshold of the chemical activation energy of the guest molecules so transferring this energy to them, via a resonating non-thermal process. If this process occurs at the same rate for every CD, they oscillate with the same frequency so realizing a macroscopic collective oscillation covering a very large region. The output of the involved chemical reactions, both in terms of new chemical species and energy emission, could in turn modify the oscillation frequency of the CDs giving rise to a new cycle of oscillation characterized by a different time period. This will generally depend on the rate of external energy

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Luigi Maxmilian Caligiuri

storage, the activation energy of reactions and rate of discharge as we’ll discuss more carefully in chapter 3 of this book.

THE QUANTUM TUNNELING – COUPLING INTERACTION BETWEEN WATER COHERENT DOMAINS The chemo-electrodynamical approach so far proposed to explain supercoherence doesn’t consider any direct interaction between CDs but just the possibility for them to oscillate in phase with each other. Nevertheless, as we’ll see by discussing our model, it is possible for two or more CDs to directly interact with each other through the e.m. evanescent fields leaking out from them by considering the quantum tunneling of virtual photons, associated to such evanescent modes.

Quantum Dynamics of “Evanescent” Electromagnetic Fields and the Tunneling of Virtual Photons From the standpoint of classical physics, evanescent waves usually originate by the “total reflection” of a light beam at the boundary between two media characterized by different values of refraction index If the transmitted wave is refracted at an angle

and

.

in the medium with

lower density, then we have, from Snell’s law, for the wave number of the transmitted component of the incident wave orthogonal to the interface, the expression

(29)

where is the frequency of the incident wave and is the reflection angle at the interface between the two media. When the condition

Quantum (Hyper)Computation by Means …

23

(30)

is satisfied,

becomes imaginary and the transmitted e.m. field will show

an exponential decay with the distance to an “evanescent” field.

from the interface, so giving rise

The condition (30) also defines a “critical” angle namely the angle for which, if

, the incident beam is totally reflected

inside the first medium and doesn’t propagate in the second one. Nevertheless, even in this case, a “transmitted” wave component is present in the second medium, namely just the evanescent wave. In fact, if a third medium (denser than the second) is placed after the second one at a distance from the first interface, the wave number becomes real again in this medium and the total reflection amplitude is reduced, giving rise to a transmitted component, through the gap, that “escapes” into the last medium. This phenomenon is called “frustrated total internal reflection” (FTIR) in which the transmitted amplitude decreases exponentially with the gap . It is important to note this exponentially decreasing behavior makes the wave practically detectable within a limited distance from the propagation point so we can define a “propagation depth” of evanescent wave, as the distance where its amplitude is reduced to the

of its initial value, namely

(31) where is the wavelength of the incident wave. It has been shown the optical refraction index plays the same role of the barrier potential in the quantum-tunneling phenomenon, so suggesting the idea that evanescent waves could actually represent tunneling modes of e.m. waves [21, 24, 25]. The spreading of e.m. waves across “impenetrable” opaque optical barriers is then analogous to the phenomenon of quantum tunneling experienced by particles through a potential barrier.

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Luigi Maxmilian Caligiuri

The tunneling of evanescent modes of e.m. waves are just interpreted, from a quantum viewpoint, as the tunneling of virtual photons through the corresponding potential barrier. Such tunneling photons are characterized by a negative square mass in the ordinary metric [21, 23, 25] or, equivalently, by a superluminal group velocity. The tunneling time of evanescent e.m. waves (or virtual photons) through opaque optical barriers has been accurately measured in several experiments [21, 25], in particular involving the double prisms configuration (see Figure 3) that, better than any other, can be considered as the optical analogous of the quantum tunneling through a potential barrier. In all the cases the experimental results confirm that the reflected and the transmitted (tunneled) waves (virtual photons) arrive at the detectors always at the same time, independently of the barrier length, and this behavior holds for all fields (photons or massive particles) characterized by wave solutions with purely imaginary wave numbers. This “universal” tunneling time of evanescent wave (or tunneling photons) is given by

where

is the carrier frequency of

the signal (or the associated quantum particle) undergoing the process and is substantially independent from the barrier length. An analogous relation can be obtained in the case of the tunneling of a particle wave packet for which the transmission time is given by

where

is the energy of

the particle. Such “universal” time relation implies a nearly “instantaneous” spreading of the signal or wave-packet across the barrier so that the measured tunneling time corresponds to a very high tunneling velocity, even greater than . This superluminal energy and signal velocity in quantum tunneling has been confirmed in many experiments [21, 25] and its theoretical consequences has been already studied in deep, also by us, in a number of previous publications [21, 23].

Quantum (Hyper)Computation by Means …

25

Figure 3. Tunneling of e.m. modes through double prims configuration and its quantum analogous.

Quantum Coupling between Water Coherent Domains and Their Interaction through Virtual Photons Evanescent modes can be considered as free e.m. field basing on the idea of refractive index of a passive, macroscopically continuous media [26]. From this standpoint such modes can be considered as the result of the spatial phase modulation, at the interface between incident and reflected wave. Evanescent modes, considered as classical c-number fields, can then interact with matter no differently from ordinary homogeneous e.m. waves. On the other hand, as Feynman has shown, virtual particles appears as intermediate states of an interaction process. Then, despite they are not directly observable, virtual particles represent necessary intermediate states between real observable states having a direct influence on them (see Figure 4). They also take part not only in the microscopic interaction processes but also in the macroscopic range. In our model the coupling among two CDs can be pictured in terms of the overlapping between their evanescent field tails [21, 27, 28] that allows the tunneling of virtual particles (tunneling photons) between them.

26

Luigi Maxmilian Caligiuri

Figure 4. Time-space diagram of evanescent mode or photonic tunnelling.

Figure 5. Overlapping between the evanescent fields associated to two nearby CDs.

When two CDs are sufficiently close each other, their respective e.m. evanescent fields, given by (21), could overlap. The width of the resulting overlapping zone, through which tunneling of virtual photons takes place, will depend on the distance between the two MTs and on the spreading of the two evanescent fields outside the respective CDs, whose “extension,” described by their tales, is a function of shown in Figure 5. If the separation average value of

. The situation is schematically

between the two CDs is much greater than the characterizing the tails of evanescent e.m. field

“escaping” from the respective CDs (Figure 5), we can assume no useful overlapping exists between the two evanescent fields and consequently a negligible probability of interaction between them.

Quantum (Hyper)Computation by Means … When instead

27

, the overlap between these two evanescent e.m.

fields takes place allowing a “non-local” interaction between the CDs. As we have seen from (23), the closer are two CDs the more stable is the system, since they increase their “binding” energy (being the largest when

), so favoring in turn their interaction.

The value of

can be approximatively estimated by observing that the

interaction between CDs may be described as the tunneling of evanescent e.m. waves (or virtual photons) through opaque optical barriers. As already discussed in details [21] the tunneling probability is inversely proportional to the barrier’s length so it appears as near-field phenomenon whose “detection” is limited to near-field zone. All the experimental evidences available about this process [24, 25, 29, 30] show this near field region to have a spatial extension up to (32) where

is the wavelength of the incident field at the interface where the

total reflection occurs. We can then assume, in our case

(33)

where we have used the (3). On the other hand, the tunneling time of the evanescent wave (or virtual photons) is related to the frequency of the signal (or the associated quantum particle) undergoing the process, namely, for

(34) that is independent of the barrier length for not too long barriers where evanescent field attenuation makes no interaction possible.

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Luigi Maxmilian Caligiuri

It is important to note this quantum - tunneling “interaction” depends on the existence of excited levels in the CDs whose excitation-deexcitation determine the tunneling of virtual coherent photons between the CDs, since in the fundamental state, the CGS, the CD is stable. We could also think this interaction to occur between the quasi-free coherent electrons, respectively pertaining to different excited levels in the two CDs, as the result of the exchange of a virtual photon through quantum tunneling. From a microscopic viewpoint, the energy transfer between two interacting CDs can be explained considering the exchange of virtual photons through a sort of a “resonance energy transfer” process (RET). In QED, the interaction between two material components (in our case the coherent quasi-free electrons belonging to the two excited CDs) that we respectively indicate as a source/donor (i.e., an excited quasi-free electron of the first CD) and an acceptor (i.e., an excited quasi-free electron of the second CD) can be described by the Hamiltonian: (35)

where (

is a complete set of quantum parameter characterizing the two CDs for the first CD and

particle Hamiltonian,

for the second one),

is the single-

is the Hamiltonian describing the interaction

between the free e.m. field and the single particle and Hamiltonian. In the electric-dipole approximation the term

is the free-field is given by

[31]: (36)

where

is the electric dipole operator acting on matter component

and

is the transverse electric field operator acting on radiation states

Quantum (Hyper)Computation by Means … as a function of the matter component position vector of the plane-wave expansion of

29

. We can make use

to write

(37)

where the sum is made over all wave-vectors action of

and polarizations

. The

then corresponds to creation or annihilation of a photon. We

then consider a generic energy transfer process from an initial state to a final state

, where

represents the excited states of the particles

and

and and

state, obeying the energy conservation constraint

respectively their ground .

The transfer process corresponds then to: a) the creation of a virtual photon at and its subsequent annihilation at and b) the opposite process. The consideration of these two paths ensures the consistence with the Heisenberg’s uncertainty relation for short times during the intermediate states. After the whole system will be in the final state, the virtual photon is annihilated and the energy conservation is restored so ensuring the observability of the final physical states while the virtual photon is not. The quantum amplitude for the coupling between electric dipoles can be written as [32] (38) where the summation over repeated indices is assumed, transferred from tensor

to

,

is given by

is the energy

and the electromagnetic coupling

30

Luigi Maxmilian Caligiuri

(39)

We see that, for short distance between and energy transfer , namely in the near-field region

and not too high , the dominant

term in (39) is the first one proportional to . This energy transfer occurs over very short times and it cannot be localized in either or showing a radiation-less resonance behavior [31, 32] whose rate depends on that is just what characterize, on macroscopic regions, the typical evanescent e.m. field feature. Macroscopically, if we consider the exchange of one photon, the “tunneling” coupling between two CDs can be described by the following total Hamiltonian (40) where

and

respectively indicate the “free” and the “interaction”

Hamiltonian of the system. We further suppose the free system (the isolated water CD) can be considered as a two-levels system characterized by only two accessible eigenstates (normalized to unity), named

and

,

satisfying the stationary Schrodinger’s equation (41)

with

and

energy eigenvalues of the system. According to our

assumption, we can decompose the Hamiltonians as (42)

Quantum (Hyper)Computation by Means …

31

where represent the “strength” of the interaction. We then consider the complete eigenvalue problem (43)

where

and

is the eigenvalue of the eigenstate

. We can then

write (44)

satisfying the normalization condition

. The eigenvalues

problem (43) has the solution

(45)

and

with

eigenvalues

satisfying

. The time-evolution of the system is described by (46)

if we now suppose the initial state of the system is just have, by considering the orthogonality condition

we ,

(47)

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Luigi Maxmilian Caligiuri

where we have also introduced the parameterization (48) By using (47) and after some algebra, we obtain the transition probability from the initial state

at

to the state

at time

as

(49)

that is the so-called “Rabi equation.” This solution is equivalent to a Rabi oscillation in which a photon wave packet of given wave number is coupled back and forth between the two CDs. An even more interesting treatment of time evolution of such system can be formalized by using the Pauli’s matrices. In this notation the Hamiltonian is (50)

where is the

identity matrix,

and

(51)

the time-evolution operator then becomes

(52)

Quantum (Hyper)Computation by Means … On the other hand, any arbitrary unitary operator a rotation of an angle

33

can be expressed as

about an axis defined by the real unit vector

, namely

(53)

where the rotation operator

can be expanded as

(54)

The (52) is identical to (53) if we assume

(55)

We have then proven a very important result namely the time-evolution operator of system of two interacting CDs through tunneling coupling can be expressed as the product of a rotation and an overall phase shift. By effect of the coupling, the two CDs cannot be further considered as independent entities but as two interconnected parts of the same system. We now remember the wavefunctions describing the coherent domains, respectively

and

, are eigenstates of the

phase operator (whose eigenvalue is related to the coherent frequency of oscillation including the angular frequency of the excited vortices) and also describe the state of tunneling photon as “being” in the one CD or in the other, through the shift in the value of their relative phase. The tunneling interaction is obviously not limited to the case of just two CDs but can be extended to a very higher number of interacting domains.

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Luigi Maxmilian Caligiuri

CONCLUSION In this chapter, after reviewing the main features of the theory of QED coherence in matter and its application to liquid water, we have examined in more detail some of the most interesting predictions of this theory such as the formation of the so-called “coherent domains,” and the dynamics of an evanescent e.m. field generated across the boundaries of the latter. We have also considered the effect of thermal fluctuations on the coherent fraction of molecules in liquid water and how the presence of hydrophilic surfaces is able to meaningfully reduce such disruptive effect by stabilizing the water coherent fraction, even at room temperature, around about the unitary value. The study of water coherent dynamics has allowed us to speculate the arising of a so far unrecognized interaction between water molecules in their coherent state, based on exchange, by quantum tunnel effect, of virtual photons, belonging to the evanescent modes of coherent e.m. field associated to the interacting water coherent domains. A very remarkable and fascinating consequence of such dynamics is that the quantum timeevolution of interacting water coherent domains could be described in terms of a Rabi- like equation and that the corresponding quantum operator can be expressed as the composition of a rotation and a global phase change in the space of quantum states of water coherent domains. Such property will let us to propose a novel model of quantum computation, based on the coherent dynamics of liquid water, whose computational speed could be made very high, opening the door, in principle, to the realization of the fastest computational framework, namely the so-called “hypercomputing.”

REFERENCES [1]

Umezawa, H. 1993. Advance field theory: micro, macro and thermal physics. New York: American Institute of Physics.

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Preparata, G. 1995. QED Coherence in Matter. Singapore, London, New York: World Scientific. [3] Caligiuri, Luigi Maxmilian. 2015. “The origin of inertia and matter as a superradiant phase transition of quantum vacuum.” In Unified Field Mechanics: Natural Science beyond the Veil of Spacetime, edited by: R. L. Amoroso, L. H. Kauffman, and P. Rowlands, 374-396. Singapore, London, New York: World Scientific. [4] Del Giudice, E., and G. Preparata. 1994. “Coherent dynamics in water as possible explanation of membrane formation.” J. of Biol. Phys. 20, 105-116. [5] Arani, R., Bono, I., Del Giudice, E., and G. Preparata. 1995. “QED coherence and the thermodynamics of water.” Int. J. of Mod. Phys. B 9, 1813-1842. [6] Del Giudice, E., and G. Preparata. 1998. “A new QED picture of water: understanding a few fascinating phenomena” in Macroscopic Quantum Coherence. Singapore, London, New York: World Scientific. [7] Voeikov, V. L., and E. Del Giudice. 2009. “Water respiration: the basis of the living state.” Water 1, 52-75. [8] Del Giudice, E., Spinetti, P. R., and A. Tedeschi. 2010. “Water Dynamics at the root of Metamorphosis in Living Organisms.” Water 2, 566-586. [9] Bono, I., Del Giudice, E., Gamberale, L., and M. Henry. 2012. “Emergence of the coherent structure of liquid water.” Water 4, 510532. [10] Clegg, J. S. 1982. “Alternative views on the role of water in cell function.” In Biophysics of Water, edited by F. Franks, and F. S. Mathias, 365-385. New York: John Wiley and Sons. [11] Drost-Hansen, W. 2006. “Vicinal hydratation of biopolymers: Cell biological consequence.” In Water and the Cell, edited by G. H. Pollack, I. L. Cameron, and D. N. Wheatley, 175-217. Berlin: Springer-Verlag.

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[12] Antonenko, Y. N., Pohl, P., and E. Rosenfeld. 1996. “Visualisation of the reaction layer in the immediate membrane vicinity.” Arch. Biochem. Biophys. 333, 225-232. [13] Pollack, G. H., and J. Clegg. 2008. “Unexpected linkage between unstirred layers, exclusion zones and water.” In Phase, Transitions in Cell Biology, edited by: G. H. Pollack, and W. C. Chin, 143-152. Berlin: Springer Science and Business Media. [14] Zengh, J. M., et al. 2006. “Surface and interfacial water: evidence that hydrophilic surfaces have long range impact.” Adv. Colloid Interface Sci. 23, 19-27. [15] Trepat, X., Deng, L., and S. An. 2007. “Universal physical response to stretch in the living cell.” Nature 447, 592-595. [16] Buzzacchi, M., Del Giudice, E., and G. Preparata. 2001. “Coherence of the glassy state.” International Journal of Modern Physics B 16 (25), 3771-3786. [17] Del Giudice, E., Tedeschi, A., Vitiello, G., and V. Voeikov. 2013. “Coherent structures in liqui water close to hydrophilic surfaces.” Journal of Physics: Conference Series 442, 012028. [18] Del Giudice, E., Voeikov, V., Tedeschi, A., and G. Vitiello. 2015. “The origin and the special role of coherent water in living systems.” In Fields of the Cell, edited by: D. Felds, M. Cifra, and F. Scholkmann, 95-111. Trivandrum: Research Signpost. [19] Del Giudice E., and A. Tedeschi. 2009. “Water and Autocatalysis in Living Matter.” Electromagnetic Biology and Medicine 28, 46-52. [20] Brizhik, L. S., Del Giudice, E., Tedeschi, A., and V. L. Voeikov. 2011. “The role of water in the information exchange between the components of an ecosystem.” Ecological Modelling 222, 2869-2877. [21] Caligiuri, L. M., and T. Musha. 2016. The Superluminal Universe: from Quantum Vacuum to Brain Mechanism and beyond. New York: Nova Science Publishers. [22] Caligiuri, L. M. 2018. “Super-Coherent Quantum Dynamics of ZeroPoint Field and Superluminal Interaction in Matter.” In Unified Field Mechanics II: Formulation and Empirical Tests, edited by: R. L.

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[28]

[29] [30] [31]

[32]

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Amoroso, L. H. Kauffman, P. Rowlands, and G. Albertini, 331-343. Singapore, London, New York: World Scientific. Caligiuri, L. M. 2019. “A new quantum-relativistic model of tachyon.” Journal of Physics: Conference Series 1251, 012009. Stahlhofen, A. A., and G. Nimitz. 2006. “Evanescent modes are virtual photons.” Europhys. Lett., 76 (2), 189-185. Nimitz, G. 2009. “On virtual phonons, photons and electrons.” Found. of Phys. 39, 1246-1355. Carniglia, C. K., and L. Mandel. 1971. “Quantization of evanescent electromagnetic waves.” Phys. Rev. D 3, 280-296. Caligiuri, L. M., and T. Musha. 2016. “Superluminal Photons Tunneling through Brain Microtubules Modeles as Metamaterials and Quantum Computation,” in Advanced Engineering Materials and Modeling, edited by: A. Tiwari, N. Arul Murugan, and R. Ahula, 291333. New Jersey: Wiley Scrivener Publishing LLC. Caligiuri, L. M. 2015. “Tunneling of super radiant photons through brain microtubules modeled as metamaterials.” Paper presented at the 2nd International Conference on Rheology and Modeling of Materials, Miskoic, Hungary, October 5-9. Haibel, A., Nimitz, G., and A. A. Stahlhofen. 2001. “Frustrated total reflection: the double-prims revisited.” Phys. Rev. E 63, 047601. Enders, A., and G. Nimitz. 1992. “On superluminal barrier traversal.” J. Phys. I, France 2, 1693. Andrews, D. L. 2005. “The photon: a virtual reality.” Paper presented at the conference Optics and Photonics 2005, San Diego, California, USA, July 31 August 4. Andrews, D., and D. S. Bradshaw. 2014. “The role of virtual photons in nanoscale photonics.” Ann. Phys. 3-4, 173-186.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 2

THE QUANTUM PHASE OPERATOR AND ITS ROLE IN QUANTUM COMPUTING Luigi Maxmilian Caligiuri* Foundation of Physics Research Center (FoPRC), Cosenza, Italy

ABSTRACT The definition of a quantum phase operator, whose eigenstates are quantum states of well-defined phase, is an important as well as difficult problem in Quantum Field Theory, especially in connection to the study of quantum coherent states as those represented by the coherent domains arising from the QED coherence in liquid water described in the previous chapter of this book. In this chapter we’ll give a suitable definition of a quantum phase operator and of its eigenstates showing that the latter conceptually coincide with the coherent domains previously found. Furthermore, we have proven the time evolution of such coherent states is ruled by a Hamiltonian operator whose action on these states causes a change in their phase. This feature allows for the possibility to manipulate the phase of coherent domains by letting them interact for a suitable time interval, that just represents the condition required to implement, in principle, any type of quantum computation by means of matter coherent *

Corresponding Author’s Email: [email protected].

40

Luigi Maxmilian Caligiuri domains like those emerging within the QED coherent dynamics of liquid water.

Keywords: quantum phase, coherent states, time evolution

INTRODUCTION As known, in quantum mechanics complex numbers are not just a calculation tool, like for example in classical electrodynamics, but they are strictly related to fundamental properties of physics systems. In particular, the phase of a complex number corresponds, in this framework, to a very peculiar feature of quantum mechanics, namely the arising of the interference phenomenon between quantum amplitudes. At a deeper level, the phase of the complex wavefunctions, that are solutions of Schrodinger’s equation, contains physical information. The trouble in interpreting the physical meaning of phase in quantum mechanics is mainly due to the ascertainment that no experiment generally measures an imaginary number, and then a “phase,” but, on the other hand, an imaginary number can be interpreted in terms of an actual physical process. The importance of quantum phase has been established in gauge transformations and gauge theories in which it relates to the concept of exponential scale transformation. Furthermore, the realization of an imaginary electromagnetic scale transformation leads to Planck-Bohr quantization condition and then to the concept of gauge invariance in quantum mechanics. The concept of quantum phase also plays an important role in the modern (non-Abelian) gauge theories of particle physics and, even more interesting, in the “miraculous” Aharanov-Bohm effect. From an experimental viewpoint, the increase of interest in quantum phase has been initially stimulated by the development of laser techniques (in particular with phase-sensitive measurements) although, from a theoretical side, it can be traced back to at least some decades before its “observation.” In fact, yet in the early stages of quantum mechanics, one of its “fathers,” Erwin Schrodinger, discovered the so-called “coherent” states

The Quantum Phase Operator and Its Role in Quantum Computing

41

of harmonic oscillator, derived as a minimum – uncertainty position – momentum states and characterized by a well-defined value of phase. Nevertheless, coherent states get a more defined meaning only within the so-called “second quantization” and the development of Quantum Field Theory (QFT) where the relation between the “number” and “phase” operators acquires a special significance when related to the number-phase uncertainty relation. Coherent states play an important role in quantum optics especially in laser physics, where they describe the state associated to coherent e.m. radiation field of a laser beam. More precisely, coherent states have properties similar to those characterizing “classical” coherent light and, in fact, they can be considered as the most closely quantum mechanical approximation of these classical states. So, as laser light can be represented, under certain conditions, as idealized classical coherent light, in the same way, so laser light can be quantum-mechanically described by quantum coherent states. The connection between coherent states and light appears also natural since, as well as different states of light are distinguished by intensity and phase, coherent state, being defined by a complex number, is determined by its amplitude and phase. Nevertheless, while light intensity has a simple quantum mechanical operator equivalent (i.e., the number operator ), in the case of phase, it is generally very more complex to define a correspondent quantum operator. In fact, as we’ll see, some important problems arise in attempting to find a suitable definition of phase operator [1-3]. Some proposals have been made [4-9] so far, often stimulating a great debate among researchers. Despite such known difficulties, the definition of a suitable quantum phase operator is very important not only when referred to laser physics, in which the coherent field is just the e.m. radiation field due to the stimulated emission in atoms, but it becomes fundamental if we consider the QED coherence (CQED) in condensed matter [10, 11] whose application to liquid water has been discussed in the previous chapter of this book. We have seen the coherent states predicted by this theory are collections of “coherent domains” (CDs), namely macroscopic spatial domains in which quantum

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Luigi Maxmilian Caligiuri

matter field and an autogenerated (from quantum vacuum fluctuations) e.m. field oscillate in phase with each other, giving rise to the “coherent ground state” (CGS), more stable with respect the “perturbative” non-coherent ground state (PGS) characterized by uncoupled and non-phased zero-point oscillations of matter and e.m. field. Such coherent states resemble the wavefunctions representing superfluid and superconductor physical systems. These macroscopic quantum states are all characterized by a welldefined value of matter and e.m. field common phase of oscillation making them the natural candidates to be eigenstates of a suitable quantum phase operator. The introduction of the latter for such coherent states is fundamental since the information about their physical structure (including the spectrum of its excited energy states) is contained in the phase. In this chapter we’ll discuss the definition and main features of a generic coherent state as well as that of a suitable quantum phase operator acting on them, also discussing how the latter is able to describe their time evolution. We then argue the quantum phase operator is actually a fundamental player in quantum dynamics of coherent domains (CDs) emerging from CQED in condensed matter. The results discussed in this chapter will be of fundamental importance for our model of quantum computation through water CDs that is the subject of the next chapter of this book.

COHERENT QUANTUM STATES In QFT the quantization of the e.m. field is realized by replacing the complex amplitudes

and

of the Fourier decomposition of radiation

fields with the photon annihilation and creation operators and respectively. This establishes a correspondence between the classical field amplitude

and the non-hermitian operator

that, as we’ll see

in the following, demands for a proper description of quantum phase. Coherent states are then defined as those states characterized by a welldefined value of phase and, from a conceptual viewpoint, they are quite

The Quantum Phase Operator and Its Role in Quantum Computing

43

analogous to those describing the quantum coherent domains predicted by coherent QED in condensed matter (see chapter 1 of this book). As mentioned above, the mathematical definition of “coherent” quantum states of quantized electromagnetic field was initially referred to laser light and, by considering the correspondence between classical amplitude

and

the operator , it can be given in terms of annihilation and creation operators as (1)

that implies

, namely

(2) By virtue of (2) a coherent state, also called “Glauber state,” is then defined as eigenstate of the amplitude operator, i.e., the annihilation operator , with eigenvalue

. Since

is non-hermitian

(with

) is a complex number that corresponds to the complex wave amplitude in classical optics. Thus, coherent states are also wave-like state of electromagnetic oscillator. The solution of eigenvalue equation (2) can be given as expansion of a coherent state

in terms of the occupation states

of Fock space as [1-3, 8, 11]

(3)

We note such coherent states involve a linear combination of an infinite number of eigenvectors of the number operator

and consequently

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Luigi Maxmilian Caligiuri

an infinite number of quanta. Furthermore, in a coherent state the dispersion of the number operator is (4) namely the number of photons is Poisson distributed around its mean number with a distribution probability given by

(5)

with

. An interesting property of coherent states is that they are

not orthogonal since we have, for any two

and

of them:

(6)

i.e., they form a so-called “overcomplete” system. Although the coherent states are not orthogonal, they satisfy the completeness relation (7)

so, it is possible to expand any of them in terms of a complete set of states. The non-orthogonality feature also means that any coherent state can be expanded in terms of all the other coherent states so they are not linearly independent

(8)

The Quantum Phase Operator and Its Role in Quantum Computing

45

THE QUANTUM PHASE OPERATOR The first to propose an expression for a quantum phase operator was P. Dirac [4] basing on the position-momentum commutation relation (we adopt natural units

): (9)

that is transposed in the second-quantized boson formalism by defying the operators (10)

called “quadrature” operators which obey the commutation relation (9). This suggests it is natural to require the existence of a quantum “phase” operator acting on the phase “component” of the annihilation operator could be expressed in the form

if the latter

(11) where is the “number” operator of QFT and is the searched quantum phase operator. In order to define such operator, we start by considering a quantum “phase” operator whose action on a coherent state would be the change of its phase. We start considering the action of the operator, called “phase shifting” operator (12)

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Luigi Maxmilian Caligiuri

where

is a real parameter. The action of

on a coherent state

is

given by

(13)

and we have

(14)

and then

(15)

The action of the operator generate a phase shift

on a coherent state

is then to

of the state namely (16)

Since is a function of only, this means the number operator generates a phase shift of the quantum state and then phase is a quantity canonically conjugated to photon number, namely we can write, in the phase representation,

The Quantum Phase Operator and Its Role in Quantum Computing

47 (17)

and suppose, similarly to the case of position and momentum existence of a quantum phase operator relation

and , the

, satisfying the commutation

(18) This commutator implies the following fundamental commutation relation

(19)

that, for a very high number of quanta (

) as occurs in the coherent

states, implies the phase is well-defined . This characterizing feature of coherent states suggests the consideration of a “phase” operator of which coherent states could be eigenvectors and that could be used to describe macroscopic quantum state as those associated to water CDs studied in chapter 1. Nevertheless, for some theoretical reasons, it is more natural to consider not just the operator

but rather the operator (20)

whose commutator with

is given by (21)

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Luigi Maxmilian Caligiuri

The need that be unitary also suggests a possible factorization of the annihilation operator in the form (11), corresponding to a polar decomposition of . From the above factorization and remembering that , we find

so we can write (22)

The definition (22) of the operator shows the validity of the following required properties for the quantum phase operator, namely: 1) it satisfies the commutation equation

;

2) it shifts the photon number according to (22). We are now in position to state the existence of the eigenvectors of such phase operator giving a possible expression for them (23)

We observe such state is not normalizable like the eigenstates of position and momentum of a single particle. The action of the operator

on

then determines its phase shift

(24)

We can then summarize the following important points:

The Quantum Phase Operator and Its Role in Quantum Computing

49

a) we can define a quantum phase operator (22) with suitable properties that satisfies the fundamental commutator (21); b) there exist eigenvalues of phase operator, expressed by equation (23), each characterized by a well-defined phase and by an infinite number of quanta (being expressed as an infinite superposition of eigenvectors of number operator); c) when the operator

acts on the eigenvectors

of phase

operator it determines a phase shift of the state of an amount

.

We finally observe the phase operator , defined by (22), is not unitary being, as we can easily verify, so, at first sight, it wouldn’t be able to represent a physical quantity within QFT. This trouble was well known already to Dirac and a certain number of suggestions to solve it has been proposed during the past years [5-9]. Among these, particularly interesting is that advanced by Pegg and Barnett [7,9] we briefly discuss in the following and that is the most suitable to be applied within our theory that considers quantum coherent domains. Their proposal is based on the truncation of the space of number states so that the number of quanta is above limited by a number

. This

assumption is particularly suitable in our case since, as we have seen in the previous chapter, the number of involved quanta is related to that of the microscopic entities (atoms and/or molecules) participating in the collective coherent oscillation of a given coherent domain. We can re-define the quantum phase operator

as

through the following rule

(25)

that now satisfies the condition

.

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Luigi Maxmilian Caligiuri

PHASE - SHIFT OPERATOR AND ITS EFFECT ON THE COHERENT DOMAINS The application of phase-shift operator given by (3), as well as to an eigenstate determines a phase shift of the amount

to a coherent state

of phase operator given by (23) . Coherent states (23) are

introduced to obtain macroscopic radiation waves from quantum electrodynamics, basing on the idea that these macroscopic radiations, whose overall behavior is characterized by classical - like properties expressed in terms of average field values, may be formed by condensation of many photons. As a consequence, that states are not eigenstates of number operator but, as we have seen, states with large uncertainty in particle numbers. This is just what happens (see chapter 1) in condensed matter when the conditions for the occurrence of superradiant phase transition from PGS to CGS are verified, namely when a large ensemble of interacting matter and e.m. quanta (considered as elementary quantum oscillators) perform a runaway towards the more stable coherent state leading to the formation of CDs. The states we are interested in are then just those corresponding to the evolution of the system of interacting matter and e.m. field, around classical paths whose stationary solution, represented in particular by the CGS, has just a space-structure that can be visualized as an array of CDs, namely as a macroscopic quantum structure resulting by the condensation of many quanta (of matter and e.m. field coherently interacting each other) [10, 11]. Even more interestingly, the true vacuum state of such coherent systems is the CGS in which a very high number of elementary quantum components oscillate in phase [10, 11]. We can see that the same result is obtained if we search for the vacuum state associated to a single oscillator that, nevertheless, contains many quanta of Fock space. If we indicate this vacuum state of the Fock space as

defined as

The Quantum Phase Operator and Its Role in Quantum Computing

51 (26)

and consider the Bogoliubov transformation for the coherent states [12] (27)

where

is a c-number. We also indicate the vacuum state of

as

so that (28) In this way we can write (29)

and the state

is the coherent eigenstate of

with eigenvalue

. If we

define the operator (30) and (31) so that (32)

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Luigi Maxmilian Caligiuri

we can then write (33)

(34)

The equation (34) shows the vacuum state

is a superposition of

states characterized by many particles of the coherent state

and then it

can be considered as condensation of the particles composing it. From the above discussion we derive the very important conclusion that the coherent state of matter and e.m. condensed field, including a very high number of quanta, oscillating in phase and giving rise to the CDs arrays in condensed matter, is conceptually analogous to the coherent state

above

considered. It is now clear that the state we have obtained as solution of the coherent equations (10) of chapter 1 must be an eigenstate of the phase operator since it is characterized by a well-defined common oscillation frequency of matter and e.m. field.

PHASE OPERATOR, TIME-EVOLUTION AND QUANTUM COMPUTING The Hamiltonian operator of a quantum system completely describes, at least in principle, the dynamic evolution of a quantum system and then it also contains the interaction terms of the system with force fields. On the other hand, the physical implementation of quantum computing must provide a feasible mechanism for applying computational steps to a quantum register. The i-th step can be considered as the result of application of a unitary transformation

The Quantum Phase Operator and Its Role in Quantum Computing

53 (35)

defined by the Hamiltonian

applied for a time interval

. Such

Hamiltonian must be controlled in order to act on specific single qubits and pair of qubits in a precise way to implement the desired quantum gates [13]. The chance to use water coherent domains to implement quantum computation, as we’ll show in the chapter 3 of this book, then assumes the possibility to control the physical state of CDs and their dynamical evolution. As we have seen, such state is completely specified by the value of the quantum phase associated to them. Consequently, how to quantummechanically “measure” this phase and how to modify its value through the action of suitable Hamiltonian is a central question within our model. In the Schrodinger representation, the time-evolution of a quantum state , satisfying the Schrodinger equation between the time instants (with

and

)

(36)

is given by the action of a unitary evolution operator (37)

so that (38)

By comparing the (38) with the (12) we see the time-evolution operator becomes identical to the phase shift operator if we formally assume

54

Luigi Maxmilian Caligiuri (39)

with

and

so we can write

(40)

and we have (41)

If we consider the application of the operator (40) to a (water) CD, we see its time evolution coincides with a change of its overall phase. This is perfectly understandable if we consider the CD is a macroscopic quantum object whose components have lose their individuality and share among each other only their common oscillation frequency

.

We also suppose the interaction between the CD and its surrounding environment is fully described by the Hamiltonian (39) since the phase shift it determines could be consider as the result of the action of vector and scalar potentials, in turn related to the e.m. fields generating the forces that affect the CD, as we’ll discuss in the chapter 3. It is also clear the phase shift induced

on the CD quantum state

(and associated to an interaction of CD with its surroundings) modifies, by virtue of (22), the number of quanta joining the given CD and, in turn, as shown in chapter 1, the common oscillation frequency

.

CONCLUSION In thi chapter we have given a suitable definition of a quantum phase operator acting on coherent states, having properties similar to classical

The Quantum Phase Operator and Its Role in Quantum Computing

55

states of light, and that represents the best quantum - mechanical approximation of these states. Starting from the definition of a “phase shifting” operator, we have then obtained a definition of a quantum phase operator whose action on a coherent state determines a change in the number of its quanta. As required, the eigenstates of phase operator are characterized by a well-defined value of phase but a very large, substantially undetermined, number of quanta. They are such that the action of phaseshifting operator on them produces a change of their phase. The quantum phase operator also satisfies a fundamental phase-number uncertainty relation but it is non - unitary. In order to solve this trouble a modified version of the operator has been proposed that restores the unitarity and is suitable to describe the coherent systems we are dealing with. We have also shown that coherent states associated to classical-like “macroscopic” state of radiation can be considered as the result of the “condensation” of many quanta from the vacuum state, a process in allsimilar, from a conceptual point of view, to that characterizing the arising of QED coherence in condensed matter leading to the formation of coherent domains (CDs) in water, as discussed in chapter 1. This feature, as well as the property of having a well-defined value of phase, allow us to consider the water CDs as the eigenstates of quantum phase operator as defined in this chapter. Finally, and very interesting, we have shown that time evolution of coherent states is generated by the action of the phase-shifting operator on such states so implying the corresponding Hamiltonian, when acting on the CDs, also determines a change of their phase. We then claim the action of a given Hamiltonian on a given CD, which in turn rules its interaction with the “environment” (typically described in terms of the vector potentials of e.m. field) is able to modify its coherence structure as well as its time evolution. This property is fundamental in order to use coherent domains for performing quantum computation since, as we’ll see in the next chapter, the possibility to dynamically change the CD quantum phase allows us to realize, in principle, a set of universal quantum gates representing the ultimate elements of the quantum circuits through which implementing any type of quantum computation.

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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13]

Lynch, R. 1995. “The quantum phase problem: a critical review.” Physics Reports 256, 367- 436. Bachor, Hans-A., and T. C. Ralph. 2019. A Guide to Experiments in Quantum Optics (3rd edition). Weinheim: Wiley-VCH. Walls, D. F., and G. J. Milburn. 1994. Quantum Optics. Berlin: Springer-Verlag. Dirac, P. A. M. 1927. “The Quantum Theory of the Emission and Absorption of Radiation.” Proc. R. Soc. Lond. A114, 243-265. Susskind, L., and J. Glogower. 1964. “Quantum mechanical phase and time operator.” Physics 1, 49-61. Carruthers, P. and M. M. Nieto. 1968. “Phase and Angle Variables in Quantum Mechanics.” Rev. Mod. Phys. 40(2), 411-440. Pegg, D. T., and S. M. Barnett. 1988. “Unitary phase operator in quantum mechanics.” Europhys. Lett. 6, 483. Freyberger, M., Heni, M., and W. P. Schleich. 1995. “Two mode quantum phase.” Quantum Semiclass. Opt. 7, 187. Pegg, D. T., and S. M. Barnett. 1997. “Quantum Optical Phase.” Journal of Modern Optics 44:2, 225-264. Preparata, Giuliano. 1995. QED Coherence in Matter. Singapore, London, New York: World Scientific. Caligiuri, Luigi Maxmilian. 2015. “The origin of inertia and matter as a superradiant phase transition of quantum vacuum.” In Unified Field Mechanics: Natural Science beyond the Veil of Spacetime, edited by: R. L. Amoroso, L. H. Kauffman, and P. Rowlands, 374-396. Singapore, London, New York: World Scientific. Umezawa, H. 1993. Advance field theory: micro, macro and thermal physics. New York: American Institute of Physics. Nielsen, M. A., and I. L. Chuang. 2016. Quantum Computation and Quantum Information (10th Anniversary Edition). Cambridge: Cambridge University Press.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 3

QUANTUM (HYPER)COMPUTATION BY MEANS OF WATER COHERENT DOMAINS – PART II: THE COMPUTATIONAL LEVEL Luigi Maxmilian Caligiuri* Foundation of Physics Research Center (FoPRC), Cosenza, Italy

ABSTRACT In chapter 1 of this book we have considered the occurrence of the evanescent-tunneling-coupling interaction between coherent domains in liquid water. Now we discuss a novel and very fascinating idea, namely to make use of such interaction to perform, in principle, any kind of quantum computation. We also show that, the use of metamaterials to enclose water molecules in order to form suitable waveguide for the evanescent photons generated inside water coherent domains, could allow for the implementation of a superfast network of interacting coherent domains able to represent a basic architecture for a novel kind of quantum hypercomputer based on the coherent dynamics of liquid water.

*

Corresponding Author’s Email: [email protected].

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Luigi Maxmilian Caligiuri

Keywords: water coherent domains, supercoherence, tunneling-coupling interaction, universal quantum computing, quantum hypercomputing

INTRODUCTION In chapter 1 of this book, we have analyzed the main features of the evanescent-tunneling-coupling interaction arising between liquid water coherent domains (CDs) as a consequence of the exchange, among them, of virtual photons belonging to the evanescent modes of e.m. fields produced inside the CDs themselves. The exchange of virtual photons is made possible by the excited energy levels of the involved CDs made up of “vortices” of quasi-free electrons. As we have seen, in order the above coupling interaction to be effective, it must be needed to have both a high grade of coherence in water and a sufficient overlapping between evanescent e.m. fields belonging to the coupled CDs, namely the reciprocal distance between two CDs should allow a suitable overlapping between evanescent fields, according to their respective propagation depths

.

We’ll prove, in this chapter, that both these issues can be solved by enclosing water molecules in suitable waveguides whose walls are constituted by substances known as “metamaterials” (MTMs), characterized by a purely imaginary value of refraction index. The waveguides so obtained can then interact each other for a very long time, through the tunneling of virtual photons, due to the stability of coherent water against the thermal fluctuations and until a distance far longer than in the case of “naked” water CDs. The most remarkable feature of such devised waveguides is, however, the very fascinating, and so far, unexplored, possibility to use their mutual interaction to realize a physical system able to perform any type of quantum (hyper)computation in water CDs according to our theoretical model discussed in this chapter. We also argue that the peculiar features of such interaction make it able to overcome, in principle, some of the main issues which have hitherto prevented the realization of an actually useful and large-

Quantum (Hyper)Computation by Means of Water Coherent …

59

scale applicable quantum computer. One of the most remarkable results of our proposed model is it allows to potentially achieve a very high computational speed (close to the ultimate theoretical computational limit) by means of the implementation of a “network” of superfast interacting water CDs enclosed in MTM-made waveguides. We argue the discussed model doesn’t represent “just another way” to quantum computation but actually a new paradigm in this field and the first step towards the realization of an actual quantum hypercomputer based on the coherent dynamics of liquid water.

THE AMPLIFICATION AND STABILIZATION OF TUNNELLING – COUPLING INTERACTION BETWEEN WATER COHERENT DOMAINS According to the results discussed in chapter 1, in order the tunnelingcoupling interaction between water coherent domains (CDs) to occur, some conditions must be satisfied. First of all, we must ensure that the conditions for the runaway towards the coherent state (critical density and temperature) are satisfied. This can be easily verified since the runaway towards the QED coherent state (the “Coherent Ground State” or CGS) is a spontaneous process since this state represents, in its non-excited configuration, the true ground state, namely the state of minimum energy and highest stability. So, it would enough to ensure, at a given time, the density of water

, where

is its

“critical” value, and a sufficiently low temperature to start the superradiant phase transition (from this point on we’ll refer to chapter 1 for the main definitions and concepts used in this chapter). Once the coherent state has been attained, there are other two issues to address:

60

Luigi Maxmilian Caligiuri A. a suitable amplification of the evanescent e.m. field generated by coherent domains; B. the stabilization of the coherent fraction of liquid water with respect to the disruptive thermal fluctuations. About the first point, in the case of water,

and

so we obtain, respectively through (33) and (34) of chapter 1, for the “penetration depth” of evanescent field and the tunneling time (1)

meaning that it is sufficient for a couple of CDs to be at a mutual distance in order to have their respective evanescent fields overlapping. Furthermore, the coupling tunneling interaction could occur even if two CDs they are placed far apart each other, since any CD could interact with its relative first neighbors and so on, to form a spatially extended network of interacting domains. Nevertheless, the evanescent field decays quickly with the distance so that it could happen that, even for short distances, the field intensity could be too low for ensure the needed coupling between CDs. About the second point, as we have seen in chapter 1, interfacial water results to be fully coherent which will come in very handy for our purposes. We now prove to be able to solve both the issues A) and B) by considering to enclose water molecules inside the inner volume of suitable (even symmetrical) waveguides whose walls are composed of metamaterials (MTMs), namely a kind of material characterized by a purely imaginary value of refraction index as, for example, plasmas or ferromagnetic materials. We firstly note that, by assuming such configuration, we can consider all the water molecules enclosed in the inner space of a given waveguide as

Quantum (Hyper)Computation by Means of Water Coherent …

61

a unique coherent domain, at least with respect to the generation of the evanescent field inside the enclosed CDs [1]. If we further consider (and it is not difficult to make it so) the inner surface of the waveguide volume to be a hydrophilic surface we can consider, as shown in chapter 1, almost all the water to be coherent (namely characterized by a coherent fraction

) even at room temperature

so cancelling the decoherence effect due to the thermal fluctuations that represents one of the most critical aspect to solve for the physical realization of a quantum computer. For a rectangular shaped waveguide with length and width , the propagation of an e.m. wave along the -axis can be described by the stationary solutions of Maxwell’s equations corresponding to the socalled

modes.

The corresponding wave number

is

(2)

where

(3)

is the cutoff frequency whose minimum value, for a given waveguide, is given by (4)

with

.

Luigi Maxmilian Caligiuri

62

If we consider, for simplicity, a cylindrical-shaped waveguide (although the conclusions are valid in general) of length , we can assume

and radius of cross-section

, and (2) can be also rewritten as (5)

with

. This means the frequency of the e,m, wave

propagating inside waveguide is smaller than the corresponding value in free space, being always

. It is just what happens within a CD where

, so preventing the “release” of the coherent electromagnetic energy towards the outside of the CD. We see from (2) that, if

, the wave vector becomes

imaginary, implying no wave propagates across the waveguide but we can only have an evanescent e.m. field and we say that the wave guide is in cutoff. More specifically, being

the smallest value of

condition for a waveguide to be in cutoff is obviously

, a sufficient .

For a water CD associated to the considered electron transition we have , so we can assume the waveguide to be in cutoff regime with respect to the coherent field generated by water CDs contained inside it, if (6)

and (7)

Quantum (Hyper)Computation by Means of Water Coherent … where

63

is the typical length of a water CD in its fundamental state and

we have used the calculated value of

and

(see chapter 1).

We incidentally note the range of the possible values of waveguide radius must then satisfy the condition (8)

where

is given by (20) of chapter 1.

This result further confirms our preliminary conjecture about the possibility to think of such a water-filled MTM waveguide as a whole long cylindrical coherent domain in which, in turn, a lot of CDs are closely packed and whose overall evanescent field tunnel across the waveguide’s boundaries. It also proves a cylindrical e. m. waveguide satisfying the (8) is always in cutoff regime with respect to the coherent e.m. field generated in water molecules, at least if we don’t consider too high excited energy levels. As we have seen, the interaction between CDs can occur through the tunneling of evanescent modes (or virtual photons) between them. On the other hand, the interaction between two waveguides, by means of evanescent e.m. fields, has been already confirmed from the standpoint of classical physics and also applied in the construction of many devices (for example using optical fibers), just called “evanescent-field-optic-couplers”. In this case, the coupling strength between the two waveguides is quantified by the rate of energy transfer between them which is proportional to the overlap of the evanescent e.m. modes belonging to the two waveguides. The coupling between waveguides represents the classical counterpart of the quantum coupling occurring through the exchange of virtual photons, and this analogy further confirms the plausibility of our theory (see chapter 1). The physical configuration we have conjectured, apart from stabilizing the coherent fraction of water enclosed inside waveguides making it fully coherent, has the further noticeable effect to amplify, as we’ll see in the following, the evanescent field produced by each CDs, so increasing the

64 value of

Luigi Maxmilian Caligiuri . This feature has been already studied in previous publications

[1, 2] according to which a “near-perfect” tunneling and amplification of evanescent e.m. field is theoretically possible and experimentally proven in a waveguide even in cutoff regime, provided that it is partially filled with a MTM. This amplification process has been described by considering a MTMfilled waveguide section sandwiched between two empty sections operating in cut-off and two input/output waveguides above the cutoff. We have shown [1] that in this way the evanescent e.m. field described by (21) of chapter 1 would be amplified by passing through the MTMs structure. The overall result of this amplification process will be then an important increase of the evanescent penetration depth

.

The amplification and the tunneling of an evanescent e.m. wave outside the waveguide are possible when [1, 3] (9)

where

is the waveguide width,

and

respectively indicate the

dielectric permittivity of the MTM and the magnetic permittivity of vacuum. In this way, it would be sufficient to use a MTM with

in order to

allow the amplification and tunneling of the evanescent e.m. field into “free” space up to over four times the original field amplitude after the interaction with the MTM-filled portion of the waveguide. Metamaterials just consist of recurring structures, like several nano-resonators, characterized by a period small if compared to the optical wavelength and able to excite electromagnetic oscillations when hit by photons of suitable frequency. For frequencies different than the resonance one, this interaction can result in a negative value of permeability relation

and then, due to the

Quantum (Hyper)Computation by Means of Water Coherent …

65 (10)

where

and

respectively indicate the relative electric and magnetic

permittivity, it determines purely imaginary wave number which can produce evanescent wave enhancement.

UNIVERSAL QUANTUM (HYPER) COMPUTATION BY WATER COHERENT DOMAINS According to the results of chapter 1 and of the previous section, the coupling tunneling interaction between water CDs can then act even if they are placed far apart each other. This could occur even when a CD directly interacts only with its own first neighbors CDs and each of the latter interacts, in turn, with its own first neighbors CDs and so on, or when two far apart CDs directly interact each other due to the evanescent field amplification. In the following we’ll discuss how to use such interaction to realize, in principle, any type of quantum (hyper)computation in water CDs across a spatially extended network of interacting CDs.

The Concept of Universal Quantum Computation Quantum computation manipulates quantum information through dynamic transformation of quantum systems. A quantum transformation is a mapping from the state space of quantum system in itself. As well-known, from a mathematical viewpoint such transformations are described by unitary linear operators (or matrices) since geometrically they correspond to rotations of the complex vector space associated with the quantum state space.

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Just as for classical computation, quantum computation is based on the concept that arbitrarily complex computations can be achieved by composing simple elements. We call a quantum gate any quantum state transformation acting on only a small number of qubits More specifically, any quantum state transformation performed on an -qubits system can be realized by using a sequence of one and two-qubits quantum state transformations or gates. So, in order to implement a quantum computer able to perform arbitrary quantum transformations, it would be appropriate to have only a finite number of gates that can be physically realizable, provided they can generate all the possible unitary transformations. Nevertheless, this is theoretically impossible since a finite set of unitary transformations can only generate a countable set of quantum transformations. On the other hand, it can be proven that all the (unitary) quantum transformations could be efficiently simulated (in the sense of giving a close approximation of a given transformation) by means of a suitable finite sets of quantum gates. A certain number of such set exists but it is useful to choose the one including all one-qubit gates with a suitable two-qubits gate as we’ll see in the following. It is well-known that, in classical computation, a small set of gates (like, for example, AND, OR, NOT gates) can be used to compute an arbitrary classical function and then such set is said to be universal for classical computation. A similar result holds for quantum computation where a set of gates can be found to be used to realize universal quantum computation. This result is based on the fact that arbitrary unitary transformations can be implemented from a set of primitive transformations, including the twoqubits CNOT (controlled-NOT) gate in addition to three kinds of single qubit gates. One goal of quantum computation is just the choice of the suitable sets of unitary transformations that can be performed efficiently. The starting point is to show that two-level unitary gates are universal for quantum computation, namely that every unitary matrix can be decomposed into a product of two-levels unitary matrices [4]. A two-level unitary matrix is a unitary matrix which acts non-trivially only on two or less vector components. More precisely, it has been shown that an arbitrary

Quantum (Hyper)Computation by Means of Water Coherent …

67

unitary matrix on a -dimensional Hilbert space can be written as a product of two-level unitary matrices. In turn an arbitrary two-level unitary operation on the space state of qubits may be realized by implementing single qubit and CNOT gates. Nevertheless, there is no general method to construct such set of single qubit and CNOT gates in a way to be resistant to quantum computation errors. However there exist a discrete set of this gates that allow for correcting such errors (so being errors resistant) by using quantum error-correcting codes, namely a set of gates able to approximate, within an acceptable level of error, the unitary operators required. If U and operators on the same quantum space where be implemented and

are two unitary

represents the operator to

is the unitary operator that is really used to

approximate it, the error in using

instead of

is given by (11)

where the max is taken over all the normalized states The (11) means that if measurement on the state

is sufficiently small, then any will give approximatively the same

measurement statistic of that performed on the state state

in the state space.

for any initial

.

If we indicate as measurement result

and

respectively the probabilities to obtain a

(from a suitable POVM – Positive Operator-Value

Measure) by using the operator

or

, we have (12)

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68

so, if the error is small enough, the two measurements have the same outcomes probabilities. Furthermore, if we use a sequence of n gates to approximate a sequence of gates

, the overall

error is (13)

If we consider a quantum circuit composed of gates, then the (12) and (13) tell us that the probability of the different measurement outcomes achieved with the “approximated” circuit (made of the gates instead of the acutal circuit (made of the gates smaller than a given quantity

)

) can be made

if, for every

(14)

As we have said, different set of universal gates can be used to realize, within a given approximation, every quantum computation. In the following we consider the particular set composed by the following unitary transformations: (a) the Hadamard gate (b) the

gate

(not to be confused with the Hamiltonian);

;

(c) the phase gate ; (d) the CNOT (Controlled-NOT) gate. As already shown [4] the choice of this set is particularly suitable since it is possible to give, for these gates, a fault-tolerant construction. The Hadamard gate is a single qubit transformation defined by the unitary operation

Quantum (Hyper)Computation by Means of Water Coherent …

69

(15)

where basis

and

are the

and

Pauli matrices. In the computational

is given by the matrix

(16)

and realizes an even superposition of

and

from either of the standard

basis elements, namely the transformation

(17)

The gate and the phase gate are particular cases of a wider class of transformations that generate a relative phase between two basis states. These are obtained by considering the transformation

(18)

where

is the

- Pauli matrix and

operator (18) generates a relative phase

is a real number. The unitary . The

(or

) gate is given

by

(19)

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70 and the phase gate

by

(20)

The CNOT gate is a quantum gate with two input qubits, known as the control qubit and target qubit in which the control qubit remains unchanged by the transformation while the target qubit is flipped if the control qubit is equal to 1. In matrix notation, when expressed in the computational basis of two-qubits

, it is given by

(21)

The specific set of gates above discussed can be then used to simulate every generic unitary transformation; in particular it can be shown [4, 5], basing on the results that every single qubit unitary operator can be written (apart an overall phase factor) in the form (22)

where

is the rotation operator around the three - dimensional unit

vector by the angle , then the error in approximating the exact one-qubit operator is given by

(23)

Quantum (Hyper)Computation by Means of Water Coherent … where

is the Hadamard gate and , ,

(23) means that, for every gate is smaller than

are three positive integers. The

, there exist suitable ,

the error in approximating the operator

and , such as

with an Hadamard and a

. This important result allows us to approximate a

generic quantum circuit composed by any Hadamard and

71

gates by using CNOT,

gates only (the use of phase gate

is introduced to

make the circuit fault-tolerant) with a total error less than approximate, as seen above, every gate within an error less than

if we .

. We’ll return on the realization of one-qubit gates in the following, now we consider how to realize a CNOT gate, which is a fundamental element of the set of universal quantum gates, by starting from the square-root of SWAP gate

.

This gate is defined so that when applied twice it realize the SWAP gate given, in the computational basis, by the matrix

(24)

The gate

is then represented, in the same basis, as

(25)

We can prove this fundamental result, by considering at first the socalled two- qubits matrix

gate (namely the “controlled-Z” gate), defined by the

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72

(26)

that applies on the target qubit the transformation bit is set to

if the control

. The meaningfulness of this transformation in our discussion

is due to the fact it can be used to construct, by means of

, the CNOT

gate. It is easy to show, for example, by using the matrix representation in the computational basis, that

(27)

where the superscript

(

) indicates the operator

acts on the first

(second) qubit. The importance of such decomposition is that we can now be able to write the CNOT gate as [6] (28)

where

indicates the tensor product between the quantum subspaces

respectively spanned by the qubits and . Since the CNOT gate can be considered, as we have seen, a universal two-qubits gate for quantum computing, the

gate can then represent the building block of any

quantum computer when properly joined with the “universal” sequence of single qubit gates above considered. The remaining step is then to prove we can use the evanescent tunneling coupling interaction between water CDs to construct the suitable quantum one- and two-qubits gates able to realize universal quantum computation.

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73

The Realization of Two-Qubits Quantum Gates through Water Coherent Domains The computational features of the water CDs can be uncovered by going back to analyze the evanescent coupling between two water CDs. It is then appropriate to study the tunneling exchange of virtual photons between two water-filled-MTM waveguides 1 and 2 by using the dual-rail representation for the photon, according to which we would adopt the following encoding for the two logic states

and

(29)

Where

and

respectively represent the state in which a photon is absent

and present as spatial mode in the i-th waveguides. If we consider the exchange of one photon, it can be shown the Hamiltonian of the system has the same form of (42) of chapter 1, then if we assume

and

, where the states

and

are those

defined in (41) of chapter 1, we have

(30)

In our theoretical framework the propagation of a photon in an optical fiber is likened to the presence of a photonic spatial mode, due to tunneling, in one or the other waveguide as sketched in Figure 1. This can be shown if we consider, assuming a not too strong coupling, the total Hamiltonian for the system (we limit ourselves to the photon dynamics) [7] (31)

Luigi Maxmilian Caligiuri

74 where

(

) is the annihilation (creation) operator for the mode

i-th waveguide (

) acting on the vacuum state

and

in the is a real

parameter quantifying the coupling strength. With this notation ,

and, if

is approximatively

independent of the value of , the total Hamiltonian has the same form of as (42) of chapter 1 and we can deduce the (31).

Figure 1. Evanescent coupling and virtual photon exchange between two waveguides filled with water CDs.

We then easily note that, if we start the systems in the state time

, after a

given by

(32)

it has the maximum probability to be in the state

as results from (49)

of chapter 1. For a given value of , by suitably choosing the values of can obtain

then realizing the transformation

, such as

we

Quantum (Hyper)Computation by Means of Water Coherent …

75 (33)

that switches the quantum state of the system and then acts like a SWAP gate on its logic state, so we can assume Even more interestingly, for

. , namely

(34)

we can define a new operator

by

(35)

namely (Figure 2) (36)

whose action on the computational basis is given by

(37)

The huge importance of such result will be discussed in deep in the following sections in which we’ll show how to use it to realize, in principle,

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76

the building block of any quantum computer by using the virtual photonevanescent coupling interaction between water CDs.

Figure 2. Realization of quantum SWAP gate by means of virtual photons exchange in waveguides filled with water CDs.

The Realization of Single Qubit Quantum Gates through Water Coherent Domains In chapter 1 we have shown a very important result, namely the timeevolution operator of a system of two interacting (through evanescenttunneling coupling) CDs can be expressed as the product of a rotation and an overall phase shift. This means the evanescent tunneling coupling dynamics above described can be used to realize any single qubit quantum gate. We are then primarily interested in realizing the quantum gates representing the Hadamard, the and the phase gates in the form given by (19). Our goal is to find the values of the parameters , real numbers

,

and

and

(i.e., the three

) such as the unitary operator (see chapter 1)

(38)

Quantum (Hyper)Computation by Means of Water Coherent … respectively represents the Hadamard, the

77

and the phase gate and then

express them as a function of the physical quantities that characterizes the dynamics of our systems namely, in our approximation, In the case of the (38)

,

and .

gate we have, from its definition (15) and from

(39)

In order the (39) to be an identity the imaginary part of the l.h.s. must be zero so we have the condition

(40)

that is satisfied if

and

. Using these values, the (39) for the real

parts becomes

(41)

that holds if we assume

and

. In order to construct the

gate, we note, from its definition (19), that So, comparing this latter with (38) we see they are equal if

.

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78

(42)

that, in turn, implies

. Finally, we can simply realize the

phase gate by writing

(43)

and then, following the same reasoning leading to (42), we have in this case

(44)

and, similarly,

.

We also note a very interesting properties of such transformations, namely they all require

that is just what the coupling dynamics

predicts if we consider the (51) of chapter 1. The values of parameters , and

used to realize the

,

and

gates are summarized in Table 1.

Table 1. Values of the parameters of operator U to implement quantum gates Quantum gate Hadamard

1 2

0

1 2

T-gate

0

0

1

S-gate

0

0

1

We have proven a very important result namely that, by a suitable selection of the physical parameters

,

,

and by letting the system

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79

evolving for an appropriate time , the system composed by interacting water CDs can realize, in principle, any generic quantum gate. One of the most important features in quantum computation, as well as in classical computation, is the possibility to realize, general controlled operations (such as CNOT) in quantum circuits. An important theorem in quantum computing states that a generic unitary operation on a single qubit can be also written as a combination of simple rotations around “standard” axes in the form (45)

where

are suitable real numbers. Furthermore, there also exists

unitary operators

acting on a single qubit such that

and (46)

where

and

(47)

The fundamental importance of the decomposition given by (46) is that is allows the realization of any controlled unitary operation

on qubits,

indicated as , by using simple unitary transformations on single qubit only. For a controlled operation on a single qubit the general circuit representation of the equation (46) is represented in Figure 3. By comparing the general decomposition (45) with (46) we can calculate the values of the “new” angles (generally different than in the (38)), , and

such that we obtain the unitary transformation given by (38). In this

way, we would be able to realize any unitary operation on a single qubit through controlled unitary operations on a single qubit. In turn, by

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Luigi Maxmilian Caligiuri

considering the (46) and the values of Table 1 we’ll able to realize quantum circuits realizing the operations on single qubit by suitably setting the physical parameters of a system composed by evanescent coupled water CDs.

Figure 3. Circuit representation of controlled unitary operation on single qubit.

By equating the matrix representations of (46) and (38) we obtain, after some simple but tedious algebraical manipulations, the following four systems (one for each matrix element in (46) and (38):

(48)

(49)

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81

(50)

(51)

By dividing side by side the two equations in each of the above systems we obtain the new system

(52)

whose solution can be obtained as

(53)

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82 Finally, the value of

can be calculated by using any of the systems

(48)-(51); for example, if we consider the (48), we obtain

(54)

where is given by (52). The previous discussion can be easily generalized to the realization of multiple - qubits conditioned unitary operations in the following way. Suppose to have control qubits and on the

input qubits, where

is the number

that of target qubits, and a unitary operator

acting

qubits.

We define the controlled operation

, that uses the - th qubit as

control, as the operator (55)

where [4] the exponent in the operator , namely the operator

stands for the product of bits

is applied to the target

qubits if all the

control qubits are set to one, otherwise all the qubits remain unchanged.

Controlling Quantum Gates Composed of Water Coherent Domains As we have seen in the previous sections, if we consider the QED coherent dynamics of water and, in particular, by exploiting the evanescenttunneling coupling interaction between water CDs arising from their energy excited states we can realize, in principle, any kind of quantum computation. This very fascinating and so far unknown result comes from the universality of the set composed by

,

,

and

quantum gates all

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83

achievable by making use of the above mentioned interaction between the water CDs. We have also seen, in order this possibility to be realized, we can stabilize the coherent fraction of water (to counteract thermal collisions) and amplify the evanescent e.m. fields generated by CDs by enclosing water molecules inside metamaterial-made waveguides. In addition, as codified by Di Vincenzo [4-6], the realization of a feasible quantum computing scheme must satisfy, in particular, further requirements (apart the implementation of a set of universal quantum gates and the long decoherence time, both of which are satisfied within the model of computation proposed by us) as, in particular: (a) the physical realization of well-defined qubits; (b) the initialization in a well-defined state, representing the qubit

.

Both these features require the ability to control the physical system implementing the quantum computation. In our framework, this means the skill to control the quantum state of water CDs which are fully characterized, being macroscopically coherent, by their quantum phase . Consequently, by controlling this phase with precision allows us to exactly controlling the quantum state of the system including its excited states i.e., the physical parameters driving the evanescent tunneling interaction between CDs and then the quantum gates based on them. We know from the general theory that the interaction of a particle of charge and mass with an e.m. field is described by the Schrodinger equation:

(56)

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Luigi Maxmilian Caligiuri

where the state vector of the system is amplitude and

its phase. For non-linear systems

,

its

so that the

relative Schrodinger equation becomes non-linear as well, containing the self-interaction terms. This means the phase factor in the state vector cannot be removed by a global phase transformation (phase invariance) in the Schrodinger equation, with deep consequence on the system dynamics. As already recognized [1, 8-10], in such complex systems, phase acquires an important physical meaning (ceasing to be a pure mathematical entity) able to drive its dynamics since the electromagnetic potentials of the different components of the system determines its phase evolution that, in turn, drives the system evolution. This can be easily seen by remember the classical fields given by

and

(57)

are invariant under gauge transformation of potentials:

(58)

where is some scalar function and observing that, under this transformation, the (56) assumes the invariant form

(59)

if the wavefunction transforms as (60)

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85

from which is clear that the modification of potentials (through the gauge ) affects only the phase of wavefunction and not its amplitude that depends by the “physical” fields (that, in fact, remains unaffected by gauge transformation). A change in phase then modifies the coherent oscillation frequency and vice versa. Even more interesting, such a change would also affect the electromagnetic potentials

of the system.

If describes an non - coherent state then its phase-shift due to a change in potentials has no effect on its evolution, on the contrary, for a coherent state, being an eigenstate of the phase operator, a phase modification would determine a substantial change in its physical state and evolution. The possible interaction among two or more CDs or between a CD and the surrounding “environment” can be then considered as mediated by the action of phase operator . As we have seen, in the coherent quantum state of water, the wavefunction (both of particles and e.m. field) acquires a macroscopic meaning and a wave-like behavior, a situation quite similar to that occurring in laser or in superconductivity where the entire collection of microscopic objects (respectively photons and super-electrons) are described in terms of an electromagnetic field with a well-defined phase. By exploiting such analogy, we can state a relationship between phase and potentials in a coherent quantum state of water through the Josephson’s equations

(61)

(62)

where the quantum phase includes, in the excited states, the information about rotational frequency of the associated cold vortices.

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86

The above equations show that e.m. potentials are able to modify the phase of the coherent state, including the frequency of excited vortices, and vice versa even in absence of the “physical” fields as occurs in the Aharonov-Bohm effect [11]. The e. m. potentials can then establish a long range as well as superfast (actually faster than light being, driven by a phase velocity) interaction. We consider the consequences of this process from the standpoint of transmission and storage of quantum information in a forthcoming publication. The manipulation of vector potentials can be then used to “program” and control the quantum gates implemented by means of the evanescent tunneling coupling between water CDs. It is conceptually simple to see that, by means of a tuning of the vector potentials, we can set the state of a water CD to a well-defined value of phase just reflecting, in turn, a well-defined value of its coherent energy state. We can then easily satisfy the previous requirement b) by setting a correspondence between a suitable value of phase and the initial quantum state of computation, namely (63)

The requirement a) is obviously satisfied within the coherent dynamics, since any coherent state corresponds to an eigenstate of quantum phase operator and then is always well-defined. As we have seen, controlling the “water-CD” quantum gates substantially amounts to drive their dynamics evolution and then, ultimately, by excite and de-excite them when required. As suggested in chapter 1, the energy stored in the coherent excited states of water CDs can be (non-thermally) released outward through the chemical activation of suitable macromolecules surrounding the CDs. These molecules, in turn, can be automatically selected by previously “programming” the frequency of coherent oscillation to attract a certain number of “guest” molecules able to resonate with the CD itself. The time

required by a CD to complete an oscillation is given by

Quantum (Hyper)Computation by Means of Water Coherent …

87 (64)

where

is the time required to excite the CD that is related to the rate of

energy storage from the “environment”;

is the time required to activate

the co-resonating “guest” molecules that is inversely proportional to the height of the activation energy barrier required by the possible reactions; is the time required by the chemical reactions to occur (after activation). The last point deserves some attention since the energy output of such reactions: (a) could be used to energetically “charge” new water CDs and then for subsequent quantum calculations; (b) should be removed from the system, if it were in the form of thermal energy, in order to not reduce the coherent fraction of water molecules. In the case (a) it would be possible to design a suitable chain of chemical reactions able to perform a given set of quantum (phase) transformations of water CDs and so a desired computational scheme (quantum algorithms). In the case (b) we must provide for a suitable method to throw out such energy from the system. These aspects, as well as the proposal of further (non - chemical) ways to obtain water CDs “discharge”, will be discussed in more details in forthcoming works. We again stress that the interaction that determines the change of phase, driven by vector potentials, propagates with superluminal speed, determining the transmission of information, involving only a local supply of energy arising from quantum vacuum fluctuations. The time required to obtain a given quantum state of water CDs, after the action of vector potential has taken place, can be vey short (we recall that the onset of the coherent state of water would occur in about ), while the time required

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88

to set the quantum gates, both for one- and two-qubits transformations, can be roughly extimated by considering, for each of them the relations between the physical parameters and the computational features of the system. More specifically, for one-qubit quantum gates, the time required to setup a given gate

,

or

), can be calculated through (55) of chapter 1

and the values of Table 1, where the values of parameters and depend on the particular gate required. Actually, what cares for the dynamical evolution of the physical system (in order to realize quantum gates) is just the value of

defined by (55) of chapter 1, since the unitary operator (38)

is always defined up to an unimportant global phase . We can see this by inserting the (51) in (55) of chapter 1, so that we have

(65)

showing that

depends only on the values of the difference

and

. The same consideration holds for the physical probability given by (49) of chapter 1, showing the dynamical evolution of the system depends only on these quantities as well as it occurs, by virtue of (32), for the realization of the SWAP gate. In summary, for given values of interval

,

and , a suitable choice of time

, completely specifies all the possible one-qubit gates.

For example, in the case of Hadamard gate we obtain, by using (65) and Table 1

(66)

and similarly for the other gates.

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89

On the other hand, by (34), the time evolution of a couple of evanescenttunneling interacting CDs, also determines, through (27) and (28), the realization of any two-qubits quantum gate and consequently, in principle, of all the possible controlled and multiple-qubits gates. The overall oscillation time of an interacting water CD, used to perform a quantum gate operation, that we could define as the “operational time”, is then given by (67)

where

is given by (64) and

is the time required to set up a determined

quantum gate (namely a suitable combination of the time intervals get from (65) for one-qubit gates and from (34) for two-qubits and multiple gates). In summary we could take control of the whole process “turning-on” and “turning-off” the quantum gates, by adjusting the terms in (64).

Speed and Memory of Quantum Computation through Tunneling - Coupled Water Coherent Domains When speaking about computation, two main questions primarily arise: (a) how fast does the computation carry out? (b) how much memory does the system have? Both these questions obviously relate to the dynamics and specific features of the system to be considered for computation, but it has been shown [12, 13] it is possible to get some bounds to these quantities as imposed by quantum mechanics itself that, nevertheless, could be reviewed when considering “non-standard” type of computation, like the co-called “hypercomputation” [1, 3, 14, 15] already studied by us in earlier papers. The question of computational speed may be simplified by asking the maximum number of distinct states a system could pass through per unit of time. For a quantum system, the distinctness of states coincides with the

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90

mathematical concept of orthogonality so that if two states are distinct the scalar product of their kets must be zero. The speed at which a system can evolve passing through its orthogonal dynamical states can be measured by the minimum time it needs to evolve between two of these states. Some authors have suggested [12, 13] this time is expressed in terms of the spread of the energy (namely its standard deviation) namely

of the system,

(68)

that, in the computational realm can be interpreted as the number of logical operations per unit of time performed by the system. Nevertheless, as properly pointed out [13], this wouldn’t establish an actual limit because, for a given bounded value of average energy of a system its spread can be arbitrarily high. A more physically sounded bound can be expressed in terms of the maximum value of energy of the system as

(69)

where

is the maximum energy eigenvalue. This bound is easily

understandable by considering that, in the energy basis, a generic quantum state can be expressed as a superposition of frequencies so that, if the energy spectrum is upper limited by maximum frequency

then we must have a corresponding from which we can deduce the (69) if we

interpret the frequency as the number of orthogonal states per unit of time the system passes through.

Quantum (Hyper)Computation by Means of Water Coherent … A more accurate estimate of the minimum time

91

required by any

state of a given system to evolve into an orthogonal state [13] leads to a number of logical operations for unit of time bounded by

(70)

in which

is the average energy of the system that we suppose, for

simplicity, to have a discrete energy spectrum ordered eigenvalues with

of non-decreasing

.

In particular, it can be shown the limit imposed by (70) holds exactly when the energy spectrum includes the energy state can be constructed

. In this case the following

(71)

that evolves, after a time

, into the state

(72)

that is orthogonal to

.

We also note that, after again a time

,

, so that the

system oscillates between these two orthogonal states. In this case we have and then, apart from (70), also the following one

(73)

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92

In the case of a dynamical evolution through a long sequence of orthogonal states instead, we have [13]

(74)

that, in the case of nearly orthogonal states, implies In a two-levels system having the spectrum

. we have

(75)

that, if we set

, imply

so that the two bounds (70) and (73)

numerically coincide. The overall computational time that can be attained within our model should then include: (a) the operational time ; (b) the time required by each quantum gate to “communicate” with any other; (c) the decoherence time of the system; (d) the delay time of memorization and readout of the information; (e) the implementation of parallel and/or serial computational schemes. The operational time

includes the two contributions

and

.

About the first one, we firstly note that, having the dimension of an energy, the term can be related to the interaction energy of the system composed by two evanescent-tunneling interacting water CDs. Consequently, the quantity

Quantum (Hyper)Computation by Means of Water Coherent …

93

(76)

could be interpreted as the square of the energy spread

of the

interacting system of two water CDs. This can be easily seen by noting that, for a two-levels system characterized by two orthogonal states with energy under the Hamiltonian the system is just

and

evolving

, the energy spread of .

We also note, due to the distinctive features of excited states of water CDs (explained in chapter 1), we could have either an energy eigenstate equal to

in the excited spectrum, or

, so that both the bounds on

calculation speed given by (70) and (73) would be simultaneously satisfied. This result is very interesting since it shows the quantum gates realized in such a way could operate very closely the theoretical limit, whatever expressed in the form of average energy (70) or energy spread (73), during the time needed to perform the elementary quantum operation both for one and two-qubits gates. Furthermore, for a given value of , we could reduce the time required to implement a gate by increasing the difference

. In the case of

excited water CDs, it would be always possible, at least until the energy gap per molecule and per CD is not exceeded. The maximum energy gap per molecule is about and then, by assuming an average number of water molecules per CD of about

, we have a single water CD could

gather in its own excited states up to about

of energy.

It is also interesting to analyze the relationship between the interaction “strength” and the tunneling time between two interacting CDs.

of the evanescent (virtual) photons

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94

It is immediate to note that, the shorter is

, the higher is the value

of . As we have seen in chapter 1, the tunneling time is independent (at least for not too long barriers) from the barrier length and depends on the energy of tunneling particle only so that

. In our case, we then

have (77)

since, for the i-th excited state of water CD, it is always

. Even for

the tunneling time, we could then increase the energy of the CD excited level in the order to decrease the tunneling time. As an indication of an upper bound for this tunneling time we could assume is (

the

number

) and

of

water

molecules

where included

in a

CD

is the energy gap per molecule characterizing the

coherent state, so obtaining (78)

The calculation of

also allows us to obtain an estimation of the

tunneling “velocity” (the group velocity of evanescent waves or, equivalently, that of the virtual photons undergoing tunneling) as

(79)

that is superluminal if

Quantum (Hyper)Computation by Means of Water Coherent …

95

(80)

Now recalling that

we see in our case the maximum value

of tunneling photons is superluminal, i.e.,

, namely about ten times

the value of light velocity in vacuum even for a non-excited CD. An alternative estimation of tunneling velocity for the tunneling photon can be obtained as [1, 14, 15]

(81)

where is the particle energy and is the tunneling distance that further confirms at least for not too long tunneling barriers. An important question to consider for the assessment of concerns the method used to “initialize” the quantum state of a given water CD before and after the gate running operation. If such method would involve, as proposed so far, the consideration of the resonance between the water CD and some “guest” macromolecules and the resulting chemical reactions, we should calculate by using the expression of given by (64). On the other hand, the coherent states are eigenstates of phase operator, entirely characterized, in terms of their dynamical evolution and macroscopic behavior, by the phase value whose modification, through e.m. vector potentials, could in turn modify their coherent frequency of oscillation

. Such “phase interaction”, as already discussed above, being

pure information, would propagate between any two points of a coherent environment in a zero time, this posing no difficulty from the standpoint of Special Theory of Relativity [1, 16] since no energy propagation is associated to phase shift, but only a local supply of energy to CD.

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By implementing this possibility, we could eventually reduce to nearly zero the time required to set the state of a quantum gate (namely the physical state of the involved water CDs) as well as the delay time occurring during the communication between two quantum gates. The study of such implementation, implying an interplay between vector potentials and quantum phase of water CDs, is currently in progress. As we have seen, the energy accumulated in the water excited coherent vortices can be reach even very high values, so that the energy cost of computation performed through quantum gates realized through water CDs, since they would operate nearly at the lower computational limit imposed by quantum mechanics, could be several orders of magnitude smaller than that of conventional computer systems. In an ideal system this energy for computational step is not dissipated during each step but could be available for the next ones. In general, a quantum computer could operate close to this limit if the available energy is equated to the energy range spanned by the Hamiltonian eigenvalues, namely the internal degrees of freedom of the system. In the coherent systems, as water CDs, a characterizing feature is just the possibility to accumulate a lot of environmental energy at high entropy into the low entropy CD inner degrees of freedom, even using its metastable excited levels. It is commonly believed that quantum computing could be intrinsically much more powerful than classical computation using traditional silicon processors. Nevertheless, interconnection delays inside logical gates cannot be never fully eliminated form any real electronic components even in quantum computers. This effect could be compensated, by introducing, as in the model discussed above, quantum tunneling processes in order to meaningfully speed up computation, bringing the overall transmission rate very close to its ultimate speed limit [12, 13] by boosting the speed of signal propagation and, consequently, computers performances. Moreover, one of the most important problem affecting the actual realization of usable quantum computers, often called the “enemy” of quantum computing, is the question of the decoherence of quantum states, according to which, in order to quantum computation to take place

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effectively, it is required that during the calculation time the quantum system remains in a superposition of its quantum states. So, the time required to perform quantum calculations should be much lesser than the decoherence time due to the system-environment interaction. In our model, quantum qubits can be codified by the logical states and

, corresponding to the photon spatial modes, in turn associated to

some excited levels of CDs. As we have shown, in the coherent system composed by water CDs enclosed inside MTs, the matter-field dynamics stabilized the coherent fraction, even at room temperature, at the value of so being able to prevent environmental decoherence. Such excited states (as well as the whole CD structure) cannot in fact decay thermally so they are theoretically unaffected by such “environmental” decoherence. On the other hand, the energy release from excited coherent domains through non-thermal channels could pose a not negligible instability issue if it would be subsequently converted in heat and not adequately taken off from the system or at least reused to someway increase the internal coherence. In general, one could think the overall calculation speed is also influenced by the computational configuration, namely if the logical operations are performed in serial or in parallel. For a quantum computer running close to the computational limits known so far, if

is the spread in energy and

the calculation rate

of the l-th gate, the maximum number of operations per unit of time of the system for

total gates, can be written as

(82)

According to the commonly accepted conception, this limit would be independent from the computational architecture. In fact, if the total energy of the system is allocated to few logic gates (logic operations

Luigi Maxmilian Caligiuri

98

performed in series), then we expect an increase of the values of for each gate , but also a small value of

and

and vice versa if the

energy would be allocated in a larger number of logic gates (parallel computing) so that the overall quantity would be approximatively the same. Nevertheless, in our proposed model of computation, it could happen a very interesting circumstance. From the general theory of QED coherence in matter, briefly reviewed in chapter 1, we know the onset of the coherence determines an energy gap that determines the coherent state to have an energy smaller than the non-coherent one. A further increase of the coherence of the system, as those occurring in supercoherence (on which our model of calculation is based), would in turn increase the stability of the system and the correlated energy gap. In this case the increase in the number of possible logic gates (proportional to the number of CDs) would be accompanied by an increase of the energy available for each gate (due to a wider range of possible excited energy levels for each CD). Such dynamics opens, as we’ll discuss in a forthcoming publication, a very remarkable perspective to further boost the computational power of the proposed model of quantum computation based on water CDs. Just to cite one of these perspectives, the implementation of an extended network of correlated (through the tunneling and/or phase interaction) water CDs oscillating in phase with each other, could reduce to nearly zero the overall parallel computational time, when increasing the number of the tuned oscillating elementary systems (water molecules and even other chemical species in the form of macromolecules). In fact, in this case, we could rewrite (82) in the form

(83)

in which, when we have

increases, each term of the sum increases as well, so that

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99 (84)

if

we recover, in principle, the concept of “hyper-computing”,

already analyzed, in a preliminary form, in several our previous publications [1, 3, 14, 15]. Finally, in a real computer architecture, it is very important also to consider the question of scalability, namely the possibility to perform quantum operations on a large number of qubits. Unlike liquid systems so far considered for implementing quantum computation (like liquid state nuclear magnetic resonance NMR) in which every molecule virtually represents a qubit and the huge number of molecules involved (of order of in NMR) makes it difficult to scale to larger number of qubits, in coherent liquid water, despite each CD can contain millions of molecules, it has to be considered, due to the coherent dynamics, as a single quantum object. The last important question concerns the memorization and readout of quantum information in the specific physical system used to compute, that is in our case the (coherent) network of water coherent CDs, and the related time required to perform such operations. The amount of information that can be processed is theoretically limited by the number of physical distinct states accessible to the system that, in our model, is a consequence of the QED coherent dynamics as well. All these features are strictly related to the specific processes used to implement memorization and readout of quantum information that, in our model, are linked to the mutual phase interaction between water CDs and to the interaction between them and their surrounding electromagneticthermodynamics environment which they “probe” through e.m. vector potentials. A more detailed study of role of these interactions in the storage and retrieval of quantum information is not trivial and it will be carried out in a forthcoming work.

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CONCLUSION In this chapter, starting from the results of QED coherence in matter applied to liquid water, we have seen how the formation of energy-excited states in water coherent domains can be used to realize universal quantum gates for one and two-qubits able, in principle, to perform any type of quantum computing. Such gates can operate at a very high speed (in terms of number of logical operations for unit of time and for gate) and very closely to the ultimate limit imposed by the fundamental principles of quantum mechanics so far generally accepted, by adjusting the energy levels of the water coherent domains (corresponding to its excited levels) used to realize them and the parameters ruling the tunneling interaction between them. One of the most important features of the proposed model is that if we consider a system composed by a very high number of interacting water coherent domains, in turn oscillating in phase with each other (i.e., they are “supercoherent”), we could realize an extended network of quantum gates acting in parallel whose computational speed would be proportional to number of the matter components that join the common coherent oscillation and then much higher than the operational speed of a single quantum gate. Despite different questions (like, for example, how to store and retrieve the quantum information, how this information “propagates” between water coherent domain, the possible existence of other interaction mechanisms apart the tunneling of virtual photons already considered) would need further in-depth studies that will be covered in future researches, the proposed model opens very fascinating perspectives towards a new conception of quantum computing using water molecules as physical substrate. The computational layer would be realized by considering a so far unexplored interaction arising between the evanescent e.m. fields generated by water coherent domains, involving the exchange of virtual photons by tunneling effect as well as a “phase” interaction ruled by the electromagnetic vector potentials to set the communication between quantum gates and the physical states used for computation.

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The possible applications of such model, if further improved, could be hugely interesting both from a theoretical and applicative standpoint, allowing the development of unimaginable technologies that would require extraordinarily high computational power (from the discover of new physical laws to artificial intelligence, from the simulation of high complex systems to the elaboration of forecasts in environmental, health and epidemiological fields, etc.).

REFERENCES [1]

[2]

[3]

[4]

[5] [6] [7]

Caligiuri, L. M. & Musha, T. (2016). The Superluminal Universe: from Quantum Vacuum to Brain Mechanism and beyond. New York: NOVA Science Publishers. Baena, J. D., Jelinek, L., Marwues, R. & Medina, F. (2005). “Nearperfect tunneling and amplificationof evanescent electromagnetic waves in a waveguide filled by a metamaterial: Theory and experiments”. Phys. Rev. E, 72, 075116. Caligiuri, L. M. & Musha, T. (2016). “Superluminal Photons Tunneling through Brain Microtubules Modeles as Metamaterials and Quantum Computation”, in Advanced Engineering Materials and Modeling, edited by: A. Tiwari, N. Arul Murugan, and R. Ahula, 291333. New Jersey: Wiley Scrivener Publishing LLC. Nielsen, M. A. & Chuang, I. L. (2016). Quantum Computation and Quantum Information (10th Anniversary Edition). Cambridge: Cambridge University Press. Stolze, J. & Suter, D. (2008). Quantum Computing. Weinheim: WileyVCH. Le Bellac, M. (2006). A short Introduction to Quantum Information and Quantum Computation. Cambridge: Cambridge University Press. Franson, J. D., Jacobs, B. C. & Pittman, T. B. “Quantum computing using single photons and the Zeno effect”. Phys. Rev. A, 70, 062302.

102 [8] [9]

[10]

[11] [12] [13] [14]

[15]

[16]

Luigi Maxmilian Caligiuri Preparata, G. (1995). QED Coherence in Matter. Singapore, London, New York: World Scientific. Caligiuri, L. M. (2015). “The origin of inertia and matter as a superradiant phase transition of quantum vacuum”. In Unified Field Mechanics: Natural Science Beyond the Veil of Spacetime, edited by: R. L. Amoroso, L. H. Kauffman, and P. Rowlands, 374-396. Singapore, London, New York: World Scientific. Brizhik, L., Del Giudice, E., Jorgensen, S. E., Marchettini, N. & Tiezzi, E. (2009). “The role of electromagnetico potentitals in the evolutionary dynamics of ecosystems”. Ecological Modelling, 220, 1856-1869. Aharonov, Y. & Bohm, D. (1959). “Significance of electromagnetic potentials in the quantum theory”. Phys. Rev. E, 115 (3), 485-491. Llyod, S. (2000). “Ultimate physical limit to computation”. Nature, 406, 1047-1054. Margolus, N. & Levitin, L. B. (1998). “The maximum speed of dynamical evolution”. Physica D, 120(1-2), 188-195. Caligiuri, L. M. & Musha, T. (2019). “Quantum hyper-computing by means of evanescent photons”. Journal of Physics: Conference Series, 1251, 012010. Caligiuri, L. M. & Musha, T. (2019). “Accelerated Quantum Computation by means of Evanescent Photons and its Prosepcts for Optical Quantum Hypercomputers and Artificial Intelligence”. Paper presented at the 2019 International Conference on Engineering, Science, and Industrial Applications (ICESI), Tokyo, Japan, August 22-24. doi: 10.1109/ICESI.2019.8862999. Caligiuri, L. M. (2019). “A new quantum-relativistic model of tachyon”. Journal of Physics: Conference Series, 1251, 012009.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 4

COMPUTING HYPERINCURSIVE DISCRETE RELATIVISTIC QUANTUM MAJORANA AND DIRAC EQUATIONS AND QUANTUM COMPUTATION Daniel M. Dubois* Centre for Hyperincursion and Anticipation in Ordered Systems (CHAOS), Institute of Mathematics, University of Liege, Liège, Belgium

ABSTRACT Nobody understands quantum mechanics, said Richard Feynman. So, this paper will begins by a step by step presentation of the second order hyperincursive discrete harmonic oscillator that bifurcates to two incursive discrete oscillators with the conservation of a constant of motion. Then, we extend this formalism to the hyperincursive discrete Klein-Gordon equation bifurcates to the Majorana real 4-spinors and to the Dirac complex 4-spinors. Naturally, the hyperincursive discrete equations defines the *

Corresponding Author’s Email: [email protected].

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Daniel M. Dubois relativistic quantum mechanics. When the time and space intervals of the discrete systems tend to zero, all these systems tend to 4 first order differential equations, representing spinors. In the Dirac generic equation, one discovers the Pauli spin matrices. The Pauli matrices X, Y, Z, are used as quantum gates for which the square are equal to the unit matrix I. The Pauli X-gate acts on a single qubit and is the quantum equivalent of the NOT gate for the classical computer. The square root of NOT defines also a quantum gate. More interesting is the Hadamard matrix that is the normalized sum of the X and Z Pauli matrices. Indeed, with the addition of the Hadamard gate to the classical computations the full quantum computation power is obtained.

Keywords: quantum computing, Majorana real spinors, Dirac complex spinors, hyperincursive discrete equations, incursive discrete equations

1. INTRODUCTION This chapter deals with the continuous and discrete equations of the Harmonic Oscillator, and the Relativistic Quantum Majorana and Dirac equations. We begin in section 2 with the presentation step by step of the two incursive discrete harmonic oscillator following my fundamental paper (Dubois, 1995) up-dated in my recent paper (Dubois, 2019f). I define a generalized forward-backward discrete derivative, depending on a weight with 3 values, applied to the time-dependent position and velocity of the harmonic oscillator. I deduce the first and the second incursive discrete harmonic oscillators, and the hyperincursive harmonic oscillator. Then I obtain what I called “the second order hyperincursive discrete harmonic oscillator” depending only on the time-dependent position. The section 3 introduces the two dimensionless incursive discrete harmonic oscillators. Then I present the analytical synchronous solutions of these incursive discrete harmonic oscillators that are related to their constants of motion (Dubois, 2019f). The section 4 deals with a rotation on the position and velocity of the incursive discrete harmonic oscillators, which gives rise to recursive discrete

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harmonic oscillators (Dubois, 2019c). This rotation matrix, with an angle of 𝜋/4, defines the second order Hadamard matrix, which is a fundamental gate in quantum computer. The two recursive discrete harmonic oscillators are then transformed to differential equations for small value of the interval of time. In defining a complex vector, we obtain the complex harmonic oscillator, with the Pauli matrix 𝜎𝑦 , which corresponds to the second Pauli quantum gate in quantum computer. Finally, we develop the chiral representation of this complex harmonic oscillator. With the unitary matrix U, the 2 recursive discrete harmonic oscillators are transformed to a complex recursive discrete harmonic oscillator. The same development was applied to the quantum Majorana equation (Dubois, 2019d). The hyperincursive discrete equations were applied to various quantum systems (Dubois, 2016, 2018). The section 5 deals with the bifurcation of the hyperincursive second order discrete Klein-Gordon equation to the discrete Majorana quantum relativistic equations and the real 4-spinors Majorana differential equations are obtained when the spacetime intervals tend to zero (Dubois, 2019a). Then we demonstrate, with an original method based on real 2-spinors matrices that the Majorana real 4-spinors equations bifurcate simply to the Dirac real 8-spinors equations, which are transformed to the original Dirac complex 4-spinors equations (Dubois, 2019b). We present the 4 complex hyperincursive discrete Dirac equations. Let us notice that the real 2-spinors matrices are related to the three Pauli gates defined in technology of quantum computer. The section 6 shows that the natural number of discrete wave functions of the hyperincursive second order discrete Klein-Gordon equation is equal to 16 discrete spacetime wave functions, instead of the classical 4 functions of Majorana and Dirac equations. My hyperincursive second order discrete Klein-Gordon equations are in agreement with the 16 wave functions of Dirac by Proca (Dubois, 2019b). Proca (1932) classified the 16 equations in 4 groups of 4 functions. There are 4 fundamental equations and the other 3x4 equations are similar to these 4 equations. But formally, only a theory with 16 solutions is the correct one, confirming the power of my hyperincursive second order discrete equations formalism.

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Then the section 7 deals with the chiral representation of the Majorana equations in 2 components (Dubois, 2019d) with the same Hadamard matrix and Unitary matrix U used for the harmonic oscillator (Dubois, 2019c). Next the section 8 gives the solutions of the non-relativistic chiral Majorana equation compared to the solution of the non-relativistic quantum Dirac equation (Dubois, 2019d). Section 9 deals with the 2 coupled Majorana equations in one spatial dimension (1D), with the 3 Pauli matrices (Dubois, 2019e). Then, the section 10 gives a remarkable relation between the Majorana and the Dirac equations in 1D (the y component), with just the inversion of the Dirac matrices 𝛼𝑦 and 𝛽, based on the Pauli matrices 𝜎𝑦 and 𝜎0 . Next, the section 11 deals with the relation between the solutions of the non-relativistic Majorana and Dirac equations, which is given by a transformation relation given, surprisingly, by an invariant function (Dubois, 2019e) depending on the Pauli matrix 𝑥 . Finally, the section 12 deals with a survey of the reversible gates used in quantum computation. The quantum Pauli gates X, Y, Z, that operate on one-qubit, are given by X = 𝑥 = (

0 1

1 0 ) , = 𝑦 = ( 0 i

−i 1 0 ) , and 𝑍 = 𝑧 = ( ) 0 0 −1

More interesting is the rotation matrix R1 (θ) = (

sin(θ) cos(θ) ), cos(θ) −sin(θ)

that generates the Pauli X, Z gates and the Hadamard 𝐻2 gate: π

π

R1 (0) = X , R1 ( 2 ) = Z , and R1 ( 4 ) =

1 +1 ( √2 +1

+1 ) = 𝐻2 −1

with the addition of the reversible logic Toffoli gate to the Hadamard gate, the full quantum computation power of a quantum computer is obtained.

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2. PRESENTATION STEP BY STEP OF THE TWO INCURSIVE DISCRETE HARMONIC OSCILLATORS The harmonic oscillator can be represented by the two ordinary differential equations: d𝑥(𝑡)⁄d𝑡 = 𝑣(𝑡) and d𝑣(𝑡)⁄d𝑡 = − 𝜔2 𝑥(𝑡)

(2.1-a-b)

where x(𝑡) is the position and v(𝑡) the velocity as functions of the time 𝑡, and the pulsation 𝜔 is related to the spring constant 𝑘 and the mass 𝑚 by 𝜔2 = 𝑘/𝑚. The solution is given by 𝑥(𝑡) = x(0)cos(𝜔𝑡) + (𝑣(0)/𝜔) sin(𝜔𝑡), 𝑣(𝑡) = − 𝜔𝑥(0) sin(𝜔𝑡) + 𝑣(0) cos(𝜔𝑡)

(2.1-c-d)

with the initial conditions 𝑥(0) and 𝑣(0). he period of oscillations is given by 𝑇 = 2𝜋/𝜔. The energy 𝑒(𝑡) of the harmonic oscillator is constant and is given by 𝑒(𝑡) = 𝑘 𝑥 2 (𝑡)⁄2 + 𝑚 𝑣 2 (𝑡) /2 = 𝑘 𝑥 2 (0)⁄2 + 𝑚 𝑣 2 (0) /2 = 𝑒(0) = 𝑒0 (2.1-e) In the discrete form, there are the discrete current time t and the interval of time ∆t = h. The discrete time is defined as t k = t 0 + kh, k = 0,1,2, … where 𝑡0 is the initial value of the time and 𝑘 is the counter of the number of intervals of time ℎ. The discrete position and velocity variables are defined as x(k) = x(t k ) and v(k) = v(t k ). In my paper (Dubois, 1995), up-dated in my recent paper (Dubois, 2019f), I defined a generalized forward-backward discrete derivative D𝑤 = 𝑤 Df + (1 − 𝑤) Db

(2.2)

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where 𝑤 is a weight taking the values between 0 and 1, and where the discrete forward and backward derivatives on a function f are defined by Df (f) = ∆+ f / ∆t = ( f(k + 1) − f(k) ) / h, Db (f) = ∆− f / ∆t = ( f(k) − f(k − 1) ) / h The generalized incursive discrete harmonic oscillator is given by (Dubois, 1995) as: (1 − 𝑤) x(k + 1) + (2𝑤 − 1) x(k) − 𝑤 x(k − 1) = h v(k) 𝑤 v(k + 1) + (1 − 2𝑤) v(k) + (𝑤 − 1) v(k − 1) = − h ω2 x(k) (2.3-a-b) When 𝑤 = 0, D0 = Db, this gives the first incursive equations: x(k + 1) − x(k) = h v(k) v(k) − v(k − 1) = − h ω2 x(k)

(2.4-a-b)

When 𝑤 = 1, D1 = Df, this gives the second incursive equations: x(k) − x(k − 1) = h v(k) v(k + 1) − v(k) = − h ω2 x(k) When 𝑤 = 1/2, D1/2 = Ds = [Df + Db ]/2, hyperincursive equations: x(k + 1) − x(k − 1) = + 2h v(k) v(k + 1) − v(k − 1) = − 2h ω2 x(k)

(2.5-a-b) this

gives

the

(2.6-a-b)

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where Ds (f) = D1⁄2 (f) = [ f(k + 1) − f(k − 1)]⁄2h defines a timesymmetric derivative, Ds . In putting the velocity, v(k), of the first equation (2.6-a), v(k) = [x(k + 1) − x(k − 1)]/2h, to the second equation (2.6-b), one obtains x(k + 2) − 2x(k) + x(k − 2) = − 4h2 ω2 x(k)

(2.7-a)

what I called “the second order hyperincursive discrete harmonic oscillator” (Dubois, 2019f), corresponding to the second order differential equation of the harmonic oscillator, from equations (2.1-a-b), given by: 𝑑2 𝑥(𝑡)⁄𝑑𝑡 2 = −𝜔2 𝑥(𝑡)

(2.7-b)

3. THE TWO DIMENSIONLESS INCURSIVE DISCRETE HARMONIC OSCILLATORS A series of papers were published on the incursive and hyperincursive discrete harmonic oscillator (Antippa and Dubois, 2004, 2006a, 2006b, 2007, 2008a, 2008b, 2010a, 2010b, 2010c). For the discrete harmonic oscillator, let us use the dimensionless variables, X and V, for the variables, x and v, as follows (Antippa and Dubois, 2010c) : X(k) = (𝑘/2)1/2 x(k), V(k) = (𝑚/2)1/2 v(k), with the dimensionless time, 𝜏 = 𝜔𝑡, where the pulsation is given by 𝜔 = (𝑘/𝑚)1/2 and with the dimensionless interval of time given by ∆𝜏 = 𝜔 ∆𝑡 = 𝜔 h = H. So, the equations (2.4-a-b) and (2.5-a-b) of the two incursive discrete harmonic oscillators are given respectively by the following two dimensionless incursive discrete equations

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Daniel M. Dubois X1 (k + 1) = X1 (k) + HV1 (k)

(3.1-a)

V1 (k + 1) = V1 (k ) − HX1 (k + 1)

(3.1-b)

V2 (k + 1) = V2 (k) − HX 2 (k)

(3.2-a)

X 2 (k + 1) = X 2 (k) + HV2 (k + 1)

(3.2-b)

and the equations (2.6-a-b) of the hyperincursive discrete harmonic oscillator are given by the following dimensionless hyperincursive discrete equation X(k + 1) = X(k − 1) + 2HV(k)

(3.3-a)

V(k + 1) = V(k − 1) − 2HX(k)

(3.3-b)

Let us recall that this hyperincursive discrete harmonic oscillator is a recursive computing system that is separable into the two incursive discrete harmonic oscillators (Dubois, 2019f). It was demonstrated (Dubois, 2019f) that the following expression K1 (k) = X1 (k)X1 (k + 1) + V1 (k)V1 (k) = X12 (k) + V12 (k) + HX1 (k)V1 (k)

(3.4)

is a constant of motion of the first incursive equations (3.1-a-b), and that the following expression K 2 (k) = X 2 (k)X2 (k) + V2 (k + 1)V2 (k) = X 22 (k) + V22 (k) − HX 2 (k)V2 (k) is a constant of motion of the second incursive equations (3.2-a-b).

(3.5)

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These constants of motion differ with the inversion of the sign of the discrete time interval, 𝐻. The analytical synchronous solutions of the equations (3.1a-b) and (3.2-a-b) are given by X1 (k) = cos(2k/N) and V1 (k) = − sin((2k + 1)/N)

(3.6-a-b)

X 2 (k) = cos((2k + 1)/N) and V2 (k) = − sin(2k/N)

(3.6-c-d)

where N is the number of iterations for a cycle of the oscillator, with the index of iterations 𝑘 = 0, 1, 2, 3, … , for which the interval of discrete time H depends of N, H = 2 sin(π/N).

4. ROTATION OF THE INCURSIVE DISCRETE OSCILLATORS TO RECURSIVE DISCRETE OSCILLATORS In the recent paper (Dubois, 2019c), it was demonstrated that rotations on the position and velocity variables give rise to a pure quadratic expression of the constants of motion (3.4, 3.5), similarly to the constant of energy of the classical continuous harmonic oscillator. The constant of motion (3.4) is an expression of a quadratic curve Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

(4.1)

with A = 1, B = H, C = 1, D = 0, E = 0, F = −K1, x = X1 (k), y = V1 (k) The discriminant, ∆ = 𝐵2 − 4𝐴𝐶= INV, is an invariant under rotations. The discriminant of the constant of motion (3.4): ∆= 𝐵2 − 4𝐴𝐶 = 𝐻 2 − 4 < 0 , defines an ellipse.

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This inequality gives the maximum value of the discrete interval of time, 𝐻 = 𝜔 ∆𝑡 < 2, with H = 2 sin(π⁄N). The equations for the rotation are given by X1 (k) = cos(θ) u1 (k) − sin(θ) v1 (k)

(4.2-a)

V1 (k) = sin(θ) u1 (k) + cos(θ) v1 (k)

(4.2-b)

or, in matrix form, the rotation matrix R1 (θ) is given by V (k) u (k) sin(θ) cos(θ) u1 (k) ( 1 ) = R1 (θ) ( 1 ) = ( )( ) (k) v (k) X1 cos(θ) −sin(θ) v1 (k) 1

(4.3-a)

with 𝐴 = 𝐶, 𝜃 = 𝜋⁄4, so cos(𝜋⁄4) = 2−1/2 = ρ, sin(𝜋⁄4) = 2−1/2 = ρ. So the equations (4.2-a-b) of the rotation are transformed to X1 (k) = (u1 (k) − v1 (k))/√2 and V1 (k) = (u1 (k) + v1 (k))/√2 (4.2-c-d) or, in matrix form, the rotation matrix R1 (𝜋⁄4) = H2 , is given by V (k) u (k) u (k) 1 +1 +1 ( 1 ) = H2 ( 1 ) = 2 ( )( 1 ) √ +1 −1 v1 (k) v1 (k) X1 (k)

(4.3-b)

with the 2 × 2 Hadamard matrix 𝐻2 , for which, 𝐻2 𝐻2 = 𝐼2 , +1 ), −1 1 +1 +1 +1 +1 +1 +0 H2 H2 = ( )( )=( ) = 𝐼2 = 1 +0 +1 2 +1 −1 +1 −1 H2 =

1 +1 ( √2 +1

where 𝐼2 is the 2-Identity matrix. So, with equations (4.2-a-b), the constant of motion (3.4) becomes a pure quadratic form

Computing Hyperincursive Discrete … u12 (k) + v12 (k) + H(u12 (k) − v12 (k))/2 = K1 (k) = K1

113 (4.4-a)

where u1 (k) and v1 (k) are defined by adding and subtracting the equations (4.2-c-d) u1 (k) = (X1 (k) + V1 (k))/√2 and v1 (k) = (V1 (k) − X1 (k))/√2, or, in matrix form, V (k) V (k) u (k) 1 +1 +1 ( 1 ) = H2 ( 1 ) = ( )( 1 ) 2 √ v1 (k) (k) X1 +1 −1 X1 (k)

(4.3-c)

Now let us make the rotation to the first incursive oscillator (3.1-a-b) (u1 (k + 1) − v1 (k + 1)) = (u1 (k) − v1 (k)) + H(u1 (k) + v1 (k)) (u1 (k + 1) + v1 (k + 1)) = (u1 (k) + v1 (k)) − H(u1 (k) − v1 (k)) − H 2 (u1 (k) + v1 (k)) In adding and subtracting these two equations, the first incursive discrete oscillator becomes: u1 (k + 1) = u1 (k) + H v1 (k) − H 2 (u1 (k) + v1 (k))/2

(4.5-a)

v1 (k + 1) = v1 (k) − H u1 (k) − H 2 (u1 (k) + v1 (k))/2

(4.5-b)

defining the first recursive discrete oscillator. For the second incursion, the constant of motion (3.5) is obtained by inversion the sign of H: u22 (k) + v22 (k) − H(u22 (k) − v22 (k))/2 = K 2 (k) = K 2

(4.4-b)

that is also a pure quadratic function. Indeed, with a similar rotation X 2 (k) = sin(θ) u2 (k) + cos(θ) v2 (k)

(4.6-a)

V2 (k) = cos(θ) u2 (k) − sin(θ) v2 (k)

(4.6-b)

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or, in matrix form, the rotation matrix R 2 (θ) is given by X (k) u (k) sin(θ) cos(θ) u2 (k) ( 2 ) = R 2 (θ) ( 2 ) = ( )( ) v2 (k) V2 (k) cos(θ) −sin(θ) v2 (k)

(4.7-a)

For 𝜃 = 𝜋⁄4, so cos(𝜋⁄4) = 2−1/2 = ρ and sin(𝜋⁄4) = 2−1/2 = ρ X 2 (k) = (u2 (k) + v2 (k))√2 and V2 (k) = (u2 (k) − v2 (k))/√2 (4.6-c-d) or, in matrix form the rotation R 2 (𝜋⁄4) = H2 is given by X (k) u (k) u (k) 1 +1 +1 ( 2 ) = H2 ( 2 ) = 2 ( )( 2 ) √ (k) v (k) V2 +1 −1 v2 (k) 2

(4.7-b)

with the Hadamard matrix. So, by adding and subtracting the equations (4.6c-d), we obtain u2 (k) = (X 2 (k) + V2 (k))/√2 and v2 (k) = (X 2 (k) − V2 (k))√2, or, in matrix form X (k) X (k) u (k) 1 +1 +1 ( 2 ) = H2 ( 2 ) = 2 ( )( 2 ) √ v2 (k) (k) V2 +1 −1 V2 (k)

(4.7-c)

Now let us make the rotation to the second incursive oscillator (3.2-a-b) (𝑢2 (k + 1) − v2 (k + 1)) = (𝑢2 (k) − 𝑣2 (k)) − 𝐻(u2 (k) + v2 (k)) (u2 (k + 1) + v2 (k + 1)) = (u2 (k) + v2 (k)) + 𝐻(u2 (k) − v2 (k)) − H 2 (u2 (k) + v2 (k)) and the sum and the difference of which give the second recursive discrete oscillator

Computing Hyperincursive Discrete …

115

u2 (k + 1) = u2 (k) − H v2 (k) − H 2 (u2 (k) + v2 (k))/2

(4.8-a)

v2 (k + 1) = v2 (k) + H u2 (k) − H 2 (u2 (k) + v2 (k))/2

(4.8-b)

These equations are the same as the equations of the first oscillator by inversion of the sign of H. The discrete equations (4.5-a-b, 4.8-a-b) can be transformed to differential equations for small value of the interval of time ∂t u1 (t) = +ωv1 (t) and ∂t v1 (t) = −ωu1 (t)

(4.9-a-b)

∂t u2 (t) = −ωv2 (t) and ∂t v2 (t) = +ωu2 (t)

(4.10-a-b)

where ∂t u(t) = ∂u(t)/ ∂t is the time derivative. And the conversion to the original variables, with the equations (4.3-c) and (4.7-c), are given by ∂t (X1 (t) + V1 (t)) = ω(V1 (t) − X1 (t)), ∂t (V1 (t) − X1 (t)) = −ω(X1 (t) + V1 (t)) ∂t (X 2 (t) + V2 (t)) = −ω(X2 (t) − V2 (t)), ∂t (X 2 (t) − V2 (t)) = ω(X 2 (t) + V2 (t) then, the sum and the difference of these equations give ∂t V1 (t) = −ωX1 (t) and ∂t X1 (t)) = +ωV1 (t)

(4.11-a-b)

∂t X 2 (t) = +ωV2 (t) and ∂t V2 (t) = −ωX2 (t)

(4.12-a-b)

In defining the complex variables u(t) = (u1 (t) − iu2 (t))/√2 and v(t) = (v2 (t) + iv1 (t))/√2 (4.13-a-b)

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the 4 real equations are reduced to 2 complex equations ∂t (u1 (t) − iu2 (t)) = ω(v1 (t) + iv2 (t)) and ∂t (v2 (t) + iv1 (t)) = ω(u2 (t) − iu1 (t)), or ∂t u(t) = +iωv ∗ (t) and ∂t v(t) = −iωu∗ (t)

(4.14-a-b)

where the star sign corresponds to the complex conjugate u∗ (t) = (u1 (t) + iu2 (t))/√2 and v ∗ (t) = (v2 (t) − iv1 (t))/√2 We then obtain the second time derivative of the complex harmonic oscillator ∂2t u(t) = −ω2 u(t) and ∂2t v(t) = −ω2 v(t)

(4.15-a-b)

In defining u(t) w(t) = ( ) v(t)

(4.16)

we obtain, with the Pauli matrix 0 −i σy = ( ), +i 0 the two equations ∂t w(t) = −ωσy w ∗ (t) and ∂t w ∗ (t) = +ωσy w(t) and the second time derivative is given by

(4.17-a-b)

Computing Hyperincursive Discrete … ∂2t w(t) = −ω2 w(t)

117 (4.17-c)

the solution of which being w(t) = cos(ωt) w(0) − sin(ωt) 𝜎𝑦 w ∗ (0)

(4.17-d)

In defining a unitary matrix U = UR + iUI, we can write the transformations of the position and velocity of the discrete harmonic oscillator as follows V1 +1 X2 1 𝑈𝑅 ( ) = ( 0 2 V2 0 +1 X1 +u1 1 +v ( 2 ) = 𝑊𝑅 √2 −v2 +u1

0 +1 −1 0

0 −1 +1 0

V1 +X1 + V1 +1 X +X 1 0 ) ( 2 ) = ( 2 − V2 ) = V2 √2 −X2 + V2 0 +1 X1 +X1 + V1

(4.18-a)

V1 V1 −X 2 − V2 0 −1 −1 0 X2 X 1 0 −1) ( 2 ) = 1 (−X1 + V1 ) = UI ( ) = 2 (+1 0 V2 +1 0 0 −1 V2 √2 −X1 + V1 0 +1 +1 0 X1 X1 +X 2 + V2 −u2 +v1 1 (4.18-b) ( ) = 𝑊𝐼 √2 +v1 +u2 The chiral representation is related to the unitary matrix U 1

U = UR + iUI = ( 2

σ0 + σy i(σ0 − σy)

−i(σ0 − σy ) ) σ0 + σy

with the property UU ∗ = U ∗ U = 1. So, we obtain the complex chiral representation

(4.19)

118

Daniel M. Dubois V1 +1 X2 1 U ( ) = ( +i 2 +i V2 +1 X1

−i −i +1 −1 −1 +1 +i +i

V1 +u1 − iu2 +u +1 −i ) (X2 ) = 1 ( +v2 − iv1 ) = ( +v ) = 𝑊 −v ∗ −i V2 √2 −v2 − iv1 +u∗ +u1 + iu2 +1 X1

(4.18-c) where we define the general function 𝑊 separated in the left wL and right wR chiral functions w wL 𝑊 = (w ) = (−𝑖𝜎 w ∗ ) 𝑦 R +u −v ∗ wL = w = ( ) and wR = −𝑖𝜎𝑦 w ∗ = ( ∗ ) +v +u

(4.20-a)

(4.20-b-c)

The analytical solutions of the first incursive discrete equations are given by X1 (k) = cos(2k/N) and V1 (k) = − sin((2k + 1)/N)

(4.21-a-b)

so, with the relations u1 (k) = (X1 (k) + V1 (k))/√2 and v1 (k) = (V1 (k) − X1 (k))/√2 (4.22-a-b) the functions u1 (k) and v1 (k) become u1 (k) = [+cos(2k⁄N) − sin((2k + 1)⁄N)]⁄√2 = +√2 cos(π⁄4 + π⁄2N) sin(π⁄4 − 2k⁄N − π⁄2N)

(4.23-a)

v1 (k) = [−sin((2k + 1)⁄N) − cos(2k⁄N)]⁄√2 = −√2 sin (π⁄4 + π⁄2N) cos(π⁄4 − 2k⁄N − π⁄2N)

(4.23-b)

and the analytical solutions of the second incursive discrete equations are given by

Computing Hyperincursive Discrete … X 2 (k) = cos((2k + 1)/N) and V2 (k) = − sin(2k/N)

119 (4.21-c-d)

so, with the relations u2 (k) = (X 2 (k) + V2 (k))/√2 and v2 (k) = (X 2 (k) − V2 (k))/√2 (4.22-c-d) the functions u2 (k) and v2 (k) become u2 (k) = [cos((2k + 1) ⁄N) − sin(2k⁄N)]⁄√2 = +√2 cos(π⁄4 − π⁄2N)sin(𝜋⁄4 − 2k⁄N − π⁄2N)

(4.23-c)

v2 (k) = [cos((2k + 1)⁄N) + sin(2k⁄N)]⁄√2 = +√2 sin(π⁄4 − π⁄2N) cos(π⁄4 − 2k⁄N − π⁄2N)

(4.23-d)

Finally, with u(k) = u1 (k) − iu2 (k) and v(k) = v2 (k) + iv1 (k)

(4.24-a-b)

the discrete recursive harmonic oscillators are written as follows u1 (k + 1) − iu2 (k + 1) = u1 (k) − iu2 (k) + H (v1 (k) + i v2 (k)) − H 2 (u1 (k) − iu2 (k) + v1 (k) − iv2 (k))/2 (4.25-a) v2 (k + 1) + iv1 (k + 1) = v2 (k) + iv1 (k) + H (u2 (k) − iu1 (k)) − H 2 (u2 (k) + iu1 (k) + v2 (k) + iv1 (k))/2 (4.25-b) So we obtain the complex discrete recursive harmonic oscillator

120

Daniel M. Dubois u(k + 1) = u(k) + iHv ∗ (k) + iH 2 (iu(k) + v(k))/2

(4.26-a)

v(k + 1) = v(k) − iHu∗ (k) − H 2 (iu(k) + v(k))/2

(4.26-b)

In conclusion, we have demonstrated the transformation, by rotation with the Hadamard and unitary matrices, of the incursive discrete harmonic oscillators to recursive discrete harmonic oscillators. The same rotation will be applied to the Majorana equation (Dubois, 2019d).

5. THE HYPERINCURSIVE DISCRETE KLEIN-GORDON EQUATION BIFURCATES TO THE MAJORANA AND DIRAC RELATIVISTIC QUANTUM EQUATIONS The Klein-Gordon equation (Oskar Klein, 1926, Walter Gordon, 1926) of the function 𝜑 = 𝜑(𝐫, t)in three spatial dimensions 𝐫 = (x, y, z) and time t is given by − ħ2 ∂2t 𝜑(𝐫, t) = − ħ2 𝑐 2 ∇2 𝜑(𝐫, t) + m2 c 4 𝜑(𝐫, t)

(5.1)

where 𝜕𝜇 𝜑 = 𝜕𝜑/𝜕𝜇, or, in the explicit form of the nabla operator ∇, − ħ2 ∂2t 𝜑(𝐫, t) = − ħ2 𝑐 2 ∂2x 𝜑(𝐫, t) − ħ2 𝑐 2 ∂2y 𝜑(𝐫, t) − ħ2 𝑐 2 ∂2z 𝜑(𝐫, t) + m2 c 4 φ

(5.2)

where 𝜕𝜇2 𝜑 = 𝜕 2 𝜑/𝜕𝜇2 , ħ is the constant of Plank, 𝑐 is the speed of light, and m the mass. From the Klein-Gordon equation, the relativistic quantum Dirac and Majorana equations can be deduced (Dirac, 1928, Majorana, 1937). As we will consider the discrete Klein-Gordon equation, we make the following usual change of variables

Computing Hyperincursive Discrete … 𝑞(𝐫, t) = 𝜑(𝐫, t)  𝑎 = 𝜔 = 𝑚𝑐 2 /ħ

121  (5.4)

where  is a frequency, so the Klein-Gordon equation (5.2) becomes 𝜕 2 q(𝐫, t)/ ∂t 2 = +𝑐 2 ∂2 q(𝐫, t)/ ∂𝑥 2 + 𝑐 2 ∂2 q(𝐫, t)/ ∂𝑦 2 + 𝑐 2 ∂2 q(𝐫, t)/ ∂𝑧 2 − a2 q(𝐫, t)

(5.5)

From the Klein-Gordon equation (5.5), the second order hyperincursive discrete Klein-Gordon equation (32, 35) is given by q(x, y, z, t + 2∆t) − 2q(x, y, z, t) + q(x, y, z, t − 2∆t) = +B + 2∆x, y, z, t) − 2q(x, y, z, t) + q(x − 2∆x, y, z, t)] 2 [q(x, +C y + 2∆y, z, t) − 2q(x, y, z, t) + q(x, y − 2∆y, z, t)] +D2 [q(x, y, z + 2∆z, t) − 2q(x, y, z, t) + q(x, y, z − 2∆z, t)] − A2 q(x, y, z, t) 2 [q(x

(5.6)

where the following parameters A, B, C, and, D, A = a (2∆t),B = c (2∆t)/(2∆x), C = c (2∆t)/(2∆y), D = c (2∆t)/(2∆z)

(5.7)

depend on the discrete interval of time ∆t, and the discrete intervals of space, ∆x, ∆y, ∆z, respectively. As usually made in computer science, let us now introduce the discrete time t k , and the discrete spaces xl , ym, zn, as follows t k = t 0 + k∆t, k = 0,1,2, …,

(5.8)

where k is the integer time increment, and xl = x0 + l∆x, l = 0,1,2, …, ym = y0 + m∆y, m = 0,1,2, …, zn = z0 + n∆z, n = 0,1,2, … (5.9) where l, m, n, are the integer space increments.

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So, with these time and space increments, the second order hyperincursive discrete Klein-Gordon equation (5.6) becomes q(l, m, n, k + 2) − 2q(l, m, n, k) + q(l, m, n, k − 2) = +B 2 [q(l + 2, m, n, k) − 2q(l, m, n, k) + q(l − 2, m, n, k)] +C 2 [q(l, m + 2, n, k) − 2q(l, m, n, k) + q(l, m − 2, n, k)] +D2 [q(l, m, n + 2, k) − 2q(l, m, n, k) + q(l, m, n − 2, k)] − A2 q(l, m, n, k)

(5.10)

This equation without spatial components, corresponding to a particle at rest, is similar to the harmonic oscillator. As presented in my recent paper (Dubois, 2019a), where the functions q̃j = q̃j (x, y, z, t) = q̃j (l, m, n, k), j = 1,2,3,4, define discrete Majorana functions, the 4 discrete hyperincursive equations of the functions q̃j , j = 1,2,3,4, are obtained as ̃[q̃ 4 (l + 1, m, n, k) − q̃4 (l − q̃1 (l, m, n, k + 1) = q̃1 (l, m, n, k − 1) + B ̃ [q̃3 (l, m, n + 1, m, n, k)] − C̃[q̃1 (l, m + 1, n, k) − q̃1 (l, m − 1, n, k)] + D ̃ q̃ 4 (l, m, n, k) 1, k) − q̃3 (l, m, n − 1, k)] − A ̃[q̃3 (l + 1, m, n, k) − q̃3 (l − q̃2 (l, m, n, k + 1) = q̃2 (l, m, n, k − 1) + B ̃ [q̃4 (l, m, n + 1, m, n, k)] − C̃[q̃2 (l, m + 1, n, k) − q̃2 (l, m − 1, n, k)] − D ̃ q̃ 3 (l, m, n, k) 1, k) − q̃4 (l, m, n − 1, k)] + A ̃[q̃2 (l + 1, m, n, k) − q̃2 (l − q̃3 (l, m, n, k + 1) = q̃3 (l, m, n, k − 1) + B ̃ [q̃1 (l, m, n + 1, m, n, k)] + C̃[q̃3 (l, m + 1, n, k) − q̃3 (l, m − 1, n, k)] + D ̃ q̃2 (l, m, n, k) 1, k) − q̃1 (l, m, n − 1, k)] − A ̃[q̃1 (l + 1, m, n, k) − q̃1 (l − q̃4 (l, m, n, k + 1) = q̃4 (l, m, n, k − 1) + B 1, m, n, k)] + C̃[q̃4 (l, m + 1, n, k) − q̃4 (l, m − 1, n, k)] − ̃ q̃1 (l, m, n, k) ̃ [q̃2 (l, m, n + 1, k) − q̃2 (l, m, n − 1, k)] + A D (5-11-a-b-c-d) with

Computing Hyperincursive Discrete …

123

̃ = A = a(2∆t), B ̃ = B = c ∆t/∆x, A

(5-12-a-b)

̃ = D = c ∆t/∆z C̃ = C = c ∆t⁄∆y , D

(5-12-c-d)

where ∆𝑡 and ∆x, ∆y, ∆z are the discrete intervals of time and space respectively. From the discrete equations, when the spacetime intervals tend to zero, we obtained the following 4 first order partial differential equations (Dubois, 2019a) ̃1 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̃ 4 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̃1 ⁄𝜕𝑦 + 𝑐 𝜕Ψ ̃ 3⁄𝜕𝑧 − (𝑚𝑐 2⁄ħ)Ψ ̃4 + 𝜕Ψ ̃ 2 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̃ 3⁄𝜕𝑥 − 𝑐 𝜕Ψ ̃ 2⁄𝜕𝑦 − 𝑐 𝜕Ψ ̃ 4⁄𝜕𝑧 + (𝑚𝑐 2⁄ħ)Ψ ̃3 + 𝜕Ψ ̃ 3 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̃ 2⁄𝜕𝑥 + 𝑐 𝜕Ψ ̃ 3⁄𝜕𝑦 + 𝑐 𝜕Ψ ̃1⁄𝜕𝑧 − (𝑚𝑐 2⁄ħ)Ψ ̃2 + 𝜕Ψ ̃ 4 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̃1⁄𝜕𝑥 + 𝑐 𝜕Ψ ̃ 4 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̃ 2⁄𝜕𝑧 + (𝑚𝑐 2⁄ħ)Ψ ̃1 + 𝜕Ψ (5.13-a-b-c-d) which are identical to the original Majorana equations (Majorana, 1937), e.g., equations (4-a-b-c-d) in Pessa (Pessa, 2006). Recently, we demonstrated that Majorana 4-spinors equations bifurcate simply to the Dirac real 8-spinors equations (Dubois, 2019b). First, let us consider the inverse parity space, in inversing the sign of the space variables in the Majorana equations (5.13-a-b-c-d), ̃1 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̃ 4 ⁄𝜕𝑥 + 𝑐 𝜕Ψ ̃1 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̃ 3 ⁄𝜕𝑧 − (𝑚𝑐 2⁄ħ)Ψ ̃4 + 𝜕Ψ 2 ̃ 2 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̃ 3⁄𝜕𝑥 + 𝑐 𝜕Ψ ̃ 2⁄𝜕𝑦 + 𝑐 𝜕Ψ ̃ 4⁄𝜕𝑧 + (𝑚𝑐 ⁄ħ)Ψ ̃3 + 𝜕Ψ 2 ̃ 3 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̃ 2 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̃ 3 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̃1⁄𝜕𝑧 − (𝑚𝑐 ⁄ħ)Ψ ̃2 + 𝜕Ψ 2 ̃ 4 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̃1⁄𝜕𝑥 − 𝑐 𝜕Ψ ̃ 4 ⁄𝜕𝑦 + 𝑐 𝜕Ψ ̃ 2⁄𝜕𝑧 + (𝑚𝑐 ⁄ħ)Ψ ̃1 + 𝜕Ψ (5.14-a-b-c-d)

124

Daniel M. Dubois In defining the 2-spinors real functions,

φa = (

̃1 ̃ Ψ Ψ ) , φb = ( 3 ), ̃2 ̃4 Ψ Ψ

(5-15-a-b)

the two equations (5.14-a-b) and (5.14-c-d) are transformed to the two 2spinors real equations + ∂φa ⁄∂t = −c 1 ∂φb⁄∂𝑥 + c 0 ∂φa ⁄∂𝑦 − c 3 ∂φb⁄∂𝑧 + (𝑚𝑐 2 ⁄ħ)2 φb + ∂φb⁄∂t = −c 1 ∂φa ⁄∂𝑥 − c 0 ∂φb⁄∂𝑦 − c 3 ∂φa ⁄∂𝑧 + (𝑚𝑐 2 ⁄ħ)2 φa (5.16-a-b) where the real 2-spinors matrices 1 , 2 , 3 , are defined by 0 1 0 −1 1 1 = ( ), 2 = ( ), 3 = ( 1 0 1 0 0 1 0 and 2-Identity 0 = ( ) = I2 0 1

0 ), −1

(5.17-a-b-c)

(5.17-d)

With the inversion between 0 and 2 , in introducing the tensor product ̃j by −2 , the functions Ψ Ψ ̃ j = ( j,1 ) , 𝑗 = 1,2,3,4, Ψ Ψj,2

(5.18)

bifurcate to two functions Ψj,1 0 −2 Ψj = −2 ( ) = −( Ψj,2 1

+Ψj,2 −1 Ψj,1 )( )= ( ) , 𝑗 = 1,2,3,4 Ψ −Ψj,1 0 j,2 (5.19)

Computing Hyperincursive Discrete …

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So the Majorana real 4-spinors equation bifurcates into the Dirac real 8spinors equations + 𝜕Ψ1,1⁄𝜕𝑡 = −𝑐 𝜕Ψ4,1⁄𝜕𝑥 − 𝑐 𝜕Ψ4,2⁄𝜕𝑦 − 𝑐 𝜕Ψ3,1⁄𝜕𝑧 + (𝑚𝑐 2 ⁄ħ)Ψ1,2 + 𝜕Ψ2,1⁄𝜕𝑡 = −𝑐 𝜕Ψ3,1⁄𝜕𝑥 + 𝑐 𝜕Ψ3,2⁄𝜕𝑦 + 𝑐 𝜕Ψ4,1⁄𝜕𝑧 + (𝑚𝑐 2 ⁄ħ)Ψ2,2 + 𝜕Ψ3,1⁄𝜕𝑡 = −𝑐 𝜕Ψ2,1⁄𝜕𝑥 − 𝑐 𝜕Ψ2,2⁄𝜕𝑦 − 𝑐 𝜕Ψ1,1⁄𝜕𝑧 − (𝑚𝑐 2 ⁄ħ)Ψ3,2 + 𝜕Ψ4,1⁄𝜕𝑡 = −𝑐 𝜕Ψ1,1⁄𝜕𝑥 + 𝑐 𝜕Ψ1,2 ⁄𝜕𝑦 + 𝑐 𝜕Ψ2,1⁄𝜕𝑧 − (𝑚𝑐 2 ⁄ħ)Ψ4,2 (5.20-a-b-c-d) + 𝜕Ψ1,2⁄𝜕𝑡 = −𝑐 𝜕Ψ4,2⁄𝜕𝑥 + 𝑐 𝜕Ψ4,1⁄𝜕𝑦 − 𝑐 𝜕Ψ3,2⁄𝜕𝑧 − (𝑚𝑐 2 ⁄ħ)Ψ1,1 + 𝜕Ψ2,2⁄𝜕𝑡 = −𝑐 𝜕Ψ3,2⁄𝜕𝑥 − 𝑐 𝜕Ψ3,1⁄𝜕𝑦 + 𝑐 𝜕Ψ4,2⁄𝜕𝑧 − (𝑚𝑐 2 ⁄ħ)Ψ2,1 + 𝜕Ψ3,2⁄𝜕𝑡 = −𝑐 𝜕Ψ2,2⁄𝜕𝑥 + 𝑐 𝜕Ψ2,1⁄𝜕𝑦 − 𝑐 𝜕Ψ1,2⁄𝜕𝑧 + (𝑚𝑐 2 ⁄ħ)Ψ3,1 + 𝜕Ψ4,2⁄𝜕𝑡 = −𝑐 𝜕Ψ1,2⁄𝜕𝑥 − 𝑐 𝜕Ψ1,1 ⁄𝜕𝑦 + 𝑐 𝜕Ψ2,2⁄𝜕𝑧 + (𝑚𝑐 2 ⁄ħ)Ψ4,1 (5.21-a-b-c-d) These 2 x 4 = 8 real first order partial differential equations represent real 8-spinors equations that are similar to the Dirac complex 4-spinors equations (Dirac, 1964). In defining the wave function

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Daniel M. Dubois Ψj (𝑥, 𝑦, 𝑧, t) = Ψj = Ψj,1 + iΨj,2 , j = 1,2,3,4,

(5.22)

with the imaginary number i, we obtain the original Dirac equations (Dirac, 1928): + 𝜕Ψ1 ⁄𝜕𝑡 = −𝑐 𝜕Ψ4 ⁄𝜕𝑥 + i𝑐 𝜕Ψ4 ⁄𝜕𝑦 − 𝑐 𝜕Ψ3 ⁄𝜕𝑧 − i(𝑚𝑐 2 ⁄ħ)Ψ1 + 𝜕Ψ2 ⁄𝜕𝑡 = −𝑐 𝜕Ψ3⁄𝜕𝑥 − i𝑐 𝜕Ψ3 ⁄𝜕𝑦 + 𝑐 𝜕Ψ4 ⁄𝜕𝑧 − i(𝑚𝑐 2 ⁄ħ)Ψ2 + 𝜕Ψ3 ⁄𝜕𝑡 = −𝑐 𝜕Ψ2⁄𝜕𝑥 + i𝑐 𝜕Ψ2 ⁄𝜕𝑦 − 𝑐 𝜕Ψ1 ⁄𝜕𝑧 + i(𝑚𝑐 2 ⁄ħ)Ψ3 + 𝜕Ψ4 ⁄𝜕𝑡 = −𝑐 𝜕Ψ1⁄𝜕𝑥 − i𝑐 𝜕Ψ1 ⁄𝜕𝑦 + 𝑐 𝜕Ψ2 ⁄𝜕𝑧 + i(𝑚𝑐 2 ⁄ħ)Ψ4 (5.23-a-b-c-d) Let us define the discrete Dirac wave function Q j (l, m, n, k) = Q j = Q j,1 + i Q j,2 , j = 1,2,3,4,

(5.24)

corresponding to the Dirac wave function (5.22). The 4 hyperincursive discrete Dirac equations of the discrete wave function are then given by Q1 (l, m, n, k + 1) = Q1 (l, m, n, k − 1) − B[Q 4 (l + 1, m, n, k) − Q 4 (l − 1, m, n, k)] + iC[Q 4 (l, m + 1, n, k) − Q 4 (l, m − 1, n, k)] −D[Q 3 (l, m, n + 1, k) − Q 3 (l, m, n − 1, k)] − i AQ1 (l, m, n, k) Q 2 (l, m, n, k + 1) = Q 2 (l, m, n, k − 1) − B[Q 3 (l + 1, m, n, k) − Q 3 (l − 1, m, n, k)] −i C[Q 3 (l, m + 1, n, k) − Q 3 (l, m − 1, n, k)] +D[Q 4 (l, m, n + 1, k) − Q 4 (l, m, n − 1, k)] − i AQ 2 (l, m, n, k) Q 3 (l, m, n, k + 1) = Q 3 (l, m, n, k − 1) − B[Q 2 (l + 1, m, n, k) − Q 2 (l − 1, m, n, k)]

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+i C[Q 2 (l, m + 1, n, k) − Q 2 (l, m − 1, n, k)] − D[Q1 (l, m, n + 1, k) − Q1 (l, m, n − 1, k)] + i AQ 3 (l, m, n, k) Q 4 (l, m, n, k + 1) = Q 4 (l, m, n, k − 1) − B[Q1 (l + 1, m, n, k) − Q1 (l − 1, m, n, k)] − i C[Q1 (l, m + 1, n, k) − Q1 (l, m − 1, n, k)] +D[q2 (l, m, n + 1, k) − Q 2 (l, m, n − 1, k)] + i AQ 4 (l, m, n, k) (5.25-a-b-c-d) with A = 2ω∆t, B = c ∆t/∆x, C = c ∆t⁄∆y , D = c ∆t/∆z

(5.26)

where ∆𝑡 and ∆x, ∆y, ∆z are the discrete intervals of time and space respectively.

6. THE HYPERINCURSIVE DISCRETE KLEIN-GORDON EQUATION BIFURCATES TO THE 16 PROCA EQUATIONS Let us show that there are 16 complex functions associated to this second order hyperincursive discrete Klein-Gordon equation. This equation without spatial components, corresponding to a particle at rest, is similar to the harmonic oscillator. For a particle at rest, the Klein-Gordon equation (5.10), with the function q(t) depending only on the time variable, is given by 𝜕 2 q(t)/ ∂t 2 = − 𝑎2 q(t)

(6.1)

with the frequency, 𝑎 = 𝜔 = 𝑚𝑐 2 ⁄ħ, given by the equation (5.4). This equation (6.1) is formally similar to the equation of the harmonic oscillator for which q(t) would represent the position x(t), and 𝜕𝑞(𝑡)/𝜕𝑡 would represent the velocity 𝑣(𝑡) = 𝜕𝑥(𝑡)/𝜕𝑡.

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So, with only the temporal component, the second order hyperincursive discrete Klein-Gordon equation (5.10) becomes q(k + 2) − 2q(k) + q(k − 2) = −A2 q(k)

(6.2)

that is similar to the second order hyperincursive discrete equation of the harmonic oscillator, as shown in section 2. This hyperincursive equation (6.2) is separable into a first discrete incursive oscillator depending on two functions defined by q1 (k), q2 (k), and a second incursive oscillator depending on two other functions defined by q3 (k), q4 (k), given by first order discrete equations. So the first incursive equations are given by: q1 (2k) = q1 (2k − 2) + Aq2 (2k − 1) q2 (2k + 1) = q2 (2k − 1) − Aq1 (2k)

(6.3-a-b)

where q1 (2k) is defined of the even steps of the time, and q2 (2k + 1) is defined on the odd steps of the time. And the second incursive equations are given by: q3 (2k) = q 3 (2k − 2) − Aq4 (2k − 1) q4 (2k + 1) = q4 (2k − 1) + Aq3 (2k)

(6.4-a-b)

where q3 (2k) is defined of the even steps of the time, and q4 (2k + 1) is defined on the odd steps of the time. The second incursive system is the time reverse of the first incursive system in making the discrete time inversion T 𝑻: ∆t → − ∆t which gives an oscillator and its anti-oscillator.

(6.5)

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In defining the following 2 complex functions, where i is the imaginary number, 𝑞13 (2k) = q1 (2k) + i q3 (2k) 𝑞24 (2k + 1) = q2 (2k + 1) − i q4 (2k + 1)

(6.6-a-b)

the 4 real incursive equations (6.3-a-b) and (6.4-a-b) are transformed to 2 complex incursive equations 𝑞13 (2k) = 𝑞13 (2k − 2) + A𝑞24 (2k − 1) 𝑞24 (2k + 1) = 𝑞24 (2k − 1) − A𝑞13 (2k)

(6.7-a-b)

So the hyperincursive equation for a particle at rest shows a temporal bifurcation into an oscillatory equation and an anti-oscillatory equation. For a moving particle, the 3 discrete space-symmetric terms in equation (5.10) q(l + 2, m, n, k) − 2q(l, m, n, k) + q(l − 2, m, n, k) q(l, m + 2, n, k) − 2q(l, m, n, k) + q(l, m − 2, n, k) q(l, m, n + 2, k) − 2q(l, m, n, k) + q(l, m, n − 2, k) are similar to the discrete time-symmetric term q(l, m, n, k + 2) − 2q(l, m, n, k) + q(l, m, n, k − 2) The two complex functions bifurcate for even and odd steps of space x, giving 4 complex functions depending on 4 discrete incursive equations. These 4 complex functions bifurcate for even and odd steps of space y, giving 8 complex functions depending on 8 discrete incursive equations. Finally, these 8 complex functions bifurcate for even and odd steps of space z, giving 16 complex functions depending on 16 incursive discrete equations.

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Daniel M. Dubois But if we consider the space variable as a set of the 3 space variables 𝒓 = (𝑥, 𝑦, 𝑧)

(6.8)

the two complex functions bifurcate for even and odd steps of the space variable 𝒓 = (𝑥, 𝑦, 𝑧), giving 4 complex functions depending on 4 discrete incursive equations, which correspond to a discrete parity inversion 𝑷 𝑷: ∆𝒓 → −∆𝒓

(6.9)

In conclusion, with the discrete time inversion and the parity, we define a group of 4 incursive discrete equations with 4 functions. This is in agreement with the thesis of Proca. Indeed, as demonstrated by Proca (Proca, 1930, 1932) in 1930 and 1932, the Klein-Gordon equation admits in the general case a total of 16 functions. Classically, for the well-known Dirac equation, there are 4 complex wave functions. Proca demonstrated that there are 4 fundamental equations of 4 wave functions for the Dirac equation 𝜑𝑟,𝑠 𝑓𝑜𝑟 𝑟 = 1,2,3,4, 𝑎𝑛𝑑 𝑠 = 1

(6.10)

and the other 3 x 4 other equations are similar to these 4 equations. Proca classified the 16 equations in 4 groups of 4 functions: 1. 2. 3. 4.

4 equations of the 4 functions 𝜑𝑟,𝑠 𝑓𝑜𝑟 𝑟 = 1,2,3,4, 𝑎𝑛𝑑 𝑠 = 1 4 equations of the 4 functions 𝜑𝑟,𝑠 𝑓𝑜𝑟 𝑟 = 1,2,3,4, 𝑎𝑛𝑑 𝑠 = 2 4 equations of the 4 functions 𝜑𝑟,𝑠 𝑓𝑜𝑟 𝑟 = 1,2,3,4, 𝑎𝑛𝑑 𝑠 = 3 4 equations of the 4 functions 𝜑𝑟,𝑠 𝑓𝑜𝑟 𝑟 = 1,2,3,4, 𝑎𝑛𝑑 𝑠 = 4

In each group, the 4 equations depend on 4 functions which are not separable except in particular cases. In this chapter we restricted our analysis to the first group of 4 functions in studying the case of the Majorana and Dirac equations.

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7. CHIRAL REPRESENTATION OF THE MAJORANA EQUATIONS IN 2 COMPONENTS In the preceding section 4, we have presented the rotation of the incursive discrete harmonic oscillators by the Hadamard matrix and unitary matrix U. The incursive discrete equations are transformed to recursive discrete equations, what is a remarkable result (Dubois, 2019c). The rotation of the relativistic quantum Majorana equations with the same Hadamard matrix and unitary matrix U, gives rise to the transformation of the Majorana equations in 2 components (Dubois, 2019d). Indeed, we will give the Chiral representation of Majorana equations from the unitary matrix, U = UR + iUI, 1

U = UR + iUI = ( 2

σ0 + σy −i(σ0 − σy ) ) i(σ0 − σy ) σ0 + σ y

(7.1)

which can be defined with the Pauli matrix σy and with the unit matrix, σ0 = 𝐼2 = 1, with the property UU ∗ = U ∗ U = 1. An excellent introduction to the properties of the unitary matrix is given by Palash (Palash, 2011). So the real and imaginary parts of this unitary matrix are applied to the Majorana real 4-spinors as follows +1 1 ̃ 𝑈𝑅 Ψ = ( 0 2 0 +1

0 +1 −1 0

0 −1 +1 0

+1 0) 0 +1

(

̃1 Ψ ̃2 Ψ ̃3 Ψ ̃4 Ψ

=

1 2

)

(

̃1 + Ψ ̃4 +Ψ ̃2 − Ψ ̃3 +Ψ ̃ ̃3 −Ψ2 + Ψ ̃1 + Ψ ̃4 +Ψ

=

1 √2

)

(

̌11 +Ψ ̌21 +Ψ ̌21 −Ψ ̌11 +Ψ

)

(7.2-a) 0 1 +1 ̃= ( UI Ψ 2 +1 0

−1 0 0 +1

−1 0 0 +1

0 −1) −1 0

(

̃1 Ψ ̃ Ψ2 ̃3 Ψ ̃4 Ψ

= )

1 2

(

̃2 − Ψ ̃3 −Ψ ̃ ̃ +Ψ1 − Ψ4 ̃1 − Ψ ̃4 +Ψ ̃ ̃3 +Ψ2 + Ψ

= )

1 √2

(

̌12 −Ψ ̌ +Ψ22 ̌22 +Ψ ̌12 +Ψ

)

(7.2-b)

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So the application of the unitary matrix to the Majorana real 4-spinors is given by +1 1 +i ̃ UΨ = ( 2 +i +1

−i +1 −1 +i

−i −1 +1 +i

+1 −i ) −i +1

(

̃1 Ψ ̃2 Ψ ̃3 Ψ ̃1 Ψ

= )

(

̌11 − iΨ ̌12 +Ψ ̌21 + iΨ ̌22 +Ψ ̌ ̌22 −Ψ21 + iΨ ̌11 + iΨ ̌12 +Ψ

= )

(

̌1 Ψ ̌2 Ψ ̌3 Ψ ̌4 Ψ

̌ =Ψ )

(7.2-c) ̌ can be separated in the top left chiral function, The general function Ψ ̌ L , and in the bottom right chiral function, Ψ ̌ R chiral function, Ψ ̌ ̌ ̌ ̌ = ( ΨL ), Ψ ̌ L = (Ψ1 ) and Ψ ̌ R = (Ψ3 ) Ψ ̌R ̌2 ̌4 Ψ Ψ Ψ

(7.4-a-b-c)

and the bottom right function can be deduced directly from the top left function as follows ̌∗ ̌ R = −𝑖𝜎𝑦 Ψ ̌ L∗ = (−Ψ2 ) Ψ ̌1∗ +Ψ

(7.5)

with the rotation 2𝑥2 Hadamard matrix, 𝐻2 , 𝐻2 =

1 +1 ( √2 +1

+1 ) −1

(7.6)

let us transform the Majorana 2-spinors, as follow

(

̌11 ̃ ̃ +Ψ ̃4 Ψ Ψ 1 Ψ ) = 𝐻2 ( 1 ) = 2 ( 1 ) ̃4 ̃1 − Ψ ̃4 ̌ 22 √ Ψ Ψ Ψ

(

̌12 ̃ ̃ +Ψ ̃3 Ψ Ψ 1 Ψ ) = 𝐻2 ( 2 ) = 2 ( 2 ) ̃ ̃ ̃3 ̌ √ Ψ3 Ψ2 − Ψ Ψ21

(7.7-a-b)

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and let us apply these rotations to the Majorana equations as follow ̌11 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̌11 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̌ 22 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌ 21 ⁄𝜕𝑧 + + 𝜕Ψ ̌ 22 (𝑚𝑐 2 ⁄ħ)Ψ ̌12 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̌12 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̌ 21 ⁄𝜕𝑦 + 𝑐 𝜕Ψ ̌ 22 ⁄𝜕𝑧 − + 𝜕Ψ ̌ 21 (𝑚𝑐 2 ⁄ħ)Ψ ̌ 21 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̌ 21 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̌12⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌11 ⁄𝜕𝑧 + + 𝜕Ψ ̌12 (𝑚𝑐 2 ⁄ħ)Ψ ̌ 22 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̌ 22 ⁄𝜕𝑥 − 𝑐 𝜕Ψ ̌11⁄𝜕𝑦 + 𝑐 𝜕Ψ ̌12 ⁄𝜕𝑧 − + 𝜕Ψ 2⁄ ̌ (𝑚𝑐 ħ)Ψ11 (7.8-a-b-c-d) Again with the Hadamard matrix, let us transform the 2-spinors (7.7-a-b) as follows

(

̌4 ̌ ̌ + iΨ ̌12 Ψ Ψ 1 +Ψ ) = 𝐻2 ( 11 ) = 2 ( 11 ), ̌1 ̌12 √ ̌11 − iΨ ̌12 Ψ iΨ +Ψ

(

̌2 ̌ ̌ + iΨ ̌ 22 Ψ iΨ 1 +Ψ ) = 𝐻2 ( 22 ) = 2 ( 21 ) ̌3 ̌ 21 √ ̌ 21 + iΨ ̌ 22 Ψ Ψ −Ψ

(7.9-a-b)

which are the same transformations as in the unitary matrix (7.2-c). ̌1 Let us give the partial differential equations of the 2 left chiral functions Ψ ̌ 2 , as and Ψ ̌1 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̌1 ⁄𝜕𝑥 + i 𝑐 𝜕Ψ ̌ 2 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌ 2 ⁄𝜕𝑧 + i (𝑚𝑐 2 ⁄ħ)Ψ ̌ 2∗ + 𝜕Ψ ̌ 2 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̌ 2⁄𝜕𝑥 − i 𝑐 𝜕Ψ ̌1 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌1 ⁄𝜕𝑧 − i (𝑚𝑐 2 ⁄ħ)Ψ ̌1∗ + 𝜕Ψ (7.10-a-b) Let us write the chiral left Majorana equation with the left chiral ̌ L (7.4-b): function, Ψ ̌ L = +c𝑧 𝜕𝑥 Ψ ̌ L − 𝑐𝑦 𝜕𝑦 Ψ ̌ L − 𝑐𝑥 𝜕𝑧 Ψ ̌ L − (𝑚𝑐 2⁄ħ)𝑦 Ψ ̌ L∗ (7.10-c) 𝜕𝑡 Ψ

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̌R The chiral right Majorana equation with the right chiral function, Ψ (7.4-c), is easy to write. For a particle at rest, the left non-relativistic Majorana equation is given by ̌ L = −(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̌ L∗ 𝜕𝑡 Ψ

(7.11)

In conclusion, in this section, we have considered the chiral representation of the Majorana equations.

8. SOLUTIONS OF THE NON-RELATIVISTIC QUANTUM MAJORANA AND DIRAC EQUATIONS This section is written following our recent paper (Dubois, 2019d). In the non-relativistic limit 𝑝 ≪ 𝑚𝑐, the particles are at rest, with a momentum 𝑝 ≅ 0. In this limit, the Majorana equations (7.10-a-b) are given by ̌1 ⁄𝜕𝑡 = + i (𝑚𝑐 2 ⁄ħ)Ψ ̌ 2∗ + 𝜕Ψ ̌ 2 ⁄𝜕𝑡 = − i (𝑚𝑐 2 ⁄ħ)Ψ ̌1∗ + 𝜕Ψ ̌ (t) = ( With Ψ

̌1 (t) Ψ ) ̌ 2 (t) Ψ

(8.1-a-b)

(8.2-a)

these equations (8.1-a-b) become ̌ = −(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̌∗ 𝜕𝑡 Ψ

(8.3-a)

0 −1 where 𝜕𝑡 = 𝜕/𝜕𝑡, and 𝑦 = i ( ), is a Pauli matrix. 1 0 The complex conjugate of equation (8.3-a) is given by ̌ ∗ = +(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̌ 𝜕𝑡 Ψ

(8.3-b)

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With the two equations (8.3-a-b), one obtains a second order equation ̌ (t) = −(𝑚𝑐 2 ⁄ħ)2 Ψ ̌ (t) 𝜕𝑡2 Ψ

(8.4)

that is the temporal Klein-Gordon equation. The solution of equation (8.4) is given by ̌ (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌ (0) − sin(𝑚𝑐 2 t⁄ħ) 𝑦 Ψ ̌ ∗ (0) Ψ

(8.5)

or in explicit form ̌1 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌1 (0) + i sin(𝑚𝑐 2 t⁄ħ)Ψ ̌ 2∗ (0) Ψ

(8.6-a)

̌ 2 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌ 2 (0) − i sin(𝑚𝑐 2 t⁄ħ)Ψ ̌1∗ (0) Ψ

(8.6-b)

Now let us consider the following Dirac 2-spinors ̂ (t) = (Ψ1 (t)), Ψ Ψ4 (t)

(8.7)

for which the temporal non-relativistic Dirac equation is given by ̂ (t) = −𝑖(𝑚𝑐 2⁄ħ)𝑧 Ψ ̂ (t) 𝜕𝑡 Ψ

(8.8)

1 0 where 𝜕𝑡 = 𝜕/𝜕𝑡, and 𝑧 = ( ), is a the Pauli matrix. 0 −1 The analytical solution of the non-relativistic Dirac equation (8.8) is given by ̂ (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̂ (0) − i sin(𝑚𝑐 2 t⁄ħ) 𝑧 Ψ ̂ (0) Ψ

(8.9)

or in explicit form Ψ1 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ1 (0) − i sin(𝑚𝑐 2 t⁄ħ)Ψ1 (0)

(8.10-a)

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Daniel M. Dubois Ψ4 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ4 (0) + i sin(𝑚𝑐 2 t⁄ħ)Ψ4 (0)

(8.10-b)

In conclusion, in this section, we have considered the solutions of the non-relativistic chiral Majorana equation and the Dirac equation.

9. THE GENERIC MAJORANA 4-SPINORS EQUATION ̃j = Ψ ̃ j (x, y, z, t), j = 1,2,3,4, we With the Majorana wave functions, Ψ have given the 4 Majorana partial differential equations (5.13-a-b-c-d). Let us define the two Majorana bi-spinors wave functions ̃ ̃ ̃ a = (Ψ1 ), Ψ ̃ b = (Ψ3 ), Ψ ̃2 ̃4 Ψ Ψ

(9.1-a-b)

The Majorana equations (5.13a-b-c-d), for the bi-spinors, become: ̃a = +𝑐𝑥 𝜕𝑥 Ψ ̃ b − 𝑐0 𝜕𝑦 Ψ ̃ a + 𝑐𝑧 𝜕𝑧 Ψ ̃ b − 𝑖(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̃b 𝜕𝑡 Ψ ̃b = +𝑐𝑥 𝜕𝑥 Ψ ̃a + 𝑐0 𝜕𝑦 Ψ ̃ b + 𝑐𝑧 𝜕𝑧 Ψ ̃ a − 𝑖(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̃a 𝜕𝑡 Ψ (9.2-a-b) where 𝜕𝜇 = 𝜕/𝜕𝜇 and, 𝑥 , 𝑦 , 𝑧 , are the Pauli 2x2 matrices 0 1 0 −i 1 𝑥 = ( ), 𝑦 = ( ), 𝑧 = ( 1 0 i 0 0

0 1 0 ), 0 = ( ) = I2 −1 0 1 (9.3-a-b-c-d)

and where 0 , is the 2x2 unit matrix I2 . Let us define the Majorana 4-spinors wave function from the two bi-spinors (9.1-a-b):

Computing Hyperincursive Discrete … ̃ ̃ = (Ψa ) Ψ ̃b Ψ

137

(9.1-c)

The Majorana equations (9.2-a-b) for the 4-spinors become the following generic Majorana equation: ̃ = 𝑐α𝑥 𝜕𝑥 Ψ ̃ − 𝑐β𝜕𝑦 Ψ ̃ + 𝑐α𝑧 𝜕𝑧 Ψ ̃ − 𝑖(𝑚𝑐 2 ⁄ħ)α𝑦 Ψ ̃ 𝜕𝑡 Ψ

(9.4-c)

where the 4x4 matrices, α𝑥 , α𝑦 , α𝑧 , are defined with the Pauli matrices by 0 α𝑥 = ( 𝑥

0 𝑥 ), α𝑦 = ( 0 𝑦

𝑦 0 ), 𝑧 = ( 0 𝑧

𝑧 ) 0

(9.5-a-b-c)

and β is defined with the unit matrix by  β=( 0 0

0 ) −0

(9.5-d)

In the next section we will give the generic Dirac 4-spinors equation and its relation to the Majorana equation.

10. THE GENERIC DIRAC 4-SPINORS EQUATION In defining the Dirac wave function by Ψj = Ψj (𝑥, 𝑦, 𝑧, t), j = 1,2,3,4, we have given the 4 Dirac partial differential equations (5.23-a-b-c-d). Let us define the Dirac bi-spinors wave functions Ψa = (

Ψ1 Ψ ), Ψb = ( 3 ), Ψ2 Ψ4

The Dirac equations (5.23a-b-c-d), for the bi-spinors, become:

(10.1-a-b)

138

Daniel M. Dubois 𝜕𝑡 Ψa = −𝑐𝑥 𝜕𝑥 Ψb − 𝑐𝑦 𝜕𝑦 Ψb − 𝑐𝑧 𝜕𝑧 Ψb − 𝑖(𝑚𝑐 2 ⁄ħ)0 Ψa

𝜕𝑡 Ψb = −𝑐𝑥 𝜕𝑥 Ψa − 𝑐𝑦 𝜕𝑦 Ψa − 𝑐𝑧 𝜕𝑧 Ψa + 𝑖(𝑚𝑐 2 ⁄ħ)0 Ψb (10.2-a-b) where 𝜕𝜇 = 𝜕/𝜕𝜇 and 𝑥 , 𝑦 , 𝑧 , are the Pauli 2x2 matrices (9.3-a-b-c), and where 0 , is the 2x2 unit matrix I2 (9.3-d). Let us define the Dirac 4-spinors from the two bi-spinors (10.1-a-b): Ψ Ψ = ( a) Ψb

(10.3)

The Dirac equations (10.2-a-b) for the 4-spinors become the following generic Dirac equation: 𝜕𝑡 Ψ = −𝑐α𝑥 𝜕𝑥 Ψ − 𝑐α𝑦 𝜕𝑦 Ψ − 𝑐α𝑧 𝜕𝑧 Ψ − 𝑖(𝑚𝑐 2 ⁄ħ)βΨ

(10.4)

where the 4x4 matrices, α𝑥 , α𝑦 , α𝑧 , were defined in equations (9.5-a-b-c), and β was defined in equation (9.5-d). In comparing the Dirac equation (10.4-c) with the Majorana equation (9.4-c), we see that there is an inversion of the two matrices, β, and, α𝑦 , with an inversion of signs of the space variables, 𝑥, and, z. This is in agreement with my demonstration, given in the preceding section 5, of the bifurcation of the Majorana real equation to the Dirac complex equations (Dubois, 2019b). Let us remark that the Pauli matrices represent logical quantum gates in quantum compution. Let us first recall the properties of the Pauli 2x2 matrices, 𝑥 , 𝑦 , 𝑧 : 1 𝜎𝑥2 = 𝜎𝑦2 = 𝜎𝑧2 = 𝐼2 = ( 0

0 ) 1

The square of the Pauli gates are equal to the 2x2 unit gate. The square of the unit gate is equal to itself:

(10.5)

Computing Hyperincursive Discrete …

139

𝜎02 = 𝐼2 = 𝜎0

(10.6)

The Pauli gates do not commute, and show the following properties: 𝜎𝑦 𝜎𝑧 − 𝜎𝑧 𝜎𝑥 = 𝑖𝜎𝑥 𝜎𝑧 𝜎𝑥 − 𝜎𝑥 𝜎𝑧 = 𝑖𝜎𝑦 𝜎𝑥 𝜎𝑦 − 𝜎𝑦 𝜎𝑥 = 𝑖𝜎𝑧

(10.7-a-b-c)

With the Kronecker product, , it is possible to create the 4x4 matrices α𝑥 , α𝑦 , α𝑧 and, β, with the product of two Pauli 2x2 matrices, as follows: 0 α𝑥 = σx σx = (0 0 1

0 0 1 0

0 1 0 0

1 0 0 0), α = σ σ = ( 0 0 𝑦 x y 0 0 −𝑖 0 𝑖 0

0 0 1 0 α𝑧 = σx σz = ( 0 0 0 −1) 1 0 0 0 0 −1 0 0

0 𝑖 0 0

−𝑖 0) 0 0

(10.8-a-b-c) ,

1 β = σz σ0 = (0 0 0

0 1 0 0

0 0 0 0 ) −1 0 0 −1

(10.8-d)

The square of the matrices, α𝑥 , α𝑦 , α𝑧 , β, are equal to the unit matrix, 𝐼4 :

α2x

=

α2y

=

α2z

1 = β = 𝐼4 = (0 0 0 2

0 1 0 0

0 0 1 0

0 0) 0 1

(10.9)

The matrices, α𝑥 , α𝑦 , α𝑧 , β, show the following important properties: 𝛼𝑦 𝛼𝑧 + 𝛼𝑧 𝛼𝑥 = 0

140

Daniel M. Dubois 𝛼𝑧 𝛼𝑥 + 𝛼𝑥 𝛼𝑧 = 0 𝛼𝑥 𝛼𝑦 + 𝛼𝑦 𝛼𝑥 = 0

(10.10-a-b-c)

𝛼𝑦 𝛽 + 𝛽𝛼𝑦 = 0 𝛼𝑧 𝛽 + 𝛽𝛼𝑧 = 0 𝛼𝑥 𝛽 + 𝛽𝛼𝑥 = 0

(10.11-a-b-c)

The next section deals with a fundamental invariant related to the Pauli matrix, 𝑥 .

11. A NEW INVARIANT OF THE NON-RELATIVISTIC QUANTUM MAJORANA AND DIRAC WAVE FUNCTIONS This section gives the comparison of the solutions of the non-relativistic quantum Majorana and Dirac equations after (Dubois, 2019e). In the limit, 𝑝 ≪ 𝑚𝑐, the particles are at rest, with a momentum 𝑝 ≅ 0. In the preceding section, we have given the following 2-components chiral Majorana equations (7.10-a-b); ̌1 ⁄𝜕𝑡 = +𝑐 𝜕Ψ ̌1 ⁄𝜕𝑥 + i 𝑐 𝜕Ψ ̌ 2 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌ 2 ⁄𝜕𝑧 + i (𝑚𝑐 2 ⁄ħ)Ψ ̌ 2∗ + 𝜕Ψ ̌ 2 ⁄𝜕𝑡 = −𝑐 𝜕Ψ ̌ 2⁄𝜕𝑥 − i 𝑐 𝜕Ψ ̌1 ⁄𝜕𝑦 − 𝑐 𝜕Ψ ̌1 ⁄𝜕𝑧 − i (𝑚𝑐 2 ⁄ħ)Ψ ̌1∗ + 𝜕Ψ (11.1-a-b) In the non-relativistic limit, these 2-components Majorana equations are given by ̌1 ⁄𝜕𝑡 = + i (𝑚𝑐 2 ⁄ħ)Ψ ̌ 2∗ + 𝜕Ψ

(11.1-c)

̌ 2 ⁄𝜕𝑡 = − i (𝑚𝑐 2 ⁄ħ)Ψ ̌1∗ + 𝜕Ψ

(11.1-d)

Computing Hyperincursive Discrete …

141

with ̌ ̌ (t) = (Ψ1 (t)), Ψ ̌ 2 (t) Ψ

(11.2)

these Majorana equations become ̌ = −(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̌∗ 𝜕𝑡 Ψ

(11.3-a)

0 −1 where 𝜕𝑡 = 𝜕/𝜕𝑡, with the Pauli matrix, 𝑦 = i ( ) 1 0 The complex conjugate of equation (11.3-a) is given by ̌ ∗ = +(𝑚𝑐 2 ⁄ħ)𝑦 Ψ ̌ 𝜕𝑡 Ψ

(11.3-b)

These 2 equations (11.3-a-b) transform to the following second order equation ̌ (t) = −(𝑚𝑐 2 ⁄ħ)2 Ψ ̌ (t) 𝜕𝑡2 Ψ

(11.4)

which is identical to the second order derivative of the Klein-Gordon equation for a particle at rest, with a 2-spinors complex Majorana function ̌ (t). Ψ The analytical solution of the equation (11.4) is given by ̌ (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌ (0) − sin(𝑚𝑐 2 t⁄ħ) 𝑦 Ψ ̌ ∗ (0) Ψ

(11.5)

or, in explicit form ̌1 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌1 (0) + i sin(𝑚𝑐 2 t⁄ħ)Ψ ̌ 2∗ (0) Ψ

(11.6-a)

̌ 2 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌ 2 (0) − i sin(𝑚𝑐 2 t⁄ħ)Ψ ̌1∗ (0) Ψ

(11.6-b)

142

Daniel M. Dubois Now let us consider the following Dirac 2-spinors ̂ (t) = (Ψ1 (t)), Ψ Ψ4 (t)

(11.7)

The non-relativistic Dirac equation is given by ̂ (t) = −𝑖(𝑚𝑐 2⁄ħ)𝑧 Ψ ̂ (t) 𝜕𝑡 Ψ

(11.8)

1 0 where 𝜕𝑡 = 𝜕/𝜕𝑡, and with the Pauli matrix, 𝑧 = ( ) 0 −1 The analytical solution of the Dirac equation (11.8) is given by ̂ (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ ̂ (0) − i sin(𝑚𝑐 2 t⁄ħ) 𝑧 Ψ ̂ (0) Ψ

(11.9)

or, in explicit form Ψ1 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ1 (0) − i sin(𝑚𝑐 2 t⁄ħ)Ψ1 (0)

(11.10-a)

Ψ4 (t) = cos(𝑚𝑐 2 t⁄ħ)Ψ4 (0) + i sin(𝑚𝑐 2 t⁄ħ)Ψ4 (0)

(11.10-b)

Now, we will show the relation between the solutions of the Dirac equations from the solutions of the Majorana equations with the method of Lamata et al. (Lamata et al., 2012). So, let us consider the sum of the forward and backward solutions (11.9) of the Dirac equation ̂ (+t) + Ψ ̂ (−t)]/2 = cos(𝑚𝑐 2 t⁄ħ) Ψ ̂ (0) [Ψ and the difference of the forward and backward solutions ̂ ∗ (+t) − Ψ ̂ ∗ (−t)]/2 = i sin(𝑚𝑐 2 t⁄ħ) 𝑧 Ψ ̂ ∗ (0) [Ψ

Computing Hyperincursive Discrete … In multiplying by the Pauli matrix, 𝑥 = (

143

0 1 ), the relation becomes 1 0

̂ ∗ (0) = sin(𝑚𝑐 2 t⁄ħ)𝑦 Ψ ̂ ∗ (0) i sin(𝑚𝑐 2 t⁄ħ)𝑥 𝑧 Ψ so we obtain the following relation between the solution of the Dirac equation and the solution of the Majorana equation ̌ (t) = [Ψ ̂ (+t) + Ψ ̂ (−t)]/2 − 𝑥 [Ψ ̂ ∗ (+t) − Ψ ̂ ∗ (−t)]/2 Ψ

(11.11)

that is equal to the solution (11.5) of the Majorana equation. We obtain the same result as Lamata et al, but they have not given the inverse equation for obtaining the Majorana solution from the Dirac solution. Let us now make the inverse in giving the Dirac solution as a function of the Majorana solution, after (Dubois, 2019e). So, let us start from the solution (11.5) of the Majorana equation. Let us consider the sum of the forward and backward solutions (11.5) ̌ (t) + Ψ ̌ (−t)]/2 = cos(𝑚𝑐 2 t⁄ħ)Ψ ̌ (0) [Ψ and the difference of the forward and backward solutions ̌ ∗ (t) − Ψ ̌ ∗ (−t)]/2 = sin(𝑚𝑐 2 t⁄ħ) 𝑦 Ψ ̌ (0) [Ψ Let us multiply this relation by the Pauli matrix, 𝑥 , ̌ (0) = i sin(𝑚𝑐 2 t⁄ħ) 𝑧 Ψ ̌ (0) sin(𝑚𝑐 2 t⁄ħ) 𝑥 𝑦 Ψ So we obtain the relation between the solution of the Majorana equation and the solution of the Dirac equation (11.9) as follows ̂ (t) = [ Ψ ̌ (t) + Ψ ̌ (−t)]/2 − 𝑥 [Ψ ̌ ∗ (t) − Ψ ̌ ∗ (−t)]/2 Ψ

(11.12)

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Daniel M. Dubois

Surprisingly, the transformation relation is invariant, the relation (11.12), which gives the Dirac wave function from the Majorana wave function, is identical to the relation (11.11), which gives the Majorana wave function from the Dirac wave function. At our knowledge, this is a new invariant of the non-relativistic quantum Majorana and Dirac equations. This invariant is based on the Pauli matrix 𝑥 , that is the quantum gate X, which is the “spin flip” or the NOT gate, a reversible gate in quantum computation.

12. QUANTUM COMPUTATION WITH REVERSIBLE GATES In this chapter, we have used some reversible quantum gates as defined for developing quantum computers. The quantum Pauli gates X, Y, Z, that operate on one-qubit, are based on Pauli matrices: X = 𝑥 = (

0 1

1 ) 0

(12.1)

which is a “spin flip” or NOT gate, 0 i

−i ) 0

(12.2)

1 0

0 ) −1

(12.3)

𝑌 = 𝑦 = ( 𝑍 = 𝑧 = (

that is a phase shift gate with 𝜑 = 𝜋. Only the X and Z are necessary, because the Y can be deduced from them: 0 𝑌 = iXZ = ( i

−i ) 0

The square of each Pauli gate is the identity matrix 𝐼

(12.4)

Computing Hyperincursive Discrete …

145

𝐼 2 = 𝑋 2 = 𝑌 2 = 𝑍 2 = −𝑖𝑋𝑌𝑍 = 𝐼

(12.5)

The quantum Hadamard gate 𝐻2 is defined by 𝐻2 =

1 +1 ( √2 +1

+1 ) −1

(12.6)

which is a rotation gate, that gives a basis change. The Hadamard gate can be deduced from the X and the Z gates: 𝐻2 =

1 (X √2

+ Z) =

1 +1 ( √2 +1

+1 ) −1

(12.7)

In the section 4, the Hadamard matrix was deduced from the rotation matrix R1 (θ) = (

sin(θ) cos(θ) ) cos(θ) −sin(θ)

(12.8)

for the angle θ = π/4, as sin(π/4) cos(π/4) 1 +1 +1 R1 (π/4) = ( ) = 2( ) = 𝐻2 √ +1 −1 cos(π/4) −sin(π/4)

(12.9)

In this section 4, we have demonstrated a remarkable result: by the rotation of the position and velocity of the two incursive discrete equations of the harmonic oscillator, with the Hadamard matrix gate, we have transformed the incursive discrete equations to recursive discrete equations of the harmonic oscillator. Let us remark that the X ans Z gates can be deduced from this rotation matrix for the angles θ = 0 and θ = π/2 respectively sin(0) cos(0) 0 R1 (0) = ( )=( cos(0) −sin(0) 1

1 )=X 0

(12.10)

146

Daniel M. Dubois sin(π/2) cos(π/2) 1 0 R1 (π/2) = ( )=( )=Z cos(π/2) −sin(π/2) 0 −1

(12.11)

In the technology of quantum computers, many quantum gates are also defined, for example, the phase gate, the square root of the NOT gate, the CNOT gate and the CCNOT gate. The phase gate is given by 1 0 𝑆=( ) 0 i

(12.12)

This phase gate can also be deduced from the Z gate, 2 1 0 𝑆 = √𝑍 = ( ) 0 i

(12.13)

indeed, it is the square root of Z, 1 𝑆𝑆 = 𝑍 = ( 0

0 1 0 1 )( )=( i 0 i 0

0 ) −1

(12.14)

The square root of the NOT gate is written as 1 1+i 1−i 2 ) √X = √NOT = 2 ( 1−i 1+i

2

(12.15)

The XOR (exclusive OR) gate, the Controlled NOT gate CNOT, is a two-qubit operation defined by 1 𝐶𝑁𝑂𝑇 = (0 0 0

0 1 0 0

0 0 0 1

0 0) 1 0

(12.16)

And finally, the reversible Toffoli gate, the Controlled-Controlled NOT gate CCNOT, is a three-qubit operation defined by

Computing Hyperincursive Discrete … 1 0 0 𝐶𝐶𝑁𝑂𝑇 = 0 0 0 0 (0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0)

147

(12.17)

The Hadamard and Toffoli gates are quantum universal gates (Aharonov, 2003).

REFERENCES Aharonov Dorit 2003 A simple proof that Toffoli and Hadamard are quantum universal, arXiv preprint quant-ph/0301040. Antippa Adel F and Dubois Daniel M 2004 “Anticipation, Orbital Stability, and Energy Conservation in Discrete Harmonic Oscillators” Computing Anticipatory Systems: Conf. Proc. of CASYS’03–Sixth Int. Conf. (11-16 August 2003 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 718 pp 3–44. Antippa Adel F and Dubois Daniel M 2006a “The Dual Incursive System of the Discrete Harmonic Oscillator” Computing Anticipatory Systems: Conf. Proc. of CASYS’05–Seventh Int. Conf. (8-13 August 2005 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 839 pp 11–64. Antippa Adel F and Dubois Daniel M 2006b “The Superposed Hyperincursive System of the Discrete Harmonic Oscillator” Computing Anticipatory Systems: Conf. Proc. of CASYS’05–Seventh Int. Conf. (8-13 August 2005 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 839 pp. 65–126. Antippa Adel F and Dubois Daniel M 2007 “Incursive Discretization, System Bifurcation, and Energy Conservation” Journal of Mathematical Physics 48 1 (012701).

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Antippa Adel F and Dubois Daniel M 2008a “Hyperincursive Discrete Harmonic Oscillator” Journal of Mathematical Physics 49 3 (032701). Antippa Adel F and Dubois Daniel M 2008b “Synchronous Discrete Harmonic Oscillator” Computing Anticipatory Systems: Conf. Proc. of CASYS’07–Eighth Int. Conf. (6-11 August 2007 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 1051 pp 82–99. Antippa Adel F and Dubois Daniel M 2010a “Discrete Harmonic Oscillator: A Short Compendium of Formulas” Computing Anticipatory Systems: Conf. Proc. of CASYS’09–Ninth Int. Conf. (3-8 August 2009 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 1303 pp 111–120. Antippa Adel F and Dubois Daniel M 2010b “Time-Symmetric Discretization of The Harmonic Oscillator” Computing Anticipatory Systems: Conf. Proc. of CASYS’09–Ninth Int. Conf. (3-8 August 2009 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 1303 pp 121–125. Antippa Adel F and Dubois Daniel M 2010c “Discrete Harmonic Oscillator: Evolution of Notation and Cumulative Erratum” Computing Anticipatory Systems: Conf. Proc. of CASYS’09–Ninth Int. Conf. (3-8 August 2009 Liege Belgium) ed D M Dubois (Melville, New York: American Institute of Physics) AIP CP 1303 pp 126–130. Dirac P A M 1928 “The Quantum Theory of the Electron” Proc of the Royal Society A Mathematical, Physical and Engineering Sciences Vol 117 No. 778 pp 610-624. Dirac P A M 1964 Lectures on Quantum Mechanics (New York: Academic Press). Dubois Daniel M 1995 “Total Incursive Control of Linear, Non-linear and Chaotic Systems” Advances in Computer Cybernetics ed G E Lasker (Canada: The International Institute for Advanced Studies in Systems Research and Cybernetics) Volume II pp 167-171 ISBN 0921836236. Dubois Daniel M 2016 “Hyperincursive Algorithms of Classical Harmonic Oscillator Applied to Quantum Harmonic Oscillator Separable Into Incursive Oscillators” Unified Field Mechanics, Natural Science

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Beyond the Veil of Spacetime: Proc. of the IXth Symp Honoring Noted French Mathematical Physicist Jean-Pierre Vigier (16-19 november 2014 Baltimore USA) Ed R L Amoroso, L H Kauffman et al. (Singapore: World Scientific) pp 55-65. Dubois Daniel M 2018 “Unified Discrete Mechanics: Bifurcation of Hyperincursive Discrete Harmonic Oscillator, Schrödinger’s Quantum Oscillator, Klein-Gordon’s Equation and Dirac’s Quantum Relativist Equations” Unified Field Mechanics II, Formulations and Empirical Tests: Proc. of the Xth Symp Honoring Noted French Mathematical Physicist Jean-Pierre Vigier (25-28 July 2016 Porto Novo Italy) Ed R L Amoroso, L H Kauffman et al. (Singapore: World Scientific) pp 158177. Dubois Daniel M 2019a “Unified discrete mechanics II: The space and time symmetric hyperincursive discrete Klein-Gordon equation bifurcates to the 4 incursive discrete Majorana real 4-spinors equations” Journal of Physics: Conf. Ser. 1251 012001, open access; https://iopscience.iop.org/article/10.1088/1742-6596/1251/1/012001. Dubois Daniel M 2019b “Unified discrete mechanics III: the hyperincursive discrete Klein-Gordon equation bifurcates to the 4 incursive discrete Majorana and Dirac equations and to the 16 Proca equations” Journal of Physics: Conf. Ser. 1251 012002, open access, https://iopscience.iop.org/article/10.1088/1742-6596/1251/1/012002. Dubois Daniel M 2019c “Rotation of the Two Incursive Discrete Harmonic Oscillators to Recursive Discrete Harmonic Oscillators with the Hadamard Matrix,” Proc. of the Symp. on Causal and Anticipative Systems in Living Science, Biophysics, Relativistic Quantum Mechanics, Relativity: held as part of the 31st Int. Conf. on Systems Research, Informatics and Cybernetics (July 29-August 2, 2019 Baden-Baden Germany) Ed D M Dubois and G E Lasker (IIAS) Volume I pp 7-12 ISBN 978-1-897546-41-3. Dubois Daniel M 2019d “Rotation of the Relativistic Quantum Majorana Equation with the Hadamard Matrix and Unitary Matrix U” Proc. of the Symp. on Causal and Anticipative Systems in Living Science, Biophysics, Relativistic Quantum Mechanics, Relativity: held as part of

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the 31st Int. Conf. on Systems Research, Informatics and Cybernetics (July 29-August 2, 2019 Baden-Baden Germany) Ed D M Dubois and G E Lasker (IIAS) Volume I pp 13-18 ISBN 978-1-897546-41-3. Dubois Daniel M 2019e “Relations between the Majorana and Dirac Quantum Equations” Proc. of the Symp. on Causal and Anticipative Systems in Living Science, Biophysics, Relativistic Quantum Mechanics, Relativity: held as part of the 31st Int. Conf. on Systems Research, Informatics and Cybernetics (July 29-August 2, 2019 Baden-Baden Germany) Ed D M Dubois and G E Lasker (IIAS) Volume I pp 19-24 ISBN 978-1-897546-41-3. Dubois Daniel M 2019f “Review of the time-symmetric hyperincursive discrete harmonic oscillator separable into two incursive harmonic oscillators with the conservation of the constant of motion” Journal of Physics: Conf. Series 1251 012013 - doi:10.1088/1742-6596/1251/ 1/012013, article online open access - https://iopscience.iop.org/ article/10.1088/1742-6596/1251/1/012013. Gordon Walter 1926 “Der Comptoneffekt nach Schrödingerschen Theorie” “The Compton effect according to Schrödinger's theory”. Zeitschrift für Physik 40 p 117. Klein Oskar 1926 “Quantentheorie und fünfdimensionale Relativitätstheorie” [“Quantum theory and five-dimensional relativity theory”] Zeitschrift für Physik 37 p 895. Lamata L, Casanova J, Egusquiza I L, and Solano E, 2012 “The nonrelativistic limit of the Majorana equation and its simulation in trapped ions” Physica Scripta, Volume 2012, T147 arXiv:1109.0957v2 (quant-ph). https://doi.org/10.1088/0031-8949/2012/T147/014017. Majorana Ettore 1937 “Teoria simmetrica dell’elettrone e del positrone” [“Teoria simmetrica dell’elettrone e del positrone”] Il Nuovo Cimento, 14 p 171. Palash B Pal, 2011 “Dirac, Majorana and Weyl fermions” American Journal of Physics, 79 485 arXiv: 1006.1718v2. (hep-ph), https://doi.org/10.1119/1.3549729. Pessa Eliano 2006 “The Majorana Oscillator” Electronic Journal of Theoretical Physics EJTP 3 10 pp 285-292.

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Proca Al 1930 “Sur l’équation de Dirac” J. Phys. Radium 1 (7) pp 235-248. Proca Al 1932 “Quelques observations concernant un article “sur l’équation de Dirac” J. Phys. Radium, 3 (4) pp 172-184.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 5

QUANTUM COMPUTING AND THE QUANTUM MIND: A NEW APPROACH TO QUANTUM GRAVITY Paola Zizzi1, and Massimo Pregnolato2 1

Department of Brain and Behavioural Sciences, 2 Department of Drug Sciences, Univeristy of Pavia, Pavia, Italy

ABSTRACT We’ll illustrate some subtle features of quantum computers, which are less popular in current literature, with respect to some more technical ones (like for example, quantum algorithms). In particular, we’ll focus on those aspects of quantum computers which are related to quantum gravity (via entanglement) and to the quantum side of the mind (via quantum computational logic and quantum meta-language). We expect that, in the quantum computing framework, Quantum Gravity and the Quantum Mind may appear strictly interconnected. Just to use a metaphorical language, we can say the following: General Relativity teaches how space-time deals 

Corresponding Author’s Email: [email protected].

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Paola Zizzi and Massimo Pregnolato with matter and vice-versa, Quantum Mechanics teaches (or better, it should) how reality deals with measurements. Time (the problem of time) is in the middle, together with the classical-minded observer and Quantum Gravity is still a chimera. But…Quantum computing exploits the inner quantum-computational side of quantum gravity. Quantum gravity (or better its quantum space-time background) seems to have a meta-logical structure (a quantum metalanguage) quite similar to that of the quantum brain when the latter is described by a (dissipative) Quantum Field Theory. In more suggestive words, empty quantum space-time tells to the Quantum Mind how to quantum meta-think.

Keywords: quantum computers, quantum logic, quantum mind, quantum metalanguage, quantum gravity

1. INTRODUCTION “We are the dreams of which the void is made.”

In this review chapter, we present a structured set of collected articles concerning the various (sometimes unexpected) relationships between quantum computing, quantum gravity and quantum brain/mind. The main theme seems to be the quantum metalanguage, the meta-glue that gives a common meaning (a common semantics) to the logic underlying the apparently different arguments. In general, a formal metalanguage is a formal language talking about another formal language, called the object language (in general the latter is a logic, or a program). In particular, we’ll consider a formal metalanguage based on sequent calculus, introduced by Gentzen (1935, 405–431). In this formalism, the metalanguage appears to be much simpler than the logic it “controls”. In fact a logic has: propositions, connectives, structural rules (which give the structure of the rules), logical rules for the logical connectives (for example the formation rule of a logical connective), and theorems. Nevertheless, a logic itself is simple syntax. Then, without a metalanguage, which encodes the semantics, a logic has no “meaning”. The simpler structure of a metalanguage consists of assertions (propositions which are asserted) and two metalinguistic links between

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assertions. Moreover, the metalanguage has the identity axiom, and a metarule, the “cut.” It was believed that the cut was the only possible meta-rule of sequent calculus, but quite recently, it was found (Zizzi 2010, 5976) that in the case of a quantum metalanguage, there is a second meta-rule, the EPRrule. Finally, a quantum metalanguage can have meta-theorems. The only one known since now is the “teleportation” (TEL) theorem, presented for the first time in (Zizzi 2018b). The most direct relation between a metalanguage and its object language is the “definitional equation” (Sambin et al. 2000, 979-1013), which relates assertions to propositions, and metalinguistic links to logical connectives. Everything about Quantum Metalanguage seems quite formal (as it actually is) and rather harmless (but it isn't). The power of a Quantum Metalanguage seems to be its propensity to attach itself to anything labelled “quantum.” So we discover that quantum space-time has a quantum metalanguage as quantum brain functions are dictated by a quantum metalanguage. Then we found that the characteristics of quantum computation of quantum gravity and quantum mind actually derive from the same quantum metalanguage that governs the logic of quantum computers. At this point, it may be worth clarifying some points. First of all, when we talk about quantum brain and quantum mind, we mean two different things. The first is physical and can be described as a (dissipative) quantum field theory (DQFT) (Vitiello 2000) and is formalized by a quantum metalanguage. The latter is a quantum logic, derived from the Quantum Metalanguage which formally describes the first. Second, the quantum logic of the quantum mind is the same logic of a quantum computer. It follows that QFTs (for example quantum gravity and DQFT of the brain) “possess” their own quantum metalanguage. Their derived logics are the same as the logic of a quantum computer, a quantum system that follows the rules of Quantum Mechanics (QM). The quantum computer cannot have its own Quantum Metalanguage, in fact we remind that a Quantum Metalanguage is not Turing-computable, this is the reason why a quantum computer could not use it. A quantum computer is a machine and must be “programmed”, either by a human possessing a quantum brain, or more generally, by QFT. This leads to the understanding a deeper difference between QFT and QM: they are found on

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two different logical levels, as well as on two different physical levels. Finally, it is just the fact that a (Quantum) Metalanguage is not Turingcomputable, which, together with Gödel’s incompleteness theorem, highlights the existence of the non-algorithmic side of the mind (Zizzi 2012). The non-computability of the (quantum) metalanguage, is for the same reasons discussed above, a challenge for a (future) quantum gravity theory which might not be complete, in the sense of Gödel’s Incompleteness Theorem (Gödel 1931, 173-98). This fact would skip the (unknown) Theory of Everithing (TOE), but on the other side it would shorten the ontological distance between human beings and the fundamental structure of quantum space-time. In fact, in this view, empty quantum space-time seems to be the seed of the quantum metalanguage of the human mind. Quantum gravity and the human mind escape completeness to hold a “meaning”.

2. THE “BIG WOW” AND THE QUANTUM MIND “We were in the mind of the Cosmos.”

The early inflationary universe can be described in terms of quantum information. More specifically, the inflationary universe can be viewed as a superposed state of quantum registers. Actually, during inflation, one can speak of a quantum superposition of universes. At the end of inflation, only one universe is selected, by a mechanism called self-reduction, which is consistent with Penrose’s objective reduction (OR) model. The quantum gravity threshold of (OR) is reached at the end of inflation, and corresponds to a superposed state of 109 quantum registers. This is also the number of superposed tubulins – qubits in our brain, which undergo the Penrose– Hameroff orchestrated objective reduction, (Orch OR), leading to a conscious event. Then, an analogy naturally arises between the very early quantum-computing universe, and our mind. In fact, we argue that at the end of inflation, the universe underwent a cosmic conscious event, the so-called

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“Big Wow,” which acted as an imprinting for the future minds to come, with future modes of computation, consciousness and logic. The post-inflationary universe organized itself as a cellular automaton (CA) with two computational modes: quantum and classical, like the two conformations assumed by the cellular automaton of tubulins in our brain, as in Hameroff’s model. In the quantum configuration, the universe quantum-evaluates recursive functions, which are the laws of physics in their most abstract form. To do so in a very efficient way, the universe uses, as subroutines, black holes – quantum computers and quantum minds, which operate in parallel. The outcomes of the overall quantum computation are the universals, the attributes of things in themselves. These universals are partially obtained also by the quantum minds, and are endowed with subjective meaning. The units of the subjective universals are qualia, which are strictly related to the (virtual) existence of Planckian black holes. Further, we consider two aspects of the quantum mind, which are not algorithmic in the usual sense: the self, and mathematical intuition. The self is due to a reversible self-measurement of a quantum state of superposed tubulins. Mathematical intuition is due to the paraconsistent logic of the internal observer in a quantum-computing universe.

3. THE NON-ALGORITHMIC SIDE OF THE MIND “There are thoughts that are not such.”

The original conjecture of Penrose about the existence of nonalgorithmic aspects of the mind regarded mainly consciousness. However, we reported (Zizzi et al. 2012, 1- 8) that conscious, rational human thought consists of a very rapid sequence of decoherence processes from the quantum computational mode to the classical one. More specifically, in the Penrose-Hameroff Orch-Or theory (Hameroff et al. 1996, 507-540) superposed tubulins/qubits decohere to classical bits at a fast rate. Accordingly to this theory, it looks like consciousness is made of “flashes” of classical computation. Consciousness cannot be identified with the

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classical mode of the mind, because that would lead to an absurd conclusion: a classical Turing machine, which persists in the classical mode, would be more “conscious” than a human mind. The problem is that the static conscious state of a classical computer is totally useless for any kind of aware reasoning, which is dynamical by definition. More precisely, our philosophical point of view is that there are three different ways by which fundamental high-level mental activities manifest themselves (Zizzi 2012). Two are algorithmic (Turing-computable): the classical computational mode, and the quantum computational mode. The third is non-computable. Each of the three modes of the mind can be formalized in a mathematical way (the first two by a logic, the third by a metalanguage) and also acquires a physical interpretation, and a psychological status. The quantum mode concerns extremely fast mental processes of which humans are mostly unaware of, and is logically described by the logic of quantum information and quantum computation, called Lq (Zizzi 2010, 5976). The atomic propositions of Lq are interpreted as the basis states of a complex Hilbert space, while the compound propositions are interpreted as qubit states. Therefore, the physical model of the quantum mode of the mind is Quantum Information. The classical mode concerns those mental processes, which humans are aware of. It arises from the decoherence of the quantum computational state, and is logically described by a sub-structural, nonclassical logic called Basic Logic (BL) (Sambin et al. 2000, 979-1013). In a sense, the quantum mode “prepares” the classical mode, which otherwise would take very long to perform even the easiest tasks, but most of the quantum information remains hidden. The classical mode comes into play by “flashes” of decoherence, which occur so often that humans get the impression of a “continuous awareness.” The third mode, which is nonalgorithmic, concerns metathought (intuition, intention and control) and is described by a quantum metalanguage (QML) (Zizzi 2010, 5976), which “controls” the logic Lq of the quantum mode. The assertions of the QML are physically interpreted as the field states of a dissipative quantum field theory (DQFT) of the brain (Vitiello 1995, 2001). The atomic propositions of the quantum object-language (QOL) (Zizzi 2010, 5976) are asserted, in the quantum metalanguage (QML) with an assertion degree, which is a complex

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number. We show that this fact requires that the atomic propositions in the QOL are endowed with a fuzzy modality “Probably” (Hajek 1998) and have fuzzy (partial) truth-values (Zadeh 1996), which sum up to one. The QML is the language of metathought. The very importance of metathough, which deals with intuition, intention and control, resides in the fact that it distinguishes humans from machines. In fact, the language of metathough, which is non-algorithmic, being described by a metalanguage, cannot be acquired independently by a machine, which is endowed only with an object-language. The physical interpretation of this impossibility, is the irreversibility of the reduction process (Zizzi 2011) from the DQFT of the brain to the quantumcomputational theory of the mind. The question of machine implementation of (human-like) mental processing is very old and dates to the early days of Artificial Intelligence. As well known, in late 1950 A. M. Turing adopted a purely behavioural criterion (instantiated through his famous test) (Turing 1950, 433-460) to establish whether a machine can be considered intelligent or not. Within this approach, a machine is recognized as endowed with a mind when its behaviour is indistinguishable from the one of a human being performing (supposed) mental operations. Later this attitude gained a wide popularity when Cognitive Psychology adopted the Computational Symbolic Approach (Newell 1976) speaking of the (functional) equivalence between mind and a digital computer. In the eighties the philosophical considerations already made by Searle and others (1980, 417-458), began to cast serious doubts about the validity of this definition of mind. An adequate logic of reasoning should take into account the fact that humans have basic logical rules, and in general structural rules are by-passed. In other words, the logic of reasoning should be much more concrete and weaker than other abstract and structural logics, like for example Aristotelian classical logic. This requires a sub-structural logic, which can be viewed as the general platform for any other logic. All these requirements were met in BL (Sambin et al. 2000, 979-1013) in the classical case (or classical mode). A quantum version of BL, called Lq, was introduced by us (Zizzi 2010, 5976). In Lq, two new logical connectives were introduced, the connective “quantum superposition” (the quantum version of the classical conjunction) and the

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connective “entanglement.” Finally, the probabilistic feature of any quantum theory is also present in Lq, because the partial truth values, whose range is the real interval [0,1] are interpreted as probabilities. This takes into account the fuzzy and probabilistic features of some non-formalized aspects of (nonordinary) thought. In this context, we shall try to clarify Penrose’s conjecture (Penrose 1989) on the non-computational aspects of the mind in relation with Gödel’s First Incompleteness Theorem (Gödel 1931). Penrose claims that a mathematician can assert the truth of a Gödel sentence G, although the latter cannot be demonstrated within the axiomatic system, because he is capable of recognizing an indemonstrable truth due to the non-algorithmic aspect of the mind. In our opinion, the fact that the mathematician can assert the truth of G, is that he is using the non-computable mode of metathought described by the metalanguage, where assertions stand, and where Tarski introduced the truth predicate (Tarski 1944). Furthermore, the fuzzyprobabilistic features of QML, induce to modify Tarski Convention T as the Convention PT (where P stands for “Probably”).

4. THE QML IS PHYSICALLY INTERPRETED AS A DISSIPATIVE QUANTUM FIELD THEORY (DQFT) OF THE BRAIN “Matter is not such a material stuff.”

In 2012 we discussed the modalities by which we humans should (and in fact, do) compute (Zizzi et al. 2012, 425-431). We investigated about the logical languages and the computational modes of human reasoning, and the corresponding physical interpretation. In this context, however, the classical world (physical, logical, and computational) does not seem sufficient to give a complete description of the Mind. In fact, the Mind accomplishes different tasks, where it exhibits, alternatively, both classical and quantum features. There are some novelties in two important issues: the long-standing debate on the mind-body relationship, and Turing’s question about a possible

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identification of the Mind with a computer. The introduction of a quantum metalanguage (QML) for the logic of reasoning is the most important feature to deal with both issues. As far as the first issue is considered, the QML is physically interpreted as a Dissipative Quantum Field Theory (DQFT) of the brain. The corresponding quantum object-language (QOL), which is the logic of reasoning, and is controlled by the QML, is physically interpreted as the Quantum Mechanics of qubits, that is, Quantum Computing (QC). Therefore, the Mind is both language and metalanguage, and the brain is its physical interpretation. With regard to the second issue, the QML (and its physical interpretation, DQFT) represents the non-algorithmic aspect of the Mind, in the sense that it is non-Turing-computable. Nevertheless, the Mind has also two computational modes: the quantum one, whose logic is the QOL of the QML, and the classical one, the latter arising from the logical “decoherence” of the former. The quantum and classical computational modes pertain to the ordinary thought processes, while the non-algoritmic mode pertains to the metathought, which is the peculiar process of thinking about our own ordinary thought. The very importance of metathought, which deals with intuition and intention (control), resides in the fact that it distinguishes humans from machines. In fact, the language of metathought, which is non-algorithmic, being a metalanguage, cannot be acquired independently by a machine, which is endowed only with a formal language. The physical interpretation of this impossibility is the irreversibility of the reduction process from the DQFT of the brain to the QC of the mind. An interesting fact, which we expect to have relevance in QC, is the possibility to interpret the coherent assertions of the QML as coherent field states in the brain. Finally, the improvement of robustness of quantum computers against decoherence is very relevant to the implementation of quantum computers with a sufficiently high number of qubits to run efficiently the already known quantum algorithms and possibly new ones. We recall that coherent assertions of the QML can be interpreted as Glauber coherent states in the DQFT of the brain. Then, the introduction of a mental control on quantum computers through a QML could improve the robustness of quantum computers against decoherence.

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5. QUANTUM LOGIC, THE COLLECTIVE UNCONSCIOUS AND THE BIOLOGICAL BASIS OF PSYCHOPATHOLOGY “Insanity is not deprived of logic.”

In the paper “Quantum logic of the unconscious and schizophrenia” (Zizzi et al. 2012, 566–579) the hypothesis that the logic of the unconscious is coextensive with the logic of schizophrenia was proposed. It might plausibly be argued that, while healthy minds employ both the classical logic of consciousness and the quantum primary process logic of the unconscious, schizophrenic minds use primary process thinking not only in their unconscious psychodynamics but also as their dominant conscious operating mode. The logics of both the unconscious and schizophrenic thinking were formalised and it was concluded that is the same logic. At first, it was recognised that the sudden flashes of creative insight and other intuitive “leaps” arise from intermediate mental states that usually remain hidden from consciousness. These ultra-fast processes involving hidden intermediate stages are consistent with quantum computation. The logic of the normal unconscious mind and the schizophrenic consciousness may therefore be Lq, or the logic of quantum information (Zizzi 2010). For a healthy mind, the passage from the unconscious to the conscious state is determined by decoherence of qubits in the polymerised tubulins in microtubules, according to the OrchOR model of Hameroff and Penrose (1996, 507-540). This may be understood in terms of very fast switches from the quantum logic of the unconscious to the classical logic of ordinary consciousness. It has been hypothesized that in schizophrenia these switches are not fast enough, and therefore the schizophrenic mind remains trapped too long in the unconscious logical mode. In Lq, propositions are configured in qubits and the formal interpretation of the unconscious mind may potentially be understood as quantum-informational. The quantum concept of truth in the context of Lq is different from that of classical truth, to the extent that quantum truth manifests itself as many-valued (fuzzy) and is probabilistic, while on the contrary classical truth is single-valued and

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deterministic (Zizzi 2013, 86-198). The metalinguistics of primary process thinking and the related psychopathological phenomena should be formalised by the quantum metalanguage (QML) with an appropriate application to schizophrenia, in which a surplus of quantum propositions dominates the classically logical discourse (Zizzi and Pregnolato 2012, 1-8). In this context, it was possible to introduce the theoretical notion of the Internal Observer (IO) (Zizzi 2005, 287–291), a useful tool for developing a new type of therapy for schizophrenia. Depression is a mood disorder that causes a persistent feeling of sadness and loss of interest (Chand et al. 2019). The etiology of major depressive disorder is multifactorial with both genetic and environmental factors playing an important role. The underlying pathophysiology of major depressive disorder (MD) points to a complex interaction between neurotransmitter availability and receptor regulation and sensitivity underlying the affective symptoms. Disturbance in central nervous system serotonin (5-HT) activity is an important factor as well. Other neurotransmitters implicated include norepinephrine (NE), dopamine (DA), glutamate, and brain-derived neurotrophic factor (BDNF). Vascular lesions may contribute to depression by disrupting the neural networks involved in emotion regulation. Recently, from experimental basis, the molecular depression hypothesis and the involvement of interactome have been formulated (Cocchi et al. 2010, 603-613). Later we descibed the logic (the object-language) and the metalanguage of MD subjects, both at the classical level of consciousness, and at the quantum level of the unconscious, concluding that MD subjects use permanently a quantum metalanguage which is the negation of the quantum metalanguage of schizophrenic subjects (Cocchi et al. 2010, 603-613). This argument is supported by a series of experimental results about the reasoning ability of human subjects. In fact, the “NAND” can be rewritten as the disjunction of two negated propositions. Thus, MD subjects have a different logic from that of normal, bipolar and psychotic subjects. This also means that MD metalanguage is different and consists of negative assertions, which are a sign of pessimism and negative mood. When negative assertions are the only possible ones, that is when you cannot switch to positive assertions (because only the

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logical connective “NAND” is available) then MD arises. In addition, it was found that MD subjects permanently use a quantum metalanguage (Zizzi 2010, 5976) which is the negation of the quantum metalanguage permanently used by schizophrenics. The use of a (negative) quantum metalanguage was therefore suggested for the psychotherapy of MD subjects, as the use of a (positive) quantum metalanguage had been proposed for subjects with schizophrenia (Zizzi et al. 2012, 566–579).

6. ENTANGLED SPACETIME “After all there is not such a big difference between particles and pixels.”

Entanglement, a very particular property of the quantum world, is a quantum correlation that does not have an analogue in the classical world. The building blocks of quantum entanglement are the qubits (we recall that a logical qubit is the unit of quantum information, the quantum analogue of the classical bit, that is, a quantum superposition of bits 0 and 1). An entangled bipartite quantum state, in which the two parts are the qubits A and B, is not separable, that is, it cannot be written as a tensor product A  B . The physical realization of a logical qubit is for example a spin ½ particle (electron) or single atoms or ions (with two internal electronic states) or single polarized photons. Since now entanglement has been demonstrated experimentally with optical photons, neutrinos, and electrons. But, in the context of quantum gravity, the following question arises: can entanglement also concern quantum space-time itself? The answer depends on the context, but the important task is to make the very concept of space-time entanglement as clear as possible. To speak of an entangled space-time means first of all to consider a discrete spacetime, possibly given in integer multiples of Planck units of time and length.

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This request is necessary to associate to each Planckian pixel, at a given time step, a logical qubit B that could be entangled with another qubit A at the precedent time step. As we already said, an entangled bipartite quantum state is not separable. This means that the two parties lose their identity and behave as a whole. In the case of space-time entanglement, then, two pixels of area at different time steps would behave as a whole. The non-separable character of entangled space-time would lead us to review many of our established beliefs, such as locality, the arrow of time and causality. Note that the simple fact that pixels encode qubits is a necessary but not sufficient condition for spatio-temporal entanglement. An entanglement mechanism is required, and the only operation that can provide it is the transformation carried out by the quantum port CNOT (Controlled Not). The CNOT gate uses a qubit B as a control and a bit 0A (or 1A) as a target and returns a maximum entangled state (a Bell state). A question would then arise: where that bit appears from, as all pixels encode qubits, rather than classical bits? The answer is that bit 0A (or 1A) is obtained by measuring the qubit A. Then, a two-dimensional projector is required as well. Another question could be the following: why should we expect that space-time is entangled? The answer is that, in the case of discrete space-time, entanglement is what “glues” together spatial slices occurring at different time steps. Finally, one might wonder which is the physical mechanism of entanglement, that is formally simulated by the operations of the CNOT gate and the projector. As we will see in this paper, such a mechanism is led by the quantum fluctuations of the vacuum, together with the quantum fluctuations of the metric. We believe, then, that at the fundamental level of the Planck scale the answer to the question whether space-time itself can be entangled is affirmative, and in (Zizzi 2018a) we gave the motivations of our belief. But first, we would like to make the following remark. From the fact that quantum entanglement is a quantum correlation which has not a classical analogue, it follows that the appearance of entanglement in a theory under study ensures that such a theory is a quantum theory. Then,

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if a space-time theory manifests an entangled structure, we would be sure that it is a quantum space-time theory, a candidate for quantum gravity. The motivations that we gave were based on various concepts and results already illustrated in previous works, among which the discrete, quantum version of the empty space-time of the de Sitter universe (Zizzi 1999, 2333), its logical quantum-computational realization in terms of a Quantum Growing Network (QGN) (Zizzi 2008) and the quantum extension (Zizzi 2000, 39) of the holographic principle (‘t Hooft 1993; Susskind 1994). In the context of Loop Quantum Gravity (LQG) (Rovelli 2004), the formalism of “spin networks” leads to the very important result of discreteness of area and volume (Rovelli et al. 1995, 93). The application of the quantum holographic principle (QHP) (Zizzi 2000, 39) to the formalism of spin networks in LQG, lead to Computational Loop Quantum Gravity (CLQG) (Zizzi 2005, 645-653). Quite recently (Zizzi 2018a) we made a change to the QGN, by including an internal observer who, standing on the nth horizon of the de Sitter's discrete universe, observes the (n-1)th horizon by using a photon with the appropriate energy. The presence of the observer is equivalent to add a projector to node n, where there was already a Hadamard quantum gate. The apparent loss of the quantum information due to the measurement is restored by the quantum gate CNOT also added to node n, which entangles a qubit of node n with one of node n-1, by using the bit, obtained from the measurement, as target. This new quantum network will be called OQGN, where “O” stands for “Observer.” It may seem that the introduction of the CNOT quantum gate to preserve quantum information through entanglement is done by hand. Instead, it is only the logical aspect of what happens physically. In fact, the energy of the vacuum is shared between the energy of the observational photon (OP) and the cosmological constant. Since the latter is given in terms of quantum information, this energy balance ensures the conservation of quantum information. In turn, the quantum information required for the balance is given by the entanglement entropy of a Bell state. In logical terms, this is just given by the action of the quantum gate CNOT.

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This modification of the QGN, which leads to entanglement of a qubit of node n with a qubit of node n-1, can be also seen in a geometrical way, by using Wheeler’s quantum foam (Wheeler 1962) in terms of the quantum fluctuations of the metric, and the QHP. We showed (Zizzi 2018a) that the discrete quantum fluctuations of the metric on the nth spatial slice have a quantum-gravitational energy with a discrete spectrum. This quantum-gravitational energy is “borrowed” from the energy of the quantum fluctuations of the vacuum, that is, from the cosmological constant. And the energy balance guarantees the conservation of quantum information. For a certain expression of the quantumgravitational energy, equal to that of the OP, this equilibrium leads to entanglement. What get entangled in this geometric version? Space-time itself. In fact, a Planckian pixel of slice n gets entangled with one of slice n-1, because of the QHP. The entangled pixels are in fact identified with each other, through virtual wormholes, which are the maximum quantum fluctuations of the metric at the Planck scale. Thus, the two equivalent views of space-time entanglement discussed in (Zizzi 2018a) seem to fit very well with the EREPR conjecture (Maldacena et al. 2013, 781–811).

7. META-ENTANGLEMENT “Teleportation is a question of quantum spacetime.”

At the light of the results discussed in the previous Section, quantum space-time appears to behave like a quantum computer, although at a different logical level. While a physical quantum computer (QC) uses a quantum logical language, entangled space-time uses a quantum metalanguage (QML), which controls the QC logical language as a quantum object-language (QOL). The QML cannot be given to the quantum physical machine.

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The entangled quantum space-time of our model is empty; therefore it cannot support physical quantum logic gates like the CNOT etc. Entangled space-time is endowed with QML, which can be reflected into the QOL of the quantum network. This is realized by means of two meta-rules, a new theorem, the “Entanglement” (ENT)-theorem, a new meta-theorem, the “Teleportation” (TEL)-theorem, and two new structural rules, the Hadamard (H)-rule and the CNOT-rule. The meta-rules are rules that describe how other rules should be used, and thus belong to the realm of meta-logic. The meta-rules cannot be given to a machine (not even to a quantum computer). So far, only two meta-rules have been known in sequent calculus: the cut rule and the EPR rule. The latter, which has been discovered quite recently, was built by the use of the quantum logical connective “Entanglement” in a quantum version (Zizzi 2010) of Basic Logic (Sambin et al. 2000, 979-1013). Weak (sub-structural) logics are those logics that do not have at least one of the (usual) structural rules: Weakening (W), Contraction (C), and Permutation (P). These logics, like linear logic (Girard 1987, 1-102) and Basic logic, are most suitable for computer science. In fact, in logic, a weak structure leaves more room for new connectives (like the two new quantum connectives (Zizzi 2010) “quantum superposition” and “entanglement”), for meta-rules, and, as we will see, also for new structural rules, which are best suited for quantum computing. While the logical rules introduce a new logical formula either on the left or on the right of the sequent, the structural rules operate on the structure of the sequent itself. Basic logic, the logic upon which our quantum-computational logic was built, is sub-structural as it has not the structural rules C and W. The absence in Basic logic of the C and W rules corresponds to the validity of the no-cloning (Wootters et al. 1982, 802-803) and no-erase (Pati et al. 2000, 164) theorems, respectively, in quantum computing. This correspondence was found in (Zizzi 2010). The above no-go theorems just state that there is not a unitary operation (performed by a quantum gate), which can reproduce quantum copying (and erase). Then, the sequent calculus for a sub-structural logic is in agreement with unitary operations in quantum computing. Extrapolating a little further, we argue that the structural rules allowed in a quantum-computational logic,

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are those that describe quantum logic gates. In a sense, the quantum version of Basic logic is not sub-structural tout curt, but it is sub-structural only with respect to two usual structural rules, which are not in agreement with quantum computing. In this paper, we will introduce two new structural rules, which describe the action of two important quantum logic gates, the one qubit gate Hadamard (H), which creates superposition, and the two qubits gate CNOT, which creates entanglement. The cut rule can be used in sequent calculus to describe the projective measurement of a qubit. In the same way, the EPR rule can be used to describe the projective measurement of a Bell state. In this sense, the two rules cut and EPR are meta-rules, and for this reason they cannot be given to a quantum computer, which performs only unitary (reversible) operations. In particular, the EPR-rule is the statement of quantum logical connectivity (or logical topology, which illustrates how data flows within a network) in the quantum version of Basic logic. As we said, to these two meta-rules we added two other rules, which however are not meta-rules, but structural rules: the Hadamard-rule and the CNOT-rule, and we will prove that the toolkit of such four rules and the ensuing theorems (the Entanglement theorem and the Teleportation metatheorem) is sufficient to prove space-time entanglement, and to show that they represent quantum space-time as a quantum control (better, a quantum meta-language) over a quantum object-language. The structural CNOT-rule describes, in sequent calculus, the unitary (reversible) operation performed by a CNOT gate, which creates entanglement. The CNOT gate uses one classical bit (for example 0) as target, and a cat state as control. When the CNOT rule is performed in parallel, it corresponds to the simultaneous use of bits 0 and 1 as targets. The result is that there is no entanglement in the conclusion of the proof. The CNOT-rule turns to be fundamental in the proof of the new theorem “Entanglement” (ENT). The proof of the (ENT)-theorem reproduces all the steps needed in the entanglement mechanism of two pixels of Planck area belonging to two subsequent spatial slices at two successive Planck time steps, as described in (Zizzi 2018a).

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Also, the (ENT)-theorem, when demonstrated in parallel, does not give entanglement in the conclusion. The consequence, in the case of the entangled space-time, is that two pixels of Planck area cannot get entangled to each other if they belong to the same spatial slice, at a given Planck time step. This extra result can be seen as a no-go theorem, in accordance with the no self-entanglement theorem (Zizzi 2010) when interpreted in the case of space-time entanglement. There is a meta-theorem “Teleportation” (TEL), with a formal proof of it in terms of the two (branched) meta-rules cut and EPR. A meta-theorem is proven within a meta-theory, and may reference concepts that are present in the meta-theory but not in the object-theory. The theorem TEL is a meta-theorem because it deals only with two meta-rules (the cut and the EPR) rather than with logical or structural rules, i.e., it is a theorem within the meta-theory. We show that the proof of the (TEL)-meta-theorem fairly reproduces the protocol of quantum teleportation. Moreover, in the case of space-time entanglement, such a theorem suggests that two maximally entangled pixels of Planck area can convey the teleportation of the unknown quantum state of a particle. At the end, the unknown quantum state is entangled with a pixel of Planck area. The fact that the proof of the (TEL)-meta-theorem can only be performed in parallel, means that it corresponds to the operation identity. In fact we start from entanglement in the premise, and we end with entanglement in the conclusion, although one party of the original EPR pair was replaced with the unknown quantum state. Now, we will outline the logical (in the design sense) architecture of meta-entanglement in a friendly way for the reader, as follows. 



Entanglement: it is not just a quantum correlation among elementary particles, but it can also concern empty space-time at the Planck scale, where logical qubits are encoded in elementary pixels by the (quantum) holographic principle. By assuming that quantum space-time is itself entangled, one might wonder which is the (quantum) underlying logic. The answer is complex, because the question is, in a sense, ill-posed. In fact,

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entangled space-time might not be described by Quantum Mechanics (QM), but rather, by Quantum Field theory (QFT). If that was the case, then entangled space-time should be described by a Quantum Metalanguage, not by a quantum logic., in fact we believe that QFT is described by a Quantum Metalanguage, differently from QM, which is described by a quantum logic. In the case of QFT, the quantum logic deriving from the quantum metalanguage would concern the logical qubits encoded in there. This was one of the main reasons why we investigated the meta-logic of entangled space-time. A possible observation might be, nevertheless, that in the present meta-logical description of entangled space-time there is no trace of QFT.

In fact. here we deal only with the meta-logical aspect of entangled space-time, but if we could assert QFT = Quantum Meta-language, and since it is worth: Entangled Quantum space-time = Quantum Metalanguage, it will have to be worth: Entangled Quantum space-time in terms of QFT, that is, quantum gravity. 

A metalanguage is a language which talks about another language, called the object language. In logic (but also in computer-science) a metalanguage is a language used to formally define the objects and the semantics of a language, in order to study the object language.

Here we will use a quantum metalanguage in order to define the semantics of the quantum logic underlying quantum physics. 

In a quantum metalanguage, sits in fact the semantics of the quantum logic, or roughly speaking, the “meaning”. The quantum logic alone is just the syntax.

So, we have two different logical levels, the Quantum Metalanguage, and the Object Language (the quantum logic), which is the language the

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Quantum Metalanguage is talking about. The two interact with each other through the definition equation (Sambin et al. 2000, 979-1013), which can be seen as the reading of the control exercised by the quantum metalanguage on the object language. 





A quantum machine cannot have its own quantum metalanguage, because its logical language should be Turing-computable, while a quantum metalanguage is not. Thus, a quantum computer could simulate an entangled space-time, without understanding the “meaning.” Thus, we will not get any new information from the simulation. One might ask why it is necessary to reconsider well-known topics in quantum physics in terms of quantum metalanguage, because it is just another interpretation, from which nothing new can emerge. However, the attentive reader will discover that what is happening is not always just finding a new interpretation. Sometimes, Quantum Metalanguage will shed new light on the meaning of some physical phenomena, as any good Meta-Theory should do. But this is not the end of the story: the meta-logic can give suggestions to search for new results or confirm old ones. This is what happens in the case of the meta-theorem TEL. = “Teleportation”. To achieve the goal of describing entangled space-time in terms of meta-logic, it was worth formulating the meta-theorem “Teleportation” (TEL) by which one can see that two entangled Planckian pixels behave as they were entangled particles, and are able to teleport an unknown quantum state. The interpretation of the TEL meta-theorem is as follows: the entangled space-time can carry the teleportation (within it) of an unknown quantum state of a particle, from one node to another, entangle this quantum state with the destination node, and thus make it part of the space-time structure. In order to demonstrate the TEL theorem, we had to use two meta-rules, the cut, and the EPR. Since few years ago, the cut was believed to be the only meta-rule possible in sequent calculus. But then, another meta-rule, the EPR rule was introduced [to

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account for EPR pairs. EL is the only meta-theorem considered here, and most probably is the only meta-theorem of this kind of logics. The other theorem, the “Entanglement” (ENT)-theorem, is not a meta-theorem. To demonstrate ENT, we had to introduce a new structural rule, the C-NOT rule. Then, ENT shows the entangled structure of quantum space-time, and TEL gives the “meaning” of such a structure.

REFERENCES Chand SP, Arif H. 2019. Depression. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing. Cocchi M, Gabrielli F, Tonello L, Pregnolato M 2010. Interactome hypothesis of depression. Neuroquantology, 8:603-613. Gentzen, G. 1935. Untersuchungen über das logische Schließen [Studies on logical inference]. Mathematische Zeitschrift, 39,176–210, 405–431. Girard, J. Y. 1987. Linear Logic. Theoretical Computer Science 50 pp. 1102. Gödel K. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme [About formally undecidable propositions of Principia Mathematica and related systems], I. Monatshefte für Mathematik und Physik 38, 173-98 Hajek, P. 1998. Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4, Kluwer, Dordercht. Hameroff S., Penrose R. 1996. Orchestrated Reduction Of Quantum Coherence In Brain Microtubules: A Model For Consciousness? Toward a Science of Consciousness - The First Tucson Discussions and Debates, eds. Hameroff, S. R., Kaszniak, A. W. and Scott, A. C., Cambridge, MA: MIT Press, 507-540 Maldacena, J. and Susskind, L. 2013. Cool Horizons for entangled black holes. arxiv:13060533v2 [hep-th]. Fortsch. Phys. 61 (9): 781–811.

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Pati, A. K. and Braunstein, S. L. 2000. Impossibility of Deleting an Unknown Quantum State. Nature 404, pg. 164. Penrose R. 1989 Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press; The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics. Oxford University Press. Sambin G., Battilotti G., Faggian C. 2000. Basic logic: reflection, symmetry, visibility. The Journal of Symbolic Logic, 65, 979-1013. Searle J. R. 1980 Minds, brains, and programs. Behavioral and Brain Sciences, 3, 417-458. Rovelli, C. and. Smolin, L. 1995 Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442; 593. Rovelli, C. 2004. Quantum Gravity. Cambridge, UK. Univ. Pr. Tarski A. 1944. The semantic conception of truth. Philosophy and Phenomenological Research, 4, 13-47. 't Hooft, G. 1993. Dimensional reduction in Quantum Gravity. grqc/9310026. Susskind, L. 1994. The World as a Hologram. hep-th/9409089. Turing A. M. 1950. Computing machinery and intelligence. Mind, 59, 433460. Vitiello G. 1995. Dissipation and memory capacity in the quantum brain model. International Journal of Modern Physics B. 9, 973-989. Vitiello G. 2000. My double unveiled. Amsterdam: Benjamins. Wheeler, J. A. 1962. Geometrodynamics. Academic Press, New York. Wootters, W. K. and Zurek, W. H. 1982. A Single Quantum Cannot be Cloned. Nature 299, pp. 802-803. Zadeh L. A. 1996. Fuzzy Sets, Fuzzy Logic, Fuzzy Systems, World Scientific Press Zizzi, P. A. 1999. Quantum Foam and de Sitter-like Universe. Int. J. Theor. Phys. 38, 2333. Zizzi, P. 2000. Holography, Quantum Geometry and Quantum Information Theory. Entropy 2, 39. Zizzi, P. A. 2005. “A minimal model for quantum gravity,” Mod. Phys. Lett. A20; 645-653.

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Zizzi, P. 2005. Qubits and quantum spaces. Int. J. Quantum Inf. 3 (1), 287– 291. Zizzi, P. 2008. Emergence of universe from a quantum network, in: Physics of Emergence and organization, pg. 313, World scientific Publishing, I. Licata and A. Sakaji Eds. Zizzi P. 2010. From Quantum Metalanguage to the Logic of Qubits. PhD Thesis, arXiv:1003.5976. Zizzi P. 2011. Quantum Mind from a Classical Field Theory of the Brain. Journal of Cosmology, Vol 14. Zizzi P. 2012. When Humans Do Compute Quantum, in: A Computable Universe, Hector Zenil (Ed), Word Scientific Publishing. Zizzi P, Pregnolato M. 2012. Looking for the Physical, Logical, and Computational Roots of the Mind. Journal of Consciousness Exploration & Research. Vol. 3 (Issue 4) pp. 425-431 Zizzi P, Pregnolato M. 2012. The Non-Algorithmic Side of the Mind. Quantum Biosystems Vol 4, Issue 1, 1- 8. Zizzi, P., Pregnolato, M., 2012. Quantum logic of the unconscious and schizophrenia. Neuro Quantology 10 (3), 566–579. Zizzi P. 2012. When Humans Do Compute Quantum, in: A Computable Universe, Hector Zenil (Ed), Word Scientific Publishing. Zizzi, P. 2013. The uncertainty relation for quantum propositions. Int. J. Theor. Phys. 52 (1), 86–198. Zizzi, P. 2018a. Entangled Spacetime. Modern Physics Letters A, Vol. 33, No. 29, 1850168. Zizzi, P. 2018b. Meta-Entanglement, https://arxiv.org. quant-ph. Submitted to IJTP.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 6

CAN INSTANTANEOUS QUANTUM ALGORITHMS BE DEVELOPED? Richard L. Amoroso* Noetic Advanced Studies Institute, Beryl, UT, US

ABSTRACT In order to develop Instntaneous Quantum Computing Algorithms (IQCA) one must move beyond the current limits of quantum theory imposed by the Standard Model (SM) of particle physics currently described by the Copenhagen Interpretation because of limitations imposed by the uncertainty and Pauli exclusion principles. Why is this the case? It should be obvious to most, that the Einstein-Podolsky-Rosen (EPR) experients do not operate faster than the speed of light, but demonstrate instantaneous correlations. Quantum Mechanics, a local theory, is silent about both collapse of the wave function and the existence of nonlocality because it utilizes a regime of 4D dimensionality utilized by the SM. Thus, modeling IQCA requires utility of an M-theoretic form of Einstein’s long sought Unified Field Theory (UFT); which additionally entails a new set

*

Corresponding Author’s Email: [email protected]

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Richard L. Amoroso of topological transformations beyond the Galilean-Lorentz-Poincaire that is able to surmount the quantum uncertainty principle and make full use of the additional dimensionality (XD) of M-theory.

Keywords: Bohmian mechanics, instantaneous algorithms, quantum computing

INTRODUCTION Theoretical quadratic, exponential, and polynomial Quantum Computing (QC) speedup algorithms have been discussed. Recently, the new field of relativistic information processing (RIP) and introduction of a relativistic qubit (r-qubit) with additional degrees of freedom beyond the current Bloch 2-sphere qubit formalism (we consider a nonphysical maths convenience) extended theory begins to appear. Here, we propose an ultimate form of QC speedup – Instantaneous Quantum Algorithms (IQA). Note that burgeoning discussion has already occurred on passing beyond the limits of ‘locality and unitarity’ (basis of quantum theory) heretofore restricting the evolution of quantum systems to the standard Copenhagen Interpretation. In that respect, as introduced in prior work, an ontologicalphase topological QC model takes advantage of these developments. Simplistically, as well-known by EPR experiments, instantaneous connectivity, albeit experimentally primitive, exists in the nonlocal arena. We utilize Bohm’s work on a ‘super-implicate order’ where inside a wave packet, a super-quantum potential introduces nonlocal connections. From EPR experiments, we are well-versed in entangling simultaneously emitted photon pairs by parametric down-conversion. To operate an IQA, a form of parametric up-conversion is also required, as a new set of Unified Field Mechanical (UFM) M-Theoretic topological transformations beyond the current Galilean, Lorentz-Poincairé transforms of the standard model, obviating restrictions of ‘Locality and Unitarity,’ the current basis of quantum theory. Yang-Mills Kaluza-Klein (YM-KK) correspondence is shown to provide a path beyond the ‘semi-quantum limit’ to implement IQA.

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Instantaneous quantum computing algorithms (IQCA) must obviate current SM restrictions of Locality and Unitarity – The fundamental parameters of quantum theory [1]; meaning that quantum mechanics (especially the Uncertainty Principle) cannot be considered an impenetrable Planck-scale Basement of Reality. History has shown good theories are logical and have broad explanatory power. As well-known, severally the SM is incomplete; its main pillar Quantum Electrodynamics (QED) has been violated with  5 confidence the level appropriate for claiming discovery [2]; thus, a need for additional physics. Although this work is highly speculative; the avenue remains inherently viable because required approbative theoretical fundaments key to formulating any pragmatic model justifiably point the way to an imminent paradigm shift to occur in the process of discovering additional dimensionality [3-5] finally mitigating M-Theory. Laying claim to Newton’s Hypothesis non Fingo (I do not feign hypotheses) [6] best effort for correspondence to existing theory is made to proceed deducing warranted suppositions post hoc ergo non propter hoc. Seemingly obvious from the point of view of empirically demonstrated EPR instantaneity IQCA appear feasible. But the dilemmas of EPR signaling no-cloning no-deleting and no-go class theorems have handicapped that position [7-9]. EPR entanglement is generally achieved by bringing simultaneously paired parametric down converted photons from nonlocality into locality. To include a parametric up conversion cycle in order to operate an IQCA has been opaque from Euclidean bits or even Minkowski-Riemann space qubit engineering. Now by obviating the fundamental quantum principles of locality and unitarity a method for obviating uncertainty and supervening decoherence is devised [5]. The new field of Relativistic Information Processing (RIP) while adding an additional degree of processing freedom remains an untenable intermediate step [5]. The no-cloning no-go theorems state the impossibility of creating identical copies of arbitrary unknown quantum states; proving the impossibility of perfect non-disturbing measurement schemes [7-9]. The state of one system can be entangled with the state of another system; using controlled NOT and Walsh–Hadamard gates to entangle two qubits.

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Entangling is not cloning. In this basis well-defined states cannot be attributed to a subsystem of an entangled state. A cloning process results in a separable state with identical components. The no-cloning theorem describes pure states; the no-broadcast theorem generalizes this for mixed states. Also, of interest is the time-reversed dual of the no-cloning theorem called the no-deleting theorem. All these theorems relate to quantum states in isolation and have been proven inviolate. This is true in terms of SM quantum physics. But Nature is often full of surprises. A key theoretical supposition introduced relates to the so-called M-Theoretic Bulk (more detail in an ensuing section). Firstly, let us clarify how the term Bulk is used: According to M-Theory the visible 3D universe is restricted to a 3-brane (manifold where we live) inside a higher dimensional (HD) space (the bulk) a domain of large-scale extra dimensions (LSXD) in our model where various brane transitions compactifications and correlations continuously occur as a hyperstructure called a brane bouquet [10]. This is 10D in string theory 11D in M-Theory and 12D in our ontological-phase topological field theory (OPTFT) because of an additional Einsteinian unified field control factor [25]. We have designed experiments to obviate this suggested brane topology [11, 12] - There is something required by the nature of reality itself; the meshing of local temporality and nonlocal holographic instantaneity demands inherent continuously evolving causally separated copies of the 3D QED particle-in-a-box. This is a continuous-state product of Calabi-Yau dual 3-brane mirror symmetric topology. This is a naturally occurring freebie Feynman talked of the necessity of a synchronization backbone requirement for implementing universal quantum computing (UQC) [5, 13-15]. According to Huerta a brane scan classifies Green–Schwarz strings and membranes in terms of invariant cocycles on super-Minkowski spacetimes forming a brane bouquet generalizing this by consecutively forming invariant higher central extensions induced by the cocycles yields the complete fundamental bulk brane content of string theory including Dbranes and the M5-brane as well as the various duality relations between

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these culminating in the 10-and 11D super-Minkowski spacetimes of string/M-theory leading directly to the aforesaid brane bouquet [10]. We must modify Huerta’s brane bouquet to include the RandallSundrum concept of a warped-throat singular D3-brane with largescale additional dimensions (LSXD) [16, 17] into a model of uncertainty as a traversable programmable manifold of finite radius [5, 18, 19]. Current thinking for all practical purposes insists that gravity must be quantized. This arises from the current belief that the quantum mechanical stochastic foam is the impenetrable basement of reality. String theory an evolution of Kaluza-Klein theory has been developed as the theory of quantum gravity [20, 21]; but there is no a priori reason that gravity must be quantized. Maybe we should not try to quantize gravity. Is it possible that gravity is not quantized and all the rest of the world is? Now the postulate defining quantum mechanical behavior is that there is an amplitude for different processes. It cannot be that a particle which is described by an amplitude such as an electron has an interaction which is not described by an amplitude but by a probability seems that it should be impossible to destroy the quantum nature of fields. In spite of these arguments we should like to keep an open mind. It is still possible that quantum theory does not guarantee that gravity has to be quantized [22]. - RP Feynman.

Current thinking states that the KK-XD and beyond to 10D string theory are invisible because they are curled up at the 10-33 cm Planck-scale. This is not the only interpretation. Randall-Sundrum have proposed an alternative to compactification and LSXD [16, 17]. We extend the Randall-Sundrum model and modify the M-theoretic constraint of one unique compactification producing the 4D SM. There was an initial problem with Kaluza’s 5D model; Klein added cyclicality to the line element which solved the problem. The following are an extremely important key concatenation of parameters: 

Compactification cycles continuously through all dimensionality – 12D to virtual 0D.

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

 



The Planck-scale is a virtual asymptote never reached. When cyclic compactification reaches the Larmor radius of hydrogen the cycle repeats. This system is built microscopically on a least-unit [23] with correspondence to the Wheeler-Feynman-Cramer Transactional Interpretation where the present instant is a standing-wave of the future past. To that scenario is added Calabi-Yau dual mirror-symmetric 3-tori. Simplistically this tower interpretation enables an annihilationcreation dimensional (brane) subtractive interferometry of the continuous compactification cycle keeping XD-LSXD invisible. Uncertainty is a cyclically rotating manifold of finite radius Perhaps 6D at the semi-quantum limit Randall-Sundrum throat; beyond which lies he M-Theoretic LSXD 12D bulk.

Fourteen experimental models have been devised to falsify this putative M-Theoretic-UFM model [11, 12] which if successful allows experimental access XD-LSXD [24] and developing new forms of algorithms [25-27]. We do not at this time attempt to write sample M-Theoretic UFM algorithms or convert existing QC algorithms to instantaneous form only outline the semi-quantum Bohmian Implicate Order duality framework required to compose such IQCAs when technologically required. Such an attempt at this point is likely to fall short as a brane-based IQCA we assume must be written in an M-Theoretic context. Our proposed M-Theoretic-Unified basis for UQC is 12D. But the rqubit is likely to only require a 6D quaternion-octonion algebra to describe. The manifest reason at the moment is that a new set of UFM transformations beyond the current Galilean-Lorentz-Poincaré is required to understand/operate and formalize the qubit basis. Let us remind the reader that this form of topological quantum computing [5] will not only eventually lead to IQCA but intrinsic to that process is as allowed by the additional degrees of freedom provided by a 6D qubit is that the programmable

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spacetime structure enables an instantaneous P  1 surmount of the uncertainty principle which supervenes decoherence [5].

REALISTIC QUANTUM ALGORITHMS Currently qubits are algebraically designated utilizing Block 2-spheres in Hilbert space. This scenario is a convenience for mathematical manipulation; since Hilbert space is not physically real [28] neither does a 2-sphere provide a sufficiently realistic qubit basis conforming to the needs of true UQC. Therefore, our existing basis for qubits and by correspondence descriptions of QIP [algorithms] is not realistic. This is moot by current thinking and is proposed here because a key supposition is that UQC will not occur without systematic violation of the Uncertainty Principle which requires an M-Theoretic qubit basis [5]. Decoherence is considered bulk-scalable UQC final problem. It is said that topological quantum computing is the most advanced QC model generally because it is theorized to solve this problem; but the protected qubits are not yet experimentally accessible [29]. Topological quantum computing operates cryogenically as 2D quantum Hall quasiparticle-Anyon based localized Majorana zero modes that are in topologically protected braided states that do not decohere. Operationally a subspace a collective non-local property of the non-Abelian anyons is employed to encode quantum information in the topologically protected manner. This protection arises from the presence of an energy gap and from non-locality. In our IQCA model the nature of Calabi-Yau mirror-symmetric brane topology (12D quantum state copy) allows uncertainty to be surmounted and decoherence supervened not by topological protection but because the quantum states position in the topological brane-bulk is causally free of SM quantum restrictions of locality and unitarity! To repeat since the HD brane copy is causally free of the 3-space QED quantum state decoherence is inherently nonexistent during the computation cycle [5]. This is essentially a completely realistic form of topological protection.

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Quantum algorithm research remains in its infancy because although a fair number of quantum gates and qubit technology research platforms exist it is safe to say that until an actual UQC implementation capable of bulk operation occurs a complete conception of what sufficient quantum algorithms are seems unlikely; especially if much of the novel new parameters proposed are required [5]. Meaning for example that the first true bulk quantum computing system should in actuality be scalable. But a dearth of quantum algorithms to implement sufficient quadratic speedup for practical utility beyond classical computing may remain. We propose a new class of unified field mechanical (UFM) based holographic quantum algorithms with asymptotic speedup beyond the purely classical holographic reduction algorithmic process currently under development even to the point of a new class of instantaneous algorithms. There is recent talk of an end to locality and unitarity as a new basis for QC along with the new field relativistic information processing (RIP); these scenarios may cause dramatic changes in QC research. The concept of Algorithm in simplest terms a classical algorithm is a finite sequence of instructions step-by-step process or set of rules followed in calculations or other computed logical operations which always terminates. A quantum algorithm is purported to run on a realistic model of quantum information processing usually applied to algorithms that are inherently quantum using some essential feature of quantum computation such as quantum superposition or entanglement. The development of algorithms for simulating quantum mechanical systems was Feynman’s original motivation for proposing a quantum computer [13-15]. Quantum algorithms require modules that are uniformly scalable and reversible (unitary) that can be efficiently implemented; the most commonly used model has been the quantum circuit model. In 1994 Shor demonstrated that prime factorization has an efficient solution in QC [30-32]. In general input to a quantum algorithm consists of n classical bits and the output also consists of n classical bits. If the input is an n-bit string x then the QC takes input as n qubits in state x . A series of quantum operations performed ends with the state of the n qubits transformed to some

Can Instantaneous Quantum Algorithms Be Developed? superposition



y

185

 y y . If a measurement is made with as output the n-bit 2

string y with probability  y [32]. The Church-Turing thesis states that any function that can be computed by a physical system can be computed by a Turing Machine. Many mathematical functions cannot be computed on Turing Machines such as the halting function h :

 0,1 that decides whether the ith Turing Machine

halts or the function deciding if a multivariate polynomial has integer solutions. Therefore the Church-Turing thesis is a statement of belief about limits of both physics and computation Some functions can be computed faster on a quantum computer than on a classical one but as noticed by Deutsch [33-35] this does not challenge the physical Church-Turing thesis itself: a QC could even be simulated by pen and paper through matrix multiplications. Therefore, what they compute can be computed classically. Several researchers have pointed out that Quantum theory does not forbid in principle that some evolutions would break the physical Church-Turing thesis [36-38]. Technically the only limitation upon quantum evolution is that it be by unitary operators. Then as Nielsen argues it suffices to consider the unitary operator U 

 i, h(i)  b

i, b with i over integers and b over 0,1 to

have a counterexample [37]. The paradox between Deutsch’s and Nielsen’s arguments is only an apparent one as both are valid; the former applies specifically to Quantum Turing Machines and the latter to full-blown quantum theory This is not satisfactory; if Quantum Turing Machines are to capture Quantum theory’s computational power it falls short and needs amending. Unless in contrast quantum theory itself needs to be amended and its computational power brought down to the level of the Quantum Turing Machine [39, 40]. Most likely quantum theory will be amended. It was known very early on that quantum algorithms cannot compute functions that are not computable by classical computers however they might be able to efficiently compute functions that are not efficiently computable on a classical computer [5]. This scenario may evolve also.

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FOURIER TRANSFORM BASED ALGORITHMS The first QC algorithms were called the ‘black-box or ‘oracle’ framework where part of the input is a black-box implementing a function f(x) The only way to extract information about f was to evaluate it on the x inputs. These early algorithms used a special case of the quantum Fourier transform the Hadamard gate. This allowed a problem to be solved with fewer black-box evaluations of f than a classical algorithm would need [40]. Deutsch [35] formulated the problem of deciding whether a function f : 0,1  0,1 was constant If one has access to a black-box implementing f reversibly by mapping x, 0 box

does

x 0

implement

x, f ( x); one further assumes that the black a

unitary

transformation

Uf

mapping

x f ( x) . Deutsch’s problem is to output “constant” if f(0) = f(1)

and to output “balanced” if f(0)  f (1) given a black-box for evaluating f. Thus, to determine f (0)  f (1) (  denotes addition modulo (2) Outcome ‘0’ means f is constant and outcome ‘1’ means f is not constant [40]. Classical algorithms would have to evaluate f twice to solve the problem. A quantum algorithm need only apply Uf once to produce. 1 1 0 f (0)  1 f (1) . 2 2

(1)

with an end to the no-cloning Signal theorems by M-Theoretic UFM parameter-based UQC another basis change will likely occur for QC development. Under these conditions, if f(0) = f(1) applying the Hadamard gate to the first register yields 0 with probability 1 and if f(0)  f(1) then applying the Hadamard gate to the first register and ignoring the second register leaves the first register in the state 1 with probability 1/2; thus a result of

1

can only occur if f(0)  f(1) [40]. Of special interest given

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1/ 2 0  0  1/ 2 1  1 a ‘Hadamard test’ can be performed if a Hadamard gate is applied to the first qubit. A measurement will give ‘0’ with probability

1 2

 Re   0  1  [40].

EXPONENTIAL SPEED-UP OF PROCESSING The salient utility of UQC is the offering of algorithms that will provide a fully exponential speed-up over classical algorithms making them the most sought-after research avenue for unleashing the power of QCs. Following work of Aaronson for finding a general theorem for developing exponential speedups from quantum algorithms; in recent efforts he makes two advances toward such a theorem in the black-box model where most quantum algorithms operate [41]. 



First, Aaronson shows for any problem invariant under permuting inputs and outputs that has sufficiently many outputs (like collision and element distinctness problems) the quantum query complexity is at least the 7th-root of classical randomized query complexity. Earlier he found a 9th-root [42] resolving a conjecture of Watrous [43]. Second inspired by work of O’Donnell [44] and Dinur [45] he conjectured that every bounded low-degree polynomial has a ‘highly influential’ variable (A multivariate polynomial p is bounded if 0 ≤ p(x) ≤ 1 for all x in the Boolean cube). Assuming this conjecture he then showed that every T-query quantum algorithm can be simulated on most inputs by a TO(1)-query classical algorithm. Essentially one cannot hope to prove P  BQP relative to a random oracle.

Perhaps the central lesson gleaned from fifteen years of quantum algorithms research is this: Quantum computers can offer superpolynomial

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speedups over classical computers but only for certain “structured” problems. The key question of course is what we mean by “structured.” In the context of most existing quantum algorithms “structured” basically means that we are trying to determine some global property of an extremely long sequence of numbers assuming that the sequence satisfies some global regularity [41]. Aaronson offers period finding as a canonical example the core of Shor’s factoring algorithms and computing discrete logarithms [46] where black-box access to exponentially-long sequences of integers X = (x1…xN) is given; that is to compute xi for a given i. We find the period of X that is the smallest k > 0 such that xi = xi−k for all i > k with the promise that X is indeed

N (and that xi values are approximately distinct periodic with period k within each period). The requirement of periodicity is crucial: it lets us use the Quantum Fourier Transform to extract the information we want from a N

superposition of the form 1 /

N  i xi . i 1

For other known quantum algorithms X needs to be a cyclic shift of quadratic residues [47] or constant on the cosets of a hidden subgroup. By contrast, the canonical example of an ‘unstructured’ problem is the Grover search problem Black-box access is given to an N-bit string

 x1 ,..., xN   0,1

N

, and we are asked whether there exists an i such that

xi = 1. Grover formulated a quantum algorithm to solve this problem using O

 N  queries [48] as compared to the ( N ) needed classically However

Bennett et al. showed this quadratic speedup is optimal [49] For other “unstructured” problems see [50-54]. This ‘need for structure’ limits prospects for super-polynomial quantum speedups to areas of mathematics likely to produce similar periodic sequences or sequences of quadratic residues. This is the fundamental reason why successes of quantum algorithm research have been cryptographic, specifically in number-theoretic cryptography. This explains why there are no fast quantum algorithm to solve NP-complete problems or to break arbitrary one-way functions [41, 55].

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Figure 1. a) Quantum walk algorithm. Figure adapted from [55]. b) Shor-like Fourier transform algorithms. c) Boson sampling algorithm. Identical single photons sent through network of interferometers measured at output modes. Adapted from [56].

Quantum walk algorithms can achieve provable exponential speedups over any classical algorithm (in query complexity) but according to Childs et al. only for extremely fine-tuned’ graphs [55]. In the 20 years since the appearance of Shor’s factoring algorithm only a few additional quantum algorithms like Grover’s search and quantum walks have appeared. Aaronson claims that while there are a number of exponential and polynomial speedup algorithms “there just aren’t that many compelling candidates left for exponential quantum speedups” [56]. Factoring algorithms can break almost all public-key cryptosystems used today but theoretical public-key systems exist that are unaffected causing one to ask ‘Can Shor’s algorithm be generalized to nonabelian groups?’ [56]. Grover-like algorithms provide Quadratic speedup for any problem involving searching an unordered list provided the list elements can be queried in superposition. This implies subquadratic speedups for many other basic problems [49]. For black-box searching the square root speedup of Grover’s algorithm is the best possible approach [29-31].

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It was shown if a fast-classical exact simulation of boson sampling is possible then the polynomial hierarchy collapses to a 3rd-level. Experimental demonstrations with 3-4 photons were achieved [57-59].

HOLOGRAPHIC ALGORITHMS Yes, holographic algorithms (HA) already exist a concept originated by Valiant in 2004 [60] HA utilize a process called ‘holographic reduction’ mapping solution fragments ‘many-to-many’ so that the summation of solution fragments remains unchanged. Valiant coined the term HA because “their effect can be viewed as that of producing interference patterns among the solution fragments” [60]. The power of HA comes from the mutual cancellation of many contributions to a sum analogous to the interference patterns in a hologram [61]. So far HA have discovered solutions to previously unsolved polynomial problems. Although HA have some similarities to QC they are currently completely classical in nature [62]. Holographic algorithms occur in the context of what is called Holant problems which generalize counting Constraint Satisfaction Problems (#CSP). A #CSP example is the hypergraph G = (VE) also called a constraint graph. Each hyperedge is a variable and each vertex v is assigned a constraint

fv . A vertex is connected to a hyperedge if the constraint on the vertex involves the variable on the hyperedge. The counting problem is to compute

 f

 :E 0,1 vV

v

 E  , v

(2)

which is a sum over all variable assignments the product of every constraint where the inputs to the constrain fv are the variables on the incident hyperedges of v. A Holant problem is similar to a #CSP except the input must be a graph not a hypergraph. For a #CSP instance one replaces each hyperedge e of size s with a vertex v of degree s with edges incident to the vertices contained in

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e. The constraint on v is the equality function of s identifying all the variables on the edges incident to v. For Holant problems Eq (4) is called the Holant after a related exponential sum introduced by Valiant [63]. To further clarify Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as #CSP and Graph Homomorphism. An e dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and Graph Homomorphism can be viewed as subfamilies of Holant with the additional assumption that the equality constraints are always available [63]. Considering holographic reduction for a bipartite graph G = (UVE} the constraint assigned to each vertex u U is fu likewise for vertex

v V is f v . This counting problem is Holant(G, fu , f v ). Thus for a complex 2 x 2 invertible matrix T, there is a holographic reduction between

Holant(G, fu , f v ) and Holant(G, fu , T

 deg u 

, (T 1 )

 deg v 

Thus, Holant(G, fu , f v ) and Holant(G, fu , T

f v ).

 deg u 

,(T 1 )

 deg v 

fv )

have precisely the same Holant value for all constraint graphs essentially defining the same counting problem which can be proved by holographic reduction. Valiant’s original application of holographic algorithms used holographic reduction which has since been used in polynomial time algorithms and proofs of #P-hardness [64].

ONTOLOGICAL-PHASE HOLOGRAPHIC IQCA To try to stop all attempts to pass beyond the present viewpoint of quantum physics could be dangerous for the progress of science and would be contrary to the lessons we may learn from the history of science. This teaches us in effect that the actual state of our knowledge is always provisional and that there must be beyond what is actually known immense new regions to discover – de Broglie [65].

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A fundamental theory is needed which would tell us from first principles when quantum speedups are possible. There is a related longstanding open problem: Is there any Boolean function with a quantum quantum/classical gap better than quadratic? A Boolian function f is simply f : 0,1  0,1 n

with n input bits and a single output bit [32] We will answer yes below. There are new results from Ben-David: If F : SN  0,1 is any Boolean function of permutations then D(F) = O(Q(F)12). If F is any function with a symmetric promise and at most M possible results of each query then R(F) = O(Q(F)12(M-1)) [66]. We need a ‘structured’ promise if we want an exponential quantum speedup. Exponential quantum speedups depend on structure, for example, abelian group structure glued-trees structure or relational structure. The term Semiclassical in common usage means: intermediate between a classical Newtonian description and one based on quantum mechanics or relativity. Semiclassical physics refers to a theory in which one part of a system is described quantum-mechanically whereas the other is treated classically. For example, external fields will be constant or when changing will be classically described. In general it incorporates a development in powers of Planck’s constant resulting in the classical physics of power 0 and the first nontrivial approximation to the power of (−1). In this case there is a clear link between the quantum mechanical system and the associated semiclassical and classical approximations. Now for UFM, we create a new term semi-quantum where one part will be quantum and the other part UFM. This is a small regime of finite radius called the Manifold of Uncertainty (MOU). This is the 1st step in the realization that the central pillars of quantum field theory spacetime locality and unitarity are to be superseded. In assuming the universe is a huge information processor in terms of unitarity and locality (phenomenal) each distinct point is like a central processing unit (CPU) but in the move to nonlocality and holographic (ontological) ballistic processing there is no CPU; there is a simultaneity of information at each tessellated node. Clearly, I am trying to say this scenario is not classical or quantum but a unified field mechanical ontology. It is hard to fathom what kind of algorithm from a new

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class of holographic ontological algorithms able to operate without decohering the wavefunction this leads to. Creative thinking has already begun to skirt this empyrean realm: All we experience is nothing but a holographic projection of processes taking place on some distant surface that surrounds us - Brian Greene.

Locality is the idea that particles can interact only from adjoining positions in space and time and unitarity states that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The usual picture of space and time with particles moving around in them is a construct “Locality and unitarity emerge hand-in-hand from the positive geometry of the amplituhedron” [67-69]. What’s beyond the end to locality and unitarity as we know it? The QC paradigm until now has been local and semiclassical. Aaronson himself said ‘UQC will require a new discovery in physics’ Our hypothesis non fingo is that this putative discovery in physics is in fact an empirical Gödelization beyond quantum mechanics (unitarity and locality) into the 3rd regime of reality dubbed UFM [5]. We have seen that holographic computing algorithms are classical; we are not just looking for a quantum holography (already exists in NMR spectroscopy) we are proposing a special new class of UFM algorithms. In the course of preparing this paper, our opinion on this matter has evolved. We thought that the existing body of QC research would suffice; and what we had to add to the mix was ontological measurement without collapse and violation of the no-cloning theorem. We hope it is obvious that opinion has changed. If one has the stamina to read this whole volume one sees we expend a lot of effort skirting around issues without doing much of the math. This is our excuse; NASA flew around the moon a couple times before actually landing on it. Since the framework of quantum mechanics seems to rest on unitarity most physicists will tend to look for possible ways to get around such a drastic modification. In quantum physics unitarity is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event is always 1. Giving up space and time

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as fundamental constituents of nature and figuring out how the cosmological evolution of the universe arose out of pure geometry is a fascinating opportunity. In a sense we would see that change arises from the structure of the object. But it’s not from the object changing. The object is basically timeless. The revelation that particle interactions the most basic events in nature may be consequences of geometry significantly advances a decadeslong effort to reformulate quantum field theories of elementary particles and their interactions. Interactions previously calculated by formulas thousands of terms long are now described by computing a volume of the corresponding jewel-like ‘amplituhedron’ yielding an equivalent one-term expression [67-69]. In the quantum world probabilities were expressed as complex numbers with both a quantity and a phase and these so-called amplitudes were squared to produce probability. This was the mathematical procedure necessary to capture the wavelike aspects of particle behavior. Probability amplitudes were normally associated with the likelihood of a particle’s arriving at a certain place at a certain time [71]. Feynman said he would associate the probability amplitude ‘with an entire motion of a particle’-with a path He stated the central principle of quantum mechanics: ‘The probability of an event which can happen in several different ways is the absolute square of the sum of complex contributions one from each alternative way.’ These complex numbers amplitudes were written in terms of classical action; Feynman calculated the action for each path as a certain integral [44-55].

BOHM AND ISTANTANEOUS ALGORITHMS Who might have guessed there might be a class of QC algorithms better than polynomial and exponential speed QIP? Let’s peek at the basis for possible instantaneous algorithms. It is generally known that information passes instantaneously in systems of EPR correlated photons. We know how to parametric down-convert entangled EPR pairs; what if we can learn

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parametric up-conversion utilizing the tenets of M-Theoretic UFM correspondence? Following Bohm, we assume a field  ( x, t ) will take the form of a wavepacket  c Fc ( x, t )   s Fs ( x, t )

with  c ,  s

real and positive

proportionality factors; then functions ( x, t ) orthogonal to Fc(xt) and Fs(xt) will have no effect on the factor in front of  0 meaning their variation will be the same as in the ground state. Thus, chaotic variation of the field will be modified by statistical tendencies to change around an average form of the wavepacket    ' f k qk  0 . k



In (6) the sum is over all k and no restriction made that f  k  f k

because the wave function is complex even though f(x) is real. Considering

q k  qk we write    '  f k qk  f  k qk   0

(3)

k

where



' k

indicates summation over a suitable half of the total set of k

values. With the assumption in (3) that the space average of the field f0 = 0 we write

   ' f k qk exp  ikt   0 .

(4)

k

Then write g   ' f k qk exp  ikt  giving R     gg   0 [37]. k

According to Bohm, inside this wave packet the super-quantum potential introduces nonlocal connections between fields at different points separated by a finite distance (unlike ground state). Now we write the quantum potential as

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Richard L. Amoroso Q   ' k

2 R / R. qkqk

(5)

Now we evaluate the quantum potential change from the ground state

Q  

kf q exp  ikt  f f 1 1 ' k k   ' k k  c.c.  4 k g g 2 k g

(6)

For a wave packet with only a small range of wave vectors the factor k on the right reduces to the fixed number k0 while the remaining factors reduce to unity. This term varies with time but we are only interested in the wave packets for which the spread of k makes negligible contributions. But when the qk are expressed in terms of  ( x) as in qk  1/ V  exp  ik  x  ( x)dV the q-potential reduces to

Q 

1 ' 4 k

f k f k

 F ( x, t ) ( x)dV

  F ( x, t ) ( x)dV 

.

(7)

It should be obvious the term implies nonlocal interaction between  ( x) at one point and  ( x) at other points where the integrand is substantial. Writing Q  Q  Q0 , with Q0 the quantum potential of the ground state as given in

 0  exp   ( x) ( x) f ( x  x)dVdV 

(8)

as taken from (11) with the t coordinate suppressed and where f ( x  x ) 

1/ V  ' k exp ik   ( x  x) we can write the field equation (14a) as (14b) k

 2  Q  2 Q  Q0 2 2    ; 2   2 t  t  

(9a, b)

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Using (8) to express Q in terms of  ( x) by Fourier analysis Bohm obtains [37]  2 1      ' f k f k   2 t 8 k 



f k f k

 F ( x, t ) ( x)dV

  3/2

  F ( x, t ) ( x)dV 



1/2

 c.c.

(10)

Remember from the ground state the field is static because the effect of the quantum potential cancels out the Laplacian   in the field equation 2

 2 is the Laplacian or divergence of the gradient of a function f ( p ) on a point p in Euclidean space. Now with (10) in the excited state there is an additional term causing the wavepacket to move and as happens with the quantum potential the field equation is nonlocal and nonlinear [65]. The point we have been building up to in this section is that the nonlocality represents an instantaneous connection of the field at different points in space. However, as Bohm reminds us this is significant only over the extent of the wavepacket. In the usual interpretation the spread of the wavepacket applies to a region within which according to the uncertainty principle nothing whatsoever can be said regarding what is happening. Therefore, the de Broglie-Bohm-Vigier causal interpretation [84] attributes nonlocality only to situations in which the usual interpretation cannot attribute well-defined properties [65]. It is of key importance to note that a wavefunction of the form

  qk exp  ikt  0 does not correspond to the usual picture of an oscillation. This is shown by (15) because the term   is absent. This result 2

follows because stationary wavefunctions usually correspond to static situations contradicting intuitive expectations of a dynamic state of motion [65]. But Bohm was only thinking from a 4D Standard Model perspective in terms of an amplituhedronic-type (volume) for a Wheeler-DeWitt wavefunction of the universe H  0 instantaneous EPR-holographic algorithms should prove possible with sufficient UFM insight.

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When considered in terms of our UFM brane topological additions to the structure of matter and Bohm’s superimplicate order, full utility of nonlocal information as hinted by EPR correlations hints at the possibility of instantaneous algorithms. Unified Theory postulates that spacetime topology is ‘continuously transformed’ by the self-organizing properties of the long-range coherence of the unified field [85,86] In addition to manipulating conformational change in HD brane topology from the experimental results we attempt to calculate the energy Hamiltonian required to manipulate Casimir-like boundary conformation in terms of the unified field equation F( N )   /  (simple unexpanded form). This resonant coupling produced by the teleological action of the unified field driving its hierarchical self-organization has local nonlocal and global (complex LSXD) parameters. The Schrödinger equation extended by the addition of the de Broglie-Bohm quantum potential-pilot wave mechanism has been used to describe an electron moving on a manifold; but this is not a sufficient extension to describe HD unified aspects of the continuous-state symmetry breaking of spacetime topology requires further extension to include action of the unified field in XD.

EVOLUTION OF M-THEORY Every Calabi-Yau manifold with mirror symmetry or T-duality admits a hierarchical family of supersymmetric toroidal 3-cycles showing possible duality couplings illustratitng he compactification–boost hierarchy. In type-II string theories closed strings are free to move through the 10D bulk of spacetime but the ends of open strings attach to D-branes. In typeIIA their dimensionality is odd – 1,3,5,7 and even in type IIB – 0,2,4,6. Through different gauge symmetry conditions various types of strings or branes are related by S-duality which relates the strong coupling limit of one type to the weak coupling limit of another type T-duality relates strings/branes compactified on a circle of radius R to strings/branes compactified on a circle of radius 1/R.

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Following work by Sundrum [87] for 5D General Relativity where the 0 Einstein action is    or 5GrMN  x   0 for large-scale XD fluctuations

ds 2  Gr55  dx 5   2

Gr55 R2 d 2  Gr55(0)  x   dynamical XD radius.

Randall and Sundrum [16, 17] have found an HD method to solve the hierarchy problem by utilizing 3-branes with opposite tensions  residing at the orbifold fixed points which together with a finely tuned cosmological constant from sources for 5D gravity for a spacetime with a single S1/Z2 XD orbifold [88-90].

(a)

(b)

Figure 2. Conceptualized string (S) and brane (B) couplings in Advanced-Retarded spacetime arising from a least-cosmological unit D0 S-0 a) String-brane duality couplings from 0 to 12D for odd-even Fermi-Bose topologies b) Ising model spin-glass rotations which driven by an internal Lorentz-like force or external resonances for vacuum engineering.

Figure 3. Sundrum’s view of the dynamic oscillations of bulk large size XD readily making correspondence to the continuous-state dimensional reduction parameters inherent in the multiverse cosmology paradigm. Redrawn from [91].

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3-branes with opposite tensions residing at the orbifold fixed points along with a model of a finely tuned cosmological constant serve as sources for 5D gravity.

LORENTZ CONDITION IN COMPLEX 8-SPACE AND TACHYONIC SIGNALING In order to examine as the consequences of the relativity hypothesis with time a 4th dimension, and that we have a particular form of transformation called the Lorentz transformation, we must define velocity in the complex space. That is, the Lorentz transformation and its consequences, the Lorentz contraction and mass dilation etc., are a consequence of time as the 4th dimension of space and are observed in three spaces [30]. These attributes of 4-space in 3-space are expressed in terms of velocity as in the form   1   2 

1/ 2

for   vRe / c with c always real.

If complex 8-space can b projected into 4-space what are the consequences? We can also consider a D slice through the complex 8D space. Each approach has its advantages anddisadvantages. In projective geometries, information about the space is lost. What is the comparson of a subset geometry formed from a projected geometry or a subspace formed as a slice through an XD geometry? What does a generalized Lorentz transformation “look like”? We will define complex derivatives and therefore we can define velocity in a complex plane [92]. Consider the generalized Lorentz transformation in the system of xRe and tIm for the real time remote connectedness case in the xRe , tIm plane. We define

our

substitutions

from

4-to

8-space

before

us

x  x  xRe  ixIm ; t  t   tRe  itIm and we represented the case for no imaginary component of xRe or xIm  0 where the xRe , tRe plane comprises the ordinary 4-space plane.

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Let us recall that the usual Lorentz transformation conditions is defined in 4D real space. Consider two frames of reference  at restand  ' moving at relative uniform velocity v We call v the velocity of the oigin of  '

moving relative to  . A light signal along the x direction is transmittd by x = ct or x - ct = 0 and also in  ' as x’ = ct’ or x’-ct’ = 0 since the velocity of liht in vacuo is constant in any frame of reference in 4-space. For the usual 4D Loretz transformation we have as shown that

x  xRe , t  tRe and vRe  xRe / tRe and x' 

x  vt 1  v2 / c2

z '  z; t ' 

   x  vt  ; y '  y;





t  v / c2 x

  v   t   2  c 1 v / c 2

2

(11)  x 

for   (1   2 )1/2 and   v / c. Here x and t stand for xRe and tRe and v is the real velocity. We consider the xRe , tIm plane and write the expression for the Lorentz conditions for this plane Since again tIm like tRe is orthogonal to xIm and

t 'Im is orthogonal to x 'Im ; we can write x' 

x  ivtIm 1  v2 / c2

z '  z; t ' 

  v  x  vtIm  ; y '  y;



(12)



t  v / c2 x

  v   v t   2 x   c  1 v / c 2

2

where  v represents the definition of  in terms of velocity v; also

v Im  vIm / c where c is always taken as real [93] where v can be real or imaginary. In

Eq

(12)

for

simplicity,

we

let

x ', x, t ' and t

denote

' ' xRe , xRe , tRe and tRe and we denote script v as vIm . For velocity

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v is vRe  xRe / tRe and v  vIm  iIm / itIm ; where the i drops out so that v  vIm  xIm / tIm is a real value function In all cases the velocity of light c is c. We use this alternative notation here for simplicity in the complex Lorentz transformation. The symmetry properties of the topology of the complex 8-space gives us the properties that allow Lorentz conditions in 4D, 8D and ultimately 12D space. The example we consider here is a subspace of the 8-space of

xRe , tRe , xIm and tIm . In some cases we let xIm  0 and just consider temporal remote connectedness; but likewise we can follow the anticipatory calculation and formulate remote nonlocal solutions for xIm  0 and

tIm  0 or tIm  0. The anticipatory case for xIm  0 is a 5D space as the space for xIm  0 and tIm  0 is a 7D space and for tIm  0 as well as the other real and imaginary spacetime dimensions we have our complex 8D space. It is important to define the complex derivative in order to define velocity vIm. In the xRetIm plane then we define a velocity of vIm = dx/ditIm. In the next section we detail the velocity expression for vIm and define the derivative of a complex function in detail [85]. For

vIm  dx / idtIm  idx / dtIm  ivRe , for vRe as a real quantity we substitute into our xRe , tIm plane Lorentz transformation conditions as

(13) These conditions are valid for any velocity vRe = - v. Let us examine the way this form of the Lorentz transformation relates to the properties of mass dilation We will compare this case to the ordinary

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mass dilation formula and the tachyonic mass formula of Feinberg [94] which nicely results from the complex 8-space. In the ordinary xRe, tRe plane we have the usual Einstein mass relationship

m

m0 2 1  vRe / c2

for vRe  c

(14)

and we can compare this to the tachyonic mass relationship in the xt plane m

m0* 1 v / c 2 Re

2



im0 1 v / c 2 Re

2



m0 v / c2 1

(15)

2 Re

for vRe now vRe  c where m* or mIm stands for m* = im and we define m as mRe

m

m0 1  v2 / c2

(16)

For m real (mRe) we can examine two cases on v as v < c or v > c so we will let v be any value from   v  , where the velocity v is taken as real or vRe . Consider the case of v as imaginary (or vIm) and examine the consequences of this assumption. Also, we examine the consequences for both v and m imaginary and compare to the above cases. If we choose v *2 2 2 2 imaginary or v* = iv (which we can term vIm) the v / c  v / c and

1  v*2 / c2 becomes 1  v*2 / c 2 or m

m0 2 1  vRe / c2

(17)

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Richard L. Amoroso We get the form of this normal Lorentz transformation if v is imaginary

(v*  vIm ) . If both v and m are imaginary as v* = iv and m* = im then we have

m

m0* 1 v / c *2

2



im0 1 v / c 2

2



m0

(18)

v / c2 1 2

or the tachyonic condition. If’ we go “off” into xRe tRe tim planes then we have to define a velocity “cutting across” these planes and it is much more complicated to define the complex derivative for the velocities. For subliminal relative systems



and

 ' we can use vector addition such as W  vRe  ivIm for vRe  x, vIm  c and W < c. In general, there will be four complex velocities. The relationship of these four velocities is given by the Cauchy-Riemann relations in the next section. These two are equivalent. The actual magnitude of v may be expressed as v  [vv*]2 vˆ (where vˆ is the unit vector velocity) which is formed using 1

either of the Cauchy-Riemann equations. It is important that a detailed analysis not predict any extraneous consequences of the theory. Any new phenomenon hypothesized should be formulated in such a manner as to be easily experimentally testable. Feinberg suggests several experiments to test for the existence of tachyons [94] in the following experiment – consider in the laboratory atom A at time t0 is in an excited state at rest at x1 and atom B is in its ground state at x2. At time t1 atom A descends to the ground state and emits a tachyon in the direction of B Let E1 be this event at t1 x1. Subsequently at t2  t1 atom B absorbs the tachyon and ascends to an excited state; this is event E2 at t2 x2. Then at t3  t2 atom B is excited and A is in its ground state. For an observer traveling at an appropriate velocity v < c relative to the laboratory frame events E1 and E2 appear to occur in the opposite order in time. Feinberg describes the experiment by stating that at t 2'

atom B

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spontaneously ascends from the ground state to an excited state emitting a '

tachyon traveling toward A Then at t1 , atom A absorbs the tachyon and drops to the ground state. It is clear from this that what is absorption for one observer is spontaneous emission for another. But if quantum mechanics is to remain intact so that we are able to detect such particles then there must be an observable difference between them: The first depends on a controllable density of tachyons, the second does not. In order to elucidate this, point we should repeat the above experiment many times. The possibility of reversing the temporal order of causality sometimes termed ‘sending a signal backwards in time’ must be addressed [96]. Is this cause-effect statistical in nature? In the case of Bell’s Theorem these correlations are extremely strong whether explained by v > c or v = c signaling.

Figure 4. Cramer’s Transactional Interpretation. a) Offer-wave - confirmation-wave combined into a resultant transaction. b) taking the form of an HD future-past advanced-retarded standing or stationary wave. Figs Adapted from Cramer [95].

Bilaniuk formulated the interpretation of the association of negative energy states with tachyonic signaling [85, 94, 97]. From the different frames of reference thus to one observer absorption is observed and to another emission is observed. These states do not violate special relativity. Acausal experiments in particle physics have been suggested by a number of researchers [98, 99]. Another approach is through the detection of Cherenkov radiation which is emitted by charged particles moving through a substance traveling at a velocity v > c. For a tachyon traveling in free space with velocity v > c Cherenkov radiation may occur in a vacuum causing a tachyon to lose energy becoming a tardon [11, 12, 94].

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In prior work [85, 94] in discussions on the arrow of time, we have developed an extended model of a polarized Dirac vacuum in complex form that makes correspondence to Calabi-Yau mirror symmetry conditions, which extends Cramer’s Transactional Interpretation [85, 95] of quantum theory to cosmology. Simplistically, Cramer models a transaction as a standing wave of the future-past (offer wave-confirmation wave). However, in the broader context of the new paradigm of Holographic Anthropic Multiverse cosmology it appears theoretically straight forward to ‘program the vacuum.’ The coherent control of a Cramer transaction can be resonantly programmed with alternating nodes of constructive and destructive interference of the standing-wave present. It should be noted that in UFM cosmology the de Broglie-Bohm quantum potential becomes an eternity-wave,  or super-pilot-wave force of coherence associated with the UF ordering the reality of the observer or the locus of the spacetime arrow of time. To experimentally test for the existence of Tachyon/Tardyon interactions, an atom is placed in a QED cavity or photonic crystal. Utilizing the resonant hierarchy, by interference, a reduced eternity wave,  is focused constructively or destructively as the experimental mode may be, and according to the parameters illustrated by Feinberg above, temporal measurements of emission are taken.

INSTANTANEITY IN COMPLEX 8-SPACE Utilizing the Cauchy-Riemann relations we formulate the hyperdimensional velocities of propagation in the complex plane in various slices through a hyperdimensional complex 8-space. In this model finite limit velocities v > c can be considered. In some Lorentz frames of reference instantaneous signaling can be considered. It is the velocity connection between remote nonlocal events and temporal separated events or anticipatory and real time event relations.

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It is important to define the complex derivative so that we can define the velocity vyIm . In the xit plane then, we define a velocity of v  dx / d (i ). We now examine in some detail the velocity of this expression. In defining the derivative of a complex function, we have two cases in terms of a choice. With the differential increment considered, Consider the orthogonal coordinates x and itIm ; then we have the generalized function

f ( x, tIm )  f ( z ) for z  x  itIm and f(z) = u ( x, tIm )  iv( x, tIm ) where u ( x, tIm ) and v( xIm , tIm ) are real functions of the rectangular coordinates x and tIm of a point in space P( x, tIm ) . Choose a case such as the origin

z0  x0  it0Im and consider two cases one for real increments h  x and imaginary increments h  itIm . For the real increments h  tIm , we form the derivative f '( z0 )  df ( z ) / dz z0 which is evaluated at z0

(19a) or

(19b) Again, x  xRe , x0  x0Re and vx  vx Re . Now for the purely imaginary increment h  itIm we have

(20a) and

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f '  z0   iut Im  x0 , t0Im   vt Im  x0 , t0Im 

(20b)

for uIm  ut Im and vIm  vt Im then

ut Im 

u v and vt Im  . tIm tIm

(20c)

Using the Cauchy-Riemann equations

u v u v  and  x tIm tIm x (21) assuming all principle derivations are definable on the manifold and letting

h  x  itIm , we use f '  z0   lim h  0

f  z0  h   f  z0  h



df  z  dz

z0

(22a)

and ux  x0 , t0Im   ivx  x0 , t0Im  

u  x0 , t0Im  x

i

v  x0 , t0Im  x

(22b)

with vx for x and tRe that is uRe  u x Re , with the derivative form of the charge of the real space increment with complex time we can define a complex velocity as

f '  z0  

dx 1 dx  d  itIm  i dtIm

(23a)

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209

If x(tIm ) where xRe is a function of tIm and f(z) and using h  itIm then

f '  z0   x '  tIm  

dx dx  dh idtIm

(23b)

If we define a velocity where the differential increment is in terms of

h  itIm . Using the first case as u ( x0 , t0Im ) and obtaining dt0Im / x (with i’s) we take the inverse. If ux which is vx in the h  itIm case have both ux and vx one can be zero. Like the complex 8D space, 5D Kaluza-Klein geometries are subsets of the supersymmetry models. The complex 8-space deals in extended dimensions, but like the TOE models, Kaluza-Klein models also treat n > 4D as compactified on the scale of the Planck length 10-33 cm [85, 94]. In 4D space event point P1 and P2 are spatially separated on the real space axis as x0Re at point P1 and x1Re at point P2 with separation

xRe  x1Re  x0Re . From the event point P3 on the tIm axis we move in complex space from event P1 to event P3. From the origin t0Im we move to an imaginary temporal separation of tIm to t2Im of tIm  t2Im  t0Im . The distance in real space and imaginary time can be set so that measurement along the tIm axis yields an imaginary temporal separation tIm subtracts out from the spacetime metric the temporal separation xRe . In this case occurrence of events P1 and P2 can occur simultaneous that is the apparent velocity of propagation is instantaneous. For the example of Bell’s Theorem, the two photons leave a source nearly simultaneously at time t0Re and their spin states are correlated at two real spatially separated locations x1Re and x2Re separated by

xRe  x2Re  x1Re . This separation space-like and forbidden by special relativity; however in complex space the points x1Re and x2Re appear to be contiguous for the proper path ‘travelled’ to the point.

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CONCLUSION The UFM Transformation is different from the Galilean and LorentzPoincairé group transformations in that in its simplest form it is stationary in ds0 with no temporal components (still ontological displacement). This is because its transformation process, which is cyclic, moves upwards through the HD/LSXD brane topology bulk. As a further challenge, it also has an ontological component with no energy transferred; only topological information. This is the duality: Usual localized phenomenological quantal field mediation up to the semi-quantum limit, and ontological holographic nonlocally. The UFM transform is a topological-phase interaction with an energyless UFM force of coherence mediated by topological charge in conjunction with a nilpotent zero-totality (defined in [5, 85]). However, all new theory must make correspondence to existing theory; when referring to a Standard Model particle in motion &c, the TN would include all pertinent aspects of Galilean and Lorentz-Poincairé transforms. This is no different than the additions made to CM with the discovery of QM. There is always risk in proposing new theory. I still remember vividly Tom Toffoli chastising Vlasov, a young Russian postdoc at the time, at Physcomp96, regarding his paper putting forth a relativistic qubit; now 20 years later finally there is talk of r-qubits and a new field of relativistic information processing (RIP) is well under way. In any case, we have ‘put up’ viable protocols [11, 12] much simpler and more revealing than those putatively to be processed by the CERN LHC. Now here’s the rub; let’s consider a general case of n = 500 electron qubits in a linear superposition of all 2500 possible classical states, much larger than the number of particles estimated in the classical universe (1080):



x0,1

n

x x .

(24)

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211

This exponentially huge superposition is ‘the private world’ of the electrons involved and measurement only allows us to find the n bits (500) of information

 x . If our UFM model proves successful in surmounting 2

uncertainty, then measurement does not change the system, leaving all 2500 possible superposed states intact. This also leads to violation of the nocloning theorem. Profoundly, ‘ontological-phase eversion cycles for IQCA,’ provide the explorable framework for producing instantaneity. The framework for the imminent age of discovery we term unified field mechanics (UFM); a 3rd regime of reality in the progression Classical Mechanics  Quantum Mechanics  UFM. Just as infinities (ultraviolet catastrophe) in the Raleigh-Jeans Law describing blackbody radiation led to Planck’s 1900 formulation of the process of energy absorption and emission, becoming known as the quantum hypothesis - any energy radiating atomic system can theoretically be divided into a number of discrete ‘energy elements’  , with each element proportional to the frequency ν individually radiating energy by:   hv . There is an obvious parallel today in the renormalization of the troublesome infinities in quantum field theory. It is quite curious that in this case, a reversal occurs, and quantization is undone again by entry into the 3rd regime. Since quantum mechanics can no longer be considered the ‘basement of reality,’ an initial discovery popping out of the UFM prize bucket, is that QM uncertainty is a ‘complex manifold of finite radius.’ The new set of UFM transformations beyond the Galilean-Lorentz-Poincairé naturally cancel the infinities from fundamental principles, not in an ad hoc manner. Physicists still ‘believing’ in a quantum universe, where the Planckscale is the ‘basement of reality,’ adamantly proclaim the impossibility of violating the quantum uncertainty principle. Here is the manner in which science fiction writer Isaac Asimov put it: “You can’t lick the uncertainty principle man, any more than you can live on the sun, there are physical limits to what can be done” [100].

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The assumed Physical Limits apply of course, to the tools available to us currently to the semi-quantum limit. Suggested experimental protocols [11, 12], when proven successful, begin the awaited UFM paradigm. In HD space, each particle is comprised additionally of a mirror symmetric brane topology of conformal scale-invariant components, Huerta’s brane bouquet [10]. A Dirac-like KK spherical rotation occurs cyclically, separating each mirror symmetric half, and reconnecting it again in continuous dimensional reduction compactification cycle [85]. If a measurement is taken when the symmetry is reconnecting, the cross-section is revealed; like the 3-sphere able to ‘see’ the insides of circles in Abbot’s book Flatland. This ontological-phase moment, is the pragmatic gateway to LSXD. The conceptual framework for this discovery occurred while pondering solutions to the inherent problem of anyonic ‘topologically protected’ states in TQC. I realized I would need to develop, what I decided to term, an ‘Ontological-Phase Topological Field Theory (OPTFT) that entailed a dynamic duality of quantum mechanics and unified field mechanics. When the great innovation appears, it will almost certainly be in a muddled incomplete and confusing form. To the discoverer himself it will be only half understood; to everybody else it will be a mystery. For any speculation which does not at first glance look crazy there is no hope Freeman Dyson [101].

OPTFT, essential for UFM bulk UQC, will end up taking us far into the future; with it one leaves polynomial and quadratic algorithmic speedup in the dust, as it will soon enough be possible to develop ‘instantaneous algorithms’ by utilizing the full EPR aspects of nonlocal holography. The reduction of all information about any fermion state to the instantaneous direction of its spin vector; the specification of locality as occurring within the fermion bracket and nonlocality as outside it; full derivation of spin helicity and zitterbewegung. The description of all known boson states (for the first time) in terms of fermion combinations and the derivation of the SU(2) structure of the weak interaction. The first explicit representation of

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213

baryon wavefunctions with the consequent explanation of baryon mass and the SU(3) structure of the strong interaction; the first explanation of vacuum as the dual structure to a fermion which maintains a nilpotent zero totality. Now we get to important reasoning for extending the de Broglie-BohmVigier approach. The Copenhagen interpretation claims that any pathdetermining measurement will destroy the interference pattern; however, the key idea is that the causal interpretation predicts that interference will persist if future techniques allow a sufficiently subtle non-demolition measurement to be performed. There is an extensive body of literature referring to the evolution of the Elitzur-Vaidman Interaction Free-Measurement scenario [5, 85]; which is abandoned as unnecessary in our experimental approach for surmounting uncertainty [5, 85] demonstrating the incompleteness of the Copenhagen description of reality, beckoning new physics. Vigier proposed that nonlocal interactions are not absolutely instantaneous, but causal and superluminal; they are mediated by the de Broglie-Bohm pilot-wave quantum potential, and carried by superluminal phase waves in a covariant Dirac-type ether consisting of superfluid states of particle-antiparticle pairs [5, 85]. We have noticed a duality between Newton’s and Einstein’s gravity (instantaneous versus luminal) [3] in addition to a complex ‘manifold of uncertainty’ bridging the gap between quantum mechanics and UFM; it turns out experimentally, that this duality is a real condition, a principle of nature. Vigier writes: In my opinion the most important development to be expected in the near future concerning the foundations of quantum physics is a revival in modern covariant form of the ether concept of the founding fathers of the theory of light … it now appears that the vacuum is a real physical medium which presents some surprising properties.

Considerable effort was expended to review the causal-stochastic approach to hint at its foundation for UFM. Whether one is inclined to accept anything to do with the parameters of the de Broglie-Bohm-Vigier interpretation at all; one must even if myopically, agree that the

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interpretation has sufficient richness to point in the direction we want to take it. Bias was so strong against heliocentricity that it literally took thousands of years before its pieces could finally be placed into the fabric of reality. While progress seems to be inevitable, it can be thwarted for lengthy periods. The de Broglie-Bohm-Vigier scenario has been waylaid for nearly 100 years; but finally, the day of reckoning persistently looms… A big question is, does an ontological measurement change the basis for quantum algorithms? Would such a scenario (other than putatively removing the need for error correction cycles) provide another category of speedup? We have considered that UFM based UQC is primarily a boon to measurement, and possibly in that case, classes of quantum algorithms might remain the same. Let’s not call it ‘parallel QC” but rather, could we discover a class of ‘holographic UQC’ with asymptotically infinite speedup? As we devise, it is not called infinite; but with nonlocal EPR-like dualAmplituhedron connectivity, it is termed a class of ‘instantaneous’ ontological algorithms! The Larmor cyclotron radius of the circular motion of a charged particle in the presence of a uniform magnetic field. This scenario, in conjunction with an incursive harmonic oscillator, applied to the finite hyperspherical radius of uncertainty at the semi-quantum limit, is the gateway to the regime of Einstein’s long sought unified field theory. This preliminary study, only evaluated putative frameworks/regimes suggesting IQCA viability once empirical access opens to LSXD utilizing incursive harmonic resonance surmounting the finite semi-quantum radius of uncertainty. A preliminary step has appeared [102]. We pointed out attempts to write a IQCA is futile until the UFM transform is understood, because passing beyond the nonphysical Block 2-sphere qubit to likely 6D hyperspherical r-qubits remains opaque without the new transform.

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[67] Arkani-Hamed N and Trnka J 2013. The amplituhedron (arXiv:13122007). [68] Bai Y He S and Lam T 2015. The amplituhedron and the one-loop Grassmannian measure (arXiv:151003553). [69] Arkani-Hamed N Hodgesb A and Trnka J 2013. Positive amplitudes in the amplituhedron (arXiv:14128478v1 [hep-th]). [70] Aaronson S 2013. The unitarihedron: The jewel at the heart of quantum computing (http://wwwscottaaronsoncom/blog/?p=1537). [71] De Angelis SF. 2013 (http://wwwenterrasolutionscom/2013/09/arespace-and-time-realhtml). [72] D’Ariano G M Van Dam W and Mosca M 2007 General optimized schemes for phase estimation. Physical Review Letters 98 9 090501. [73] Papageorgiou A and Traub JF 2014 Quantum algorithms for continuous problems and their applications. Adv Chem Phys 154 151178 Hoboken Wiley. [74] Harrow AW Hassidim A and Lloyd S 2009 Quantum algorithm for linear systems of equations. Phys Rev Lett 103 150 502. [75] Portugal R and Figueiredo C M H 2006 Reversible Karatsubas algorithm. J Univ Comp Sci 12 5 499-511. [76] Cao Y Papageorgiou A Petras I Traub JF and Kais S 2013 Quantum algorithm and circuit design solving the Poisson equation. New J Phys 15 013021. [77] Draper TG 2000. Addition on a quantum computer (arXiv quantph/0008033). [78] Wegner P 1997. Why interaction is more powerful than algorithms Communications ACM 40. [79] Wegner P 1998 Interactive foundations of computing. Theor Comp Sci 192 2 315-351. [80] Wegner P and Goldin D 2003. Computation beyond turing machines Comm ACM 46 4. [81] Hopcroft JE and Ullman JD 1969 Formal Languages and Their Relation to Automata Reading Addison-Wesley.

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[94] Rauscher E A and Amoroso R L 2011. Orbiting the Moons of Pluto: Complex Solutions to the Einstein Maxwell Schrödinger and Dirac Equations. Singapore World Scientific. [95] Cramer JG 1986 Transactional interpretation quantum mechanics. Rev Mod Phys 58 3 647-687. [96] Farmelo G 2009. The Strangest Man The Hidden Life of Paul Dirac Mystic of the Atom. Perseus. [97] Zeh H-D 1989. The Physical Basis of the Direction of the Arrow of Time New York Springer. [98] Kafatos M Roy S and Amoroso R L 2000. Scaling in cosmology and the arrow of time R Buccheri and M Saniga (eds) Studies on the Structure of Time: From Physics to (Psychpatho)logy. Dordrecht Kluwer. [99] Asimov I 1989. The dead past Hartwell D G (ed). The World Treasury of Science Fiction Little Brown. [100] Dyson F 1958 Innovation in physics. Sci Am 199 Coll from Eros to Gaia 1993. [101] Sainadh U S et al. 2019 (https://arxiv.org/abs/1707.05445).

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 7

SENTIENT ANDROIDS Richard L. Amoroso* Noetic Advanced Studies Institute, Beryl, UT, US

ABSTRACT An android is meant to look and act like a human being even to the extent of being indistinguishable. Generally, the simplistic distinction between a humanoid robot, a computerized machine capable of replicating a variety of complex human functions automatically, and an android is one of appearance. While one day a yottaflop (10 24 bits per second) hypersupercomputer could have a sufficient holographic database and processing power to be truly indistinguishable from a human being, the issue of the applicability of sentience (self-awareness) to an android comes to the forefront. The currently dominant cognitive model of awareness, closely aligned to the AI model, states that mind equals brain and that once correct algorithms are known all of human intelligence could be replicated artificially. This is the so-called mechanistic view: ‘The laws of physics and chemistry are sufficient to describe all living systems; no additional life principle is required’. In this work we develop the point of view that the regime of Unified Field Mechanics (UFM) supplies an inherent action principle driving both the evolution of complex Self-Organized Living *

Corresponding Author’s Email: [email protected].

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Keywords: algorithm, android, quantum computing, life principle, sentience

INTRODUCTION The nature of consciousness, for which we include sentience, is often called the oldest and most difficult problem facing human understanding. Current thinking that Mind = Brain has not helped. For example, cognitive scientists ask – How can the brain, a physical thing, generate a non-physical essence of mind? This ghost in the machine is at the central issue of the riddle of non-physicality. That there might be a Cartesian mind-stuff (res cogitans) and body-stuff (res extensa), a dualism of mind and body is currently politically incorrect. Thus, cognitive scientists claim a mind stuff is nonphysical and violates laws of thermodynamics in order to destroy any efficacy; but Descartes did not claim his res cogitans was nonphysical, he claimed it was immaterial. Even today a valid definition of immaterial is spiritual; an extremely unacceptable term to most scientists. When Sir John Eccles was still alive (Nobel for synapse) it was said, how can one argue with a Nobel Laureate – Eccles was the last great Cartesian dualist. The nature of awareness has been called the hard problem. In the history of science, whenever a hard problem has existed, later discovery has shown incorrect fundamental questions had been asked. We provide empirical protocols to test for a Cartesian action principle. Universal Quantum Computing (UQC) will essentially provide infinite power because superposed qubit states scale by a factor of 2N, such that a mere 400 qubits has a number of possible states tantamount to the number of atoms in the universe, suggesting androids indistinguishable from humans will be constructed provided an appropriate system of algorithms is

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developed. This brings the issue of android sentience (self-awareness) to the forefront. The dominant cognitive approach to consciousness, aligned with the AI model, states that mind equals brain and once the specificity of algorithms is known, all of human intelligence could be replicated artificially. This so-called mechanistic view: The laws of physics and chemistry provide a sufficient description of living systems; with no additional life principle required is untenable. Our point of view entails an inherent Cartesian action principle driving the evolution of complex Self-Organized Living Systems (SOLS) and the physical processes of awareness – necessitating a physical distinction between mind and brain because human rationality contains something beyond the design criterion of machine intelligence. Based on the fundamental premise that awareness is associated with an Einsteinian Unified Field as an inherent aspect of the nonlocal fabric of the physical universe. The architecture of a quantum computer designed to embody the physical elements of natural intelligence could allow consciousness to emerge within its core because the utility of the missing parameters of mind contained in the deeper ontology could function as a carrier to simulate a platform for the extracellular containment of natural intelligence. In the mind = brain panoply it is argued that if we knew the correct algorithms all of human intelligence could be duplicated on todays existing computers. Current computing platforms are called Turing Machines as invented by Alan Turing in 1936. Generally. A turing machine consists of a tape of infinite length on which read and writes operation can be performed. The tape consists of infinite cells each containing symbols. It also consists of a head which points to the cell currently being read; it is linear and can move in both directions. It is argued that turing machines can not achieve sentience or free will because they cannot escape their linear programming. Turing proposed a test, now called the Turing Test: if a computer can pass for a human, it is intelligent. To clarify this issue philosopher John Searle created the Chinese Room argument: Imagine a native English speaker who knows no Chinese locked in a room with boxes of Chinese symbols (data base) and a book of instructions for manipulating the symbols (program). People outside the room send in Chinese symbols which,

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unknown to the person in the room, are questions in Chinese (input). By following instructions in the program, the man in the room passes out Chinese symbols that are correct answers to the questions (output) enabling the person in the room to pass the Turing Test for understanding Chinese albeit he does not understand any Chinese. With a UQC of essentially infinite computing power/database, it seems obvious androids will easily outperform humans. For example, IBM Supercomputer Watson defeated two of the Jeopardy Quiz shows greatest champions. Watson is massively parallel, employing a cluster of ninety IBM Power 750 servers, each using a 3.5 GHz POWER7 eight-core processor, with four threads per core. In total, the system has 2,880 POWER7 processor threads and 16 terabytes of RAM. Watson processes 500 gigabytes, the equivalent of a million books, per second. All content needed to be stored in RAM because data stored on hard drives would be too slow to be competitive with human Jeopardy champions. Since June 2018, the US Summit is the world’s most powerful supercomputer, reaching 143.5 petaFLOPS. This is nothing compared to the theoretically infinite processing UQC will provide. Watson was designed for a specific purpose; it is easy to imagine a bulk UQC encompassing all of human intelligence, and thus being able to pass the Turing Test. Does this mean an android equipped with such a brain would be sentient? We suggest no, because it will still be following a program and thus could not have free will in an absolute sense. What we cannot answer now, is wether by a UQC modeled after Cartesian dualism incorporating the Extracellular Containment of Natural Intelligence could achieve sentience. Generally, the simplistic distinction between a humanoid robot, a computerized machine capable of replicating a variety of complex human functions automatically, and an android is one of appearance; an android is meant to look and act like a human being even to the extent of being indistinguishable. Qubits reportedly scale at 2N, such that only a few hundred qubits are tantamount to the number of atoms in the universe. With crystalline holographic databases coupled with the processing power of bulk Universal Quantum Computing (UQC) [1], it seems logical that androids could be constructed truly indistinguishable from a human

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being provided a proper system of algorithms could be developed. This scenario however, brings the issue of the applicability of android sentience (self-awareness) to the forefront. The currently dominant cognitive model of awareness, closely aligned to the AI model, states that mind equals brain and that once the specificity of algorithms is known, all of human intelligence could be replicated artificially. This is the so-called mechanistic view: The laws of physics and chemistry are sufficient to describe all living systems; no additional life principle is required [2]. In this work we develop the point of view that the regime of Unified Field Mechanics (UFM), as Einstein himself claimed, supplies an inherent action principle describing life, or as we state: driving both the evolution of complex Self-Organized Living Systems (SOLS) and the physical processes of awareness. This Cartesian model (distinction between mind and brain), where UFM parameters in conjunction with conscious QC (QC modeled with physical parameters of mind-body interaction), putatively could lead directly to the possibility for construction of sentient (or sentient-like) Androids. Constructing sentient robotic devices in our model requires three precursors: 1. Utility of the 3rd regime of Natural Science-Unified Field Mechanics (UFM) which includes an inherent life principle with experimental access to a physically real ‘light of the mind’. 2. Development of the fundamental principles of awareness (solving the Mind-Body problem); and 3. Implementing a special class of universal ‘conscious quantum computer’ (QC) modeled after the naturally occurring mind-body interface.

EXTRACELLULAR CONTAINMENT OF NATURAL INTELLIGENCE We delineate the framework for discovery of the mind and the requirements for general universal quantum computing incorporating those

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elements into a class of ‘conscious’ quantum computing. Our approach to defining awareness does not adhere fully to the standard Cognitive approach where mind equals brain but rather to a Cartesian interactionist model where Descartes res cogitans (mind stuff) is considered a physically real coherent action of the Unified Field; with instead of a ‘flashing stream of positrons’ as Asimov suggested, rather a stream of ‘noeons’ the proposed exchange unit of an Einsteinian UFM [1]. Epistemology progressed from myth and superstition to the age of logic and reason. When logic failed Galileo was credited with founding the age of empirical science currently in effect. This evolution in modern times centered at first on the 3D Euclidean space of Newtonian or Classical Mechanics. Then at the turn of the 20th Century Quantum Mechanics and Relativity were created in a 4D Minkowski-Riemann spacetime. Now as we develop the ‘Age of Mind’ we enter a 3rd 12D String/M-Theoretic regime of UFM [1-3]. It is postulated that this UFM regime contains an inherent new action or life principle driving the evolution of complex Self-Organized Living Systems (SOLS) and mind or mentation (stream of qualia) [1, 4]. Therefore, sentient life is a form of complex self-organized autopoietic system within which awareness is an evanescent process between local phenomenological and nonlocal ontological domains of Descartes res extensa (body stuff) interacting with physically real res cogitans (mind stuff). We will discuss several experimental protocols under development that test these noetic hypotheses [5, 6]. It is in this guise that we are able to propose that the mind-body interface is a form of naturally occurring ‘conscious quantum computer’ [7]; which under the right conditions could lead to the ‘extracellular containment’ of natural intelligence or awareness in an android. In addition, our QC model is radically different from those currently studied; it relies on a relativistic model of the qubit (r-qubit) and relativistic topological quantum field theory in conjunction with salient aspects of UFM. The r-qubit adds additional degrees of freedom causing the standard Bloch sphere representation of a qubit to be obsolete [2]. The additional degrees of freedom require development of a new class of quantum logic gates and algorithms. QC operation becomes a duality,

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partially quantum mechanically (current thinking) and partially within the 3rd regime of UFM (new physics) in conjunction with brane dynamics correlated with Calabi-Yau mirror symmetry attributed to M-Theory [2, 3]. This is key to surmounting the uncertainty principle which also puts an end to the major problem of decoherence [5, 6, 8]. We suspect Bulk Scalable Universal QC cannot be achieved without these proposed improvements in QC modeling. Finally we discuss the current state of the art for the ‘extracellular containment of awareness’ and timeline for implementing the physical principles of mind and processing in a first sentient android prototype - not as Asimov suggested with a ‘flashing stream of positrons but rather with a noeon flux, the putative exchange unit of the Unified Field synonymous with a life principle and stream of qualia.

NEW DIRECTION FOR MIND-BODY RESEARCH In contrast to current thinking we can no longer accept reasoning that ‘the Planck scale is the fundamental basement level of the universe’ (reality), or that spacetime geometry is the fundamental domain where the psychophysical bridge occurs; because spacetime is an emergent property associated with the regime of quantum mechanical uncertainty which we now know has a finite radius [3] beyond which lies the domain of UFM. A sufficient basis for defining awareness requires parameters of UFM beyond this virtual veil of uncertainty. In the same way a distinction between Classical and Quantum was discovered with each domain being a physical regime with its own laws and methods of investigation; mind is also comprised of physically real matter that exists and operates in another arena hidden until now. Recall that UFM [3] is just being formalized providing the long anticipated 3rd regime of reality. Thus, our understanding of the physical world evolves from Classical to Quantum to Unified (CQU). The current description of our universe, called the Standard Model, is presently governed by the rules of the Copenhagen Interpretation of quantum theory, electromagnetism and Special/General Relativity cast in a Big Bang

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cosmology. A top down description that reduces to an impenetrable barrier, a so-called stochastic quantum foam at the 10-33 cm Planck scale representing the lower limit of a reality where we (mind, awareness) as ‘observer’ are embedded in and made out of its emergent material properties. This Planck scale is not the ‘basement of reality’ as Hameroff calls it [9], only a temporarily closed door [10] imposed by the Copenhagen interpretation of quantum theory that can now be opened and past through with parameters of Noetic Field Theory (NFT): The Quantization of Mind [1-5, 8]. This CQU progression is neither top-down nor bottom-up but entails what is described as a ‘continuous-state’ free fall-like cycling [1-3, 8].

Figure 1. a) Macroscopic movie theatre metaphor of anthropic awareness (like Plato’s analogy of the cave or virtual reality) and the observer’s (self) place in the theatre. Discrete frames (film) pass through the projector (spacetime) lit by coherent energy of the UF streaming through the observer embedded in the theatre and appearing as the continuous flow of reality (awareness) on the screen. b) Microscopic details of transduction of the UF through the complex spacetime raster into every point, atom and thus molecule of Self-Organized Living Systems (SOLS). c) Showing relativistic injection of the noetic field into spacetime points. d) Coherent interaction of the UF bridging the stochastic quantum barrier coupled to a brain dendron of radius R correlated with an underlying array forming one Eccles Psychon unit within the brain.

Classical Mechanics describes an event between two coordinate systems by what is called the Galilean transformation for uniform motion at velocities less than the speed of light in 3D Euclidean space. Events of quantum mechanics and with relativistic velocities are described by the Lorentz-Poincairé group of transformations in a 4D Einstein-Minkowski spacetime. In order to cross the Psycho-Physical Quantum Bridge noetic cosmology utilizes an extension of M-Theory requiring a new 12D set of

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transformations called the Noetic Transform because it includes properties of an inherent teleological anthropic principle described by the evolution of UF dynamics [1-6] in the 3rd regime of Large-Scale Additional Dimensionality (LSXD). To achieve this result, we utilize a battery of new physical assumptions (developed in ensuing sections): 

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The LSXD regime of UF dynamics is a ‘sea’ of infinite potentia from which the 4D reality of the 3D observer cyclically emerges as a nilpotent resultant (Figures 4, 5). Nilpotency - technically meaning ‘sums to zero’ [2, 11], is a required basis for the noetic cosmologies infinite potentia simplistically like the entangled alive-dead quantum state of Schrödinger’s cat before a realized local event occurs. Action of the UF mediated by noeon ‘flux’ (noeon is the exchange unit of the UF) is the life principle both animating SOLS and supplying psychon energy for the physical evolution of qualia [1-6]. The UF does not operate as a usual phenomenal field (mediated by an energetic exchange quanta like the photon of the electromagnetic field) but as an energyless field by a process called ‘topological switching’ transferring a force of coherence ontologically between M-Theoretic branes [4, 12]. Note: This property of UF dynamics removes the problem of violation of the 2nd law of thermodynamics or the conservation of energy from Cartesian interactive dualism. The key process for the topological transformation of noeon exchange is a holophote action (like a lighthouse beacon) providing a gating mechanism acting as the psychophysical bridge between the potentia of the UF 12D space and the localized 4D spacetime and 3D matter it embeds [1, 4].

MIND-BODY PROBLEM-NATURE OF SENTIENCE To solve the mind-body problem the scientific perspective must evolve beyond the usual Copenhagen Interpretation of quantum theory to the new

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physics required to explain, utilize and design experimental access to the UF regime where physical parameters able to explain psychophysical-bridging reside. 

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The Planck scale cannot be considered the most fundamental level of reality. Three regimes of reality must be addressed: Classical 

Quantum  Unified Field; all of which cycle continuously [1-4, 8]. Qualia are not quantum phenomena per se but unified field phenomena. Quale ‘rest on’ the quantum regime (tip of iceberg) only as part of the sensory transduction apparatus (Mind-body interaction). The Planck scale is not an impenetrable barrier [3] even though considered so as an empirical fact demonstrated by the quantum uncertainty principle - UFM can be utilized to surmount uncertainty.

Fourteen empirical protocols have been proposed [5, 6] (the 1st reviewed here) for demonstrating, gaining access to and leading to a variety of experimental platforms for first hand investigation of the physical basis of awareness (qualia) breaking down the 1st person 3rd person barrier as called for by Nagel [13]. String theory has one parameter, string tension, TS; but has been fraught with the dilemma of a Googolplex (10googol) or infinite number of vacuum possibilities. By utilizing the Eddington, Dirac, and Wheeler large number hypothesis [1, 8] we found an alternative derivation of TS leading to one unique vacuum and what we call the ‘continuous-state hypothesis’ an alternative to expansion/inflation parameters of Big Bang cosmology [8]. Simplistically the perceived inflation energy of Big Bang cosmology postulates a Doppler expansion from a primordial temporal singularity. But the noetic continuous-state hypothesis proposes a localized ‘eternal present’ as if in permanent ‘gravitational free-fall’ [1, 4, 8]. Since we are relativistically embedded in and made out of matter this condition means that all objects (in our 3D virtual reality) are embedded in LSXD in gravitational ‘free-fall’. This is better explained by two other interpretations

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of quantum theory generally ignored by the physics community because they are myopically considered to add nothing. That of the de Broglie-Bohm Causal Interpretation [14] and the Cramer Transactional Interpretation [15]; where spacetime and the matter within it (matter is made of de Broglie waves) is created-annihilated and recreated over and over as part of the perceived arrow of time and creation of our 3D reality as a resultant from LSXD infinite potentia as a ‘standing-wave’ (Figure 2) [1-3, 10]. This can be understood conceptually by a movie theatre metaphor (Figure 1).

Figure 2. a) Conceptualized Cramer transaction (present state or event) where the present (simplistically) is a standing-wave of future-past potential elements. A point is not a rigid singularity (although still discrete) as in the classical sense, but has complex structure like a mini-wormhole where R1 & R2 (like frets holding a wire of a stringed instrument) represent opposite ends of its diameter. b) How observed (virtual) 3D reality arises from the infinite potentia of HD space (like a macroscopic transaction). The ‘standing-wave-like’ (retarded-advanced future-past) mirror symmetric elements C4+/C4- (where C4 signifies 4D potentia of complex space distinguished from the realized 3D of visible space) of continuous-state spacetime show a central observed Euclidian, E3, Minkowski, M4 space resultant. Least Cosmological Units (LCU) governing evolution of the ‘points’ of 3D reality are represented by circles. The Advanced-Retarded future-past 3-cubes in HD space guide the evolution of the central cube (our virtual reality) that emerges from elements of HD space. c) Transactional model with offer-wave & confirmation-wave combined into a resultant transaction d) A future-past advanced-retarded standing/stationary wave. Figures adapted from Cramer [15].

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Figure 3. Conceptualization of the cosmological Least-Unit (LCU) tessellating space which like quark confinement cannot exist alone. a) Current view of a ‘fixed’ point particle or metric x,y,z vertex. The three large circles are an LCU array slice. A form of close-packed spheres forming a 3-torus; missing from the illustration are an top and bottom layers covering the x,y,z vertex and completing one fundamental element of an LCU complex. Field lines emanating from one circle to another represent the de Broglie-Bohm concept of a ‘pilot wave or potential’ governing evolution. b) Similar to a) but drawn with a central ‘Witten string vertex’ [16] and relativistic quantum field potentials (lines) guiding its evolution in spacetime. A Witten vertex is not a closed singularity and because of its open structure provides a key element to the continuousstate process and rotation of the Riemann sphere cyclically from zero to infinity representing rotational elements of the LSXD brane topology. c) Hysteresis loop energy of the hypervolume, R is the scale-invariant rotational radius of the action and the domain wall (curves) string tension, T0 .

The problem has to do with the nature of a point or 3D vertex in physical theory [2, 11]. What extended versions of de Broglie-Bohm and Cramer suggest is a basis for defining a fundamental ‘point’ that instead of being rigidly fixed classically (Figure 3a) is continuously transmutable (Figure 3b) as in string theory. This elevates the so-called wave-particle duality for quanta to a Principle of continuous-state cosmology canceling the troubling infinites in the standard model of particle physics in a natural way rather than by use of a mathematical gimmick called renormalization. We build the continuous-state hypothesis from an object in string theory called the Witten Vertex [16] (Figure 3b after noted M-Theorist E. Witten). This means that when certain parameters (compactification, dimensional reduction, etc.) associated with the Riemann sphere reach a zero-point; the Riemann sphere relativistically rotates back to infinity and so on continuously (Reminiscent of how water waves operate). The LSXD branes of so-called Calabi-Yau mirror symmetry are forms of Riemann 3-

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spheres or Kahler manifolds [2, 10]. Instead of the insurmountable Plank foam, the gate keeper in this scenario is an array of least cosmological units (LCU) [1-3, 10] of which part (like the tip of an iceberg) resides in our virtual 4-space and the other part resides in the LSXD (12D) regime of M-Theory. These LCU gates govern mediation of the UF in the coherent ordering of the life principle of SOLS. Accessing the UF basis centers on defining what is called a Least Cosmological Unit (LCU) [1, 8] tiling the spacetime backcloth. An LCU (Figure 3) conceptually parallels the unit cell building up crystal structure. The LCU entails the next evolutionary step for the basis of a point particle [10, 11] and has two main functions: It is the raster from which matter arises, and is a central mechanism that mediates the syntropic gating of life principle parameters of the UF. Syntropy is the negentropy process expelling entropy by the teleological action of SOLS. There is a major conceptual change from Quantum Mechanics to UFM. The energy of the UF is not quantized and thus radically different from other known fields, troubling Nobelist R Feynman: “...maybe nature is trying to tell us something new here, maybe we should not try to quantize gravity...Is it possible that gravity is not quantized and all the rest of the world is?” [17].

Not only is gravity not quantized but neither is the noeon energy of the UF related to gravity [1-3]. Here is one way to explain it. In a usual field like electromagnetism (easiest to understand sense we have the most experience with it) field lines connect to adjacent point charges. The quanta of the fields force is exchanged along those field lines (in this case photons). We perceive this as occurring in 4-space (4D). This is phenomenological as the phenomenon of fields. For topological charge as in the UF with properties related to consciousness; the situation is vastly different. The fields are still coupled and there is tension between them but no phenomenological energy (i.e., field quanta) is exchanged. This is the situation in the ontological case. The adjacent branes “become” each other as they overlap by a process called

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‘topological switching’ [1, 12]. This is not possible for a 4-space field because they are quantized resultants of LSXD topological field components. The LSXD ‘units’ (noeons) are free to “mix” ontologically as they are not resolved into fixed points.

Figure 4. a) Complex HD Calabi-Yau mirror symmetric 3-forms,  C4 complex dimensions become embedded in Minkowski space, M4. This resultant UF energy is projected into brain dendrons as a stream of evolving (evanescing) superradiant qualia as a continuous quantum state evolution considered a nilpotent Bloch sphere representing the lower portion that embeds in local spacetime. There is an additional duality above this projection embedded in the infinite potentia of the UF from which it arises. What a UF LSXD reality means is that the usual consideration of a Bloch 2sphere (representing the lower portion only that embeds in local spacetime) vectorbased qubit is insufficient for bulk QC. b) From 8D to 12D, illustrating full extended rendition of additional parameters for a relativistic LSXD continuous-state dual CalabiYau mirror symmetric cosmology as far as currently understood. The Bloch 2-sphere qubit representation is replaced with the new extended r-qubit Riemann 3-sphere resultant representation that has sufficient parameters to surmount the uncertainty principle and therefore operate quantum computers.

If the UF is not quantized how can a force mediate quantal exchange? Firstly the UF does not provide a 5th force as one might initially assume; instead the ‘presence’ of the UF provides a ‘force of coherence’ which based on ‘topological charge’ [1, 8] is ontological or ‘energyless’. Consider this perceptually: If one looks along parallel railroad tracks they recede into a

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point in the distance, a property of time and space. For the unitary evolution of consciousness [1, 4] this would break the requirement of coherence. For the UF which is outside of local time and space, a cyclical restoring force is applied to matter waves putting it in a mind mode where railroad tracks do not recede into a point - The Riemann sphere flips (our perception) beforehand. The familiar 3D Necker cube (center of Figure 2b is like a Necker cube) when stared at central vertices topologically reverse. This is called topological switching. In the LCU spacetime background this topological switching represents the gate which is the lighthouse with the rotating light on top.

Figure 5. Locus of nonlocal HD mirror symmetric Calabi-Yau 3-tori (here technically depicted as quaternionic trefoil knots) spinning relativistically and evolving in time. Nodes in the cycle are sometimes chaotic and sometimes periodically couple into resultant (faces of a cube) quantum states in 3-space depicted in the diagram as Riemann Bloch 2-spheres.

PHYSICAL BASIS OF QUALIA Qualia, plural of quale, is defined as ‘the subjective quality of experience; a qualitative feel associated with an experience’. The physics of noetic cosmology with an inherent ‘life principle’ based on UF mechanics also provides for the first time a physical basis for representing quale in a rigorous empirically testable manner. If experience has a specific subjective nature; if one removed the viewpoint of the subjective observer; what would be left? The remaining properties might be those detectable by other beings,

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the physical processes themselves or states intrinsic to the experience of awareness. This changes the perspective of qualia to the form “there is something it is like to undergo certain physical processes”. “If our idea of the physical ever expands to include mental phenomena, it will have to assign them an objective character” [13]. This breaks down the 1st person3rd person barrier: A major step in implementing extra-cellular sentience. These are questions an integrative Noetic Science now answers theoretically and empirically. Standard definitions of qualia are an inadequate philosophical construct describing only the subjective character. In the physical sense of Noetic Field Theory (NFT): The Quantization of Mind components describing qualia from the objective sense distinguish the phenomenology of qualia from the underlying ontological ‘nonlocal noumenon’ or physical existence of the fundamental thing in itself. NFT suggests that a comprehensive definition of qualia is comprised of triune form considered physically real because the noetic unified field on which the NFT is based is physically real. The proposed triune basis of quale is as follows: Type I. The Subjective - The what it feels like basis of awareness. Phenomenological mental states of the qualia of experience. (This is the current philosophical definition of qualia, Q-I) Type II. The Objective - Physical basis of qualia phenomenology independent of the subjective feel that could be stored or transferred to another entity breaking down the 1st person-3rd person barrier. Noumenal nonlocal UF elements and related processes evanesce qualia by a form of superradiance, Q-II. Type III. The Cosmological - SOLS by being alive represent a Qualia substrate of the anthropic multiverse, acting as a ‘blank slate’ carrier (like a television set turned on but with no broadcast signal) from within which Q-II are modulated into the Q-I of experience by a form of superradiance. Note: Q-III has sub-elements called quanemes addressed elsewhere [1, 4].

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Standard images require a screen or reflective surface to be resolved; but if the foci of two parabolic mirrors (Casimir-like vacuum plates) coincide, the two images superpose into a real 3D holographic image not needing a screen. A toy called the magic mirage demonstrates this effect of parabolic mirrors. Objects placed in the bottom appear as solid objects at the top of the device. In 12D LSXD reality Calabi-Yau brane topology performs the same function for the locus of qualia propagation. The ‘light-house’ (flashing) action of UF life principle energetics arises from harmonic oscillations of boundary conditions tiling the spacetime backcloth and pervading all SOLS. The inherent beat frequency of this continuous action produces the Q-III carrier wave that is an empty slate modulating cognitive data of Q-II physical parameters into Q-I awareness states as a superposition of the two (Q-III and Q-II). This modulation of qualia occurs in the HD QED cavities of the individual’s psychosphere cognitive domain. The QED cavities are a close-packed tiling of LCU noetic hyperspheres; the Casimir surfaces of which reflect quaneme subelements. The best reflectors of em-waves are polished metal mirrors; charged boundary conditions also reflect em-waves the same way radio signals bounce off the ionized gases of the Kennelly-Heaviside layers in the ionosphere. This reflective ‘sheath’ enclosing the cognitive domain is charged by the Noeon radiation (exchange particle of the noetic field) of the life principle, the phases of which are ‘regulated’ in the complex LSXD space. How does noetic theory describe more complex aspects of qualia? Like a rainbow, light quanta (water drop) are microscopic in contrast to the macroscopic sphere of awareness (rainbow). It thus seems reasonable to assume that scale-invariant properties of the LCUs modulating awareness would apply. Like phonemes as fundamental sound elements for audible language qualia-nemes or quanemes are proposed for subelements of awareness; all based on the physical modulation of Q-II states by the geometric structural-phenomenology of the Q-III carrier base of living systems. The quaneme is a singular Witten point in the raster of mind like a locus of points forming a line. Each of these ‘quaneme points’ of noeon entry through the LCU gating array are like an individual raindrop that summate

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into a rainbow or thought train of awareness. This again takes us back to the movie theater metaphor of Figure 1 where the discrete frame of film (LCU gated) is projected continuously on the screen, in this case the mind.

Figure 6. 2D rendition of an HD holographic process. a) An object (small black circle) placed inside two parabolic mirrors (Casimir-like domain walls) produces a virtual image (white circle) representing creation of a point in spacetime or stream of elements producing qualia. b) Our virtual holographic reality is produced in a similar fashion by Cramer futurepast standing-wave parameters from the LSXD Calabi-Yau mirror symmetric infinite potentia of the UF. As in Figure 1 this same process produces qualia with each lit point like a raindrop producing a rainbow by the ‘light’ of the UF.

Figure 7. a) The physical basis of the continuous superradiant generation of qualia from the three components of mind: eternal Elemental Intelligence, Brain-Body (Descartes res extensa), and the superradiant qualia (Descartes res cogitans) mediated by the spacetime raster (quaneme locus) that gates ‘the light of the mind’ or UF energy. The term quaneme is derived to parallel the phoneme component of sounds. b) LCU construct hidden nonlocally behind a local 3-space singularity (black cross vertex).

For cognitive theory all intelligence/consciousness resides in the brain (mind equals brain). The situation is radically different for Cartesian

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interactive dualism requiring a life principle. Cognitive theory requires only one component - body stuff or matter; but interactive dualism requires three components: 1) Matter (Cartesian res extensa). 2) Mind stuff (res cogitans) and 3) Nonlocal Elemental Intelligence. Because of space constrictions these critically important aspects are only mentioned here [1].

PSYCHON UNIT MEASURING ENERGY OF MIND NFT elevates the concept of qualia from the traditionally philosophical concept used in cognitive science to a physically real fundamental noumenon. Noumenon is defined as the ‘thing in itself’ beyond the veil of the 3D phenomenological world; in Kantian philosophy a noumenon is something existing independently of intellectual or sensory perception. This fundamental physicality allows qualia to be ‘digitized’ in some form breaking down the 1st person-3rd person barrier leading to profound new ‘conscious’ technologies. Nobelist Sir John Eccles coined a construct called the psychon, to illustrate how mental energy coupled to brain dendrons (bundle of neural dendrites) [18] to complete his Cartesian interactionist model of mind-body dualism [1-6]. Formalizing the ‘Psychon’ as a unit of measure is made possible by a comprehensive science of qualia or fundamental basis of awareness. In meditative science it is said that ‘energy follows thought’. Here we postulate that the qualia of awareness are comprised of a real physical flux of energy related to new physics of the unified field, UF [4]. In honor of Nobelist Sir J.C. Eccles (synapse discovery) we propose to quantify this mental energy in terms of a new physical unit called the Psychon. The Einstein, a physical unit of energy measure named in honor of Albert Einstein for his explanation of the photoelectric effect in terms of light quanta (photons) bears conceptual similarity and we thus use that as our starting point. The Einstein is used to measure the power of electromagnetic radiation in photosynthesis where one Einstein represents one mole or Avogadro’s number of photons (6.02 x 1023). In general physics the energy, E of n photons is E  n   n (c /  )

242 where

Richard L. Amoroso is Planck’s constant and  the frequency. The second part of the

equation is energy in terms of the wavelength,  (in nanometers, nm) and the speed of light, c. Adapting this photon energy equation to measure Einsteins is similar, E  N0   N0 (c /  ) where the energy of N0 photons is instead in Einsteins, E. In photometrics the measure used is one microeinstein per second per square meter, where one microeinstein, uE is one-millionth of an Einstein or 6.02 x 1017 photons imping a leaf for example. A similar unit of measure to quantify the mental energy of quale called the Psychon as one mole or Avogadro’s number of noeons is created. The force of all four known phenomenological fields (electromagnetic, strong, weak and gravitational) are said to have exchange quanta mediating the field’s interactions by a quantal exchange of energy. For electromagnetism the exchange quanta is the photon. This quantal mediation has been experimentally verified for all fields except gravity because the graviton has not been discovered. According to NFT the regime of unification is not quantum but instead correlates with ontological parameters of UFM [3]. The trefoil knots (Figure 5 drawn as Planck scale quaternion vertices) is holomorphic to the circle. Since energy is conserved, we may ignore the complexity of the LSXD Calabi-Yau and AdS5 Dodecahedral symmetries and use the area of the circle, in this case a resultant continuous rotation of two circles as a 2-sphere quantum state or perhaps better as a torus as the coupling area of one psychon to a dendron. This idea is further conceptualized in Figure 4 illustrating how a 3D object emerges from spacetime. In considering psychon energy it appears easier to calculate the nonlocal brane area rather than the local volume or surface area of a neural dendron or array of microtubules, etc. Recall that the intestinal villi are purported to provide the area of a football field. In any case we will not calculate here but leave it for a later publication since we still struggle with the conceptual problems relating to the geometric topology of noeon coherence. Recall that the de Broglie-Bohm interpretation entails a nonlocal pilot-wave or quantum-potential said to guide the evolution of the wavefunction

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ontologically. This concept was not very successful in 4D, but when carried to LSXD it works elegantly and the pilot-wave-quantum potential is like a Super Quantum Potential becoming synonymous with coherent aspects of the UF. Note that the UF provides the basis for gravitation [1, 8] and the life principle for living systems not just the evolutionary flow of qualia in the mind. A bit more noeon-psychon theory: A torus is generated by rotating a circle about an extended line in its plane where the circles become a continuous ring. According to the equation for a torus,





2

 x 2  y 2  R   z 2  r 2 , where r is the radius of the rotating circle and R   is the distance between the center of the circle and the axis of rotation. The

volume of the torus is 2 2 Rr 2 and the surface area is 4 2 Rr, in the above Cartesian formula the z axis is the axis of rotation. We apply this to the holophote action of noeon flux. In atomic theory electron charged particle spherical domains fill the toroidal volume of the atomic orbit by their wave motion. If a photon of specific quanta is emitted while an electron is resident in an upper (like the UF domain) more excited Bohr orbit, the radius of the orbit drops back down to the next lower energy level decreasing the volume of the torus in the emission process. Like the Einstein, the psychon is defined as a measure of one mole of noeons, purported to be the topological exchange complex of the Unified Field, UF which provides the energy that animates the stream of awareness or qualia. Using the noetic field equation,  F   /  [1, 4] we need to calculate the energy of the noeon field from its space-time hysteresis loop (Figure 3 b,c). This is a practical and conceptual challenge that is hard to meet. Imagine trying to calculate the surface area of the dendrite and synaptic boutons in a dendron, neural network or array of microtubules for example. Instead imagine a helicopter like those used to put out forest fires carrying a bucket of water retrieved from a nearby lake (UF). The volume of that bucket is known. So it is infinitely easier to work with the volume of the helicopter water bucket than to try to measure the surface area of the trees and other objects on the ground. When Eccles loosely defined the psychon-dendron

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correlation he did not consider an Avogadro’s number of noeons to enter into the picture. The question is can we correlate helicopter buckets of the UF with the volume or surface area of an array of the hysteresis loop modulating energy of coherence entering the local space-time of a dendron? For simplicity at this stage of development we use the general unexpanded form of the Noetic UF equation,  F   /  where NF is the force of coherence of the UF,



the relativistic rotational energy and



the

‘cavity’ radius (Figure 3). The cavity represents a hysteresis loop of the LSXD brane energy dynamics. The cavity relates to the volume of the Calabi-Yau mirror symmetric dual 3-tori of the lighthouse gating mechanism. The gate cycles continuously through LSXD symmetries of MTheoretic space through various compactification modes [2, 8] until it reaches a 4D standing-wave Minkowski spacetime of the standard model of observed reality, i.e., a Copenhagen domain wall of noeon energy pervading all spacetime and matter, i.e., SOLS as the life principle (in our example a dendron). This process, further described by physics of the gating mechanism which is mediated by a new set of transformations beyond the Galilean-Lorentz-Poincairé called in regard to an anthropic multiverse it is cast in - the Noetic Transform [2]. We derived our definition of the noeon (from the Greek nous, mind and noēsis/noētikos, perception-what the nous does) and the common “on” suffix in particle physics such as the phot-on as the fundamental exchange unit of the anthropic unified noetic field. Although UF dynamics entails a ‘force of coherence’ this does not seem to entail a 5th force. The ‘coherence’ implied is the resultant action; perhaps that is misleading. The UF is primary - an originator of all the other forces that brings noeons, which are then immediately returned to the sea of infinite potentia. This cyclical process energizes living systems, qualia and gravitation, etc. One sees that the anthropic principle provides all these phenomena - Life, the Light of the Mind (qualia) and Gravitation! More work has to be done on noeon dynamics. This is what the experimental protocols are designed for - rigorous investigation.

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TESTING UNIFIED FIELD THEORY NFT is empirical testable and has broad explanatory power. Viable experimentation will lead to new consciousness research platforms for studying fundamental properties of SOLS. Fourteen tests [5, 6] of NFT have been proposed; in this short paper only the main experimental protocol testing the UF ‘life-principle’ hypotheses is summarized: Mediating the ‘gating mechanism’ by which access is gained to the UF regime. This will facilitate not only mind-body but also new aspects of M-Theory and nuclear physics research. Extrapolating Einstein’s energy dependent/deformed spacetime metric,

Mˆ 4 [8] to a 12D Calabi-Yau mirror-symmetric standing-wave future-past advanced-retarded topology [2, 8] a spacetime resonance hierarchy protocol utilizing a covariant Dirac polarized vacuum is designed [5, 6] to access the UF regime [3].

Figure 8. The Dirac polarized vacuum has hyperspherical symmetry. a) Metaphor for standing-wave present showing future-past elements, R1, R2 , eleven of twelve dimensions suppressed for simplicity. b) Top view of a) a 2D spherical standing-wave. c) Manipulating the relative phase of oscillations creates nodes of destructive and constructive interference.

Motion of a 1D classical harmonic oscillator is given by q  A sin(t   ) and p  m A cos(t   ) where A is the amplitude and with



is the phase constant for fixed energy E  m 2 A2 / 2 . For state

n  0,1,2... and Hamiltonian En

harmonic oscillator becomes

 (n  1/ 2)  the

n,

quantum

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n q2 n  / 2m n (a† a  aa† ) n  En / m 2

(1)

n p2 n  1/ 2(m ) n a†a  aa†  mEn

(2)

and

a & a † are

where

q

the

annihilation

and

creation

operators,

/ 2m (a †  a ) and p  i m  / 2(a†a) . For the 3D harmonic

oscillator each equation [19] is the same with energies

Ex  (nx  1/ 2) x Ey  (ny  1/ 2)  y Ez  (nz  1/ 2) z ,

(3)

and

In Dubois’ notation classical 1D harmonic oscillators for Newton’s 2nd law in coordinates t and x(t), mass m in potential

U ( x)  1/ 2(kx 2 ) ,

in

differential form

d 2x   2 x  0 where   k / m 2 dt

(4)

which can be separated into the coupled equations [5, 19] dx(t ) dv(t )  v(t )  0 and  2x  0 dt dt

From

incursive

x(t  t ) v(t  t )

discretization,

Dubois

(5)

creates

two

solutions

providing a structural bifurcation of the system

producing Hyperincursion. The effect of increasing the time interval discretizes the trajectory. This represents a background independent discretization of spacetime [8, 19].

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BULK UNIVERSAL QUANTUM COMPUTING Quantum Computing (QC) has remained elusive beyond a few qubits. Feynman’s recommended use of a “synchronization backbone” [20] for achieving bulk implementation has generally been abandoned as intractable; a conundrum we believe arises from limitations imposed by the standard models of Quantum Theory (QT). It is proposed that Feynman’s model can be utilized to implement Universal Quantum Computing (UQC) with valid operationally completed extensions of QT and cosmology [2]. Requisite additional degrees of freedom are introduced by defining a relativistic basis for the qubit (r-qubit) in a higher dimensional (LSXD) conformal scaleinvariant context and defining a new anticipatory based cosmology (cosmology itself cast as a hierarchical form of complex self-organized system) making correspondence to unique 12D Calabi-Yau mirror symmetries of M-Theory. The causal structure of these conditions reveals an inherent new Unified Field, UF “action principle” (force of coherence) driving self-organization and providing a basis for applying Feynman’s synchronization backbone principle. Operationally a new set of transformations (beyond the standard Galilean/Lorentz-Poincaré) ontologically surmounts the quantum condition (producing decoherence during both initialization and measurement) by an acausal energyless (ontological) topological interaction [2]. Utilizing the inherent LSXD regime requires new commutation rules and corresponding I/O techniques based on a coherent control process with applicable rf-pulsed incursive harmonic modes of LSXD spacetime manifolds described by a spinexchange continuous-state spacetime resonance hierarchy. We postulate bulk universal QC cannot be achieved without surmounting the quantum uncertainty principle, an inherent barrier by empirical definition in the regime described by the 4D Copenhagen interpretation - last remaining hurdle to bulk QC. QC operations by surmounting uncertainty with probability  1 , requires redefining the basis for the qubit. Our form of M-Theoretic Calabi-Yau mirror symmetry cast in an LSXD Dirac covariant polarized vacuum contains an inherent ‘Feynman

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synchronization backbone’. This also incorporates a relativistic qubit (rqubit) providing additional degrees of freedom beyond the traditional Block 2-sphere qubit bringing the r-qubit. Review of bulk UQC prototype design able to incorporate a sentient android: 





We arbitrarily choose a class-II mesoionic xanthine crystal stable at room temperature for ~ 100 years with 10 evenly separable quantum states in its ground state configuration. The xanthine is programmed by rf-pulsed Sagnac Effect resonance to overcome I/O decoherence [2, 8]. This is the holographic ‘neural net android brain. For greater efficiency (intelligence) quantum dot ring laser arrays manufactured with internal mirrors may be utilized instead of IC arrays. The quantum dots would be arrayed on a suitable substrate rather than an IC. Another android brain model could utilize a class II mesoionic xanthine doped multilayer graphene molecule array (currently under study) where it may be possible to operate a QC by forms of Quantum Hall effects, bilayer graphene alone, or a stand-alone mesoionic xanthine configuration since several mesoionic xanthine molecules have pertinent polar properties.

Because the model surmounts the quantum uncertainty principle in a complex 12-space the current Bloch (Riemann) sphere representation of qubits (classical 2-sphere model) is a nonphysical mathematical representation too primitive and not suited for actualizing bulk universal QC. For the past several years our model was based on a relativistic (r-qubit) where the additional degree of freedom was an aid to surmounting uncertainty [2, 6, 8]. Recently we realized this 4D r-qubit, while on the right track was also insufficient. This arose from extending quantum theory to the regime of the Unified Field, UF primarily based on extended LSXD versions of Cramer’s transactional interpretation and de Broglie-Bohm interpretation of QT. This was as much a breakthrough in nilpotent cosmology as QT. We discovered there was more to a quantum state than a Copenhagen ‘particle

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in a box’; the quantum state was conformally scale-invariant requiring a representation utilizing a system of dual continuous-state Calabi-Yau mirror symmetric 3-tori (class of Kähler manifolds) [6, 8]. One surprise is that this cosmology contains an inherent ‘synchronization backbone’ [20] which ends up like getting half the QC for free; making the essential process of surmounting uncertainty almost simplistic [2, 6].

THE CASE FOR RELATIVISTIC QUBITS This summarizes the current thinking on representations of quantum states where the quantum wavefunction is the most complete description that can currently be given to a physical system:   

Physical information about a transition is encoded in a unit vector in a complex vector space. Physical process without measurement corresponds to unitary transformation of this vector. A measurement corresponds to the probabilistic choice of a covector to form an amplitude

 U 

where the probability is  U  . 2

We intend to show that this currently utilized vector algebra is not physical but rather a convenient mathematical representation. The Bloch sphere is merely a 2D representation of 4D reality. We show below a recent attempt at a 6D dual qubit as an indicium of our 12D model which we believe is required to fully represent a properly physicalized qubit! In the philosophy of physical science there is no a priori reason why nature must be described by a UF theory. The current drive in physics is to bring the four fundamental field interactions into a single unified framework as a form of quantum theory. Because of the inherent difficulty associated with renormalization and uniting gravity and quantum theory many physicists believe a framework other than a field theory such as a version of an 11D String/M-Theory may be a viable alternative avenue.

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In the usual nonrelativistic quantum theory of computation, it was necessary only to point to the number of states, 2n for a description of n qubits. In our extended relativistic theory, there are many special cases. Charged and neutral, massive and massless particles, etc. should be described differently. The problem of extending the fundamental basis of the qubit is manifold. Many physicists do not accept dimensionality beyond 4D. Those that do, predominantly string theorists, now M-Theorists, are confounded by the search for a unique string vacuum claimed to have a Googolplex or 10 possibilities. Our model has discovered a unique string vacuum [8]. Further restrictions arise from a unique form of inherent Calabi-Yau mirror symmetry. Thus, a clear avenue is provided to ‘divine’ the complex LSXD space from which our 3D virtual reality is a resultant. Fortunately, our unusual model is empirically testable [2, 5, 8].

Figure 9. a) Representation of a qubit 0  1   2 as a complex Riemann Bloch 2sphere. b) Combinatorial graph of vertices corresponding to basis vectors of a Bloch sphere for two qubits [e1, e2, e3] & [f1, f2, f3] and the edges to the corresponding bivector basis Gij. Dashed ellipses enclose induced subgraphs corresponding to “local” subalgebras of each Bloch sphere model, while the perfect matching of a Cartan subalgebra is indicated by the bolder lines of edges G11, G22, G33. Figure redrawn from [21].

The perceived required redefinition of the qubit also requires new logic gates and QC algorithms taking full advantage of the required new physics.

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Operationally the new r-qubit basis entails a new set of transformations beyond the usual Galilean-Lorentz-Poincairé which have been temporally adjoint along an axis or light cone in Euclidean and then Minkowski coordinates. We choose to call the new transformation ‘The Noetic Transformation’ because it is cast in an anthropic multiverse. What separates the Noetic Transform from its precursors (Galilean, Lorentz-Poincaré) is that it uncouples from the 3D or 4D realm of the observer and has no temporal component. This evolution now continues to a new regime of Unified Field Theory, UF. We do not wish to say ‘uncouples from reality’, rather that fundamental reality should now be considered 12D instead of the 3(4)D of the LorentzPoincaré Transformation. The elimination of the concept of time occurs by a double superluminal boost, x  t x

 wx that also occurs along the y and

z axes simultaneously x]. The infinities plaguing renormalization are indicia of this 12D reality (the same way infinities in the Raleigh-Jeans law for black body radiation were an indicium of the immanent discovery of quantum mechanics). We anticipate that the realized basis for bulk universal QC diverges from the anticipated form by current QC researchers utilizing the standard Copenhagen Interpretation (CI) of quantum theory. What this means is that the Bloch 2-sphere vector basis is archaic and not an appropriate model for bulk QC gates or algorithms. As our starting point we follow recent efforts of Makhlin [22] Zhang et al. [23] and Havel [21], (MZH) who have pointed the way to our model with a geometric algebra rendition of a dual Bloch sphere. MZH illustrates the Cartan decompositions and subalgebras of the 4D unitary group, which have recently been used to study the entangling capabilities of two-qubit unitaries. “…we show how the geometric algebra of a 6D real Euclidean vector space naturally allows one to construct the special unitary group on a twoqubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit” [21].

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The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere Since unit quaternions can be used to represent rotations in 3D space (up to sign), we have a surjective homeomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. The geometric structure of nonlocal gates is a 3-torus. The local equivalence classes of 2-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. The MZH model is based on complex Minkowski space and the Copenhagen Interpretation. Our model differs - cast in 9D M-Theoretic Calabi-Yau mirror symmetry utilizing an operationally completed form of QT by integrating LSXD forms of the de Broglie-Bohm Causal Interpretation [14] and Cramer’s Transactional Interpretation [15] still corresponding to the MZH 6D model [21-23].

Figure 10. (a) Stereographic projection model of a qubit on a complex Riemann sphere, usual q-gate with constant number of states and particles. (b) Relativistic model of a qubit (r-qubit) with interacting quantum fields entailing an extra HD degree of freedom with constant particles but variable or infinite states.

In the conventional consideration of quantum computing a qubit is any two-state quantum system defined as a superposition of two logical states of a usual bit with complex coefficients that can be mapped to the Riemann sphere by stereographic projection (Figure 10a); formally represented as:    0   1 with each ray  ,  C in complex Hilbert space and

Sentient Androids 

2

     1, where

Riemann sphere and

253

0 corresponds to the south or 0 pole of the

1 corresponds to the opposite north or

 pole of the

Riemann complex sphere. The conventional qubit maps to the complex plane of the Riemann sphere shown below as:     X ,    iY ,    Z . Unitary qubit transformations correspond to 3D rotations of the Riemann sphere; but following Vlasov [24] for relativistic considerations of a qubit (r-qubit) an additional 4D W parameter is added to the equation (6):

    X ,     iY ,     Z ,     W

(6)

In cartography and geometry, a stereographic projection is a mapping projecting each point on a sphere onto a tangent plane along a straight line from the antipode of the point of tangency (with one exception: the center of projection, antipodal to the point of tangency, is not projected to any point in the Euclidean plane; it corresponds to a “point at infinity”). One approaches that point at infinity by continuing in any direction at all; in that respect this situation is unlike the real projective plane, which has many points at infinity. This 4D r-qubit representation is only the first step; viable quantum computing requires extension to a 12D r-qubit!

THE NOETIC TRANSFORMATION The Noetic Transform extends quantum theory into the regime of UFM as a requirement for quantum computing. An event in spacetime is an idealized instant of time at a definite position in space labeled by time and position coordinates, t,x,y,z. Coordinates have no absolute significance; they are arbitrary continuous single-valued labels given invariant meaning by the

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expression for the line-element connecting two events [25, 26]. The usual expression for a line-element in Minkowski coordinates is ds 2  dt 2  dx2  dy 2  dz 2 .

(7)

For simplicity at this stage of development of the Noetic Transformation we devise the XD coordinates as orthogonal and evenly spaced. Firstly since the LSXD space is time independent we may drop the dt 2 term from the lineelement and introduce a new spatial form, dl 2 where dl 2 reduces to ds 2 and dl 2  dx 2  dy 2  dz 2  dW 2

(8)

where W  wi  w j  wk (before complex dualing to LSXD Calabi-Yau mirror symmetry) as a 9D quaternion-like trivector representation. This is like an extension of the 3-sphere of Einstein’s space where the set of points x,y,z,W are at a fixed distance R from the origin such that R 2  x 2  y 2  z 2  W 2 preserving the wanted three time independent space variables, x,y,z and where the fourth LSXD variable W W 2  R 2  r 2 where

dW 

2

is given as

r 2  x 2  y 2  z 2 such that(5) becomes

r dr r dr  W R2  r 2





1/2

(10)

So that the dual local-HD spatial line-element dl 2 becomes dl 2  dx 2  dy 2  dz 2 

r 2 dr 2 R2  r 2

(11)

where R may be used to represent the center of dual Calabi-Yau mirror symmetric 3-tori. See Figure 8. Continuing to follow Peebles [25, 26] this generalizes the usual 2D line-element to 9D where the length R is a constant

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because spacetime is assumed to be static. For r R our extended Einstein line element approaches the usual Minkowski form (11). When r = R the geometry makes correspondence to the surface of a Riemann 2-sphere which is utilized in the standard description of a qubit as a Bloch Sphere. (Figure 9a) Let’s look at the additional parameters this space allows us to add to the fundamental description of a quantum state beyond the usual inherent uncertainties of Copenhagen interpretation. Because of the conformal scaleinvariance of the Nilpotent criteria an additional duality must be incorporated into the mirror symmetric parameters of W 2 which is a further correspondence to the standing wave-like properties of the Cramer Transactional Interpretation to simplistic-ally what might be labeled, W 2 . This addition would incorporate all the additional parameters for a complete description of a quantum state as embedded in the LSXD aspects of the UF required for the r-qubit to include the additional HD conformal scaleinvariant parameters. The Pythagorean Theorem, a 2  b 2  c 2  d 2 gives the diagonal length, d of a 3D cube, a,b,c. Adding terms to the equation describes the diagonal of an nD hypercube. The locking together of the Calabi-Yau components in the resultant localized cube creates the quantum uncertainty principle which can be surmounted [2, 3, 5] if the Calabi-Yau nilpotent ‘copies’ are accessed by incursive resonance. The additional parameters of this space allows us to add to the fundamental description of a quantum state beyond the usual Copenhagen interpretation. Because of the conformal scale-invariance to the Nilpotent criteria an additional duality must be incorporated into the mirror symmetric parameters of W 2 which is a further correspondence to the standing-wavelike properties of the Cramer Transactional Interpretation to simplistically what might be labeled, W 2 . This addition as far as we currently understand would incorporate all the additional parameters for a complete description of a quantum state as embedded in the HD aspects of the UF requiring a new representation of the qubit to include the additional parameters.

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We can also attempt to describe this topological geometry with dual quaternion-like trefoil knots. The trefoil knot array (in Figure 5 drawn as Planck scale quaternion vertices) is holomorphic to the circle. Since energy is conserved, we may ignore the complexity of the HD symmetries and use the area of that circle as the Lagrangian, in this case a resultant of two trefoil knots as a 2-sphere quantum state as the coupling area. The figure also provides a conceptualized view of how one sees continuous-state evolution of conformal scale-invariant Calabi-Yau mirror symmetric topology. As QT has a semi-classical limit this might be termed semi-quantum in terms of the HD UF. There is a 2nd LSXD level ‘above’ this one postulated as the regime of full UF potentia. The cycle goes from chaotic-uncertain to coherentcertain non-commutative to commutative according to the noetic transformation. This is represented in the Dirac string trick [27]. To formalize the model a complex quaternion Clifford algebra is required to incorporate all the new LSXD UF parameters. Thus, in contrast to Havel’s 6D bivector in complex Minkowski or Hilbert space (Figure 9b) we can illustrate a LSXD r-qubit by the Philippine wine dance [27]. Each wine glass would represent one standard Bloch sphere; the dancer is like an atom and each glass represents one of the 2 possible spin states. Havel would have 2 entangled wine dancers standing near each other in MinkowskiHilbert space. What we require to completely define a quantum state physically is that the wine dancers are like puppets standing additionally in a hall of mirrors [28] (Calabi-Yau mirror symmetry). The puppet master is the super-quantum potential provided by parameters of the UF. The mirror images are restricted on each side of the Cramer future-past Calabi-Yau mirror symmetry. By the continuous-state premise of this LSXD hierarchy the left-right or future-past components become embedded in each other in the cycle [2, 6, 8]. The bottom (3D resultant) becomes the usual semiclassical phenomenological q-state we observe. At the 12D top the embedding is the causally free (ontological) quantum state copy - i.e., surmounting the quantum uncertainty principle [6, 8]. In summary Havel uses a 6D bivector to represent 2 qubits. In our model a single qubit should be represented as some form of a dual quaternion trivector. What we get with this new qubit representation is QC logic gates

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able to surmount the uncertainty principle and proper algorithms for universal QC. Normalized quaternions are simply Euclidean 4-vectors (length one) and thus fermionic vertices in spacetime or points on a unit hypersphere (this case a 3-sphere) embedded in 4D. Just as the unit sphere has two degrees of freedom, e.g., latitude and longitude, the unit hypersphere has three degrees of freedom. The coordinate fixing-unfixing mechanism is superbly illustrated by the ‘walking of the Moai on Rapa Nui’ [29]. However, a 3rd complex metric is involved making an evolution from dual quaternions to a 3rd quaternion we choose to name a trivector that acts as a baton passing mechanism between the space-antispace or dual quaternion vector space. Of paramount importance this trivector facilitates a ‘leap-frogging’ between anti-commutative and commutative modes of HD space. This inaugurates a Mobius transformation between the Riemann dual stereographic projection complex planes. Geometrically, a standard Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit 2-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. Möbius transformations are defined on the extended complex plane (i.e., the complex plane augmented by the point at infinity): ˆ 

  .

This extended complex plane can be thought of as a sphere, the Riemann sphere, or as the complex projective line. Every Möbius transformation is a bijective conformal map of the Riemann sphere to itself. Every such map is by necessity a Möbius transformation. Geometrically this map is the Riemann stereographic projection of a rotation by 90° around ±i with period 4, which takes the continuous cycle 0  1    1  0 . This is required to oscillate from anticommutivity to commutivity in order to provide the cyclic opportunity to violate 4D quantum uncertainty [2, 6]!

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QC P  1 OPERATIONAL ANDROID DESIGN In a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. if the particles are classical “spinning” particles then the distribution of their spin angular momentum vectors is taken to be truly random and each particle would be deflected up or down by a different amount producing an even distribution on the screen of a detector. instead, quantum mechanically, the particles passing through the device are deflected either up or down by a specific amount. this means that spin angular momentum is quantized (also called space quantization), i.e., it can only take on discrete values. there is not a continuous distribution of possible angular momenta. this is the usual fundamental basis of the standard quantum theory and where we must introduce a new experimental protocol to surmount it. This is the crux of our new methodology: If application of a homogeneous magnetic field produces quantum uncertainty upon measurement, then “do something else”! Of the three types of spin-spin coupling, this QC protocol relies on the hyperfine interaction for electron-nucleon coupling, specifically the interaction of the nuclear electric quadrupole moment induced by an applied oscillating rf-electric field to act on the nuclear magnetic dipole moment,  . When the electron and nuclear spins align strongly along their zcomponents the Hamiltonian is m  B , and if B is in the z direction

H   N I  B   N BI x with

m NI , N

(12)

the magnetogyric ratio

 N  e / 2m p

and

mp

the mass

of the proton. Radio frequency excitation of the nuclear magnetic moment, resonance occurs for a nucleus collectively which rotates with respect to the applied field,



 to

to some angle

B0 . This produces a torque i  B0 causing

Sentient Androids the angular momentum, frequency

L   N B0 .



259

itself to precess around

This coherent precessing of

B0 

at the Larmor also induces a

voltage in surrounding media, an energy component of the Hamiltonian utilized to create interference in the structure of spacetime [8]. Metaphorically this is like dropping stones in a pool of water: One stone creates concentric ripples; two stones create domains of constructive and destructive interference. Such an event is not considered possible in the standard models of particle physics, quantum theory and cosmology. However Noetic science uses extended versions of these theories wherein a new teleological action principle is utilized to develop what might be called a ‘transistor of the vacuum’. Just as standard transistors and copper wires provide the basis for almost all modern electronic devices; This Laser Oscillated Vacuum Energy Resonator using the information content of spacetime geodesics (null lines) will become the basis of many forms of Noetic Technologies especially QC. Simplistically in this context, utilizing an array of modulated tunable lasers, atomic electrons are rf-pulsed with a resonant frequency coupling them to the magnetic moment of nucleons such that a cumulative interaction is created to dramatically enhance the HaischRueda inertial back-reaction [8]. The laser beams are counter-propagating producing a Sagnac effect Interferometry to maximize the violation of Special Relativity. This is the 1st stage of a multi-tier experimental platform designed (according to Noetic Field Theory) to ‘open a hole’ in the fabric of spacetime in order to isolate and utilize the force

FˆU of the Unitary Field.

The interferometer utilized as the basis for our vacuum engineering QC platform is a multi-tiered device. The top tier is comprised of counterpropagating Sagnac effect ring lasers that can be built into an IC or Q-dot array of 1,000+ ring lasers. If each microlaser in the array is designed to be counterpropagating, an interference phenomenon called the Sagnac Effect occurs that violates special relativity in the small scale [8]. This array of rfmodulated Sagnac-Effect ring lasers provides the top tier of the multi-tier QC unit. Inside the ring of each laser is a cavity where quantum effects called Cavity Quantum Electrodynamics (C-QED) may occur. A specific molecule

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is placed inside each cavity (we propose a xanthine). If the ring laser array is modulated with resonant frequency modes chosen to achieve spin-spin coupling with the molecules electrons and neutrons, by a process of Coherent Control [8] of Cumulative Interaction an inertial back-reaction is produced so electrons also resonate with the spacetime backcloth to ‘open an oscillating (periodic) hole’ in it. The first step in the interference hierarchy (Figure 11) is to establish an inertial back-reaction between the modulated electrons and their coupled resonance modes with the nucleons. Following the Sakarov and Puthoff conjecture [8] the initial resistance to motion, are actions of the vacuum zero-point field. Therefore, the parameter m in Newton’s second law, f = ma is a function of the zero-point field [8]. Newton’s third law states that ‘every force has an equal and opposite reaction’. Haisch & Rueda [8] claim vacuum resistance arises from this reaction force, f = - f. This inertial back-reaction is like an electromotive force (Electromotive force, E: The internal resistance, r generated when a load is put upon an electric current, I between a potential difference V, i.e., r  ( E  V ) / I ) of a de Broglie matter-wave field in the spin exchange annihilation creation process inherent in a hysteresis of relativistic spacetime fabric. We further suggest that the energy responsible for Newton’s 3rd law is a result of a continuous-state flux of the ubiquitous noetic UF [2, 8]. For QC android implementation we let the Haisch-Rueda postulate be correct.

f 

d *  d   lim   lim *  f* dt t 0 t dt* t* 0 t*

where  

(13)

is the impulse from an accelerating agent and thus

*zp  * [8]. The cyclotron resonance hierarchy must also utilize the proper beat frequency of the continuous-state dimensional reduction spin-exchange compactification process inherent in the cyclic symmetry of noetic spacetime ‘tuned’ so the speed of light c  c . With this apparatus noetic

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theory suggests that destructive-constructive C-QED interference of spacetime occurs such that the noeon eternity wave,  of the

UF

is

harmonically (holophote) released into the detector cavity array. Parameters of the Dubois incursive oscillator are also required for aligning the interferometer hierarchy with the beat frequency of spacetime. As illustrated in Figure 11 the coherent control of the multi-level tier of cumulative interactions relies on full utilization of the continuous-state cycling inherent in parameters of Multiverse cosmology [8]. What putatively will allow noetic interferometry to operate is the harmonic coupling to periodic modes of Dirac spherical rotation in the symmetry of the HD geometry. The universe is no more classical than quantum as currently believed; reality rather is a continuous state cycling of nodes of classical to quantum to unitary, C  Q  U . Space does not permit detailed delineation of the parameters of Multiverse cosmology here; see [8]. The salient point is that cosmology, the topology of spacetime itself, has the same type of spinorial rotation and wave-particle duality Dirac postulated for the electron. Recall that the electron requires a 4D topology and 720° for one rotation instead of the usual 360° to complete a rotation in 3D. The hierarchy of noetic cosmology is cast in 12D such that the pertinent form of relativistic quantum field theory has significantly more degrees of freedom whereby the modes of resonant coupling may act on the structural-phenomenology of Dirac ‘sea’ itself rather than just the superficial zero-point field surface approaches to vacuum engineering common until now. 12D is the minimum to surmount uncertainty because the ‘mirror image of the mirror image in HD space is causally free of the 3D quantum particle! The parameters of the noetic oscillator (Figure 11) may best be implemented using a form of de Broglie fusion. According to de Broglie a spin 1 photon can be considered a fusion of a pair of spin 1/2 corpuscles linked by an electrostatic force. Initially de Broglie thought this might be an electron-positron pair and later a neutrino and antineutrino. “A more complete theory of quanta of light must introduce polarization in such a way that to each atom of light should be linked an internal state of right and left polarization represented by an axial vector with the same direction as the

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propagation velocity” [14]. These prospects suggest a deeper relationship in the structure of spacetime of the Cramer type [8, 15] (Figures 2, 8). The epistemological implications of 12D must be delineated. The empirical domain of the standard model relates to the 4D phenomenology of elementary particles. It is the intricate notion of what constitutes a particle that concerns us – objects emerging from the quantized fields defined on Minkowski spacetime. This domain e is insufficient for our purposes.

Figure 11. a) Design elements of the Noetic Interferometer postulated to constructively-destructively interfere with the topology of the spacetime manifold to manipulate the UF. The first three tiers set the stage for the critically important 4th tier which by way of an incursive oscillator punches a hole in the fabric of spacetime creating a holophote or lighthouse effect of the UF into the experimental apparatus momentarily missing its usual coupling node into a biological system. b) Conceptualized Witten vertex Riemann sphere cavity-QED multi-level Sagnac effect interferometer designed to ‘penetrate’ space-time to emit the ‘eternity wave,  ‘ of the UF. Experimental access to vacuum structure or for surmounting the uncertainty principle can be done by two similar methods. One is to utilize an atomic resonance hierarchy and the other a spacetime resonance hierarchy. The spheroid is a 2D representation of a HD complex Riemann sphere able to spin-flip from zero to infinity continuously.

For a basic description, following de Broglie’s fusion concept, assume two sets of coordinates

X 

x1  x2 , 2

x1 , y1 , z1

Y

and x2 , y2 , z2 which become

y1  y2 , 2

Z

z1  z 2 . 2

(14)

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Then for identical particles of mass m without distinguishing coordinates, the Schrödinger equation (for the center of mass) is i

 1   , t 2M

M  2m

(15)

Eqation 15 corresponds to the present and Eq. 16a corresponds to the advanced wave and (16b) to the retarded wave [15]. i

 1   , t 2M

i

 1   . t 2M

(16)

Extending Rauscher’s concept for a complex eight space differential line element

dS 2   dZ  dZ  , where the indices run 1 to 4, is the

complex eight-space metric, Z  the complex 8-space variable and where

Z   X Re  iX Im

and Z 

is the complex conjugate [8], to 12D

continuous-state spacetime; we write just the dimensions for simplicity and space constraints

xRe , yRe , zRe , tRe ,  xIm ,  yIm ,  zIm , tIm

(17)

where  signifies Wheeler-Feynman/Cramer type future-past/retardedadvanced dimensions. This dimensionality provides an elementary framework for applying the hierarchical harmonic oscillator parameters suggested in Figure 11 to operate a QC without decoherence.

CONCLUSION – CRITERIA FOR SENTIENCE Sentience is suggested to be synonymous with an entity having subjective experiences also known in Philosophy of Mind as experiencing qualia. Sentience is often considered to be distinct form other aspects of

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mind like intelligence, self-awareness or free agency. The issue of conscious machines remains difficult compounded by the ‘Chinese Room’ analogy suggesting it could also remain a challenge experimentally. The problem cannot be solved philosophically only laid bare to certain probabilities. It is possible to list salient components of awareness. We suggest four: Sentience, Intelligence Self-awareness and Free will. Must a conscious system be considered alive? We have addressed this issue elsewhere in what we have termed System-Zero: The proteinaceous unit called the prion, (responsible for neurodegenerative encephalopathies) a particle ‘below’ the virus. System-Zero propagates from normal to infectious by a conformal change in the protein structure by action of the force of coherence of the UF. Following the assumptions: 1) A physically real noetic ‘life principle’ exists synonymous with the action of the UF, 2) The mind-body interface is a form of naturally occurring ‘conscious quantum computer’ (not that the QC is conscious but modeled after such principles) and 3) Combining the two concepts leads to truly sentient androids when applied to a class of QC systems modeled after the noetic mind-body interface. The noetic QC Android model is empirically testable with experimental protocols summarized. Access to the UF action of the life principle requires surmounting the quantum uncertainty principle. Furthermore, the required universal bulk QC cannot be achieved with 4-space parameters and requires M-Theoretic principles of UFM cast in LSXD [30]. We believe implementing sentient android devices is only this far away!

REFERENCES [1]

Amoroso, R. L. (2010). Complementarity of Mind and Body: Realizing the Dream of Descartes, Einstein, and Eccles, New York: Nova Science Publishers.

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Amoroso, R. L., Kauffman, L. H. & Rolands, P. (2013). The Physics of Reality: Space, Time, Matter, Cosmos, Hackensack: World Scientific. [3] Amoroso, R. L., Kauffman, L. H. & Rolands, P. (2015). Unified Field Mechanics: Natural Science Beyond the Veil of Spacetime, London: World Sci. [4] Amoroso, R. L. & Di Biase, F. (2013). Crossing the psycho-physical bridge: elucidating the objective character of experience, J Consc Explor and Research. [5] Amoroso, R. L. (2013). Empirical protocols for mediating long-range coherence in biological systems, J Consciousness Exploration and Research. [6] Amoroso, R. L. (2010). Simple resonance hierarchy for surmounting quantum uncertainty, In Amoroso, R. L., Rowlands, P., & Jeffers, S. (eds) AIP Conference Proceedings-American Institute of Physics, Vol. 1316, No. 1, p. 185. [7] Amoroso, R. L. (1997). The theoretical foundations for engineering a conscious quantum computer, in M. Gams et al. (eds) Mind Versus Computer: Were Dreyfus and Winograd Right? Amsterdam: IOS Press, 43, 141-155. [8] Amoroso, R. L. & Rauscher, E. A. (2009). The Holographic Anthropic Multiverse: Formalizing the Complex Geometry of Ultimate Reality, Singapore: World Scientific. [9] Hameroff, S. & Powell, J. (2008). The Conscious Connection: A Psycho-physical Bridge between Brain and Pan-experiential Quantum Geometry in D. Skrbina, (ed.), Mind That Abides: Panpsychism in the New Millennium, New York: Benjamins. [10] Amoroso, R. L. (2013). “Shut the front door!”: Obviating the challenge of large-scale extra dimensions and psychophysical bridging, in R. L. Amoroso, L. H. Kauffman, & P. Rolands, P. (eds.) The Physics of Reality: Space, Time, Matter, Cosmos, Hackensack: World Scientific. [11] Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics, Singapore: World Scientific.

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[12] Toffoli, T., Biafore, M. & Leao, J. (eds.) Physcomp96, Cambridge: New England Complex Systems Institute; http://arxiv.org/abs/quantph/9701027. [13] Nagel, T. (1974). What’s it like to be a bat?, Philos Rev., 83, pp. 435-450. [14] Holland, P. R. (1995). The quantum theory of motion: An account of the de Broglie-Bohm causal interpretation of quantum mechanics, Cambridge: Cambridge University Press. [15] Cramer, J. (1986). The Transactional Interpretation of Quantum Mechanics, Rev. Mod. Phys, 58, 647-687. [16] Witten, E. (1996). Reflections on the fate of spacetime, Phys. Today, (April), pp. 24-30. [17] Feynman, R. P. (1971). Lectures on Gravitation, Pasadena: Cal Inst. Tech. [18] Eccles, J. C. (1992). Evolution of consciousness, Proc. Natl. Acad. Sci. USA, Vol. 89, pp. 7320-7324. [19] Dubois, D. M. (2001). Theory of incursive synchronization and application to the anticipation of delayed linear and nonlinear systems, in D.M. Dubois (ed.) Computing Anticipatory Systems: CASYS 2001, 5th Intl Conf., AIP Conf. Proceed., 627, pp. 182-195. [20] Feynman, R. P. (1986). Quantum mechanical computers, Found. Phys., 6, pp. 507-531. [21] Havel, T. F. & Doran, C. J. L. (2004). A Bloch-sphere-type model for two qubits in the geometric algebra of a 6-D Euclidean vector space, arXiv:quant-ph/0403136v1. [22] Makhlin, Y. (2002). Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations, Quantum Inform. Processing, 1, pp. 243–252. [23] Zhang, V. J., Sastry, S. & Whaley, K. B. (2003). Geometric theory of nonlocal two-qubit operations, Phys. Rev. A, 67, p. 042313. [24] Vlasov, A. Y. (1996). Quantum theory of computation and relativistic physics, in T. Toffoli, M. Biafore & J. Leao (eds.) Physcomp96, Cambridge: New England Complex Systems Institute;

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[25] [26] [27]

[28]

[29] [30]

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http://arxiv.org/abs/quant-ph/9701027, and additional material from private communication. Peebles, P. J. E. (1993). Principles of Physical Cosmology, Princeton: Princeton University Press. Peebles, P. J. E. (1992). Quantum Mechanics, Princeton Univ. Press. Francis, G., Kauffman, L. H. & Sandin, D. (1993). Air on the Dirac Strings (video) http://www.evl.uic.edu/hypercomplex/html/ dirac.html. Goertzel, B., Aam, O., Smith, T. F. & Palmer, K. (2007). Mirror Neurons, Mirrorhouses, and the Algebraic Structure of the Self http://www.goertzel.org/dynapsyc/2007/mirrorself.pdf. Amazing Video, Walking of the Moai on Rapa Nui (Easter Island) http://www.youtube.com/watch?v=yvvES47OdmY. Amoroso, R. L. (2017). Universal Quantum Computing - Supervening Decoherence - Surmounting Uncertainty, London: World Scientific.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 8

IMMINENT ADVENT OF UNIVERSAL QUANTUM COMPUTING (UQC) Richard L. Amoroso Noetic Advanced Studies Institute, Beryl, UT, US

ABSTRACT Intensive research continues in the field of quantum information processing (QIP). Some claim Quantum Computers (QC) already exist; but single purpose arrays of quantum logic gates does not signify a Universal Quantum Computer (UQC). Generally, these quantum processors operate at cryogenic temperatures, are room size and cost millions of dollars leaving the full potential of QCs largely theoretical. At the time of writing, Google’s Summit quantum processor performed a calculation in 3 minutes and 20 seconds that a supercomputer would require 10,000 years to perform. QCs are extremely prone to error by tiny changes in temperature, or tiny vibrations, that destroy the subtle state of a qubit. This is called decoherence and is the remaining problem hindering UQC. It has been stated that UQC will require new discovery in physics; in this chapter we introduce a path to that discovery; a methodology able to supervene decoherence by a procedure surmounting the quantum uncertainty 

Corresponding Author’s Email: [email protected].

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Richard L. Amoroso principle. In contrast to current QC research platforms, this model utilizes a version of Einstein’s long sought Unified Field Theory based on modification to string/M-theory, is tabletop, room temperature and will cost pennies.

Keywords: decoherence, quantum computing, uncertainty principle

INTRODUCTION – BITS, QUBITS AND XD SPACE QC theorists state the remaining problem to be solved before bulk UQC occurs is that of decoherence of the quantum state. Numerous improvements, have not been sufficient. Considered the most advanced model, Topological Quantum Computing (TQC) with cryogenic superconducting quantum Hall anyons on 2D graphene bilayers, overcomes decoherence with topologically protected quantum states. However, this is currently an oxymoron, because the topologically protected states are inaccessible. TQC theorists have recently discovered synthetic Additional Dimensions (XD). We believe with a better understanding of topological phase transitions in XD, TQC could be successful, however this room-sized multimillion-dollar cryogenic device would be reminiscent of the city-block sized 1946 Eniac. In contrast, our model, is tabletop and room temperature, discovers and utilizes XD by a simple resonance model able to surmount quantum uncertainty, thereby supervening decoherence. This is possible in a UFM M-theoretic model because brane matter in the bulk has an inherent causally free topological copy of the local 3-space C-QED quantum state in a box. The utility of additional degrees of freedom provided by an XD topological brane-basis of a relativistic (r-qubit) is a key starting point. Realistic r-qubits allow space-antispace vacuum programming to surmount the quantum uncertainty principle, thereby removing the problem of decoherence by inherent mirror symmetry in M-theoretic Calabi-Yau brane topology. The theoretical claim of XD began over a hundred years ago with the work of Kaluza and Klein. For the last 40-or 50-years, renditions of string

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theory have been based on XD, (currently 11D). Synthetic XD are without utility at the moment; CERN clerics seek XD by experimental designs called Gravity’s Rainbow, but it seems unlikely a supercollider of sufficient power can be built for success. Physicists like Randall-Sundrum’s XD model for a D3 brane XD throat coincides with our model of uncertainty as a semiquantum manifold of finite radius as the limit before entry into the brane bulk of 12D UFM topology. We present Sagnac Effect resonance protocols for XD access and show how the paradigm programs and operates bulk UQCs as soon as a class of M-theoretic qubit algorithms exist. “… Trying to find a computer simulation of physics, seems … an excellent program to follow … I’m not happy with all the analyses that go with just the classical theory, because nature isn’t classical, … and if you want to make a simulation of nature, you’d better make it quantum mechanical,” R. Feynman [1]. Thirty-four years have passed. The author takes liberty to update Feynman: I’m not happy with all the analyses that go with just classical and quantum theory, because nature isn’t classical or quantum, … if you want to make a simulation of nature, you’d better make it unified field mechanical, R. L. Amoroso.

A classical Turing bit (short for binary digit) is the smallest unit of digital data and is limited to the two discrete binary states, 0 and 1; but a quantum bit (qubit) can additionally enter an entangled superposition of states, in which the qubit is effectively in both states (and any in between) simultaneously. While a classical register made up of n binary bits can contain only one of 2n possible numbers, the corresponding quantum register can contain all 2n numbers simultaneously. Thus, in theory, a QC could operate on seemingly infinite values simultaneously in parallel, so that a 30qubit QC would be comparable to a digital computer capable of performing 1013 (trillion) floating-point operations per second (TFLOPS) which is comparable to currently fastest supercomputers will 100s of trillions of bits.

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Figure 1. Geometrical qubit representations. a) The qubit resides on a complex unit circle in a Hilbert space of all possible orientations of

qi

called the Hilbert space

representation. In the logical basis, the two degrees of freedom of a qubit are expressed as two angles geometrically interpreted as Euler angles. b) The Bloch sphere in spin space is a geometric qubit representation where    1   0 for orthogonal eigenstates 1 and 0 of a single qubit on opposite poles, with superpositions located on the sphere’s surface. Adapted from [2].

The qubit, a geometrical representation of the pure state space of a 2level quantum mechanical system, is described in Dirac’s ‘bra-ket notation’ by the state  0   1 where  and  are complex numbers satisfying the absolute value parameter result in state

  2

2

 1;

such that measurement would

0 with probability  and 1 with probability  . 2

2

Formally, a qubit is represented in the 2D complex vector space, 2 where the  0   1 can be represented in the standard orthonormal basis as

1  0 0    for the ground state or 1    for the excited state, or on the 0  1  Bloch sphere as in Figure 1b. A qubit is shown in Figure 1 in both its SU(2) Hilbert space representation (L), and the same qubit on the Bloch sphere in its O(3) representation (R). The SU(2) and O(3) representations are homomorphic, i.e., mapping preserves form between the two structures. Vincenzo itemized what he felt were the major requirements for implementing practical bulk QC [3]:

Imminent Advent of Universal Quantum Computing (UQC)     

273

Physically scalable - qubits sufficiently increased for bulk implementation. Qubits must be able to be initialized to arbitrary values. Quantum gates that operate faster than the decoherence time. A universal gate set for running quantum algorithms. Qubits that can be easily read correctly.

None of Vincenzo’s requirements are yet fulfilled; some are further along than others; system decoherence is among the most challenging aspects remaining. Recently, the fundamental basis of quantum information systems is undergoing an evolution in terms of the nature of reality with radical changes in the nature of the measurement problem. The recent introduction of relativistic information processing (RIP), with relativistic rqubits, brings into question the historical sacrosanct basis of locality and unitarity in terms of Bell’s inequalities, overcoming the no-cloning theorem [4, 5]. Now an end to limits of quantum mechanics looms.

OVERVIEW OF QC ARCHITECTURE The following list represents many prominent QC architectures and substrates currently under development. For a brief review of the challenges and merits of each system as distinguished by the computing model and physical substrates used to implement qubits see [7].        

Quantum Turing Machine Quantum Circuit Quantum Computing Model Measurement Based Quantum Computing Adiabatic Quantum Computing Kane Nuclear Spin Quantum Computing QRAM Models of Quantum Computation Electrons-On-Helium Quantum Computers Fullerene-Based ESR Quantum Computer

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Richard L. Amoroso             

Superconductor-Based Quantum Computers Diamond-Based Quantum Computer Quantum Dot Quantum Computing Transistor-Based Quantum Computer Molecular Magnet Quantum Computer Bose–Einstein Condensate-Based Quantum Computer Rare-Earth-Metal-Ion-Doped Inorganic Crystal Quantum Computers Linear Optical Quantum Computer Optical Lattice Based Quantum Computing Cavity Quantum Electrodynamics (CQED) Quantum Computing Nuclear Magnetic Resonance (NMR) Quantum Computing Topological Quantum Computing Unified Field Mechanical Quantum Computing

We touch on the bottom three bullets. For detail on the others see [7]. Nuclear Magnetic Resonance Quantum Computing (NMRQC) is among the 1st and most mature technologies for implementing quantum computation. It utilizes the motion of spins of nuclei in a variety of molecules such as the hydrogen and the carbon nuclei of chloroform, manipulated by rf-pulses. The spin-lattice (T1) and spin-spin (T2) relaxation processes in NMR are key factors in the ability to implement NMRQC quantum algorithms. NMRQC has taken two forms:  

Liquid-state NMRQC on molecules in solution with the qubit provided by nuclear spins within the dissolved molecule [7]. Solid-state NMR Kane quantum computers with qubits realized by the nuclear spin state of phosphorus donors in silicon [8].

NMR differs from other implementations of QC in that it uses an ensemble of systems, in this case molecules. The ensemble is initialized to be the thermal equilibrium state as given by the density matrix:

Imminent Advent of Universal Quantum Computing (UQC)



e  H , Tr  e  H 

275 (1)

where H is the Hamiltonian matrix for a single molecule with   1/ kT , k Boltzman’s constant and T temperature. Ensemble operations are performed by rf-pulses applied orthogonally to a strong, static field, by a large NMR magnet [7, 8]. It is very difficult to prepare NMRQC systems in pure spin states because of the tiny energy gap between nuclear spin states. This seriously challenges the scalability of NMRQC because the procedure for preparing the required pseudo-pure states averages all the populations but one. As long as the spin system can be described by the high temperature approximation, the population of an individual spin state is inversely proportional to the number of states. But this scenario decreases as 2-N with an increase in the number of spins, N. Detectable signal size thus limits the possible number of spins used in NMR quantum processors. The reduction of sensitivity associated with the preparation of pseudo-pure states can be avoided by using algorithms that do not require pure states to work with [7].

Figure 2. Signal amplitude loss due to preparation of pure states as a function of size of the quantum register, causing NMRQC to be difficult beyond a few qubits.

It has also been shown that liquid state ensemble NMRQC do not possess quantum entanglement as required for quantum information processing; thus, it appears NMRQC are only classical simulations of a QC.

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Quantum ensembles represent possible states of a mechanical system of particles that are maintained thermodynamically with a reservoir. The system is open in the sense that it can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system [7]. For solid-state NMRQC the brief review of Kane Nuclear Spin QC where: 1) The state has an extremely long decoherence time, on the order of 1018 seconds at mK temperatures. 2) Qubits can be manipulated by applying an oscillating NMR field [7]. Topological Quantum Computing (TQC) is based on braiding anyons in a 2D lattice at cryogenic temperatures near absolute zero. TQC is among the most promising considerations for Bulk UQC; the scenario Microsoft bet on [7]. Anyons are 2D quasiparticles neither Bosons or Fermions operating by Fractional quantum Hall effect. Common substrates are doped GaAs, Pb or Si, InSb and InAs semiconducting nanowires some of which support Majorana Zero Modes (MZM). Non-Abelian anyons are the key requirement for the anyonic model of TQC, but their existence has not yet been experimentally confirmed. But recent experimental work following theoretical predictions, has shown signatures consistent with the existence of Majorana modes localized at the ends of semiconductor nanowires in the presence of a superconducting proximity effect [7]. The topological braiding of these anyonic non-Abelian fractional quantum Hall effect quasiparticle Majorana fermions provides a high degree of error protection from decoherence by interaction with the environment (the braid state has remained experimentally inaccessible). The actual TQC is done by the edge states of a fractional quantum Hall effect. When anyons are braided the quantum information which is stored in the state of the system is impervious to small errors in the trajectories. Braiding acts as a

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matrix on a degenerate space of states. The relevant quasiparticle in the Moore-Read state is a ‘Majorana fermion’ which is its own antiparticle, ‘half’ of a normal fermion. The effect of the exchange on the ground state need not square to 1. ‘Anyon’ statistics: the effect of an exchange is neither +1 (bosons) or -1 (fermions), but a phase [7]. Freedman, Kitaev, Larsen, & Wang (FKLW) found that a conventional QC device, with an error-free operation of its logic circuits, gives a solution with an absolute level of accuracy, whereas a FKLW device with flawless operation will give the solution with only finite accuracy; but any level of precision for a readout can be obtained by adding more anyon braid twists (logic circuits) to the TQC, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer [7]. Note that this solution can be considered the same as applying the Quantum Zeno Effect (QZE) to Interaction Free Measurement (IFM), discussed in detail in [7]. Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero [9]. TQC provides a possible new architecture for QC with a low error rate by exploiting anyon braiding in the topology of quasiparticles. Anyons have different statistics than Bosons (Bose-Einstein spin 1 statistics) and Fermions (Fermi-Dirac spin 1/2 statistics). Semiconductor devices are expected to host these exotic quasiparticle states, predicting that TQC will have properties sufficient for error-free quantum computation. A more detailed analysis of TQC is given in Chaps. 9 and 10. TQC is considered a ‘toy model’ for the introduction of the Unified Field Mechanical (UFM) Ontological-Phase Topological Field Theoretic QC presented as the main purpose of this volume. See [7].

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Figure 3. Topological quantum computing schema of quasiparticle exchange. (a) Basic operations,  1 and

 2 on a system of three quasiparticles. Top: Temporal evolution of

the system from the initial state  i to the final state  1(2)  1(2) i . Bottom: diagrammatic representations of the quasiparticle exchange operations. (b) Example of logic gate operations for the basic operations

 1 and  2 shown in (a) and their

inverses 11 and  2 1 . Redrawn from [9].

Unified Field Mechanical Quantum Computing (UFMQC) is probably the newest QC model; although under theoretical development for over a decade, understanding its formulation only began to gel while writing this volume. Its Group Theory is not fully known yet; and its basis has been given the provisional name: Ontological-Phase Topological Field Theory or in terms of quantum information processing: Ontological-Phase Topological Quantum Computing (OPTQC), which we will do our best to make a case for [7]. What this currently speculative model has to offer is pointed out acutely in the subtitle of this volume ‘Surmounting Uncertainty – Supervening Decoherence.’ Those UFM scenarios, if correct totally remove conditions plaguing virtually all the other QC models outlined in this chapter. Its most redeeming factor is that it is experimentally testable; and preparations are underway to do such [10, 11].

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Computation whether classical or quantum is Boolean, utilizing a symbolic system of algebraic notation for binary variables that are used to represent logical propositions or logical operators having two possible values denoted as ‘true’ and ‘false’, 1 and 0 or



or



[7]. Information is physical and cannot

exist without a physical representation. The question is how to move from current more symbolic representations to methods of representing algorithms in a manner connoting physical reality to the extent now required by UFM? The supposition is that a purely mathematical space such as the multidimensional Hilbert space currently in use can no longer be considered adequate for implementing universal quantum information processing (QIP). It is a fairly recent idea to worry about the fact that information is physical in this respect [7,12], and that while mere binary representations have been adequate for Turing machines for the last 70 years, and even all the QIP done to date at the semi-classical limit; the scenario is not sufficient for QC at a UQC level, especially as we pass beyond the historical basis of Unitarity and Locality to the requirements for relativistic QIP effects and further to incorporate the necessary phenomena imposed by UFM and the associated OPTFT. Moore’s Law has approached unity as we speak; and as everyone knows computing at the quantum level is plagued by a lack of control of quantum degrees of freedom by interaction with the environment and vacuum fluctuations. The wavefunction, y = a 0 + b 1 complicates the concept of reality for Euclidean observers. In order to determine the state of a physical object we have to interact with it; don’t we? Two quantum systems as represented by the wavefunction above are entangled by a standard unitary operation,

0

U ent y

0 =a 0 0 +b 1 1 , y

can

be unknown and a known state, that can be extended to an N-qubit product state which can be operated on simultaneously by U:

280

Richard L. Amoroso Uent 0 0 0 0 ... 0 0 ~ 00000...00  00000...01  00000...10  N times

...  1111...00  1111...01  1111...10  1111...11  2 N terms  .

(2)

We have been at this point for a long time for all QC systems under study; all plagued by decoherence with severity increasing with the number of qubits. Error correcting techniques have been proposed for arbitrary size qubit registers [7]. Any quantum system such as electron spin whose state space can be described by a 2D complex vector space can be used to implement a qubit. By current thinking, ‘The QC must operate in a Hilbert space whose dimensions may be grown exponentially as an infinite-dimensional analog of Euclidean space such that an abstract vector space possessing the structure of an inner product allows length and angle to be measured’. This scenario has been good in principle until now, and probably retains utility in some QC mathematical and algorithm preparation processes; but since the Bloch 2-sphere representation is not physical, it can no longer be considered a sufficient description for practical implementations of UQC. Few physicists consider Large-Scale Additional Dimensionality (LSXD), but sufficiently so, that experiments at CERN are being developed to search for them. Our UFM protocol to find them is table top and low energy, which if successful will put an end to the need for supercolliders. We are formulating an Ontological-Phase Topological Field Theory (OPTFT) to address the putative parameters. Our view of a UFM fortunately makes easy correspondence to HD extensions of the Wheeler-FeynmanAbsorber Cramer-Transactional De Broglie-Bohm-Vigier causal interpretations of quantum theory as well as dual 3-tori Calabi-Yau brane mirror symmetry (thus OPTFT). Even though I’m riding a wild stallion, it is a very radical paradigm shift that blows even the author’s mind. The most difficult part for colleagues to embrace/comprehend is the ‘continuous-state’ evolution of the HD brane topology; along with the fact that ‘the Earth is not the center of the Universe,’ flagrantly meaning that we, as physical observers, must give up observation from the perspective of Euclidean 3-

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space as the primary vantage. It’s always like this in a paradigm shift; get over it, leap-frog over and beyond me and enter the ‘brave new world.’ The late Karl H. Pribram, noted Stanford neuroscientist (holographic brain model), once asked me on a beach approaching sunset, in Long Beach, CA USA “Aren’t we all in this together?”, while we were pondering the reflection of the sun on the water, arguing about how many images there were.

QUANTUM LOGIC GATES AND PROPERTIES A logic gate is the elementary building block of a computing circuit or algorithm practically applying the concept of binary Boolean bits to circuits using combinational logic. Logic gates are of recent origin. From the time that Leibniz refined binary numbers and showed that mathematics and logic could be combined in 1705, it took well over a hundred years before Babbage devised geared mechanical logic gates in 1837 for use in his proposed Analytical Engine. Another sixty years passed before the first electronic relays appeared in the late 1890s. Then it wasn’t until the 1940s that the first working computer was built. Now, with the arrival of quantum logic gates the evolution continues; and universal quantum computers (UQC) wait in the wings while finishing the absorption of required remaining discovery in physics. Linear algebra concerns vector spaces and linear mappings between such spaces. The 3D Euclidean space

3

is a vector space, where lines and

planes passing through the origin are vector subspaces in

3

. The most

n

important space in basic linear algebra is , a Euclidean space in n dimensions where a typical element is an n-element vector of real numbers. The space of infinite-dimensional vectors defines the Hilbert space,

2

( ).

Such a Hilbert space, H is a vector space endowed with an inner or dot product,

x and associated norm and metric, x  y such that every

Cauchy sequence (converges to a limit) in

n

making

n

a Hilbert space.

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Complex Hilbert spaces are used to represent the pure states of a quantum mechanical system utilizing unit vectors, called state vectors. What is the difference between classical and quantum information? In the matrices below, the classical bit is described by two nonnegative real numbers for probabilities P(0) = 1/3 and P(1) = 2/3. In contrast the quantum bit has two complex amplitudes giving the same probabilities by taking the square of the absolute value.  1 / 3  1 / 3  Classical bit:  Qubit     2 / 3 1  i / 3 

(3)

A quantum system described like this with nonzero amplitudes is said to be in a superposition of the 0 and 1 configurations. The basis of all computing is the logic gate [7]. A quantum logic gate is most often represented by a matrix. For example, a gate acting on n qubits forms a 2  2 unitary matrix. The number of qubits input and qubits output from any gate must be equal. The operation of the gate is determined by multiplying the gate’s unitary matrix by the vector representation of the quantum state. For example, the vector representation of a solitary qubit and of two qubits is represented respectively as: n

n

v  v0 0  v1 1   0  , v00 00  v01  v1 

For the 2-qubit state

v00  v  01  v10 10  v11 11   01  .  v10     v11 

(4)

ab , a represents the value of the 1st qubit and b

represents the 2nd. A single qubit wave function takes the form

   0 0  1 1 such that  0   1  1 ; with this as the case 2

2

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observation gives a result of either a 0 or a 1 with a probability for 0 as  0

2

and the probability for observing a 1 is  1 . 2

The state of a quantum computer is described by a state vector, Ψ which is a complex linear superposition of all binary states of the qubits xn {0,1}:  (t ) 



 x x1 ,..., xn ,

xn {0,1}n



2 x

 1.

(5)

x

The state’s evolution in time, t is described by a unitary operator, U on the same vector space, meaning any linear transformation is bijective and length-preserving. This unitary evolution on a normalized state vector is known as a correct physical description of an isolated system evolving in time according to the laws of quantum mechanics [13, 14]. Quantum physics is reversible because reverse-time evolution specified by the unitary operator, U 1  U † always exists; as a consequence, reversible computation could be executed within a quantum-mechanical system. Quantum physics postulates that quantum evolution is unitary (reversible); i.e., if we have an arbitrary quantum system, U taking an input state,

 that outputs a different state, U  , then we describe U as a

unitary linear transformation, defined as follows. If U is any linear transformation, the adjoint (functions related by transposition) of U, denoted





† U † , is defined in the relation Uv , w   v ,U w . In a basis, U † is the

conjugate transposition of U; as,

a b  a † U   U   c d  b

c . d

(6)

By definition U is unitary if U †  U 1 . Thus, rotations and reflections are unitary. Also, the composition of two unitary transformations is also

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unitary, for a unitary transformation U, the rows and columns also form an orthonormal basis [15]. Evolution of the state,



of a quantum system in time, t is a unitary

transform,   Uˆ  . Temporal evolution of a quantum system is linear because it does not depend on the state, combination in t of state,

 

 



and

  Uˆ  



. For example, any linear

 has the same operator

 

  Uˆ 

 Uˆ  .

(7)

Unitary operators conform to the Schrödinger equation

d dt

 i

Hˆ  t  

(8)

† with Hˆ  t   Hˆ  t  the system’s Hamiltonian.

The

general

bXˆ  cYˆ  dZˆ 

Hamiltonian

E0 n  ˆ

with

for

a

spin-1/2

system

is

Hˆ 

E0  b2  c 2  d 2 , n   nx , ny , nz  

b / E0 , c / E0 , d / E0  , nx2  ny2  nz2  1





and ˆ  Xˆ , Yˆ , Zˆ . Thus the

unitary is ˆ   iHt  E0t  ˆ  E0t  ˆ exp     cos   I  i sin   n  .      

(9)

This is a rotation about the n axis in the Bloch sphere representation with a rotation rate of transform [15].

E0 /

. For spin-1/2 this is the most general unitary

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Assuming we can turn the Hamiltonian off and on, the state can be rotated by a specific angle; again for spin-1/2 a unitary transform takes the form, Uˆ    cos  / 2  Iˆ  i sin  / 2  n  ˆ . It is important to realize that any product of unitary operators is also unitary [7] ˆ ˆ   Vˆ Uˆ UV ˆ ˆ  Iˆ   UV

ˆˆ Uˆ †Uˆ  Vˆ †Vˆ  Iˆ  UV

†

†

†

(10)

The Copenhagen Interpretation’s restriction of time evolution to unitary operators suggests that certain kinds of evolution are deemed impossible. Two such operations are the quantum no-cloning and non-erasure theorems. We show that this is a condition of the 4D standard model Copenhagen Interpretation up to the ‘semi-quantum limit’ and is no longer the case for UFM topology [7]. Unitary transformations, or quantum gates can be built from sets of unitaries, Uˆ . The simplest spin-1/2 quantum system, the qubit, has two quantum states with the basis 0   Z , 1   Z . Some examples of simple single and 2-qubit unitary transforms, or ‘quantum gates’ are: The Hadamard Gate; The Phase Shift Gate; The Swap Gate; The Pauli X (NOT) Gate; The Pauli Y Gate; The Pauli Z Gate; The CNOT (Controlled NOT) Gate; The Toffoli Gate; The Fredkin Gate; The Controlled U Gate; and The Rotation Gate.

EVADING UNCERTAINTY AND DECOHERENCE Eliminating ensemble decoherence time and uncertainty in the operation and measurement process of Quantum Information Processing (QIP) systems are remaining problems considered to be of paramount importance in the task of implementing viable bulk scalable Universal Quantum Computing (UQC). Most teams currently attempt to supervene decoherence by utilizing multimillion-dollar room sized cryogenic apparatus. If our model is correct, it will allow tabletop room temperature UQC. We

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theoretically illustrate (in a manner empirically testable) that these conditions essentially become irrelevant in terms of the radical new Unified Field Mechanical (UFM) approach to QIP introduced here. It should be noted that the recent relativistic restrictions the QC research community has imposed on QIP point the way to our model. The additional degrees of freedom obtained by leaving the 3D realm of Euclidean space associated with Newtonian Classical Mechanics and entering the 4D domain of Minkowski 4-space had a profound effect on physics during the last century. Now as we enter a 12D M-Theoretic (String Theory) dual Calabi-Yau mirror symmetric 3-torus 3rd regime associated with UFM, more surprises like the ability to surmount the quantum uncertainty principle are proffered. We review a UFM protocol for allowing uncertainty and decoherence to be routinely surmounted and supervened respectively, 100% of the time with probability, P  1 . We begin with a discussion of Interaction-Free Measurement (IFM), an interesting 4D precursor providing another indicium of the 12D brane topology model introduced here. IFM is a novel quantum mechanical procedure for detecting the state of an object without an interaction occurring with the measuring device. What we propose is a radical extension of the various experimental protocols spawned by the recent Elitzur-Vaidman IFM thought experiment. The highly speculative, at time of writing, UFM alternative to IFM protocols, is a single pass ontological method for surmounting uncertainty, without (phenomenological) quantal field interaction or collapse of the wave function. Surmounting the Quantum Uncertainty Principle with probability, P  1 is achieved through utility of the additional degrees of freedom inherent in a new cyclic interpretation of the Calabi-Yau mirror symmetric SUSY regime of string/brane theory. Just as the UV catastrophe provided a clue for the immanent transition from Classical to Quantum Mechanics, duality in the Turing Paradox (quantum Zeno Effect where an unstable particle observed continuously will never decay), suggests another imminent new horizon in our understanding of reality. IFM as mentioned provides an intermediate indicator of this developing scenario. The quantum Zeno paradox experimentally implemented in IFM

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protocols hints at the duality between the regular phenomenological quantum theory and a completed unified or ontological model beyond the usual 4D Gauge formalism of the standard Copenhagen interpretation. Utilizing extended theoretical elements associated with a new formulation for the topological transformation of a ‘cosmological least unit’ (LCU), a putative empirical protocol for producing IFM with probability, P  1 is introduced in a manner representing a direct causal violation or absolute surmounting of the putatively inviolate quantum Uncertainty Principle imposed by 4D Copenhagen restrictions. The concept of quantum non-demolition (QND) [7, 16] arose as a process for performing very sensitive measurements without disturbing an extremely weak signal. But there was a trade-off between the accuracy of a QND measurement and its inevitable back-action on the conjugate observable to that being measured. By definition an interaction (phenomenological) is any action, generally a force, mediated by an exchange particle for a field such as the photon in electromagnetic field interactions. This physical concept of a fundamental interaction regards phenomenological properties of matter (Fermions) mediated by the exchange of an energy/momentum field (Bosons) as described by the Galilean, or Lorentz-Poincairé groups of transformations. “There has been some controversy and misunderstanding of the IFM system concerning what is meant by ‘interaction’ in the context of ‘interaction-free’ measurements. In particular, we stress that there must be a coupling (interaction) term in any Hamiltonian description.” [17]

This is the distinction we are talking about. The Hamiltonian, H is generally used to express a systems energy in terms of momentum and position coordinates based on forces. While it might bring abject clarity to differentiate the differences between our model and the usual framework of Hamiltonian Mechanics; to do so is beyond the scope of this volume and will be addressed in detail elsewhere. Here we wish to introduce a new ontological type of homeomorphic transformation without the phenomenology of an exchange particle mediated by an ontological

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interactionless or ‘energyless’ topological switching process based on the concept of ‘topological charge’ in M-Theoretic brane configurations [18]. As indelibly ingrained in the current mindset; it is impossible by definition to violate the uncertainty principle, xpx  / 2 or Et x  / 2 within the framework of Copenhagen phenomenology arising from operation of a ‘Heisenberg Microscope’. This is a fundamental empirical fact demonstrated by the Stern-Gerlach experiment where space quantization is produced arbitrarily along the z axis by continuous application of a non-uniform magnetic field to an atomic spin structure [19], or as demonstrated by Young’s double-slit experiment for example. Recent work stemming from the Elitzur-Vaidman bomb-test thought experiment [7] has begun to change the interpretation of this ‘immutable law’! The ElitzurVaidman bomb-test experiment was first demonstrated experimentally in 1994 using a Mach-Zehnder interferometer; and soon led to two main procedures for improving probability outcomes: 1) Multiple recycled Measurements and 2) Multiple array of Interferometers. A Mach-Zehnder interferometer works by using pairs of correlated photons made by spontaneous parametric down-conversion from a molecular crystal such as LiIO3. Initial experiments for a 50-50 beam splitter with a 1-time measurement, the IFM probability was 25% according to the formula in Eq. (11); but for repeated measurements and/or various forms of multiple interferometers it was found IFM probability could be arbitrarily increased toward unity [7].



P( Det 2) P( Det 2)  P( Bomb)

(11)

The probability for the IFM model was suggested to occur in powers of

 / 2N by

PIFM  [1  1/ 2( / 2 N ) 2  ...]2 N where N is the number of beam

splitters in the Max-Zehnder interferometer. In his seminal paper (A thought

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experiment) Elitzur suggested a maximum IFM of 50%. Thereafter Kwiat’s team developed a method to improve the model to 80% with

PQSD  1  ( 2 / 4 N )  O(1/ N 2 ) where in this case N is the number of photon cycles through the apparatus. In regards to the Elitzur and Vaidmann consideration that their model could be explained by the ‘Many-Worlds’ interpretation Cramer proposed, “They suggest that the information indicating the presence of the opaque object can be considered to come from an interaction that occurs in a separate Everett-Wheeler universe and to be transferred to our universe through the absence of interference.” [7]

In terms of creativity in the history of scientific progress, it is interesting to note that Cramer’s suggestion, ‘the idea of a Many-Worlds interpretation to explain how IFM works,’ is an LD shadow the new HD UFM model! In the UFM model of LSXD Calabi-Yau mirror symmetry the supposition is that the 4D Cavity-QED ‘particle in a box’ state has conformal scaleinvariant Supersymmetric (SUSY) ‘mirror copies’ inherent in the HD Calabi-Yau brane topology [7]. Thus if the experiment proposed here is successful it will demonstrate that the IFM model is not suggestive of a reality with ‘many parallel worlds’ but provides instead indicia of CalabiYau mirror symmetric topological ‘copies’ extending ‘our’ reality beyond the veil of stochastic spacetime to a 3rd UFM regime with LSXD; and that these extra degrees of freedom, when properly accessed, allow the uncertainty principle to be surmounted in one pass with probability, P  1 . We delinate a putative protocol, not for another sophisticated improvement of the stepwise degrees of violating uncertainty by the several IFM protocols; but for completely surmounting the uncertainty relation directly, for every singular resonant action, with probability, P  1 . In an unexpected way our model has similarities to IFM/QSD, but instead extends quantum theory with new UFM theory fully completing the task of uncertainty violation. The HD regime of the unified protocol is like a complete IFM fun house ‘hall of mirrors’ where the whole battery of

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interferometers and multiple cycling routines is inherent in the HD mirror symmetric brane regime, so only one ‘ontological measurement’ is required to obtain P  1 . We emphasize that the methodology of this new empirical protocol is completely ontological (rather than usual phenomenological field couplings mediated by energy exchange quanta) with action in the HD SUSY regime in causal violation of the 4D Copenhagen phenomenology, not in an Everett ‘many-worlds’ sense, but in a manner that extends to completion the de Broglie-Bohm-Vigier causal interpretation of quantum theory with a so-called ‘super-quantum potential’, the ontological ‘force of coherence’ of UFM (not a 5th force). The ontological basis is realized utilizing the additional degrees of freedom of a 12D version of M-Theory along with the key supposition of conformal scale-invariance pertaining to the state of quantum informational SUSY brane mirror symmetric copies extended to Large-Scale Additional Dimensions (LSXD) [7]. Considerations of the vacuum bulk are paramount for string theory, much of its putative essential parameters used here are ignored in the avid exploration of other parameters. The P  1 model also relies heavily on the existence of a Dirac covariant polarized vacuum. Of primary concern at this point of our development is the Dirac vacuum inclusion of extended electromagnetic theory which is a key element in manipulating the structural-phenomenology of LSXD SUSY brane topology with a spinexchange resonant hierarchy [7]. The experimental design relies on a new fundamental action principle inherent in the LSXD cyclical brane topology putatively driving selforganization in spacetime as a complex system of cellular automata-like Least Cosmological Units (LCU) tessellating space. Stated more directly, space, spacetime (no longer considered fundamental but emergent) and the HD mirror symmetric Calabi-Yau brane structure is an evolutionary form of self-organized complex system. The new action principle is suggested not to be a 5th force of nature per se, but a combination of the four known forces as united in the unified field (not quantized). Initially this can appear confusing because the three known forces are phenomenological in action, i.e., mediated by the Hamiltonian for

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phenomen-ological energetic field exchange quanta, whereas the topological field is mediated by an ‘ontological charge’ of the unified field. Which is energyless by definition, albeit acting as a ‘force of coherence’ in conjunction with the driving of LSXD brane conformation dynamics. Continuous evolution of the ontology is a form of ‘becoming’ or merging of one informational aspect with another without the exchange of energy, as in the usual sense of a physical field. This key UFM aspect is difficult at first, because it is also a challenge for us to explain [7]. Topologically, HD Calabi-Yau mirror symmetric copies,  4 are in constant motion [7]. This inherent synchronization backbone (nameed by Feynman) is essential to providing a resonance hierarchy ‘beat frequency’ for surmounting uncertainty, and of paramount importance to QIP for bulk UQC. The field concept is a supporting paradigm of the entire edifice of modern physics; until now specifically for phenomenological field dynamics only. Be reminded that physically, physicists have no idea what a field is, we are only able to associate it with a metric and parametrize various phenomena. Our view of what constitutes an ontological field is radically different. We do not feel equipped to definitively define the distinction rigorously in this volume (as the whole nascent edifice of UFM has yet to even reach infancy); but realize we cannot get away with saying nothing either. We want to let experiment drive theory at this moment in development. The ontological properties of the dynamics inherent in the HD unified field theoretic topological brane world do not transfer energy, and the ‘exchange’ of information also does not occur in time; further hinting at bringing into question the historically fundamental basis of ‘locality and unitarity.’ The best metaphor we know for energyless ontological charge is the switching of central vertices of the ambiguous Necker cube when stared at [7]. There is no event relative to the perceived switching of the vertices of the metric of the cube in 3D (rather suggested to occur in 4D extensions like the spherical rotation of the Dirac electron requiring 720o to complete). For a cellular automata-like close-packed 12D dual space tessellation of an array of such hyperspherical objects, (the 3D nilpotent resultant designated as

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quantum particle/states in a box) we propose that quantum entanglement occurs in a conformal scale-invariant LSXD brane topology with inherent cyclic mirror symmetric copies of the usually considered stationary 3D Cavity-QED quantum ‘particle in a box.’ This provides sufficient degrees of freedom for allowing quantum uncertainty to be surmounted, thus avoiding problems associated with decoherence times in QC. In this scenario quantum mechanical uncertainty is a manifold of finite radius separating two regimes of infinite size dimensionality - the 3(4)D Euclidean/Minkowski and a complexified mirror symmetric 8D LSXD M-Theoretic brane world [7]. The de Broglie-Bohm-Vigier Causal and Cramer Transactional Interpretations are generally ignored by the physics community; most saliently considered to add nothing new or are incomplete interpretations. The Quantum Potential-Pilot Wave model is extended to a form of ‘Super Quantum Potential’ synonymous with a putative action of the Unified Field; the future-past parameters of Cramer’s model enhance the hierarchy of Calabi-Yau mirror symmetry annihilation-creation parameters. The two theories together form key pillars for an ontological basis of the predicted ‘Force of Coherence’ of the Unified Field which is a mandatory requirement in the new model for developing UQC [7]. We hope to show the protocol relies on symmetry conditions of new self-organized cosmological parameters amenable to a resonant hierarchy of coherently controlled topological interactions able to undergo what Toffoli calls ‘topological switching’ an energyless basis for Micromagnetics of information exchange. Finally, to complete the concatenation of concepts we utilize theoretical modeling in conjunction with the parameters associated with a covariant polarized Dirac vacuum as described from the context of extended electromagnetic theory (more heresy). In other Chaps. We show how this model relates to an M-Theoretic dual form of Calabi-Yau mirror symmetry; the conceptual mantra of which is: Continuous-state, spinexchange, dimensional reduction, compactification process. Not a unique 4D compactification to the standard model as sought by string theorists, but a continuous cyclic dimensional reduction 12D ~0D symmetry exchange through pertinent aspects all five M-Theories.

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By using the quantum Zeno effect, also known as the Turing paradox, the efficiency of an IFM can be made arbitrarily close to unity [7]. “It is easy to show using standard theory that if a system starts in an eigenstate of some observable, and measurements are made of that observable N times a second, then, even if the state is not a stationary one, the probability that the system will be in the same state after, say, one second, tends to one as N tends to infinity … continual observations will prevent motion …” – A. Turing [20]

The Turing Paradox also called the Quantum Zeno Effect is a scenario where a particle observed continuously will never decohere; in a sense the evolution of the system is frozen by frequent measurement in its initial state. More technically the Quantum Zeno Effect can suppress unitary time evolution not only by constant measurement, but applying a series of sufficiently strong fast pulses with appropriate symmetry can also decouple a system from its decohering environment or other stochastic fields [7]. Cramer has suggested that IFM can be interpreted by utilizing the Everett ‘Many Worlds Hypothesis’ to explain the subtleties of the quantum Zeno paradox. While Cramer’s hypothesis is certainly logical, we believe nature in higher dimensions (HD) is more surprising. The Standard Model of Quantum Mechanics predicts that physical reality is influenced by events that can potentially happen (Heisenberg potentia) but factually do not occur. Peise suggests that IFM exploits this counterintuitive influence to detect the presence of an object without requiring any interaction with it. “Here we realize an IFM concept based on an unstable many-particle system. In our experiments, we employ an ultracold gas in an unstable spin configuration, which can undergo a rapid decay. The object (realized by a laser beam) prevents this decay because of the indirect quantum Zeno effect and thus, its presence can be detected without interacting with a single atom. Contrary to existing proposals, our IFM does not require single-particle sources and is only weakly affected by losses and decoherence. We demonstrate confidence levels of 90%, well beyond previous optical experiments.” [21].

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Our UFM model is radically different [7, 10, 11]. There is in a sense no interaction, but not in the sense Paise suggests. His claim is based on the usual ‘quantal or phenomenological’ form of interaction. But as we shall see in later chapters, there is another UFM type of ‘energyless’ ontological interaction or exchange of information based on ‘topological charge’ in HD brane topology described by a new 3rd regime theory we call ‘OntologicalPhase Topological Field Theory’(OPTFT) [7]. This model arises in answer to recent forays into relativistic information processing calling for an end to the historically fundamental utility of ‘locality and unitarity’ as the basis for describing the nature of reality. The measurement problem is not yet solved. Finally, after making further correspondence to current thinking in terms of the dual amplituhedron we delve into the ontological topology of UFM requiring a new set of topological transformations beyond the Galilean, Lorentz-Poincairé. We hope to take a bold step at least philosophically correct into the new UFM arena.

P  1 EXPERIMENTAL DESIGN Comprehending the P  1 model from the perspective of cosmology is only necessary for more fully understanding the context from which developing the experimental protocol arises; otherwise the reader may skip to the next section, especially since no one seems to understands it very well yet anyway. When physicists last embraced a 3D Newtonian world view about a hundred years ago, the universe was believed to be a predictable mechanical clockwork. Since the advent of Quantum Theory (QT) reality has been considered to be stochastic and statistical or uncertain with a Planck scale basement. Following this line of reasoning when a Theory of Everything (TOE) is realistically discovered based on formalizing a unified field, should some form of fundamental monism be embraced? Although the fermionic point-particle is considered the basic unit of physics, this concept is embedded in the global context of cosmology. We postulate that additional cosmology is required to understand the basis for bulk Universal

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Quantum Computing (UQC) because cosmology ultimately speaks to the nature of reality and the ultimate basis for the Fermionic singularity or point particle; and we are finding out that using a nonphysical mathematical calculation space is not sufficient for UQC implementation. The three regimes stated above (classical, quantum and unified field TOE) are currently thought to have a Planck-scale ‘basement of reality’.It remains impossible to surmount uncertainty in this context; it is perceived as an inadequate view requiring a reality with an open LSXD ‘continuous-state’ process instead of an impenetrable basement barrier. Not seeing XD because they are curled up at the Planck scale is not the only interpretation. If the continuous-state process includes a form of ‘subtractive interferometry’, like discrete frames of film passing through a movie projector appearing continuous on the screen, additional dimensionality can be large scale. Experiments under development at CERN are trying to make this discovery, our proposal however, is tabletop and low energy [7]. In a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. if the particles are classical ‘spinning’ particles then the distribution of their spin angular momentum vectors is taken to be truly random and each particle would be deflected up or down by a different amount producing an even distribution on the screen of a detector. Instead quantum mechanically, the particles passing through the device are deflected either up or down by a specific amount. This means spin angular momentum is quantized (space quantization), i.e., it can only take on discrete values. There is not a continuous distribution of possible angular momenta. This is the usual fundamental basis of the standard quantum theory and where we must introduce a new experimental protocol to surmount it. This is the crux of our new methodology: If application of a homogeneous magnetic field along a Z-axis produces quantum uncertainty upon measurement, then simplistically “do something else” [7]. In NMR spectroscopy often it is easier to make a first order calculation for a resonant state and then vary the frequency until resonance is achieved. Among the variety of possible approaches that might work best for a specific quantum system, if we choose NMR for the UFM Interferometer it is

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relatively straight forward to determine the spin-spin resonant couplings between the modulated electrons and the nucleons. But achieving a critical resonant coupling with the wave properties of matter with a putative beat frequency inherent in the HD spacetime backcloth is another matter. Firstly, for UFM cosmology

is not a rigid barrier as in Standard Model Big

Bang-Cosmology; is a virtual limit of retarded-advanced elements of the continuous-state standing-wave present as it cyclically recedes into the past where the least unit [7, 22] cavities tiling the spacetime backcloth can have cyclical radii length (



the Larmor radius of the hydrogen atom. This new Planck

 Ts ), where TS is string tension, oscillates through a limit cycle

from the Larmor radius of the hydrogen atom to standard , as asymptote never reached. We utilize the original hadronic form of string tension which is variable, not the current M-Theoretic form which is fixed. This cycle is like a wave-particle duality – Larmor radius at the futureretarded moment and at the past-advanced moment that opens and closes periodically into the HD regime. The dynamics are different for futureretarded elements which have been theorized to have the possibility of infinite radius for D > 4. This scenario is a postulate of string theory. Considering the domain walls of the least-unit structure, the  -Larmor cyclical regime is considered internal-nonlocal and the Larmor-infinity regime rotation considered external-supralocal. For our review of NMR concepts for the hydrogen atom, a single proton with magnetic moment,  , angular momentum, J related by the vector    J where  is the gyromagnetic ratio and J  I where I is the nuclear spin, see [7]. Coherent precession of  can also induce a ‘voltage’ in surrounding media, an energy component of the Hamiltonian to be utilized to create interference in the structure of spacetime. Metaphorically this is like dropping stones in a pool of water: One stone creates concentric ripples; two stones create domains of constructive and destructive interference. Such an event is not considered possible in the standard models of particle physics, quantum theory and cosmology. However, UF science uses extended versions of these theories wherein a new teleological action principle is

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utilized to develop what might be called a ‘transistor of the vacuum’. Just as standard transistors and copper wires provide the basis for almost all modern electronic devices; This Laser Oscillated Vacuum Energy Resonator using the information content of spacetime geodesics (null lines) will become the basis of many forms of new UF technologies. Simplistically in this context, utilizing an array of modulated tunable lasers, atomic electrons are rf-pulsed with a resonant frequency that couples them to the magnetic moment of the nucleons such that a cumulative interaction is created to dramatically enhance the Haisch-Rueda inertial back-reaction in conjunction with the Dubois incursive oscillator. The laser beams are counter-propagating for a Sagnac Effect Interferometry to maximize the small-scale local violation of Special Relativity. This is the 1st stage of a multi-tier experimental platform designed (according to the tenets of UFT) to periodically ‘open a hole’ in the fabric of spacetime in order to isolate and utilize the force

FˆU of the UFM Field [7, 11].

The interferometer utilized as the basis for the vacuum engineering research platform is a multi-tiered device. The top tier is comprised of counter-propagating Sagnac effect ring lasers that can be built into an IC array of 1,000+ ring lasers. If each microlaser in the array is designed to be counterpropagating, an interference phenomenon called the Sagnac Effect occurs that violates special relativity in the small scale. This array of rfmodulated Sagnac-Effect ring lasers provides the top tier of the multi-tier Laser Oscillated Vacuum Energy Resonator. Inside the ring of each laser is a cavity where quantum effects called Cavity-Quantum Electrodynamics (CQED) may occur. A specific molecule is placed inside each cavity. If the ring laser array is modulated with resonant frequency modes chosen to achieve spin-spin coupling with the molecules electrons and neutrons, by a process of Coherent Control [23] of Cumulative Interaction an inertial incursive back-reaction is produced whereby the electrons also resonate with the spacetime backcloth in order to ‘open an oscillating hole’ in it. This requires a TFT compatible with the 12D version of M-theory relying on the key ‘continuous-state’ symmetry conditions of UFM cosmology in which it is cast.

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The first step in the interference hierarchy is to establish an inertial backreaction between the modulated electrons and their coupled resonance modes with the nucleons. The complete nature of inertia remains a mystery [24]. It may later be shown that the continuous-state energy in conjunction with the UF force of coherence will solve this mystery of Mach’s Principle. It is critical to realize that the Standard Model contains no fundamental ‘beat frequency’ of a spacetime annihilation-creation cycle. Physicists have come to the realization that spacetime is not fundamental, but little has been said yet of the nature of its emergence. In our cosmological model, a key breakthrough is that this beat frequency arises as an inherent property of the continuous-state cycling. But if one follows the Sakarov and Puthoff conjecture, regarding the force of gravity and inertia, the initial resistance to motion, are actions of the vacuum zero-point field. Therefore, the parameter m in Newton’s second law f = ma is a function of the zero-point field Newton’s third law states that ‘every force has an equal and opposite reaction.’ Haisch & Rueda claim vacuum resistance arises from this reaction force, f = - f. We have also derived an electromagnetic interpretation of gravity and electromagnetism that suggests this inertial back-reaction is like an electromotive force1 of the de Broglie matter-wave field in the spin exchange annihilation creation process inherent in a hysteresis of the relativistic spacetime fabric. In fact, we go further to suggest that the energy responsible for Newton’s third law is a result of the continuous-state flux of the ubiquitous UFM noetic field. For the Laser Oscillated Vacuum Energy Resonator we assume the Haisch-Rueda postulate is sufficiently correct to be adapted for use in our rf-pulsed Sagnac Effect resonance hierarchy [7].

f 

d  d *   lim   lim *  f* dt t0 t dt* t* 0 t*

with   the impulse given by the accelerating agent so

1

(12)

*zp  * [7].

Electromotive force, E: The internal resistance r generated when a load is put upon an electric current I between a potential difference, V, i.e., r  ( E  V ) / I .

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The cyclotron resonance hierarchy must also utilize the proper spacetime beat frequency according to the mantra of the continuous-state dimensional reduction spin-exchange compactification process inherent in the symmetry of UF spacetime naturally ‘tuned’ to make the speed of light c  c and not infinite. With this apparatus in place noetic theory suggests that destructive-constructive C-QED interference of the spacetime fabric occurs such that the UFM noeon wave,



of the unified field,

UF

is

harmonically (like a light house beacon or holophote) released periodically into the cavity of the detector array. Parameters of the Dubois incursive oscillator are also required for aligning the interferometer hierarchy with the beat frequency of spacetime. If the water wave conception for the ‘Dirac sea’ is correct, the continuous state compactification process contains a tower of spin states from spin 0 to spin 4. Spin 4 represents the unified field making cyclic correspondence with spin 0 where Ising lattice Riemann sphere spin flips create dimensional jumps. Spin 0, 1/2, 1, & 2 remain in standard form. Spin 3 is suggested to relate to the orthogonal properties of atomic energy levels and space quantization. Therefore, the spin tower hierarchy precesses through 0, 720º, 360º, 180º, 90º & 0 () as powers of i. Experimental access to vacuum structure or for surmounting the uncertainty principle can be done by two similar methods. One is to utilize an atomic resonance hierarchy and the other a spacetime resonance hierarchy. The spheroid is a 2D representation of a HD Ising model Riemann sphere able to spin-flip from zero to infinity in conjunction with the putative beat frequency of spacetime. Basic conceptual components of the applied harmonic oscillator: classical, quantum, relativistic, transactional and incursive are all required in order to achieve coherent control of the cumulative resonance coupling hierarchy needed to produce harmonic nodes of destructive and constructive interference in the spacetime backcloth by incursion. The coherent control of the multi-level tier of cumulative interactions relies on full utilization of the continuous-state cycling inherent in parameters of Multiverse cosmology. What putatively will allow noetic

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interferometry to operate is the harmonic coupling to periodic modes of Dirac spherical rotation in the symmetry of the HD brane geometry. The universe is no more classical than quantum as currently believed; reality rather is a continuous state cycling of nodes of classical to quantum to unified, C  Q  U . We elevate the concept of wave-particle duality to a principle of cosmology especially in terms of the HD continuous-state cycle; this is what allows the UFM ‘mantra’ to operate. The salient point is that cosmology, the HD topology of spacetime itself, has a conformal rotation like the wave-particle duality Dirac postulated for electron spin. Recall that the electron requires a 4D topology and 720° for one complete rotation instead of the usual 360° to complete a rotation in 3D. The hierarchy of noetic cosmology is cast in 12D such that a pertinent form of ontologicalphase topological field theory has significantly more degrees of freedom, whereby the modes of resonant coupling may act on the structuralphenomenology of the Dirac ‘sea’ itself rather than just the superficial zeropoint field surface approaches to vacuum engineering common until now. The separation of the parameters of a Cramer transaction in terms of de Broglie’s fusion model is suggested to allow manipulation of the harmonic tier of the UF interferometer with respect to T-Duality or Calabi-Yau mirror symmetry. The parameters of the noetic oscillator seem best be implemented by an OPTFT using a form of de Broglie fusion. According to de Broglie a spin 1 photon can be considered a fusion of a pair of spin 1/2 corpuscles linked by an electrostatic force. Initially de Broglie thought this might be an electronpositron pair and later a neutrino and antineutrino. “A more complete theory of quanta of light must introduce polarization in such a way that to each atom of light should be linked an internal state of right and left polarization represented by an axial vector with the same direction as the propagation velocity.”

This suggests a deeper relationship in the structure of spacetime of the Cramer Transaction type [25] (Figure 4).

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Figure 4. Structure of Cramer’s Transactional model (present quantum state or event). a) Offer-wave and confirmation-wave combined into the resultant transaction b) which takes the form of an HD advanced-retarded standing or stationary wave of future-past elements. Figures Adapted from Cramer [25].

The epistemological implications of a 12D OPTFT must be delineated. The empirical domain of the standard model relates to the 4D phenomenology of elementary particles. It is the intricate notion of what constitutes a particle that concerns us here – the objects emerging from the quantized fields defined on Minkowski spacetime. This domain for evaluating physical events is insufficient for our purposes. The problem is not only the additional degrees of freedom and the associated XD, or the fact that ‘particles’ can be annihilated and created but that in UFM cosmology they are continuously annihilated and recreated within the holograph as part of the annihilation and recreation of the fabric of spacetime itself. This property is inherent in the 12D Multiverse because temporality is a subspace of the atemporal 3rd regime of the UF. This is compatible with the concept of a particle as a quantized field. What we are suggesting parallels the waveparticle duality in the propagation of an electromagnetic wave. We postulate this as a property of all matter and spacetime albeit as continuous-state standing waves. For a basic description, following de Broglie’s fusion concept, assume two sets of coordinates

x1 , y1 , z1

and x2 , y2 , z2 which become

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x1  x2 , 2

Y 

y1  y2 , 2

Z

z1  z2 . 2

(13)

Then for identical particles of mass m without distinguishing coordinates, the Schrödinger equation (for the center of mass) is i

 1   , t 2M

M  2m

(14)

In terms of Figure 4, Eq. 14 corresponds to the present and Eq. 15a corresponds to the advanced wave and (15b) to the retarded wave [7]. i

 1   , t 2M

i

 1   . t 2M

(15)

Extending Rauscher’s concept for a complex 8-space differential line element

dS 2   dZ  dZ  , where the indices run 1 to 4, is the

complex 8-space metric, Z  the complex 8-space variable where

Z   X Re  iX Im

and Z  is the complex conjugate. This can be extended

to 12D continuous-state UFM spacetime; we write just the dimensions for simplicity and space constraints:

xRe , yRe , zRe , tRe ,  xIm ,  yIm ,  zIm , tIm

(16)

where  signifies Wheeler-Feynman/Cramer type future-past/retardedadvanced dimensions. This dimensionality provides an elementary framework for applying the hierarchical harmonic oscillator parameters suggested. The concept conceptualized is that although commutativity was sacrificed by Hamilton in creating a closed quaternion algebra utilizing a 12D complex quaternion Clifford algebra approach to describe the

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Fermionic singularity; the additional degrees of freedom allow anticommutativity and commutativity to cycle periodically through the constraints of the algebra. This scenario when applied to the continuousstate cycle can be utilized to periodically via the suggested resonance hierarchy protocol to surmount the quantum uncertainty principle. If the noeon interferometer resonance hierarchy is able to surmount the uncertainty principle as outlined and can isolate and manipulate the LSXD brane world, in addition to quantum computing it will lead to a new research platform for developing a whole new class of vacuum based technologies; whereas one could say virtually all electronic devices up to now are based on transistors and copper wires. The Laser Oscillated Vacuum Energy Resonator could lead to a transistor of vacuum cellular automata, where rather than copper wires, the geodesics or null lines of space would be utilized to transfer information topologically with no quantal exchange particle mediating the ‘interaction’ in this scenario distinguishing phenomenology from unified field ontology. This introduction is an overview introducing the anticipated new field of vacuum engineering as Cramer stated should revolutionize many fields of science. When the great innovation appears, it will most certainly be in a muddled, incomplete form. To the discoverer himself it will be only halfunderstood; to everyone else it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope [26]. Finally, we stress that vacuum energy is not ‘produced’ by the noeon interferometer. The interferometer manipulates the boundary conditions ‘insulating’ or ‘hiding’ the unitary geodesics of HD spacetime by constructive and destructive interference allowing vacuum energy to be ‘emitted’ as a form of cursory superradiance [27] of the dynamics of the hysteresis loop of inherent least-unit synchronization backbone energy (topological charge) in continuous-state parallel transport. The model is empirically testable hopefully making up for some of the lack of precision in our axiomatic approach or thin rigor in portions of our attempts at formalism. In addition to the protocol presented here we have described elsewhere an additional experiment to utilize the noeon  -wave

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to study the putative manipulation of prion protein conformation responsible for degenerative neuropathies [7].

MEASUREMENT WITH CERTAINTY Because of what Unified Field Mechanics (UFM) appears to tell us about the fundamental basis of matter (albeit a preliminary foray); it is postulated that a bulk UQC cannot be built without utilizing UFM parameters with an inherent ability to supervene the quantum uncertainty principle. Although no attempt has been made yet to make correspondence with M-Theoretic supersymmetry, since it remains sufficiently unfinished; the topological order envisioned for UFM additions to the structure of matter can probably readily be made to do so. Concepts required to supervene uncertainty, such as a Dirac polarized vacuum, the de Broglie-Bohm causal interpretation and Cramer’s transactional interpretation are already wellknown to physics, but generally ignored. Concepts like Large-Scale Additional Dimensions (LSXD), brane topology and the vision that spacetime is not fundamental, but emergent, are already known and under ongoing development. The three main additions we apply are the discovery of a manifold of uncertainty (MOU) with finite radius, to which the unified field provides an ontological force of coherence (not 5th force) and that the underlying bulk hidden behind the ‘veil of uncertainty’ is a tessellation of ‘Least Cosmological Units’ (LCU) annihilated and recreated with a cyclic beat frequency. Obviously, this inherent LCU beat frequency is the key factor in supervening uncertainty for measurement with certainty. The general principle of superposition of quantum mechanics applies to the states ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states … indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state ...

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The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states – Dirac [13].

Generally, the 4-space of observation is restricted to a manifold inside a HD space, called the ‘bulk’ (hyperspace) by M-Theorists. If additional dimensions are compactified compactified, then the observed universe would contain any possible extra dimensions; and no reference to a bulk is required. However, if a bulk with Large-Scale Additional Dimensionality (LSXD) does exist, a rich interacting brane-world influencing 4-space is postulated. Kaluza-Klein XD compactification in string theory differs from the particle theory version in that a closed string can be wound several times around a rolled-up dimension. A string with this property, has what is called winding mode oscillations that add additional symmetry not found in particle physics. A theory with a rolled-up dimension of size R was found to be equivalent to a theory with a rolled-up dimension of size Ls2/R with winding modes and momentum modes exchanged in XD. (Ls is the string length scale). This symmetry allows correspondence between theories with small XD to theories with LSXD, known as T-Duality. In superstring theory, KaluzaKlein compactification must be applied to a 6D space. The well-known method of doing so is to use a heterotic dual Calabi-Yau 3-torus which determines the geometric topology of the symmetries and spectrum of the particle theory. In our theory the manifold of uncertainty has a 6D topology

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compatible with this type of supersymmetry. Braneworld models generally radically differ from superstring Kaluza-Klein compactification models because they require few steps between the Planck scale and electroweak scale. This huge difference between the Planck and the electroweak scale is called the gauge hierarchy problem [7]. Sufficient theoretical insight related to a new anthropic multiverse UFM cosmology has occurred during the 17 intervening years since the prior work [7, 10, 11] to design rigorous empirical protocols for isolating and manipulating fundamental parameters related to long-range coherence in semi-quantum systems. A key premise is that the so-called Planck scale stochastic regime is not fundamental and need no longer be a barrier to the study coherent phenomena in quantum systems generally or biological systems. Since Heisenberg’s 1927 discovery, the quantum uncertainty principle (4D) has been by empirical definition a barrier to accessing certain kinds of complementary biophysical information. As will be shown, the simple solution is - Do something else! That is, use a different fundamental basis for quantum and biophysical ‘measurement’ criteria by utilizing additional degrees of freedom inherent in a noetic UFM cosmology. Nine experimental protocols are outlined for testing postulates of the model; which if successful will lead to bulk UQC, a standardized biophysical research platform and a new class of biosensors. Noetic UFM cosmology makes correspondence to 11D M-Theoretic dual Calabi-Yau symmetry, M10  M 4  K6 albeit adding a 12th dimension incorporating Unified Field, UF dynamics, M12  M 4  K8  Mˆ 4  ˆ 4  ˆ 4 . String Theory has struggled to discover one unique vacuum compactification from the googolplex, 10googol or infinite potentia provided by XD, with Standard Model Minkowski space, M4 as the sought resultant. Noetic UFM cosmology is different - All dimensionalities from 12D to 0D cycle through continuously as a Continuous-state spin exchange dimensional reduction compactification process that led to discovery of a unique string vacuum. Note: The ‘continuous-state’ LCU is radically different than a Big Bang singularity [7]. Summary of salient theoretical postulates:

Imminent Advent of Universal Quantum Computing (UQC)

 







The Unified Field, UF provides an evolutionary ‘force of coherence’ guiding evolution in quantum systems. The HD UF regime is accessible by surmounting the uncertainty principle (limitation imposed by space-quantization parameters of the Copenhagen Interpretation) by manipulating new cosmological parameters described by additional degrees of freedom related to a Large-Scale Additional Dimensionality (LSXD) version of MTheory [7]. Utilizing UF parameters provides a new action principle with an inherent force of coherence acting like a ‘super-quantum potential’ or pilot wave guiding the ‘continuous-state’ spin-exchange dimensional reduction compact-ification process of spacetime and evolution of complexity in quantum and the Self-Organized Living Systems (SOLS) it pervades. The putative unique 12D M-Theoretic regime of UF action correlates parameters of Calabi-Yau mirror symmetry with heretofore generally ignored properties of de Broglie-Bohm Causal and Cramer Transactional interpretations of quantum theory and their higher dimensional (HD) extensions utilized in the new paradigm of noetic UFM cosmology [7]. This unique string vacuum forms a conformal scale-invariant covariant polarized Dirac-Einstein energy dependent spacetime metric, Mˆ 4 



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4

which by nature of its inherent continuous-state

dimensional reduction process acts as a Feynman ‘synchronization backbone’ facilitating/simplify-ing empirical accessibility. This empirical mediation of the LSXD polarized Dirac-Einstein metric,

Mˆ 4 

4

(12D) can be performed by a specialized incursive

form of rf-modulated Sagnac Effect resonant interference hierarchy able to surmount the uncertainty principle [7].

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Since 1993 the so-called Elitzur-Vaidman Interaction-Free Measurement (IFM) paradigm, a procedure for detecting the quantum state of an object without a phenomenological interaction occurring with the measuring device that ordinarily collapses the quantum wave function,  provides an indicia of our model suggesting it may be possible in general, as proposed here, to completely override the quantum uncertainty principle with probability, p  1 through utility of additional degrees of freedom inherent in the supersymmetric regime of string/brane theory. Note: in Newtonian mechanics the universe was 3D, Einstein introduced a 4D cosmology; now the next step seems to require 12D as the minimal dimensionality for producing causal separation from

Mˆ 4 .

The disadvantages of the IFM model is that in order to improve probability towards certainty more and more Mach-Zehnder interferometers and more and more cycles through the apparatus are required [28,29]; while our apparatus acts with a single cycle because it represents a true and complete overriding of the quantum uncertainty principle by utilization UF dynamics. We emphasize our position that it is impossible to violate the uncertainty principle in 4D (by empirical fact) which the IFM method is limited to. This duality in the Quantum Zeno Paradox as experimentally implemented in IFM protocols suggests a duality between the regular phenomenological quantum theory and a completed unified or ontological model beyond the formalism of the standard Copenhagen Interpretation as proposed here Utilizing extended theoretical elements, a putative empirical protocol for producing IFM with probability p  1 is introduced in a direct causal violation or absolute surmount of the methodology of the current 4D Copenhagen quantum Uncertainty Principle [7, 10, 11].

THE PROBLEM OF DECOHERENCE If the universe has a very rich structure, with many different branes, on which there exist very different physics, living in an as yet unknown geometry. - L. Randall [7].

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Except perhaps for quantum Hall quasiparticle protected anyons, in principle, quantum systems are open and not isolated from environmental noise or coupling. Decoherence, the destroyer of quantum superstition is essentially the only remaining barrier to bulk UCQ. It is curious that although anyonic TQC apparently has solved this dilemma by the braiding of topological phase; as yet there is no known method of accessing the protected qubits. We suspect our proposed Ontological-phase Topological Field Theory (OPTFT) will provide a method of doing so; but if such is the case, cryogenic temperatures would not be needed for UQC and an anyonic TQC might only be built as an interesting proof of concept. One key to developing UQC is to have quantum states with lifetimes longer than it takes to perform a computing operation. Current records for maintaining coherence are curiously interesting. For isolated atoms in ultrahigh vacuum chambers (no collisions with environment); the record for coherence is over 10 minutes. Solid-state silicon qubit systems cooled to absolute zero have long coherence times; but the new record is 39 minutes for room temperature silicon qubits [7]. In general, the problem of decoherence is strictly connected to the emergence of classicality in a world governed by the laws of quantum mechanics; and until now, any quantum information protocol must end up with a measurement converting quantum states into classical outcomes where decoherence plays a key role in this quantum measurement process. The last statement is not true exactly in the manner stated. As we intend to show, the causally-free HD ontological-phase copy of the system may be read instead of the system itself, leaving the system itself untouched and free to continue its evolution. What this does to QC algorithms, or speedup remains to be determined [7]. The Heisenberg uncertainty principle, says there is an inherent uncertainty in the relation between position and momentum in the x direction. Matter was thought to consist of localized particles, but matter exhibits wave-like properties, which means that matter, like waves, isn't localized in space. The uncertainty principle is a direct consequence of the wave-like nature of matter, because you can't completely discretize a wave.

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We move beyond these concepts of matter to one radically extended in HD UFM topology. We concur with Randall that these dimensions can be of infinite size, which follows from the existence of branes with infinite spatial extent, a property of branes that occurs because they carry energy. If there is an energetic 4D flat brane in a 5D spacetime, the 5D space does not consist of flat, uniform, LSXD. To accommodate a flat brane requires that in addition to the tension of the brane itself, there is a bulk vacuum energy, closely aligned to the brane tension. The solution to Einstein’s equations is then described locally as an anti-de Sitter (AdS) space, a space with a negative vacuum energy, although it is fundamentally 5D [7]. In this geometry, length of a yardstick depends on position. Spacetime is ‘warped’ and HD do not have to be finite in size, because unlike the case of flat XD, the gravitational force spreads very little in the direction perpendicular to the brane. To derive this form for the gravitational force, one solves Einstein’s equations of general relativity in the presence of the brane. General relativity tells us that not only do gravitational forces affect matter, but matter determines the surrounding gravitational potential. In this case, the presence of a massive brane leads to a gravitational force highly concentrated near the brane. So, although XD can be very large (even infinite), the gravitational force is highly concentrated near the brane [7].

Figure 5. Infinite size local 3-space and LSXD Braneworld and relate to gravity.

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Randall says physics tying XD to observable low energy scales, are theories, in which the XD are tied to relatively low energy scales, have the enticing possibility that they can be observed in the next generation of LHC colliders. Our model, which may be a form of supersymmetric M-Theoretic T-Duality, is different, tabletop and low energy; if successful it will put an end to the need for supercolliders. In a variant of the original proposal, in which the second brane does not end space but resides in an infinite extra dimension (essentially combining RS1 and RS2), one would have missing energy signatures identical to those one would obtain with six large ADD-type extra dimensions ... A fivedimensional AdS space is equivalent to a four-dimensional scale-invariant field theory, in the sense that all properties of the four-dimensional theory can be computed from the five-dimensional gravitational theory, and in principle one can learn about the gravitational theory from the conformal field theory (this is known as a holographic correspondence) … These include the existence of a four-dimensional domain in a higher dimensional space … It is possible that one or several of these ideas will be relevant to the question of how string theory evolves from a higher dimensional theory to one that reproduces observed four-dimensional physics [7]. In order to surmount quantum uncertainty and empirically access the hidden 3rd regime of reality (Classical  Quantum  Unified) new physics is required. We introduce a new ontological type of homeomorphic transformation (a holomorphic-antiholomorphic duality) that Toffoli calls a ‘topological switching’ by what Stern calls ‘topological charge’ that we propose as an empirical basis for the Micromagnetics of spacetime/matter information exchange without usual phenomenological exchange quanta. Mediation occurs instead as an ‘ontological becoming’ or ‘being’ by operation of an energyless coherently controlled resonant hierarchy of the topology of LSXD brane interactions which is not a local Hamiltonian phenomenon but perhaps a new form of ontological UF Lagrangian topology. Topological switching can be represented metaphorically as the perceptual switching of the central vertices of a Necker Cube (Ambiguous cube) when stared at [7].

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For example, imagine a usual 4D qubit or quantum particle in a box. In our noetic UFM interpretation the LSXD Calabi-Yau mirror symmetric regime contains a hierarchy of conformal scale-invariant ‘copies’ of the original 4D quantum state not independent Everett-Wheeler parallels. Then in way of simplistic introduction in terms of our new operationally completed interpretation of quantum theory the ‘mirror image of the mirror image is causally free’ of the underlying uncertain 4D quantum state and is accessible by manipulating the resonance hierarchy of our empirical protocol! Many physicists have been reluctant to embrace HD or LSXD physics. We suspect success of our protocol would ease this philosophical conundrum.

Figure 6. a) Left, a 4D hypercube unfolds into a 3D cross of 8 cubes. b) Right. Dimensional reduction cycle from 4D to 1D.

In Figure 6a the suggestion is that the 3-cube (bottom left) represents the region of a Cavity-QED or 3D quantum ‘particle in a box’ that through conformal scale-invariance remains physically real when the metaphor is carried to 12D where the ‘mirror copy’ becomes like a ‘mirror image of a mirror image’ and in that sense, is causally free of the E3 quantum state thereby open to ontological information transfer in violation of Copenhagen

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uncertainty. A 5D hypercube would unfold into a cross of 4D hypercubes and so on to 12D. Beyond 4D mirror symmetry adds a complexity in that the unfolding (Figure 6b) has a knot or Dirac twist (not shown) that is part of the gating mechanism insulating quantum mechanics from the 3rd regime of UFM. In Copenhagen the ‘handcuffs’ are on but during the LSXD cycle the handcuffs are periodically off and thus accessible resonantly.

QUANTUM PHENOMENOLOGY – UFM ONTOLOGY There is a major conceptual change from Quantum Mechanics to Unified Field Mechanics (UFM). The ‘energy’ of the UF is not quantized and thus is radically different from other known fields. Here is what troubled Nobelist Richard Feynman: “... maybe nature is trying to tell us something new here, maybe we should not try to quantize gravity... Is it possible that gravity is not quantized and all the rest of the world is?” [1]

Thus, not only is gravity not quantized but neither is the UFM coherent noeon energy of the UF which is a step deeper than gravity. Here is one way to explain it. In a usual field like electromagnetism, easiest for us to understand because we have the most experience with it, field lines connect to adjacent point charges. The quanta of the fields force is exchanged along those field lines (in this case photons). We perceive this as occurring in 4-space (4D). It is phenomenological. This is the phenomenon of fields. For topological charge as in the UF with properties related to consciousness; the situation is vastly different. The fields are still coupled and there is tension between them but no phenomenological energy (i.e., field quanta) is exchanged. This is the situation in the ontological case. The adjacent branes ‘become’ each other as they overlap by a process called ‘topological switching.’ This is not possible for the 4-space field because they are quantized resultants of the HD topological field components. The

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HD ‘units’ (noeons) are free to ‘mix’ ontologically (ambiguous Necker cube vertices) as they are not resolved into points. The metric still has points, or it might be better to say coordinates; but in HD super space they are unrestricted and free to interact by topological switching which is not the case for an ‘event’ in 4-space. Whereas this singular quality (basis of our perceived reality) does not exist in the HD regime (UF) of infinite potentia! So, if the UF is not quantized how can there be a force which is mediated by the exchange of energy? Firstly, the UF does not provide a 5th force as one might initially assume; instead the ontological ‘presence’ of the UF provides a ‘force of coherence’ which is based on ‘topological charge.’ It helps to consider this in terms of perception. If one looks along parallel railroad tracks, they recede into a point in the distance, a property of time and space. For the unitary evolution of the mind of the observer this would break the requirement of coherence. For the UF which is outside of local time and space, a cyclical restoring force is applied to our res extensa putting it in a res cogitans mode. The exciplex mechanism guides rotation of the Witten vertex Riemann spheres to maintain a consistent level of periodic coherence (parallelism). It is a relativistic UF process. The railroad tracks do not recede into a point, but it is not observed because the Riemann sphere flips (our perception) by subtractive interference beforehand [7]. The UF provides an inherent force of coherence just by its cyclical presence (perhaps a form of superradiance). This means that it is ontological in its propagation of information or ‘interaction’. The railroad tracks remain parallel and do not recede to a point as (perceived) in the 3D phenomenological realm where forces are mediated by a quantal energy exchange. Another way of looking at this is that the 3D observer can only look at one page of a book at a time while the HD observer (omniscient) can see all pages continuously. The LCU space-time exciplex is a mechanism allowing both worlds to interact locally-nonlocally.

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Figure 7. Complex HD Calabi-Yau mirror symmetric 3-forms, C4 become embedded in Minkowski space, M4 and the UF energy of this resultant is projected into localized matter as a continuous stream of evolving (evanescing) Bohmian explicate order. This represents the lower portion only that embeds in local spacetime; there is an additional duality above this projection embedded in the infinite potentia of the UF from which it arises.

Figure 8. Locus of nonlocal HD mirror symmetric Calabi-Yau 3-tori (here technically depicted as quaternionic trefoil knots) spinning relativistically and evolving in time. Nodes in the cycle are sometimes chaotic and sometimes periodically couple into resultant (faces of a cube) quantum states in 3-space depicted in the diagram as Riemann Bloch spheres, possibly indicative of the emergence of observed 3D reality. An animated version of Figure 7.

Most are familiar with the ambiguous 3D Necker cube (center of Figure 6a, bottom is like a Necker cube) that when stared at central vertices topologically reverse. This is called topological switching. There is another paper child's toy called a ‘cootie catcher’ that fits over the fingers and can switch positions. What the cootie catcher has over the Necker cube is that it has an easier to visualize a defined center or vertex switching point. So, in

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the LCU Exciplex spacetime background we have this topological switching which represents the frame that houses the gate which is the lighthouse holophote with the rotating light on top.

EMPIRICAL TESTS OF UFM Viable experimentation will lead to new UFM research platforms for studying fundamental syntropic properties of quantum systems. We have proposed fourteen tests of UFM; in this chapter we summarize the main experimental protocol to test the UFM noeon, Tight-Bound States (TBS) in hydrogen and the teleological ‘life-principle’ hypotheses. Note: Not all of the experiments relate directly to mediation of the UF noeon, but all of the experiments manipulate the new physical regime of the UF or importantly mediate the ‘gating mechanism’ by which access is gained to the 3rd regime of reality, thus facilitating mind-body research in addition to M-Theory, UQC and nuclear physics. The 3-cube embedded in the dodecahedron [7] represents what we term the ‘mirror image of the mirror image’ enfolding a scale-invariant ‘causally free’ copy of the Euclidean 3-space quantum ‘particle in a box,’ accessible under a precise protocol surmounting uncertainty. If experimentation proves viable a new class of UQC biophysical research platform for studying fundamental properties of the spacetime vacuum as it relates to long-range coherence in living systems. We summarize eight derivatives of the main experimental protocol to test the LSXD continuous-state Long-Range Coherence hypotheses: 

Basic Experiment - Fundamental test that the concatenation of new OPTFT UF principles is theoretically sound. A laser oscillated rf-pulsed vacuum resonance hierarchy is set up to interfere with the periodic (continuous-state) structure of the inherent ‘beat frequency’ of a covariant Dirac polarized spacetime vacuum exciplex to detect the new coherence principle associated with a cyclical holophote entry of the UF into 4-space. This experiment ‘pokes a hole in spacetime’ in order

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to bring the energy of the UF into a detector. The remaining protocols are variations of the parameters of this experiment. Bulk Quantum Computing - Utilizing protocol (1) Bulk Scalable UQC can be achieved by superseding the quantum uncertainty principle. (see [7] for details) Programming and data I/O are performed without decoherence by utilizing the inherent mirror symmetry properties that act like a ‘synchronization backbone’ [1] whereby LSXD copies of a local 3-space quantum state are causally free (measureable without decoherence) at specific resonance nodes in the continuous-state conformal Calabi-Yau symmetry cycle hierarchy.

A final essential component of the vacuum interferometer is called an incursive oscillator which acts as a feedback loop on the arrow of time. Parameters of the Dubois incursive oscillator are also required for aligning the interferometer hierarchy with the beat frequency of spacetime by

x(t  t ) v(t  t ) . Critically the size of t correlates with the bandwidth of the ‘hole’ to be punched in spacetime which also correlates with the wavelength,  of the rf-resonance pulse. Hysteresis is an important part of understanding how to quantify the topological charge with a unit of energy measure because it relates not to residual magnetization as in common usage but to the residual UF noeon charge. When the driving force drops to zero, the material retains considerable charge (coherence) for a period. The driving (noen) field must be continuously reversed and increased (holophote action) driving the charge to zero again. A Hysteresis Loop is a history dependence of a material (atom) at saturation (driven to). When the field is removed some retention occurs for a period of time. As noeon input alternately increases and decreases, hysteresis is the loop that the output forms. A simple form of hysteresis is the lag-time between input (filling) and output (draining). An example of hysteresis is sinusoidal or harmonic input X(t) and output Y(t) separated by a phase lag,  : X (t )  X 0 sin t; ;

Y (t )  Y0 (t   ) this is

the principle of hysteresis - switching cycles that retain considerable charge (coherence as in a UF LCU noeon cycle [7].

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Figure 9. a) Design elements of the Noetic Interferometer postulated to constructivelydestructively interfere with the topology of the spacetime manifold to manipulate the unified field. The first three tiers set the stage for the critically important 4th tier which by way of an incursive oscillator ‘punches a hole’ in the fabric of spacetime creating a holophote or lighthouse effect of the UF into the experimental apparatus momentarily missing its usual coupling node into an atom or biophysical system. b) Conceptualized Witten vertex Riemann sphere cavity-QED multi-level Sagnac effect interferometer designed to ‘penetrate’ space-time to emit the ‘noeon wave,  ’ of the unified field. Experimental access to vacuum structure or for surmounting the uncertainty principle can be done by two similar methods. One is to utilize an atomic resonance hierarchy and the other a spacetime resonance hierarchy. The spheroid is a 2D representation of a HD complex Riemann sphere complex able to spin-flip from zero to infinity continuously.

In the current understanding of quantum cosmology, the Planck scale is the ‘basement of reality,’ a stochastic Zero Point Field (ZPF) where virtual quantum particles wink in and out of existence with a half-life of the Planck time. This is considered an impenetrable barrier imposed by the Uncertainty Principle. UFM has sufficient degrees of freedom to surmount uncertainty and allow a cyclic or harmonic emergence of the noeon into localized matter. This holophote mechanism can be metaphorically described as an Exciplex. In an exciplex (short for excited complex) heteronuclear molecules or molecules having more than two species are exciplex molecules that are often diatomic and composed of two atoms or molecules that would not bond if both were in the ground state - An Exciplex is a complex existing in an excited state that dissociates in the ground state.

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THE TIGHT-BOUND STATE PROTOCOL Because Euclidean space is a ‘shadow’ of HD reality, gated by the Uncertainty Principle, and the current belief that the stochastic quantum foam is the ‘basement of reality’; it has not been evident that behind this veil (provided by a manifold of uncertainty of finite radius) there is a harmonic oscillation of the unified field, that opens and closes this gating mechanism with a ‘continuous-state’ periodicity. This is the key element of this scenario: in this regard there is a ‘beat frequency’ to the cyclic creation and annihilation of spacetime from the nilpotent potentia it is reduced (shadow) from. The symmetry occurs because of its inherent Cramer-like standingwave structure with de Broglie-Bohm control parameters driving its evolution. The perceived extreme radical nature of these premises, they will be difficult to accept initially; but in our favor we have an experimental paradigm waiting in the wings to be performed. We assume that all matter emerges from spacetime. In order to perform our experiment, we need to ‘destructively-constructively’ interfere with this process of continuous emergence. In the model being developed this requires finding a cyclical beat-frequency to the creation and annihilation process of space-time and matter. We believe this is best done by utilizing HD completed forms of the de Broglie-Bohm-Vigier causal and Cramer transactional interpretations of quantum theory. Once we know the size of the close-packed LCU and apply this to our ‘zero to infinity’ rotation of the Riemann sphere (Kahler manifold) we will know the radius/time of this putative inherent beat-frequency. This is where the Sagnac Effect Dubois incursive oscillator is applied to the structure where the t hyperincursion would correspond to a specific phase in the beat-frequency of spacetime and size of the hole utilized (punched by destructive interference) to send our signal through in order to detect several new TBS spectral lines in hydrogen [10]. We set the resonance hierarchy up in this case with hydrogen (simplest case with least amount of artifact from other electrons) where we jiggle the electron tuned to resonate with the nucleus tuned with the annihilation creation vectors in the beat frequency of spacetime which putatively opens

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a hole into the HD ‘manifold of uncertainty’ cavities by a process which we have stated numerous times is a direct violation of the quantum uncertainty principle. Which as you recall occurs when a field is arbitrarily set up along the z-axis to separate the states in the Stern-Gerlach apparatus, the beautiful empirical proof of the uncertainty principle. In the current model with the Planck basement there is no understanding of how to pass through; there is no XD cavities behind the Planck basement. It is finding the LCU beat frequency in the Dirac polarized vacuum that will give us success. In summary we have the 3-level tiered Sagnac Effect resonance hierarchy of electrons nucleons and spacetime. The counterpropagating properties of the Sagnac Effect violating special relativity in the small-scale will be relevant to a resonance process. For the standing-wave oscillator, the gap between R1 & R2 in the beat frequency of spacetime we take our ‘little laser blaster’ starting at the R1 bandwidth, when we reach the right point we will get a reflected blip, which will be our first new spectral line in hydrogen. So, in a sense if you’ve been following along; you see in general how straightforward and really simple this experiment is. This is a paradigm shift and beneath this infinite as yet to the reader, concatenation of mumbo-jumbo lies the framework for performing the TBS experiment. Unfortunately, one can see that any part of these elements that I’ve been gerrymandering could each take several hours to describe properly. The continuous-state, deriving the alternative formula for string tension - any of these is in hour lecture in itself. The importance of the LCU could require thousand-page treatises. I’ve been trying to give an overview of the framework for UFM that we’re in the process of discovering. Now the reason we think the continuous-state model will work is for example if you take the Bohr model of the hydrogen atom, spectroscopic measurements are taken as a 3D volume measurement from the space between the nucleus and the electrons orbit. For hydrogen the first Bohr orbit has a radius of a .5 Angstrom, and the second or orbit a radius of ~2 Å. This is the 100-year history of spectroscopic measurements from within the fixed regime of the 4D standard model. A spectroscopic cavity is going to have different properties in a 12D holographic multiverse regime.

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Firstly, we postulate the volumes of XD both within the finite radius MOU and beyond into the regime of LSXD. We continue to mention in terms of the complex quaternion Clifford algebra required to describe the continuous state process; that the cyclicality has an inherent commutativity anti-commutativity that the algebra can handle with a 3D or 4D Euclidean/Minkowski space resultant with 8D or 9D complex cycling dimensions built on top of it. Initially for a single space anti-space doubling, the MOU represents a 4th 5th and 6th hyperspherical XD. Behind or within the veil of uncertainty these XDs open and close volumetrically from zero i.e., the usual 3D Euclidean QED cavity to the added volumetric structure of the 4th 5th and 6th XD yielding: enabling us to calculate the wavelength of three rV 1 3 D , r2V4 D , r3V5 D , r4V6 D additional spectral lines in hydrogen based on the volume of these respective hyperspherical cavities.

Figure 10. NMR apparatus designed to manipulate TBS in Hydrogen. The Figure only shows possible details for rf-modulating TBS QED resonance, not the spectrographic recording and analysis components. Conceptual model of a proposed TBS experiment where hydrogen is put in the sample tube to which resonances are applied in a manner opening the manifold of uncertainty for access to HD cavities correlated with new spectral lines in hydrogen.

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Opening the 4D resonance hierarchy cavity will be relatively easy, but to open the 5th and 6D cavities probably requires the addition of some kind of precision Bessel function to the resonance hierarchy because additional artifacts like found in the refinements of the Born-Sommerfeld model; it will be a little tricky to master the protocol to measure these additional spectral lines. I do not mean this in calculating the wavelength, but the tiniest property we do not sufficiently understand will probably keep the uncertainty principle sufficiently active to keep the 5D cavity closed! The key element in this cosmology is the Least Cosmological Unit (LCU), not fully invented by us; but an extension of the idea found within a chapter called, “The size of the least unit” in a collection edited by Kafatos [22]. But Stevens of course utilizing only the 4D of the standard model attempted to describe a Planck scale least unit. But hopefully you have realized by now that our LCU oscillates from asymptotic virtual Planck,



 TS  to the Larmor radius of the hydrogen atom relative to the nature of

its close-packing tiling the spacetime foam. The left-hand part of Figure 11 shows the current thinking of string tension but, on the right, we see a multiverse version with a variable string tension that oscillates from virtual plank to the Larmor radius of hydrogen. Notice that the symbol for the Planck constant is different, we use the original Stony [7] that preceded Planck because it is electromagnetic and correlates better with the Dirac polarized vacuum which we want available for our resonance hierarchy component of the experimental protocol. Virtual plank is the asymptotic zero point on the Riemann sphere that flips back to infinity in the continuous-state cycle. Since the Planck scale is no longer considered the basement of reality the 12D continuous-state process changes the size of the LCU in the process of Riemann sphere rotation from zero back to infinity continuously. Choice of the upper limit as the Larmor radius is somewhat arbitrary. We cannot define this rigorously yet without experiment; but assume it is in this ballpark. So just to make a note we have this oscillating Planck unit,  at the microscopic level in conjunction with an oscillating  lambda or cosmological constant at the macroscopic level.

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Figure 11. Fixed string tension in M-Theory (left) and variable (right) as in the original hadronic form of string theory and HAM cosmology that also reverts to the original Stoney,

rather that Planck’s constant,

.

Figure 12. Beyond The 4D Standard Model Lies (SXD ‘Hidden’ by Uncertainty.

We attempted to illustrate some of the underlying topology of continuous-state topology. There is a dramatic increase in the number of cubes comprising HD space as we travel rectilinearly up the XD ladder. Our 12D model must cycle through nodes of commutativity and anticommutativity where one mode is degenerate and the other closed to observation. There are not sufficient degrees of freedom to cyclically break closure otherwise. Rowlands supports an inherent necessity of 3D for reality, so we have a doubling of the 1st 3D into another triplet of HD space. This suggest indicia for the necessity of the 12D where UFM wants to lead us. Imagine a 3-blade ceiling fan symbolic of a quaternion fermion vertex. If one puts one of these fans in front of a mirror (real space) rotating clockwise the mirror image (anti-space) rotates counterclockwise with the

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blades coming occasionally into phase as in Figure (8). Now we give a key insight into the TBS experiment that Figure 8 doesn’t have. If there is a light on near the fan in real space, i.e., the rf-pulse of our TBS experiment. Periodically when the blades come into phase (Figure 8 again) meaning when a blade from real space comes into phase with a blade in the mirror antispace the light is reflected off each blade (the mirror image of the mirror image) and a pulsating, reflected flash of light occurs in the direction back towards the source/detector! This is representative of how we intend to find the new TBS spectral lines in hydrogen; that we would expect to see a flashing back, like a rotating lighthouse beacon when the resonance hierarchy is aligned properly! Rowlands suggests additional space anti-space dimensions are redundant (no new information); but that’s what we want from an infinite potentia that is nilpotent and redundant. Surmounting the quantum uncertainty principle occurs by this same process that gives us a beat frequency inherent in the spacetime backcloth. In order to demonstrate existence of new spectral lines the experiment itself requires surmounting the quantum uncertainty principle [7, 10]. When the parameters for the experiment are coordinated and the rf-pulse sent into the MOU HD QED TBS hydrogen cavity, a positive result will retrieve a spectroscopic signal like the one represented in Figure 13. A negative result would send back 0 amplitude.

Figure 13. First 4D TBS spectral line in hydrogen emerging from the 4D spherical potential well for   1 . Figure adapted from [30].

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Figure 14. Logarithmic spirals and Perfect Rolling Motion. Segments of the logarithmic spiral are put into the three spheroids on the right, A,B,C. Like the 320 o 720o spinor rotation of the Dirac electron; the speroids only return to the same configuration after several 360o rotations. a) Perfect rolling motion of logarithmic spiral components. b) Applied to left-right symmetry transformations of Calabi-Yau brane topology such that while the A,B,C tower is meant to represent the usual closed quaternionic space-antispace algebra; the A,B,C and A’B’C’ towers together when doubled again as in c) will be able to cyclically commute and anti-commute (requires an additional mirror symmetric doubling with trefoil-like involution and parameters of parallel transport cyclically breaking closure of the algebra.

In the simplistic model of doing the TBS experiment we put hydrogen in a sample tube (Figure 10) and apply a series of resonant pulses in conjunction with the beat-frequency of space-time to open the HD QEDUFM cavity, send the signal in and allow the new TBS spectral line signal to be emitted back to the detector. Remember we postulated that the HD continuous-state cycle must incorporate cycles of commutativity and anti-commutativity. This is shown metaphorically in terms of logarithmic spirals applied to perfect rolling motion (Figures 14a,b). How can we find this cycle in HD Calabi-Yau mirror symmetry? The logarithmic spirals in Figure 14a are not free to rotate (Euclidean shadow). If we take pieces of the curve as in Figure 14b and paste them together as shown; the three cycloids can cycle continuously. Perfect rolling motion in this case means a mechanical process where there is no slippage if this is applied to the mechanics of gears. As hopefully clear well before now to the reader, this represents a ‘closed’ non commuting algebra. If you’re not a mechanical engineer, you may not have guessed already that after a certain number of cycles the set of three cycloids returns to the precise original position. Now in terms of the next figure let’s apply this to

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a second doubling or duality to Rowlands’ space anti-space quaternion model which of course is going to have to include Calabi-Yau mirror symmetry. What we propose metaphorically here is that with the utility of the complex quaternion Clifford algebra we can mathematically describe how to break the closure inherent in one of the mirror symmetric partners and describe cycles relative to both mirror symmetric partners that additionally pass through cycles of commutativity and anti-commutativity with each other. We cannot surmount the uncertainty principle utilizing a closed algebra - the mathematical description of course. This is similar to the property revealed in Figure 8 with the rotating of the wind generator propellers cycling from Chaos to Order; and also similar to passing by a fruit orchard, rows of chairs in an auditorium or the tombstones in a graveyard where one’s line of sight is alternatingly blocked and alternatingly open to infinity in similitude also to wave particle duality again in terms of the rotations inherent to the cyclicality of the LCU backcloth tessellating space antispace - talking about nodes in the hyperspherical structure inherent in the HD components ‘behind’ our 3space virtual reality. We assume that all matter cyclically emerges from spacetime. In order to perform our experiment, we need to ‘destructivelyconstructively’ interfere with this process. In the model being developed this requires finding a cyclical beat-frequency to the creation and annihilation process of spacetime and matter. In summary we have the 3-level tiered Sagnac Effect resonance hierarchy of electrons nucleons and spacetime. The counter-propagating properties of the Sagnac Effect that violates special relativity in the smallscale will most likely be relevant to this process. For the standing-wave oscillator, the gap between R1 & R2 (Figure 9b) in the beat frequency of spacetime we take our ‘little laser blaster’ starting at the R1 bandwidth, when we reach the right point, we will get a reflected blip, which will be our first new spectral line in hydrogen. Some experimental evidence has been found to support this view showing the possibility that this is the same property that the interaction of these extended structures in space involve real physical vacuum couplings by resonance with the subquantum Dirac ether. Because of photon mass the

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CSI model, any causal description implies that for photons carrying energy and momentum one must add to the restoring force of the harmonic oscillator an additional radiation (decelerating) resistance derived from the em (force) field of the emitted photon by the action-equal-reaction law. The corresponding transition time corresponds to the time required to travel one full orbit around the nucleus. Individual photons are extended spacetime structures containing two opposite point-like charges rotating at a velocity near c, at the opposite sides of a rotating diameter with a mass, m =10-65 g and with an internal oscillation E = m2= hv. Thus, a new causal description implies the addition of a new component to the Coulomb force acting randomly and may be related to quantum fluctuations. We believe this new relationship has some significance for our model of vacuum C-QED blackbody absorption/emission equilibrium. The purpose of this simple experiment is to empirically demonstrate the existence of LSXD utilizing a new model of TBS in the hydrogen atom until now hidden behind the veil of the uncertainty principle. If for the sake of illustration, we arbitrarily assume the s orbital of a hydrogen atom has a volume of 10 and the p orbital a volume of 20, to discover TBS we will investigate the possibility of heretofore unknown volume possibilities arising from cyclical fluctuations in large XD Calabi-Yau mirror symmetry dynamics. This is in addition to the Vigier TBS model. As in the perspective of rows of seats in an auditorium, rows of trees in an orchard or rows of headstones in a cemetery, from certain positions the line of sight is open to infinity or block. This is the assumption we make about the continuous-state cyclicality of HD space. Then if the theory is based in physical reality and we are able to measure it propose that at certain nodes in the cycle we would discover cavity volumes of say .9, 1.4 and 1.8 Å (between the .5 and 2.00 Å existing spectral lines) for example. We propose the possibility of three XD cavity modes like ‘phase locked loops’ depending the cycle position maximal, intermediate and minimal.

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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11]

[12] [13] [14]

Feynman, R. P. (1982) Simulating physics with computers, Intl J Theor Phys 21, pp. 467-488. Vandersypen, L. M. K. & Chuang, I. L. (2004) NMR techniques for quantum control and computation, Rev Mod Phys, 76, 1037-1069. Di Vincenzo, D.P. (2000) The physical implementation of quantum computation, arXiv:quant-ph/0002077. Bell, J. S. (1966) On the problem of hidden variables in quantum mechanics, Reviews of Modern Physics, 38(3), 447. Lindblad, G. (1999) A general no-cloning theorem, Letters in Mathematical Physics, 47(2), 189-196. Zurek, W. H. (2009) Quantum Darwinism, arXiv:0903.5082v1 [quantph]. Amoroso, R L (2017) Universal Quantum Computing: Surmounting Uncertainty, Supervening decoherence, London: World Scientific Kane, BE (1998) Silicon-based nuclear spin quantum computer, Nature, 393, p. 133. Muraki, K. (2012) Unraveling An Exotic Electronic State For ErrorFree Quantum Computation, NTT Tech Rev, V 10, No 10. Amoroso, R. L. (2013) Evidencing ‘tight bound states’ in the hydrogen atom: Empirical manipulation of large-scale XD in violation of QED, in R. L. Amoroso (ed.) The Physics of Reality: Space, Time, Matter, Cosmos, Singapore: World Sci. Amoroso, R. L. (2010) Simple resonance hierarchy for surmounting quantum uncertainty, AIP Conf. Proc. 1316, 185 or http://vixra.org/pdf/1305.0098v1.pdf. Nielsen, M. A. (2002) Rules for a complex quantum world, Scientific American, Vol. 287, No. 5, pp. 66-75. Dirac, P. A. M. (1958) The Principles of Quantum Mechanics, Oxford Univ. Press. Barenco, A. et al. (1995) Elementary gates for quantum computation, Phys. Rev. 52 (5): 3457-3467; arXiv:quant-ph/9503016v1.

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[15] http://www-inst.eecs.berkeley.edu/~cs191/sp05/lectures/lecture4.pdf. [16] Thorne, K. S., Drever, R. W. P., Caves, C. M., Zimmermann, M. & Sandberg, V. D. (1978) Quantum nondemolition measurements of harmonic oscillators, Phys. Rev. Lett. 40; 667-671. [17] White, A. G., Mitchell, J. R., Nairz, O. & Kwiat, P. G. (1998) “Interaction-free” imaging, arXiv:quant-ph/9803060v2. [18] Kotigua, R. P. & Toffoli, T. (1998) Potential for computing in micromagnetics via topological conservation laws, Physica D, 120:12, pp. 139-161. [19] Gerlach, W & Stern, O. (1922) Das magnetische moment des silberatoms [The magnetic moment of the silver atom], Zeitschrift für Physik 9, 353-355. [20] Hofstadter, D. (2013) Alan Turing: Life and legacy of a great thinker, in C. Teuscher (ed.) p. 54, Springer Science & Business Media. [21] Peise, J., Lücke, B., Pezzé, L., Deuretzbacher, F., Ertmer, W., Arlt, J., Smerzi, A., Santos, L. & Klempt, C. (2015) Interaction-free measurements by quantum Zeno stabilization of ultracold atoms, Nature Communications, 6,14. [22] Stevens, H. H. (1989) Size of a least unit, in M. Kafatos (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Dordrecht: Kluwer Academic. [23] Garcia-Ripoli, J. J., Zoller, P. & Cirac, J. I. (2005) Coherent control of trapped ions using off-resonant lasers, Phys. Rev. A 71, 062309; 1-13. [24] Vigier-J-P (1995) Derivation of inertial forces from the Einstein-de Broglie-Bohm causal stochastic interpretation of quantum mechanics, Found. Phys. 25:10, 1461-1494. [25] Cramer, J. G. (1986) The transactional interpretation of quantum mechanics, Rev. Mod. Phys 58, 647-687. [26] Dyson, F. J. (1958) Innovation in Physics, Scientific American, 199, No. 3. [27] Eberly, J. H. (1972) Superradiance revisited, AJP, 40; 1374-1383. [28] Elitzur, A. C. & Vaidman, L. (1993) Quantum mechanical interactionfree measurements. Found. Phys. 23; 987-997.

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[29] Vaidman, L. (2001) The meaning of the interaction-free measurements arXiv:quant-ph/0103081v1. [30] Scheid F. (1986) Theory and Problems of Numerical Analysis. McGraw-Hill.

In: Frontiers in Quantum Computing ISBN: 978-1-53618-515-7 Editor: Luigi Maxmilian Caligiuri © 2020 Nova Science Publishers, Inc.

Chapter 9

BRAIN - QUANTUM HYPERCOMPUTING SYSTEM Takaaki Musha* Advanced Science-Tecxhnology Research Organization, Yokohama, Japan

ABSTRACT Starting from standpoint of a model of the brain based on superluminal tunneling photons, the author have described theoreticallu the possibility of a brain-like computer that would be more powerful than Turing machines, would allow non-Turing computation, and that may hold the key to the origin of human consciousness itself. According to the brain model proposed, it has been shown that microtubles in the biological brain have the possibility to achieve large quantum bit computation at room temperature which is superior in performance to conventional processors.

Keywords: superluminal particle, non-Turing computation, microtuble, tunneling photon, metamaterial, hypercomputer

* Corresponding Author’s Email: [email protected].

brain,

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INTRODUCTION It is widely believed that Moore’s law for microprocessor performance will fail to hold in the next decade due to the brick wall arising from fundamental physical limitations of the computational process. Richard P. Feynman discussed the possibility of a quantum computer in which quantum computational energy cost versus speed is limited by energy dissipation during computation by taking an example of reversible computing [1]. According to his idea, the computational speed is limited by minimum energy required to transport a bit of information irreversibly between two devices, which prevents the speeding up of quantum computation. In this chapter, the author studies the energy limit of the computer system utilizing evanescent photon (photon quantum tunneling), which is considered to be a superluminal particle called a tachyon. He studies the possibility to realize a high performance computing system by utilizing superluminal tunneling photon and he has also studied the possibility that the microtubular structure of neurons in a human brain is functioning as a quantum computational system that can attain higher efficient computation compared with conventional silicon processors.

What Is a Quantum Computer? A quantum computer is a computational device that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistors (semiconductor material with a minimum of three terminals). Whereas digital computers require data to be encoded into binary digits (a bit can have only one of two values, 0 or 1), quantum computation uses quantum properties to represent data and perform operations on these data (a quantum bit or qubit can exist in superposition of two bits). An idea of quantum computing was first introduced by Richard Feynman in 1982 (see Ref. [1]).

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Hence, a classical computer has a memory made up of bits, where each bit represents either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or any quantum superposition of these two qubit states; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8. In general, a quantum computer with n qubits can be in an arbitrary superposition of up to 2n different states simultaneously, which compares to a normal computer that can only be in one of these 2 n states at any one time shown as [2] (see Figure 1):

  0  1

.

(1)

A quantum computer operates by setting the qubits in a controlled initial state that represents the problem at hand and by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The calculation ends with measurement of all the states, collapsing each qubit into one of the two pure states, so the outcome can be at most n classical bits of information. An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: “up” and “down.” But in fact any system possessing an observable quantity A, which is conserved under time evolution such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. This is true because any such system can be mapped onto an effective spin-1/2 system. A quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. For example, to represent the state of an n-qubit system on a classical computer would require the storage of 2n complex coefficients. Although this fact may seem to indicate that qubits can hold exponentially more information than their classical counterparts, care must be taken not to overlook the fact that the qubits are only in a probabilistic superposition of all of their states. This means that when the final state of the qubits is

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measured, they will only be found in one of the possible configurations they were in before measurement. Moreover, it is incorrect to think of the qubits as only being in one particular state before measurement since the fact that they were in a superposition of states (see Figure 2) before the measurement was made directly affects the possible outcomes of the computation.

Figure 1. Qubit of the quantum computation.

Figure 2. Superposition of qubits states.

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Qubits are made up of controlled particles and the related means of control (e.g., devices that trap particles and switch them from one state to another).

Figure 3. Quantum bit consisted from three states of qubits.

For example: Consider first a classical computer that operates on a threebit register. The state of the computer at any time is a probability distribution over the 2 3  8 different three-bit strings 000, 001, 010, 011, 100, 101, 110, 111 as shown in Figure 3. If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a probabilistic computer, then there is a possibility of it being in any one of a number of different states. We can describe this probabilistic state by eight nonnegative numbers A,B,C,D,E,F,G,H (where A = probability computer is in state 000, B = probability computer is in state 001, etc.), see Figure 3. There is a restriction that these probabilities sum to 1. Large-scale quantum computers will be able to solve certain problems much more quickly than any classical computer using the best currently

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known algorithms (Figure 4), like integer factorization using Shor's algorithm or the simulation of quantum many-body systems. There exist quantum algorithms, such as Simon's algorithm, which run faster than any possible probabilistic classical algorithm [3]. Given sufficient computational resources, a classical computer could be built to simulate any quantum algorithm; quantum computation does not violate the Church–Turing thesis [4]. However, the computational basis of 500 qubits, for example, would already be too large to be represented on a classical computer because it would require 2500 complex values (2501 bits) to be stored.

Figure 4. Schematic drawing of the quantum computer, the Orion processor by DWave, which has the possibility to solve many difficult problems which can not be solved by the conventional computers.

SIGNAL PROCESSING BY USING TUNNELING PHOTONS Recently it has been discovered by the research team at the University of Maryland that a signal processing device based on single photon tunneling can be realized [5]. They conducted the experiment for the transmission of

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light through nanometer-scale pinholes in a gold film covered by a nonlinear dielectric, which revealed that transmittance of a nanopore or nanopore array at one wavelength could be controlled by illumination with a second, different, wavelength. This opens the door to optical signal processing devices, such as all-optical switches realized on a microscopic scale which can manipulate single electrons, atoms or photons. If the atoms can be localized at distances smaller than the radiation wavelength, they can be coherently coupled by photons and an entangling quantum logic gate can be realized.

Figure 5. Computer gate consisting of tunneling photons.

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By applying this technology, the computer gate consisting of a large number of small, interconnected electromagnetic ion traps created for both memory and logical processing by manipulating atoms with the tunneling photon, may be realized as shown in Figure 5. The possibility of nano-switching by using evanescent photons in optical near-field was also proposed by T. Kawazoe and his co-workers [6]. By utilizing this technology, it is considered that the computation utilizing quantum tunneling photons has the possibility to achieve much faster computational speed and the influence of energy cost due to the uncertainty principle which prevents speeding up the quantum computation can be overcome as suggested in following section.

Energy Cost for Quantum Computation Utilizing Tunneling Photons Hereafter we assume that tunneling photons which travel in an evanescent mode can move with a superluminal group speed, as confirmed by some experimenters. We consider the computer system which consists of quantum gates utilizing quantum tunneling photons to perform logical operations. Benioff showed that the computation speed was close to the limit by the time-energy uncertainty principle [7]. Margolus and Levitin extended this result to a system with an averaged energy  E  , which takes time at least  E   /(2t ) to perform logical operations [8]. Summing over all logical gates of operations, the total number of logic operations per second is no more than

t  N

 , 2E

(2)

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where t is a operational time of an elementary logical operation and N is the number of consisting gates of the computer. From Eq.(2), energy spread for the quantum tunneling photon (abbreviated QTP, hereafter) gate becomes 1 / (  1) times the energy spread for the logical gate using particles moving at sub-luminal speed including photons, then the total number of logic operations per second for QTP gates can be given by

t 

 N ,  (   1) 2  E* 

(3)

where  E*  is an averaged energy for QTP gates. As the uncertainty in the momentum of tunneling photons moving at the superluminal speed can be given by

p 

m* v v2 / c2 1



 , c

(4)

where m* is an absolute value of the mass for the tunneling photon moving

at superluminal speed and  is an angular frequency of the photon, the velocity of the tunneling photon can be estimated as

 v  c1  

  ,  t  1

(5)

from uncertanity principle for superluminal particles and E   [9, 59]. If we denote the tunneling distance by d , the time for a photon tunneling across the barrier can be roughly estimated to be t  d / v . Then the velocity of the tunneling photon can be given by

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 c c c2  v  c 1    2d  d 42 d 2 

 .  

(6)

From which, the ratio of the minimum energy required for computation by QTP gates and the conventional computation can be given, as follows, by equating Eqs.(2) and (3);

R

 E*  1  ,  E  (  1)

(7)

where

c c c2   1   2d d 4 2 d 2

.

(8)

By the Higgs mechanism in quantum field theory, the penetration depth of tunneling photons can be estimated by the formula shown as [10]

r0 

 Mc ,

(9)

where M is the effective mass, which yields M  10eV . From the above considerations, the penetration depth of the tunneling photon is estimated to be 1.5  108 m. Then the ratio of minimum energy required for the computation by QTP gates to the conventional computation processes can be estimated as shown in Figure 6, when we let the tunneling distance of the barrier be d  10n m [11].

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Figure 6. Energy ratio of minimum energy required for the computation by QTP gates.

In this figure, the horizontal axis is for the wavelength of tunneling photons and the vertical axis is for the ratio of their energy required for the computation. If the wavelength of the tunneling photon is in a far infrared region (  10 5 nm) , the energy cost of computation for the computer, which consists of QTP gates, reaches to 10 6 times smaller than that of conventional computer systems. In recent years, many studies on the quantum computation were conducted and it was recognized that the computational speed by quantum computing was much higher than that of conventional silicon processors. But the energy cost due to the uncertainty principle which prevents speeding-up of the quantum computation was not considered. From the theoretical analysis of the energy limit of the quantum computer system which utilizes tunneling photons, it can be shown that energy loss of computation by utilizing superluminal tunneling photons is much lower than that of conventional silicon processors. Moreover this superluminal effect can eventually speed up computers significantly because it can compensate interconnect delays inside logic gates which can never be fully eliminated from any real electronic components and bringing overall transmission rate closer to the ultimate speed limit [12], which actually boost

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the speed of a signal traveling on an electromagnetic wave for achieving high performance computers.

Decoherence Problem of Quantum States to Conduct Quantum Computation The problem for conducting quantum computation is decoherence of quantum states. The qubit calculations of the quantum computer are performed while the quantum wave function is in a state of superposition between states, which is what allows it to perform the calculations using both 0 & 1 states simultaneously. But this coherent state is very fragile and difficult to maintain. The slightest interaction with the external world would cause the system to decohere. This is the problem of decoherence, which is a stumbling block for quantum computation. Therefore the computer has to maintain the coherent state for making calculations. If we let T to be the relaxation time of a single qubit and t is the operation time of a single logical gate, the figure of merit  for computation can be defined as   T / t [13], which is on the order of the number of qubits times the number of gate operations. As a superposition state of the L-qubits system would cause decoherence approximately 2 L times faster than a superposition state of one qubit [14], then the relaxation time of the L-qubits system can be roughly estimated to be 2  L times the relaxation time of a single qubit computation. Thus the minimum energy required to perform quantum computation for the L-qubits system can be given by

E0 

 G L L 2 , T

where  G is the number of gate operations.

(10)

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Similar to this equation, the minimum energy required to perform quantum computation utilizing superluminal particle can be estimated as [15]

E0 

 G L 2L .  (   1)T

(11)

Supposing that E 0  E 0 , an increase of qubit size to perform computation by superluminal evanescent photon compared with the conventional computation can be given by

L 

log 2 [  (   1)] , 1  1 / L log 2

(12)

when satisfying L  L . From which, we can estimate an increase of qubit size of the quantum computation utilizing superluminal evanescent photons compared with the conventional computer system.

Possibility of Quantum Computation inside Microtubles in the Brain From the high performance capabilities of quantum computation, there are many researches that can explain the higher performance of the human brain, including consciousness. As proposed by Feynman the optical computing network is the most realizable quantum mechanical computer among many possibilities such as a superconducting computer, and Hameroff suggested that microtubles in the brain were acting as waveguides for photons and as holographic processors [16].

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The cytoskelton of biological cells, including neurons of the brain, is made up of microtubles as shown in Figure 7.

Figure 7. Tubulin structure and microtubles.

Each microtuble is a hollow cylindrical tube of tubulin proteins as shown in Figure 8, which outer core diameter is 25 nm. Microtubules are comprised of subunits of the protein, named tubulin [17]. Proteins contain hydrophobic (water repellent) pockets and these pockets contain atoms with electrons called  electrons, which means electrons in the reactive outer part (outer shell) of the atom that are not bonded to other atoms. The tubulin protein subunits of the microtubules have hydrophobic pockets within two nanometers of one another, which is close enough for the  electrons of the tubulin to become quantum entangled. Each tubulin molecule can function as a switch between two conformations which exist in quantum superposition of both conformational states.

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Figure 8. Schematic diagram of the microtuble structure.

The dimensions of centrioles are close to the wavelengths of light in the infrared and visible spectrum, such that they may act as phase coherent, resonant waveguides [18] (as shown in Figure 9) which can be mapped onto the spin of material and they may play the role of retrieval, coupling/entangling of spin states of materials [19]. Thus each tubulin molecule can function as a switch between two conformations which exist in quantum superpositions of both conformational states. According to Jibu et al. [20, 21], microtuble quantum states link to those of other neurons by quantum coherent photons tunneling through membranes in biological systems and the cytoskeletal protein conformational states are entangled by these photons which form coherent domains in their interaction with the local electromagnetic field. Hameroff and Tuszynski [22] have proposed that microtuble subunit tubulins undergo coherent excitations, which leads to the automatic sequence where quantum coherence superposition is emerged in certain tubulins and consciousness is occurred as the process shown in Figure 10.

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Figure 9. Structure of the centriole cylinder and waveguide of visible and infra-red light in the centriole cylinder.

Figure 10. Quantum computation conducted inside the microtubule.

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According to their hypothesis of quantum brain, microtuble quantum states link to those of other neurons by quantum coherent photon tunneling through membranes in biological systems, functioning in a way that resemble to an ion trap computer. On the other hand, Hameroff and Penrose [23] have constructed a theory, in which human consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated object reduction) that involves a specific form of quantum computation conducted at the level of synapses among brain neurons. They have suggested that microtubles in brain neurons function as quantum computers with tubulin proteins in macrotubles acting as a quantum bits of computation through two-state q-bits formed from tubulin monomers. Hameroff proposed that microtubules were suitable candidates to support quantum processing [24]. Microtubules are comprised of subunits of the protein, tubulins. Proteins constitute much of the driving machinery of living organisms. Proteins contain hydrophobic (water repellent) pockets. These pockets contain atoms with electrons called π electrons, which means electrons in the reactive outer part (outer shell) of the atom that are not bonded to other atoms. The tubulin protein subunits of the microtubules have hydrophobic pockets within two nanometers of one another. Hameroff claims that this is close enough for the π electrons of the tubulin to become quantum entangled. Quantum entanglement is a state in which quantum particles can alter one another's properties instantaneously and at a distance, in a way which would not be possible, if they were large scale objects obeying the laws of classical as opposed to quantum physics. In the case of the electrons in the tubulin subunits of the microtubules, Hameroff has proposed that large numbers of these electrons can become involved in a state known as a Bose-Einstein condensate. These occur when large numbers of quantum particles become locked in phase and exist as a single quantum object. These are quantum features at a macroscopic scale, and Hameroff suggests that through a feature of this kind quantum activity, which is usually at a very tiny scale, could be boosted to be a large scale influence in the brain.

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Hameroff has proposed that condensates in microtubules in one neuron can link with other neurons via gap junctions. In addition to the synaptic connections between brain cells, gap junctions are a different category of connections, where the gap between the cells is sufficiently small for quantum objects to cross it by means of a process known as quantum tunneling. Hameroff proposes that this tunneling allows a quantum object, such as the Bose-Einstein condensates mentioned above, to cross into other neurons, and thus extend across a large area of the brain as a single quantum object. He further postulates that the action of this large-scale quantum feature is the source of the gamma (40 Hz) synchronisation observed in the brain, and sometimes viewed as a correlate of consciousness. In support of the much more limited theory that gap junctions are related to the gamma oscillation, Hameroff quotes a number of studies from recent years. The Orch OR theory combines Penrose's hypothesis with respect to the Gödel theorem with Hameroff's hypothesis with respect to microtubules. Together, Penrose and Hameroff have proposed that when condensates in the brain undergo an objective reduction of their wave function, that collapse connects to non-computational decision taking/experience embedded in the geometry of fundamental space-time. The theory further proposes that the microtubules both influence and are influenced by the conventional activity at the synapses between neurons. The Orch in Orch OR stands for orchestrated to give the full name of the theory Orchestrated Objective Reduction. Orchestration refers to the hypothetical process by which connective proteins, known as microtubule associated proteins (MAPs) influence or orchestrate the quantum processing of the microtubules. It is perhaps necessary to recap the various objections to this complicated theory. Penrose's take on the Gödel theorem is rejected by many philosophers, logicians and artificial intelligence (robotics) researchers. His proposal for objective reduction is distinct from anything else in physics. The main objection to the Hameroff side of the theory is that any quantum feature in the environment of the brain would undergo wave function collapse (reduction) as a result of interaction with the environment, far too quickly for it to have any influence on neural processes.

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The interiors of neurons alternate between liquid (solution: sol) states and quasi-solid (gelatinous: gel) states. In the gel state, water molecules which are electrical dipoles, are ordered, or orientated in the same direction, along the outer edge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. The tubulins also have a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Furthermore, it was suggested that the microtubules could be pumped into a coherent state by biochemical energy. Finally, it is suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction. There is little existing evidence to directly support the Orch OR theory, although a paper in the journal Nature in 2007, claiming evidence for quantum coherence in the photosynthetic systems of plants has been seen as a possible indicator for quantum coherence in biological tissue. M. Jibu et al. [20] also proposed that the conscious process in the brain is related to the macroscopic condensates of massive evanescent photons generated by the Higgs mechanism, arising from dynamical effects of electromagnetic interaction among electric dipoles in biological systems, which was studied by Del Guidance et al. [25]. They claimed that human consciousness could be understood as arising from those creation-annihilation dynamics of a finite number of evanescent (tunneling) photons in the brain. They also considered that each microtuble was a coherent optical encoder in a dense microscopic optical computing network in the cytoplasm of each brain cell, acting as a holographic information processor. G. Vitiello [26] proposed that consciousness mechanisms in the brain was achieved by the condensation of collective modes (called Nambu-Goldstone bosons) in the vacuum. Faber et al. [27] proposed a model of the mind based on the idea that neuron microtubles could perform computations and they estimated the

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favorable condition for storage and information processing, which was found at temperatures close to the human body.

Difficulties of the Orch OR Model Proposed by Penrose The wave or superposition form of the quanta is referred to as being quantum coherent. Interaction with the environment results in decoherence otherwise known as wave function collapse. It has been questioned as to how such quantum coherence could avoid rapid decoherence in the working conditions of the brain. With reference to this question, a paper by the physicist Max Tegmark refuting the Orch OR model and published in the journal Physical Review E, is widely quoted [28]. It was also found by Unruh [29] that the time required in quantum computation must be less than the thermal time scale  / k B T , which yields

2.6 1014 sec at the room temperature ( T  300K ). This value is too small for the conventional silicon processors to conduct a single gate operation. Tegmark developed a model for time to decoherence, and from this calculated that microtubule quantum states would persist for only 10−13 seconds at brain temperatures, far too brief to be relevant to neural processing [28]. Tegmark published a refutation of the Orch-OR model in his paper [28] that the time scale of neuron firing and excitations in microtubles was slower than the decoherence time by at least a factor of 1 / 1010 . According to his paper, it is reasonably unlikely that the brain functions as a quantum computer at room temperature. In their reply to his paper, also published in Physical Reviews E, the physicists, Scott Hagan and Jack Tuszynski and Hameroff [30] claimed that Tegmark did not address the Orch OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24

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nanometers (billionths of a meter) rather than the much smaller separations stipulated for the Orch OR model. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark’s, but still well short of the 25ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch OR suggested. Hagen, Hameroff, and Tuszynski have claimed that Tegmark based his calculations on a model which was different from the Orch-Or model and Tegmark’s assumptions should be amended [30]. Subsequently to it, Hameroff and Tuszynski proposed an idea [31] that microtuble subunit tubulins underwent coherent excitations, which led to the automatic sequence, where quantum coherence superposition was emerged in certain tubulins and consciousness was occurred in the brain. But some researches suggest that the brain requires quantum computing for perception [32], which means that the human brain must work as a quantum processing system. Hameroff proposed the idea in his paper [24] that the biological information processing could be occurred by computer-like transfer and resonance among subunits of cytoskeletal proteins in microtubles. But the need to maintain macroscopic quantum coherence in a warm wet brain is certainly a serious problem for the Penrose-Hameroff model. But the need to maintain macroscopic quantum coherence in a warm wet brain is certainly a serious problem for the Penrose-Hameroff model.

High Performance Computation in the Brain Utilizing Tunneling Photons Contrary to researches mentioned above, we postulate that the human brain is a quantum computer system operated by superluminal evanescent photons.

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Figure 11 shows the table of processing speed per CPU of the PowerPC. From this data, the ratio of energy loss/processing speed can be roughly estimated as 2.9 J/s/G-Flops.

Figure 11. Power loss inside the CPU of Power PC.

Supposing that the computational speed of the human brain is 2  1016 operations/sec, the energy of which required for computation by the PowerPC can be estimated as 5.8 107 J/s from Figure 11. As the human brain consumes an estimated energy of 500 kcal per day (= 24.2 J / s) , the energy ratio of the human brain and the silicon processor become

R  4.2  107 , which is similar to the calculation obtained for QTP gates from R  E*  /  E  1 /  (   1) [33].

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Figure 12. The human brain consumes an estimated energy of about day.

Figure 13.

L

353

500 kcal

per

and the wavelength of superluminal evanescent light.

Hence, it is seen that the human brain consumes much less energy than the conventional silicon processors, and it is considered that the human brain is an efficient computational system utilizing superluminal evanescent photons.

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From Eqs.(8) and (12), it can be seen that the biological brain has the possibility to perform high efficient computation up to 20 qubits more than conventional silicon processors with the same energy dissipation when satisfying L  L , as shown in Figure 13 for the infrared light frequency, the wavelength of which is   100m , where photons would propagate losslessly in a microtubule in the infra-red spectrum region [34], if we let d  15nm , which is the same order as the extracellular space between the brain cells. This calculation result suggests that the human brain can perform computational processes more efficiently in a high temperature environment than the conventional silicon processors. Tegmark calculated that microtubles in the warm and wet brain would cause decoherence on the order of   1013 sec [28], which is slower by a factor at least 1010 than the time scale of neuron firing,   103 ~ 104 sec, from

D 2 mk BT   , Ngqe2

(13)

where D is the tubulin diameter, Boltzmann constant, g  1 / 4 0 ,

m qe

is the mass of an ion, k B is the

is an ionic charge and N  Q / qe .

Contrary to his calculation, it can be shown that a quantum processor utilizing superluminal evanescent photons can perform computation to satisfy the time scale required for quantum computation in microtubles based on the following considerations. The decoherence time of the quantum system [35] can be given by

D 

2

1  0 (E ) 2

,

(14)

Brain - Quantum Hypercomputing System where

0

355

is a certain time-scale to measure the strength of time

uncertainties. From Eqs.(10) and (11), the ratio of decoherence times of quantum computation utilizing evanescent (tunnelling) photons and the conventional quantum computation becomes

 D /  D  ( E0 / E0 )2  [ (  1)]2 .

(15)

When we let  D  2.6  10 14 , which is the time required in quantum computation at the room temperature, we have

 D   D  [  (   1)]2  0.03sec

for   100m [15].

This satisfies the decoherence time, 105 ~ 104 sec , which is required for conducting quantum computation estimated by Hagen, Hameroff and Tsuzynski [30], and which also satisfies the time scale of neuron firing given by   103 ~ 104 sec. Conventionally it is supposed that the brain seems far too warm for quantum computation apparently running into the problem of a decoherence time, which would persist for only 1013 seconds at brain temperatures. But recent evidence shows that quantum processes in biological molecules are actually enhanced at higher temperatures [36]. Thus it is considered that coherent quantum states may be preserved in microtubules at room temperature by superluminal photons, which attain high efficient computation compared with the conventional silicon processors without the mechanism of quantum gravity proposed by Penrose [37]. The warm and wet inner environment of the brain does not allow any long-time entanglement and superposition of two functional units from the conventional physical mechanism, as pointed by the Tegmark’s calculation result and thus we must search for other mechanism, for example, via superluminal photons called tachyons.

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If this mechanism is true for the brain function, we can obtain the possibility to realize much more efficient computer systems like a human brain by utilizing evanescent photons. In conclusion, on the basis of the theorem that the evanescent photon is a superluminal particle, the possibility of high performance computation in biological systems has been studied. From the theoretical analysis presented in this chapter, it is shown that the biological brain has the possibility to achieve large quantum bits computation at the room temperature compared with the conventional processors. Hence it is considered that the human brain can attain high efficient computation compared with the silicon processors as shown in Figure 13. However, still remain the questions: how to determine a qubit in the brain, how it is related to a functioning neuron and how to determine the difference between memory registers and processing units in the brain. These questions must be clarified by further researches.

HOLOGRAPHIC MEMORY IN HUMAN BIOLOGICAL SYSTEMS Daniel Pollen and Michael Trachtenberg [38] proposed the holographic brain theory [39] to help explain the existence of photographic memories in some people. They suggested that such individuals had more vivid memories because they somehow could access a very large region of their memory holograms. S. R. Hameroff suggested in his paper that a centriole cylinder composed of microtubules functions as a waveguide for the evanescent photons for quantum signal processing [24]. Georgiev also proposed the idea that consciousness can be the result of quantum computation via applied laser-like pulses in quantum gates within the brain cortex [40]. Subsequently Georgiev concluded that this mechanism cannot be used for manipulation of the qubits inside the microtubule cavities, or centrioles, because the photons wavelength is two orders of magnitude longer than the size of these centrioles [41]; super radiant photons in the microtubule cavities could have

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wavelength of   100nm or more suggested by Smith [42], incompatible with the length of a moderate sized microtubule cavity, which is about 1nm. Therefore super radiant emissions could not be used to signal qubits in a fashion similar to standing wave lasers in an ion trap computation. As an alternative mechanism the author proposes that the cylinder formed by the microtubule cavity may have a negative refractive index, similar to a metamaterial [43]. Typically metamaterials are artificial materials, engineered to have a negative refractive index, a property that is not normally found in nature [44]. They usually gain their properties from structure rather than composition, using microscopic inhomogeneities to create an effective macroscopic behavior. Negative refractive index materials appear to permit the creation of super lenses which can have a spatial resolution below that of the wavelength [45]. If the inner medium of the cylinder of microtubules possesses characteristics of a negative refractive index, the generation of evanescent photons is enhanced, and they propagate without loss inside the neurons according to the properties of a metamaterial.

Mechanism of Holographic Memory Based on Evanescent Superluminal Photons in the Microtubule Due to the high performance of quantum computation, there is considerable research into how quantum computing could explain the performance behaviors of human brains, including consciousness itself. As Feynman proposed, the optical computing network is the most realizable quantum mechanical computer and Hameroff has suggested that microtubule cavities in the brain can act as waveguides for photons and as holographic processors [16]. The cytoskeleton of biological cells, including neurons of the brain, are made up of microtubules. Microtubules are comprised of subunits of the protein, named tubulin [17]. Each microtubule is a hollow cylindrical tube of tubulin proteins as shown in Figure 14, whose outer core diameter is 25

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nm. Each tubulin molecule within each microtubule can function as a switch between two conformations that exist in a quantum superposition of both conformational states [18]. The dimensions of centrioles are close to the wavelengths of light in the infrared and visible spectrum such that they may act as phase coherent, resonant waveguides [34].

Figure 14. Structure of microtubules forming a centriole cavity.

According to Jibu et al. [10, 20, 21], microtubule quantum states link to those of other neurons by quantum coherent photons tunneling through the membranes in biological systems and the cytoskeletal protein conformational states are entangled by these photons which form coherent domains in their interaction with the local electromagnetic field. They claimed that human consciousness could be understood as arising from those creation-annihilation dynamics of a finite number of evanescent (tunneling) photons in the brain. E. Recami [46, 47] claimed in his papers that tunneling photons traveling in an evanescent mode can move with superluminal group speed, which can be shown as follows; The evanescent photon generated in a quantum domain satisfies the following Klein-Fock-Gordon equation given by

Brain - Quantum Hypercomputing System

 1 2 m02 c 2  2  2 2    2  ( x, t )  0 ,    c t where

359

(16)

c is the light speed, m 0 is an absolute value of the proper mass of the

evanescent photon and  is the Planck’s constant divided by 2 . This equation gives the solution for the photon traveling in an evanescent mode as

 Et  px   ( x, t )  A0 exp ,   

(17)

which corresponds to the elementary particle with an imaginary mass im0 that travels at a superluminal speed satisfying

E 2  p 2 c 2  m02 c 4 ,

(18)

where E is the energy of the superluminal particle and p is its momentum. Hence, it can be seen that tunneling photons traveling in an evanescent mode can move at a superluminal speed. From the assumption that the evanescent photon is a superluminal particle, the author has shown that a microtubule in a biological brain can achieve quantum bit (qubit) computations on large data sets, which would account for the high performance of the computations as compared with the conventional processors [15]. It therefore seems highly plausible that macroscopic quantum ordered dynamical systems of evanescent photons in the brain could play an essential role in realizing long-range biological order in living systems. Ziolkowski pointed out in his paper that superluminal pulse propagation which permits consequent superluminal exchange without a violation in causality is possible in electromagnetic metamaterials [48].

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Metamaterials are artificial materials engineered to have properties that may not be found in nature. They are assemblies of multiple individual elements fashioned from conventional microscopic materials such as metals or plastics, but the materials are usually arranged in periodic patterns as shown in Figure 15.

Figure 15. Structure of the electromagnetic meta-material (left-handed metamaterial configuration consisting of copper split-ring resonators and wires).

Metamaterials gain their properties not from their composition, but from their exactingly-designed structures. Their precise shape, geometry, size, orientation and arrangement can affect the waves of light or sound in an unconventional manner, creating material properties which are unachievable with conventional materials. These metamaterials achieve desired effects by incorporating structural elements of sub-wavelength sizes, i.e., features that are actually smaller than the wavelength of the waves. Negative refractive index materials appear to permit the creation of super lenses that can have a spatial resolution below that of the wavelength. If the inner medium of a cylinder of microtubules possesses the characteristics of a negative refractive index, the generation of evanescent photons is enhanced, and they propagate without loss inside the neurons according to the properties of a metamaterial. If the inner medium of a cylinder of microtubules possesses the characteristics of a metamaterial with negative refractive index, tunneling photons will propagate losslessly inside the neurons in a way that is not

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restricted by wavelength, and therefore infrared photons could be used for the manipulation of qubits in the brain.

Figure 16. Similarity between the metamaterial (left figure) and the microtubule (right figure).

Similar to Prof. Hameroff’s idea, Dr. Georgiev presented an idea that consciousness could be the result of quantum computation via short laserlike pulses controlling quantum gates within the brain cortex. However he later rejected this theory because the wavelength of super radiant photon emission in the infrared spectrum is two orders of magnitude longer than any sized microtubule cavity [40]. Thus if one proposes this mechanism, infrared photon cannot account for manipulation of the quantum qubits inside the microtubule cavity. To avoid this problem, we suggest the substance in the microtubule cylinder has the characteristics of a metamaterial composed of sub-wavelength structures. As evanescent waves inside the microtubule cavity waveguide can propagate below the cutoff frequency defined by f c  c / 2d , where d is a diameter of the waveguide, thus we can model the evanescent wave according to the theory of evanescent wave holography [49] shown as follows; The basic idea of evanescent wave holography is to use an evanescent wave as a reference wave and as a readout wave [50]. Suppose that the field distribution of a wave propagating in a microtubule is periodic inside the core, and evanescent outside this region given by

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Takaaki Musha

U e ( x, z )  U e exp( ik e x) exp( k e z ) ,

(19)

and

U p ( x, z)  U p exp[ i(k p x  k p z)] , where

Ue

is an evanescent wave and

U p is

(20)

a propagating wave from

outside of the microtubule waveguide as shown in Figure 17.

Figure 17. Recording of evanescent wave hologram in the microtubule waveguide.

As shown in Figure 17,

Up

interferes with

Ue

and this interference is

recorded. This interference in the storage material will produce a proportional variation in the index of refraction, which will be periodic along the length of each microtubule. From Eqs. (19) and (20), the intensity of the wave can be given by

 



1 2 U p  U e2 exp( 2k ez z ) 2 . *  2 Re U e U p exp( i[k px  k ex ) x  k pz z ]) exp( k ez z ) I



(21)

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363

We suggest the cylinder of the microtubule consists of a storage material, whose thickness is much smaller compared to the penetration depth of evanescent waves. The response of the recoding medium to this intensity distribution is that the dielectric constant  is a function of the local intensity given by   f ( I   ) , where  is an exposure time.

Then the total field, consisting of an illuminated wave and a diffracted wave, can be described by

   U     (r )U  0 , c 2

2

(22)

and thus the wavelet scattered by each volume element of the hologram can be obtained using Green’s function and integrated over the volume of the hologram, according to the integral equation for hologram reconstruction, given by

       U (r )  U 0 (r ) exp(in  r / r ) (r )  dr , where of

U0

(23)

is an illuminating wave, U is a wave including the information

object

wave

field

and



is

a

parameter

satisfying

[2   2 ]G(r, r )   ( r  r ) . This means that the guided readout wave is propagating in the same direction as the reference wave during recording. The readout wave is diffracted by the hologram structure and the object field can be reconstructed. This is the same mechanism of holographic memory of the brain, which was proposed by Pribram et al. [39]. Figure 18 (a) shows a schematic diagram of the reconstruction of stored memory by the propagating evanescent wave inside the microtubule structure. As shown in this figure, an illumination with the reference wave U r reconstructs the objective wave field U o by interaction with the term

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U rU o in the dielectric structure of the physical substrate for memory inside

the microtubule structure. By this mechanism, the evanescent photon propagating inside the microtubule can manipulate the storage and provide for the retrieval of stored data, or memory, as shown in Figure 18 (b). The holographic mechanism would also explain how our brains could store so many memories in so little space. If the microtubules store memories using evanescent wave holography, this suggests that impairment of consciousness such as Alzheimer’s disease may be due in part to the loss of metamaterial characteristics of the substructure inside the microtubule within the brain.

a

b Figure 18. (a) Reconstruction of wave field by the guided readout wave. (b) Microtubule structure and the reconstruction of the hologram image.

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365

Furthermore, a new computer hardware system could be constructed using a similar technique for data storage and retrieval, with memories which rely on interfering waves of photons passing through multiplexed holographic gratings similar to the brain’s neural structure; i.e., an artificial holographic brain system could now be constructed using this same mechanism. A holographic theory of human memory has now been proposed which is based on the use of evanescent superluminal photons for recording and retrieving holographically stored qubits. We have now theoretically established the microtubule structure inside the biological brain is capable of storing memories as holograms, using microtubule substrate as the storage material, if the interior of the microtubule cavities has the characteristics of a metamaterial composed of sub-wavelength structures.

Holonomic Model of the Brain Function Thus microtubules have both the ability to store memories and conduct quantum computation as shown in Figure 19. As the microtubules are not just restricted to inside the brain, but distributed over all of cells within the human body, we can consider consciousness to be distributed throughout the human body.

Figure 19. Microtuble functions, quantum computation and the memory.

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David Bohm believed that objective reality does not exist and, despite its apparent solidity, the Universe is at heart a phantasm, a gigantic and splendidly detailed hologram. Bohm was involved in the early development of the holonomic model of brain function, a model for human cognition which is drastically different from conventionally accepted ideas. Bohm developed the theory that the brain operates in a manner similar to a hologram, in accordance with quantum mathematical principles and the characteristics of wave patterns [51]. To understand why Bohm makes this startling assertion, one must first understand that a hologram is a three- dimensional photograph made with the aid of a laser. To make a hologram, the object to be photographed is first bathed in the light of a laser beam. Then a second laser beam is bounced off the reflected light of the first and the resulting interference pattern (the area where the two laser beams conflate) is captured on film. When the film is developed, it looks like a meaningless swirl of light and dark lines. But as soon as the developed film is illuminated by another laser beam, a threedimensional image of the original object appears. This is similar to the plot of the 1999 Holywood film called “Matrix.” This film depicts a future world in which reality, as perceived by most people, is actually a simulated reality called “the Matrix,” created by sentient machines to subdue the human population.

Figure 20. Is our world the hologram created by our brains?

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The premise of “The Matrix” is related to Plato's Allegory of the Cave. According to Plato's theory of Forms, the true essence of an object is not what we perceive with our senses, but rather its quality. Plato compares people uneducated in this theory to being chained in a cave. A fire glows behind them and they see the shadows of objects cast on the wall, but not the actual objects themselves. These people perceive the shadows as reality and thus do not know the true form of the objects, and therefore, are confined to this false perception. What we observe and accept as reality, may be just such a false perception.

HYPERCOMPUTING BY SUPERLUMINAL PARTICLES Supposing that there exist superluminal particles, it is possible to make an infinite computation within a finite time. An accelerated Turing machine named a Zeno machine is a hypothetical computational model which can perform the countable infinite number of computational steps within a finite time. However it cannot be physically realized from the standpoint of the Heisenberg uncertainty principle, because the energy to perform the computation will be exponentially increased when the computational step is accelerated. In mathematics and computer science, an accelerated Turing machine is a hypothetical computational model related to Turing machines which can perform the countable infinite number of computational steps within a finite time. It is also called a Zeno machine which concept was proposed independently by B.Russdel, R.Blake and H.Weyl, which performs its first computational step in one unit of time and each subsequent step in half the time of the step before, allowing an infinite number of steps can be completed within a finite interval of time [52, 53]. However this machine cannot be physically realized from the standpoint of the Heisenberg uncertainty principle, Et   , because the energy to perform the computation will be exponentially increased when the computational step is accelerated. Thus it is considered that the Zeno machine is a mere mathematical concept and there is no possibility to realize it in a physical

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world. Contrary to this conclusion, the author shows the possibility to realize it by utilizing superluminal particles instead of subliminal particles, including photons.

Computational Time Required to Perform Infinite Steps of Computation Feynmann defined the required energy per step for the computation as shown in Figure 21, given by [1]

energy per step  kBT

f b , ( f  b) / 2

(24)

where k B is Boltzmann’s constant, T is a temperature, f is a forward rate of computation and b is backward rate.

Figure 21. Computational steps necessary to perform reversible computation.

Supposing that there in no energy supply and parameters f and b are fixed during the computation, we can consider the infinite computational steps given by

E1  kE0 , E 2  kE1 , ・・ E n  kEn 1 ・・,

(25)

Brain - Quantum Hypercomputing System where we let the initial energy of computation be

k  2( f  b) /( f  b) , and

En

From the above we have

369 E0  k B T ,

is the energy for the n-th step computation.

En  k n E0 , and then the energy loss for each

computational step becomes

E1  E0  E1  (1  k ) E0 E2  E1  E2  (1  k )kE0 

(26)

En  En1  En  (1  k )k n1 E0 . According to the paper by S. Lloyd [54], it is required for the quantum system with average energy E to take time at least t to evolve to an orthogonal state given by

t 

 2 E

,

(27)

from which, the total energy for the infinite steps yields

E0

if setting

E  Ei into Eq.(27), then the total time for the computation with infinite

steps becomes 

 

n 1

2 E0

T   t n 



1

 (1  k )k n 1

n 1

.

(28)

As the infinite sum of Eq.(28) diverges to infinity when satisfying 0  k  1 , the Feynman model of computation requires infinite time to complete the calculation (see Figure 22).

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Figure 22. Time required to conduct computation at each step (for the case, and   1.0 ).

k  1/ 2

Hence an accelerated Turing machine cannot be realized for computers utilizing ordinary particles due to the uncertainty principle.

Computational Time by Using Superluminal Elementary Particles E.Recami claimed in his paper [46] that tunneling photons which travel in evanescent mode can move with superluminal group speed inside the barrier. Chu and S.Wong at AT&T Bell Labs measured superluminal velocities for light traveling through the absorbing material [55]. Furthermore Steinberg, Kwait and Chiao measured the tunneling time for visible light through the optical filter consisting of the multilayer coating about 10 6 m thick [56]. Experimental results obtained by Steinberg and coworkers have shown that the photons seemed to have traveled at 1.7 times the speed of light. Recent optical experiments at Princeton NEC have verified that superluminal pulse propagation can occur in transparent media [57]. These results indicate that the process of tunneling in quantum physics is superluminal, as claimed by E.Recami [46].

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371

From the relativistic equation of energy and momentum of the moving particle, given by

E

m0 c 2 1 v2 / c2

,

(29)

,

(30)

and

p

m0 v 1 v2 / c2

the relation between energy and momentum can be shown as

p / v  E / c2.

From which, we have [58]

vp  pv E  2 . v2 c

(31)

Supposing that the approximation v / v 2  0 holds, Eq.(31) can be simplified as

p 

v E . c2

(32)

This relation is also valid for the superluminal particle (which has an imaginary mass im* ), the energy and the momentum of which are given by following equations, respectively.

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Takaaki Musha

m* c 2

E

v2 / c2 1

,

m* v

p

v2 / c2 1

(33)

.

(34)

According to the paper by M.Park and Y.Park [59], the uncertainty relation for the superluminal particle can be given by

p  t 

 , v  v

(35)

where v and v  are the velocities of a superluminal particle after and before the measurement. By substituting Eq.(32) into (35), we obtain the uncertainty relation for superluminal particles given by

E  t 

 , (  1)

(36)

when we let v   c and   v / c . Instead of subluminal particles including photons, the time required for the quantum system utilizing superluminal particles becomes [58]

  1 T   t i   2 E0 n 1  n (  n  1)(1  k )k n 1 n 1 

,

(37)

from the uncertainty principle for superluminal particles given by Eq.(36), where  can be given by

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373

m*2 c 4 m*2 c 4 n  1  1  2n 2 , E n2 k E0

(38)

which is derived from Eq.(34). From Eqs.(37) and (38), it is seen that the computation time can be accelerated as shown in Figure 23.

Figure 23. Time required to conduct computation at each step by using superluminal particles (for the case, k  1 / 2 and   1.0 ).

By the numerical calculation, it can be shown that the infinite sum of Eq.(37) converges to a certain value satisfying 0  k  1 , as shown in Figure 24. In this figure, the horizontal line is for the parameter   m*c / E0 and 2

the vertical line is for the time to complete infinite step calculations. From these calculation results, an accelerated Turing machine can be realized by utilizing superluminal particles, instead of subluminal particles, for the Feynman’s computational model. Thus, contrary to the conclusion obtained relatively to the Feynman’s model of computation when using ordinary particles, it can be seen that

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superluminal particles permits the realization of an accelerated Turing machine.

Figure 24. Computational required time for the superluminal particles.

It is known that an accelerate Turing machines allow us to be computed some functions which are not Turing-computable, such as the halting problem [60], described as “given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever.” This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever. Halting problem for Turing machines can easily solved by an accelerated Turing machine using the following pseudocode algorithm (see Figure 25). As an accelerated Turing machines are more powerful than ordinary Turing machines, they can perform computation beyond the Turing limit which is called hypercomputation, such as to decide any arithmetic statement that is infinite time decidable. From this result, we can construct an oracle machine [61] by using a superluminal particle, which is an abstract machine used to study decision problems. It can be conceived as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation.

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Figure 25. Pseudocode algorithm to solve the Halting problem.

An oracle machine can perform all of the usual operations of a Turing machine, and can also query the oracle to obtain a solution to any instance of the computational problem for that oracle. For example, if the problem is a decision problem for a set A of natural numbers, the oracle machine supplies the oracle with a natural number, and the oracle responds with "yes" or "no" stating whether that number is an element of A. Given a device that tell you in advance whether a given computer program would halt, or go on running forever, you would be able to prove or disprove any theorem whatsoever about integers: the Goldbach conjecture, Fermat’s Last Theorem, and the famous Riemann hypothesis because it is equivalent to the following problem: 2

 1 n2  1 3  n  0 ,      36n , where  (n)   . d n d  k  ( n ) k 2 

(39)

You would simply show this “Oracle” a program that would loop through all the integers, testing every possible set of values and only halting if it came to a set that violated the conjecture. By using the superluminal particle, there is a possibility to realize the hyper-computational system. As an accelerated Turing machines are more powerful than ordinary Turing machines, they can perform computation

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beyond the Turing limit, which is called hypercomputation, such as to decide any arithmetic statement that is infinite time decidable.

HUMAN INTUITION FROM THE STANDPOINT OF SUPERLUMINAL HYPERCOMPUTATION According to the stochastic electrodynamics, it can be shown that a tachyon field can be spontaneously created from the sea of omnipresent zero-point-energy field pervading all of the Universe. If the brain has the possibility to correlate with the outer tachyon field created from other living organism and cosmic zero-point field, the totality of them, we may call it a universal brain or a cosmic consciousness, can exert an influence on the individual mind and this can explain the extraordinary ability of human biological brains. To further interpret this result, we consider S.Berkovich suggestion of a “cloud computing paradigm,” in which is given an elegant constructive solution to the problem of the organization of mind. Within his article, he defines a situation where individual brains are not stand-alone computers but collective users whom have shared access to portions of a holographic memory of the Universe [62]. He proposed that the cosmic microwave background (CMB) radiation has nothing at all to do with the residual radiation leftover from the Big Bang. Instead, he claimed that CMB radiation is nothing but noise from writing operations in the holographic memory of the Universe. Such holographic write operations would require some type of universal clocking rate for these operations. Since the virtual superluminal particle pairs are created and annihilated in the vacuum within a short, finite period of time according to the uncertainty principle, we could logically consider this duration as the clock rate for these operations. There are some papers on the hypothesis that the human mind consists of evanescent tunneling photons, which has a property of superluminal particles called tachyons [11, 15, 33, 63]. If the human consciousness

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consists of superluminal particles, as claimed by Prof. Dutheil, the superiority of the human brain to conventional silicon processors may be explained. He proposed a new set of hypothesis based on superluminal consciousness:  

The brain is nothing more than a simple computer that transmit information. Consciousness, or the mind, is composed of a field of tachyons or superluminal matter, located on the other side of the light barrier in superluminal space-time.

Professor Dutheil presented his hypothesis in his book, “L’homme superlumineux” [64], proposing that consciousness is a field of superluminal matter belonging to the true fundamental Universe, as shown in Figure 26, and that our world is merely a subluminal holographic projection of it.

Figure 26. Structure of the Universe from the standpoint of superluminal particles.

As shown in this chapter, an accelerated Turing machine can be realized by using superluminal particles instead of subluminal particles including photons, and it is considered that the tachyon field outside the human brain

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acts as an oracle, and this enables us to know the mathematical problem, whether it is true or false. From the theoretical analysis, it is seen that an accelerated Turing machine can be realized by using superluminal particles from the standpoint of quantum mechanics. Thus an extraordinary capability of human brains such as intuition compared with the ordinary silicon processors might be explained if they are composed of superluminal photons, because they have a capability to function beyond the ordinary Turing machines. Figure 27 shows the similarity between the computer system and the human brain.

Figure 27. Similarity between the computer system and the human brain.

According to the hypothesis proposed by Prof. Dutheil, the brain is nothing more than a simple computer that transmits information. Human brain receive the program to recognize the world from the Cosmic brain as the terminal computers receive their commands from the

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central processor, and thus it can be considered that we can recognize the world as the three dimensional reality. Figure 28 shows the structures of the brain cell and the Universe. It shows the similarity of their structures. From which, it is considered that there is a possibility that the Universe is a huge brain operated by superluminal particles.

Figure 28. Similar structure of the brain cell and the Universe. (Is our Universe a huge brain?)

DISCUSSION AND CONCLUSION In this chapter, it has been shown that the biological brain possibly utilizes large quantum bit computation at room temperature and thus is not synonymous with conventional processors. Due to the non-locality characteristic of superluminal photons, the organism’s coherence goes beyond the coherence of the physical biological system. This non-local coherence is what gives the human brain such high performance when compared to silicon digital processors. Very early, A. Einstein showed particles with velocities greater than the velocity of light in vacuum may produce causal anomalies. Later, in quantum mechanics CPT transformations have allowed causal loops at a microscopic scale. So there is reason to analyze the possibility of faster-than-

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light particles again. Meta-Relativity has extended the special theory of Relativity to particles beyond the light barrier (tachyons), by using complex values in the relativistic formula. Meta-Relativity assigns to any tachyon an imaginary proper mass which has no readily apparent physical interpretation. In the framework of that theory, tachyons may appear to travel backwards in time and have negative energies, but they still must also be interpreted as traveling forwards in time with positive energies (reinterpretation principle). Meta-Relativity allows a tachyon reflection or re-emission to produce a causal loop, but some authors reject this objection by postulating tachyon emission cannot be systematically repeated. So causal loops can only occur at a microscopic scale. The theory of Relativity in the spacelike region has been developed by R. Dutheil using the tensor formalism of the general theory of Relativity. He defined tachyonic referential frames (TRF) with another metric tensor and showed it leads to another Lorentz group of transformations-the superluminal Lorentz group. In this theory, tachyons always have a positive energy and a real proper mass, but their behavior must be described with tachyonic referential frames.

Figure 29. Ramanujan: The man who can communicate with the Universal mind.

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R. Dutheil argued from the isomorphism of both Lorentz groups that Zeeman's theorem can be proven to be respected by tachyons; so a sequence order is always preserved by any superluminal transformation. In his present communication, he showed time coordinates of tachyonic referential frames do not preserve causal order and do not make sense for natural observers. Nevertheless, he showed the causal order is preserved within the superluminal proper time of tachyons, which can be related to the proper time of any natural observer. As regards to Ramanujan (see Figure 29), who was an Indian mathematical genius, he found many mysterious mathematical formulas related to Number Theory shown in the right part of Figure 29. He often said that “an equation for me has no meaning, unless it represents a thought of God,” that is a Universal mind. Another case is of the twins, John and Michael, mentioned by Olver Sacks in his book, “The Man Who Mistook His Wife for a Hat.” The twins, John and Michael, were idiot savants who exhibited a mysterious human ability with primes using some unknown, unconscious algorithm. They seemed to have a peculiar passion and grasp of numbers even though they could not do simple mathematical calculations, and lacked even the most rudimentary powers of arithmetic. In front of Dr.Sacks, they exhibited an extraordinary ability to recognize an eight-digit number told to them, was a prime number, after some unimaginable internal process of testing. After some time, the twins were able to produce twenty-figure primes, which is difficult even for computers if one uses Eratosthenes’s sieve or any other algorithm. There is no simple method of calculating primes. He supposed that they visualized prime patterns instead of utilizing calculation, but the riddle of how they visualized the primes and used them for communication to each other remains unanswered. According to stochastic electrodynamics, it can be shown a tachyon field can be created from the sea of omnipresent zero-point-energy field in the Universe. If the brain can possibly correlate with a universal tachyon field created from all other living organisms and the cosmic zero-point field, the totality of which we may call a universal brain or a cosmic consciousness, then this cosmic consciousness could be exerting an influence upon an

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individual mind. This could be the explanation for the extraordinary abilities of the human mind, such as the enigmas of Ramanujan and the John and Michael twins. Thus it is considered that the true mechanism of the human electromagnetic field may be similar to that of the Universe as a whole, according to the statement, “as above, so below.” Assuming a computing device utilizing superluminal particles could actually be built, what could you do with such a superluminal computer? Such a device would fall into a class of processing machine known as hypercomputers. These are hypothetical devices, more powerful than Turing machines, that allow non-Turing computations. They were first discussed by Alan Turing in the 1930s. In theory, hypercomputers can compute certain kinds of otherwise noncomputable functions. That sounds handy but even though there are uncountable non-computable functions, it’s actually quite hard to come up with an example of one that might seem useful. O.Finkel proved that some basic questions on automata reading infinite words depended on the model of the axiomatic system ZFC [65]. One of possibilities is; “There is a model of ZFC in which the complement L( A) has a cardinal 1 with

0  1  20 ,” which means there is a possibility of computation with a higher hierarchy than ordinary Turing machines. We can suppose it is possible higher order computation might be conducted in the world of a superluminal Universe. It was shown by the hypothesis of Prof. Dutheil, human consciousness may be composed of a field of tachyons or superluminal matter located on the other side of the light barrier, in superluminal space-time, and would have the capability to conduct infinite steps of computation within a finite time. By applying quantum mechanics, the author has shown it may be possible to realize a hypercomputational system capable of functioning beyond an ordinary Turing machine, by information processing conducted inside microtubule structures of neurons, utilizing concepts of superluminal particles.

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If such a hypercomputer as shown in Figure 30 can be realized, it will change the world completely as we pass the singularity point of computer technology.

Figure 30. Is there a possibility to realize the hypercomputer like a brain?

REFERENCES [1] [2]

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[16] Jibu, M., S. Hagan, S. R. Hameroff; K. H. Pribram and K. Yasue, Quantum optical coherence in cytoskeletal microtubules: implications for brain function, Biosystems, 32 (3), 1994; 195-209. [17] Satinover, J., The Quantum Brain, John Wiley & Sons, Inc., New York, USA, 2001. [18] Albrecht-Buehler, G., Rudimentary form of cellular “vision,” Proc. Natl Acad Sci. USA, 89(17), 1992; 8288-8292. [19] Hameroff, S., J. Tuszynski, Quantum states in proteins and protein assemblies: The essence of life?, SPIE Conference, Grand Canary Island, May, 2004. [20] Jibu, M., K. Yasue, What is mind?-Quantum Field Theory of Evanescent Photons in Brain as Quantum Theory of Consciousness, Informatica, 21, 1997; 471-490. [21] Jibu, M., K. Yasue, S. Hagan, Evanescent (tunneling) photon and cellular vision, Biosystems, 42, 1997; 65-73. [22] Hameroff, S. R., J. Tuzynsli, Quantum states in proteins and protein assemblies: The essence of life?, SPIE Conference, Grand Canary island, May, 2004. [23] Hameroff, S. R., and R. Penrose, Conscious Events as Orchestrared Space-Time entanglement, Selections, Journal of Consciousness Studies, Vol.3, 1996; 36-53. [24] Hameroff, S. R., Information Processing in microtubules, J. Theor. Biol. 98, 1982; 549-561. [25] Del Guidance, E., S. Doglia and M. Milani, Electromagnetic field and Spontaneous Symmetry Breaking Biological Matter, Nuclear Physics, B257, 1986; 185-199. [26] Vitiello, G., Dissipation and memory capacity in the quantum brain model, Int. J. Mod. Phys. B, Vol.9, Issue.8,1995; 973-989. [27] Faber, J., R. Portugal and L. P. Rosa, Information processing in brain microtubles, Biosystems, 83, 2006; 1-9. [28] Tegmark, M., Importance of quantum coherence in brain processes, Physical Reviews E, Vol.61, 2000; 4194-4206. [29] Unruh, W. G., Maintaining coherence in quantum computers, Physical Review A. Vol..51, No.2, 1995; 992-997.

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[30] Hagan, S., S. R. Hameroff, J. Tuszynski, Quantum computation in brain microtubles: Decoherence and biological feasibility, Physical Review E, Vol.65, 2002; 061901-1-11. [31] Hameroff, S. R., J. J. Tuzynsli, Search for quantum and classical mode of information processing in microtubles (Bioenergitic Organization in Living Systems), World Scientific, Singapore, 2003. [32] Bialek, W., and A. Schweitzer, Quantum noise and the Threshold of Hearing, Physical Review Letters, 54(7), 1987; 725-728. [33] Musha, T., A study on the possibility of high performance computation using quantum tunneling photons, Int. J. Simulations and Process Modeling, Vol.2, Nos.1/2, 2006; 63-66. [34] Hameroff, S. R., A new theory of the origin of cancer: quantum coherent entanglement, centrioles, mitosis, and differentiation, Biosystems, Vol.77, Issues 1-3, 2004; 119-136. [35] Diosi, L., Instrict Time Uncertainties and Decoherence: Comparison of 4 models, Brazilian Journal of Physics, Vol.35, No.2A, 2005; 260265. [36] Ouyang, M., and D. D. Awschalom, Coherence spin transfer between moleculary bridged quantum dots, Science, 301, 2003; 1074-1078. [37] Penrose, R., The Large, the Small and the Human Mind, Cambrigde University Press, Cambridge, 1999. [38] Talbot, M., The Holographic Universe, Harper Perennial, New York, NY, USA, 1991; 23-24. [39] Pribram, K. H., M. Nuwer, R. J. Baron, The Holographic Hypothesis in Memory Structure in Brain Function and Perception, In Contemporary Development in Mathematical Psychology.; Atkinson, R. C.; Krantz, S. H.; Luce, R. C.; Suppes, P., Eds.; W. H. Freeman & Co., San Francisco, USA, 1974; 416-467. [40] Georgiev, D. D., Bose-Einstein condensation of tunneling photons in the brain cortex as a mechanism of conscious action. Available online: http://cogprint/3539/01/tunneling.pdf (31 Mar. 2004). [41] Georgiev, D. D., Quantum computation in the neuronal microtubules: quantum gates, ordered water and superradiance. Available online: arxiv.org/abs/quant-ph/0211080 (16 Apr. 2004).

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[42] Smith, T., Quantum Consciousness. Water, Light speed, and Microtubules. Available online: http://www.innerx.net/personal/ tsmith/QuanCon2.html (31 Mar. 2003). [43] Veselago, V. G., The Electrodynamics of substances with simultaneously negative values of  and  , Soviet Physics Uspekhi, [44] [45]

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Vol. 10, 1968; 509-514. Ung, B., Metamaterials: a Metareview. Available online: www.polymtl.ca/doc/art_2_2.pddf (23 Nov. 2009) Veselago, V., L. Braginsky, V. Shklover and C. Hafner, Negative Refractive Index Material, Journal of Computational and Theoretical Nanoscience, Vol.3; 2006, 1-30. Recami, E., A bird’s-eye view of the experimental status-of-the-art for superluminal motions, Found. of Phys. 31, 2001; 1119-1135. Recami, E., Superluminal tunneling through successive barriers: Does QM predict infinite group velocities?, Journal of Modern Optics, Vol.51, No.6-7, 2004; 913-923. Ziolkowski, R. W., Superluminal transmission of information through an electromagnetic material, Phys. Rev. E, 63(4), 2001; 046604-17. Claus, R. O., Noncontact Measurement of High Temperature Using Optical Fiber Sensors, Final Report, NAG-1-831 (NASA-CR-186975) Virginia Polytechnic Inst. And State Univ., VA, USA, 1990; 40-50. Musha, T., Holographic View of the Brain Memory Based on Evanescent Superluminal Photons, Information, 3, 2012; 344-350. Bohm, D., Wholeness and the Implicate Order, Routledge & Kegan Paul, New York, 1980. Ord, T., The many forms of hypercomputation, Applied Mathematics and Computation, 178, 2006; 143-153. Hamkins, J. D., and A. Lewis, Infinite time Turing machines, Journal of Symbolic Logic, 65(2), 2000; 567-604. Llyod, S., Ultimate physical limit to computation, Nature, vol.406, 2000; 1047-1054. Brown, J., Faster than the speed of light, New Scientist. 146, 1995; 2630.

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[56] Steinberg, A. M., P. G. Kwait and R. Y. Chiao, Measurement of the single-photon tunneling time, Physical Review Letters, 71(5), 1993; 708-711. [57] Wang, L. J., A. Kuzmich and A. Dogariu, Gain-assisted superluminal light propagation, Nature 406, 2000; 277-279. [58] Musha, T., Possibility of Hypercomputation by Using Superluminal Elementary particles, Advances in Computer Science and Engineering, 8(1), 2012, 57-67, also in; Possibility of Hypercomputation from the Standpoint of Superluminal Particles, Theory and Applications of Mathematics & Computer Science, Vol.3, No.2. 2013; 120-128. [59] Park, M., and Y. Park, On the foundation of the relativistic dynamics with the tachyon, Nuovo Cimento, Vol.111B,N.11, 1996; 1333-1368. [60] Kieu, T. D., Hypercomputation with quantum adiabatic processes, Theoretical Computation Science, 317, 2004; 93-104. [61] van Melkebeet, D., Randomness and Completeness in Computational Complexity, (Lecture Notes in Computer Science), Springer, 2000 edition, 2001. [62] Berkovich, S., Obtaining inexhaustible clean energy by parametric resonance under nonlocality clocking; www.chronos.msu.ru/ RREPORT/berkovich_prime_energy.pdf, (21.Sept. 2010). [63] Georgiev, D. D., On the dynamic timescale of mind-brain interaction, Proceeding of Quantum Mind II: Consciousness, Quantum Physics and the Brain, Tucson, Arizona, 15 (March, 2003). [64] Dutheil, R., B. Dutheil, L’homme superlumineux [The superluminous man], Sand, Paris, 1992. [65] O. Finkel, Some Problems in Automata Theory Which Depend on the Model of Set Theory, RAIRO-Theoretical Informatics and Applications, Vol. 45, Issue.04, 2011; 383-397.

In: Frontiers in Quantum Computing Editor: Luigi Maxmilian Caligiuri

ISBN: 978-1-53618-515-7 c 2020 Nova Science Publishers, Inc.

Chapter 10

PARAMETRIC R ESONANCE , PARTICLE S TOCHASTIC I NTERACTIONS WITH A P ERIODIC M EDIUM , AND Q UANTUM S IMULATIONS Mario J. Pinheiro∗ Department of Physics, Instituto Superior T´ecnico - IST, Universidade de Lisboa, Lisboa, Portugal

Abstract A non-markovian stochastic model shows the emergence of structures in the medium, a self-organization characterized by a relationship between particle’s energy, driven frequency ω and a frequency of interaction with the medium ν. The interaction determines its mass and this fine tuning results in an effective force given by FL = h ¯ ω2 n(λ)/c, similar to the interaction force between photons and atoms. Condition for the particlemedium resonance is determined, with relevance to detect dark matter axion-like particles and the parametric resonance as a pop-up mechanism to turn fields into particles. The general mechanism of nonlinear coupling between the medium modes of oscillation and the characteristic frequencies of the medium represent a fully operational mode for quantum information processing and quantum simulation. ∗

Corresponding Author’s Email: [email protected].

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1.

INTRODUCTION

1.1.

Memory Effects in Physical Systems

The study of physical systems with non-markovian statistical properties has provided a natural basis for the understanding of the role played by memory effects in such different fields as anomalous transport in turbulent plasmas [1]; Brownian motion of macroparticles in complex fluids [2]; in the vortex solid phase of twinned YBa2 Cu3 O7 single crystals [3]; simulating the stochastic character of the laser fields [4]; the rate of escape of a particle over a one-dimensional potential barrier [5, 6]. Within a classical approach of an atomic process, we show in this paper that, whenever a particle undergoes a repetitive process, like a jumping process in a surrounding medium, a new type of force is exerted on it, the Lorentz invariant force [7]. The space evolution of a massless particle through a medium incorporates the space-time structure (e.g., topological, fractal) and the nature of motion. In this study we embrace the concept of an information-rich manifold as the most reasonable heuristic framework in regard the non-Markovian propagation of a singularity in a complex manifold. The simple model introduced here consists of a particle moving in a straight line for which we make no assumption about its mass (e.g., it’s an ab initio massless particle), jumping from one site to another in a non-randomly structured field but, in the meanwhile, interacting with it in a stochastic process, and keeping memory of its ”history”. 1.1.1.

Outline of the Non-Markovian Model

In a non-markovian model the prediction about the next link (xn+1 ) is defined in terms of mutually dependent random variables in the chain (x1 , x2 ,...,xn ). Consider a particle jumping from one site to another in Euclidean space - nonMarkovian singularities in a complex manifold. The jumping sites are assumed to be equidistantly distributed along the axis. Now, add to this jumping process an oscillatory motion due to interaction with a medium and characterized by stochasticity. The frequency of oscillation around an equilibrium position between two jumps is denoted by ν, is homogeneous and isotropic (the Zitterbewegungen) and β is the probability that each oscillation in the past has to trigger a new oscillation in the present.

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Our simple dynamical process is introduced in a formal way, by relating it to the probability that one oscillation from the M = m0 + ... + mq−1 which occurred in the past generates m oscillations at the qth step, Qm [q(t)]. Since we assume β is constant, this is an infinite memory model, meaning that an oscillation which has occurred long time ago produces the same effect as an oscillation which has occurred in the near past. Lets introduce the probability density, Qn (t)dt, that the nth oscillation takes place in the interval of time (t, t + dt) at qth step. Then we have the following integral in time Qn+1 [q(t)] =

Z

q(t)

0

Qn [q(t0 )]p0 [(q(t) − q(t0 )]dq(t0 ),

(1)

where p0 (t − t0 ) is the probability per unit time that the (n + 1)st oscillation takes place in the time interval (t, t + dt) given that the nth oscillation took place at t’. Since the particle is not allowed to come back and forth, there is no entanglement in Eq. 1. Due to the hidden interactions the particle undergo with the medium, we treat the time of an oscillation as a random variable following a Poisson distribution 0

p0 (t − t ) =

(

0

0 , if (t − t ) < τ 0 νdt exp[−ν(t − t )] , otherwise.

(2)

Here, ν is the frequency of an oscillation and τ is the ”dead” time. Designing by χn (s) and π0 (s) the Laplace transforms of Qn (t) and p0 (t), resp., the convolution theorem gives χn+1 (s) = χn (s)π0 (s).

(3)

From this expression we obtain the recursive relation χn (s) = [π0 (s)]n−1 χ1 (s).

(4)

The evaluation of the transforms π0 (s) and χ1 (s) gives immediately π0 (s) =

ν exp(−(ν + s)τ ) , ν+s

(5)

ν , ν +s

(6)

and χ1 (s) =

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leading us to χn (s) = ν n

exp(−(n − 1)(ν + s]τ . (ν + s)n

(7)

The inverse transform calculated using the Laplace inverse theorem, gives the probability for the occurrence of n oscillations at time t: Qn (t) =

(

]} ν {ν[t−(n−1)τ (n−1)! 0

n−1 exp(−νt)

,t > (n − 1)τ ,t < (n − 1)τ.

(8)

To simplify, we shall put τ = 0 and the probability density that the nth oscillation takes place in the interval of time (t, t + dt) reads Qn (t)dt =

ν(νt)n−1 exp(−νt)dt. (n − 1)!

(9)

It follows the probability density of occurrence of q jumps at time t is given by Ψq (t)dt =

∞ X

ξq (n)Qn (t)dt,

(10)

n=1

or, in complete form, Ψq (t)dt =

∞ X

n=1

ξq (n)

ν(νt)n−1 exp(−νt)dt. (n − 1)!

(11)

Here, ξq (n) is the probability to occur n oscillations at qth jump. To evaluate ξq (n) we first define gM (mq ), the probability that M previous oscillations generate mq oscillations at qth step [8]. The Bose-Einstein distribution is favored since many oscillations can pertain to the same step: gM (mq ) =

(M + mq − 1)! mq β (1 − β)M . mq !(M − 1)!

(12)

Introducing the conditional probability ϕq (mq |mq−1 , ..., m0) that at qth step there are mq oscillations provided that at the previous steps mq−1 , ..., m0 oscillations have occurred, subject to the normalization condition X mq

ϕq (mq |mq−1 , ...m, m0) = 1,

(13)

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Figure 1. Particle in non-Markovian jumps in a lattice medium. then, it can be shown [8] that ξq (n) =

X m

ξ0 (m)(1 − β)qm [1 − (1 − β)q ]n−m (n − 1)! . (m − 1)!(n − m)!

1.2.

(14)

The Probability Density

Therefore, the probability density of occurrence of q-jumps is finally found to be X α (αt)m−1 Ψq (t)dt = exp(−αt) ξ0 (m) dt, (15) ν (m − 1)! m

where we put α(q) ≡ (1 − β)q ν. It must be assumed we know ξ0 (m), that is the probability to occur m oscillations from t = 0 up to the first jump. With the assumption of a Poisson distribution for ξ0 (m), the summation gives ∞ X √ (αt)m−1 1 ξ0 (m) =√ I1 ( λαt), (16) (m − 1)! λαt m=0

where I1 (x) is the first class modified Bessel function of order 1. Hence, the final result for the probability of occurrence of q-jumps between t and t + dt is given by r √ α Ψq (t)dt = exp(−αt)I ( λαt)dt. (17) 1 λν 2 t

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Eq. 17 is characterized by a√temporal argument and, in particular, for a sufficient number of steps, the limit λx → 0 is satisfied, and from the above we obtain Ψq (t)dt ≈

α exp[−αt]dt. 2

(18)

We have in view a deterministic particle system evolving according to a local mapping in a space of equidistant sites. This idealization lies in the Ehrenfest’s equation describing the quantum mechanical mean value of the particle position, and thus avoids the solution of a much more complex problem [8] which, in the problem here addressed, does not bring any further substance. Hence, we can rewrite the above equation in the form Ψ(x, t)dt ≈

α exp[−αt]dt. 2

(19)

According to the statistical interpretation of wave mechanics, the probabilities are quadratic forms of a ψ functions, Ψq (t) = |ψq (t)|2 , with ψ designating the associated ”wave”. Therefore, we can seek a representation of the transport process in terms of wave function. In fact, as we will see, this is a far-reaching representation √ of the process. Fig. 2 represents the wavefunction 1 √ Ψ = t exp(−t) ∗ I1 ( 3t) of a soliton-like wave. Inquiring for a convenient simplification of the complicated initial function lead us to a simpler wave representation in which a definite functional form as x ± vt is obtained. Reducing our representation to harmonic waves in which way the energy associated with the wave is expressed? Does the energy relation E = ¯hω and De Broglie relation hold on? Or does an appropriated modification is at stake? By expanding the temporal argument present in the exponential function in Eq. 17 and retaining only terms of magnitude β 2 (higher order terms are less important and it is harder to give them a physical meaning), we obtain (1 − β)q νt ≈ νt −

1.3.

βqΛν ¯h (βqν)2 t+ t + O(β 3 ). Λ 2 ¯hν

(20)

Analogy with a Space-Time Lattice

The above expansion suggests the identification of some mechanical properties of the particle, using the analogy with a transversal wave in a vibrant string: V ≡

νl , with n’=1,2,3,..., 2πn0

(21)

Parametric Resonance, Particle Stochastic Interactions ...

395

assuming a non-dispersive medium. We denote by l ≡ qΛ the distance travelled by the particle from a fixed point O of the x-axis after time t and ν is the number of cycles per second loosed on a given space position, both quantities as seen by an observer at rest in the lattice. The wave number is defined by K≡

β2πn , Λ

(22)

where β is the probability that each oscillation in the past has to trigger a new oscillation in the present. We also obtain ω: ω = βqν.

(23)

Notice that when the number of jumps is q = 1 and the probability is equal to the unity, β = 1, then ω = ν, otherwise, they acquire different values. Attributing physical meaning to the parameters permits to identify the third term on the right-hand side of Eq. 20 with the energy carried by the particle: E=

1.4.

¯h (βqν)2 . 2ν

(24)

Inertia and Mass

We have made so far no hypothesis about the mass of the ideal particle. But due to the jumping and interaction with the medium, using Einstein’s relationship, E = mc2 , the particle energy has the equivalent of mass, its own “mass”, and the energy content is given by E=

1 (βqν)2 ¯h. 2 ν

Therefore,

(25)

¯ ω2 h . (26) 2νc2 The above equation shows the set-up of a structural relationship between energy, driven frequency ω and characteristic frequency of interaction with the medium, ν. We may notice that the non-markovian character of the stochastic process is more intrinsically related to the nature of the medium rather than the past history of the particle. Fig. 2 shows that the perturbation of the medium, which we can imagine as an excitation by an intense laser pulse inducing perturbations in the medium, m=

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Figure 2. Wave-like soliton amplitude vs. position, in arbitrary units. can give rise to an essential reconstruction of the ‘wave function’ in a region of resonance that sharply begins but extends further in the front-rear of the solitonlike quasiparticle. We can envisage a model of nonlinear coupling between harmonic oscillator photon induced by a laser in a medium of trapped ions to process quantum information and quantum simulation [19]. As shown before, for consistence ω ≡ βqν, and then from Eq. 26 we have ¯hω 2 c E= = c 2ν

¯hω 2 c

!

λ0 .

(27)

The factor 2 in Eq. 27 appears because the particle has two degrees of freedom for the transversal vibration, see also the discussion in Ref. [9]. Hence, the frequency ν0 = 2ν can be associated to the Zitterbewegung, recently experimentally observed [10], and therefore c = ν0 λ0 , with λ0 = λn. Finally, we obtain ¯hω 2 E= . (28) ν0 Eq. 28 shows the nonlinear coupling between the internal modes of vibration of the medium and the outcome at the soliton-like object that convey information at a frequency related to E. The parameter ν might be related to the frequency and number of pump photons giving rise, due to the parametric resonance, to the new eigenstates of the trapped ions in a quantum computing device[19, 20].

Parametric Resonance, Particle Stochastic Interactions ...

1.5.

397

Parametric Resonance in Space-Time Lattice or Periodic Medium

The presence of two distinct frequencies in Eq. 28 suggests the possible occurrence of parametric resonance effect between the particle and the medium. Indeed, experiments have shown that particles possess an internal clock (the well-known hypothesis advanced by De Broglie in his double-solution theory) characterized by ω and when they interact with the medium, characterized by ν0 , possible resonance may occur. This effect was shown in a channeling experiment with ∼ 80 MeV electrons traversing a 1-µm thick silicon crystal aligned with the < 110 > direction (nuclear scattering effects are stronger than in random direction). When the frequency of atomic collisions matches the internal clock frequency, the rate of electron transmission shows a 8 % dip within 0.5% of the resonance energy [11]. This idea could serve as a critical test bed for particle physics phenomena that seems to share common points with the parametric resonance effect between particles and the space-time lattice to detect dark matter axion-like particles [13] or a periodic medium like graphene to investigate the origin of half-spin quarks [12], or even parametric resonance as a pop-up mechanism to turn fields into particles [14], just to cite a few.

1.6.

The Lorentz Invariant Force

The interaction of the singularity particle with the medium develops a resistance (inertia) and gives rise to a new type of force shown in Eq. 27, also obtained with a different approach in Ref. [7] (and named by J. P. Vigier, the Lorentz invariant force): ¯hω 2 FL = n(λ), (29) c for a medium with refraction index n(λ). This force (or energy) actuating on a particle is at the origin of the mass of the particle. The analogy with a vibrating string allows the conjecture of the existence of higher harmonics. With quantum computing devices, this force might anticipate transmission effects when regarding time evolution of information. The medium perturbation is characterized by ν, a particular property of the particle surrounding medium from where it emerges the inertia of matter by means of the coefficient m and introducing a nonlinearity that produces a different pattern that the one conceptualized by the quantum mechanical expression for a photon packet (E = ¯hω). Note that Eq. 26 is consistent with the De

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Mario J. Pinheiro

Broglie relation for free particles (planar waves), since then ω = ν0 . Otherwise, when the interaction with the surrounding medium imposes a nonlinear dynamics, a new relationship is set-up, Eq. 26. This scheme leads us to a description of quanta as embedded within a complex manifold, reminding the appearance of discrete objects as part of the medium, much like propagating soliton-like waves in a fluctuating, information-rich energy field. De Broglie [15] and David Bohm [16] were proponents of the Guided Wave Theory which proposes that particle’s mass, as also it appears in our analysis, is not an intrinsic property of matter but an outcome of its interaction with a periodic medium. This simple and apparently universal mechanism is considered in contemporary cosmology, in the initial reheating process after inflation, when an explosive particle production takes place due to induce parametric resonance [17], and may explain pion production in a nonequilibrium chiral phase transition [18].

C ONCLUSION Exploring the underlying transport mechanism of a test particle with infinite memory induces us to attribute a universal and structurally simple property to the particle energy. We make a connection between the probability density for the transport process and the associated ‘wave’, as conceived within the Ehrenfest theorem in order to explain the wave-like properties, such as the wavenumber, and the energy carried by the particle. Recognizing the soliton-like behavior of the associated waveform is crucial in making this connection. As I understand it, this interpretation for a non-Markovian transport process, on the same lines as the causal interpretation of quantum mechanics, has not been made in the technical literature. As an outcome, a closed-form condition explaining a relationship between the particle energy, driven frequency and frequency of interaction with the medium has been arrived at, allowing for the possibility of a resonance condition between the two frequencies. This fundamental property is the outcome of a balance between the driven frequency ω and a frequency of interaction in the medium, ν. This certainly has far-reaching consequences in explaining experiments where particle-medium resonance has been observed, since the parametric excitation of those vibrations may lead to a particular type of force and their materialization under the form of quasiparticles or particles. The general mechanism of nonlinear coupling between the medium modes of oscillation and characteristic frequencies of the medium represent, additionally, a fully operational mode for quantum information processing and quantum simulation.

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R EFERENCES [1] Balescu, R., Phys. Rev. E 51 4807 (1995). [2] Amblard F., Maggs A. C., Yurke B., Pargellis A. N., and Leibler S., Phys. Rev. Lett. 77 4470 (1996). [3] Valenzuela S. O., and Bekeris V., Phys. Rev. Lett. 84(18) 4200 (2000). [4] Kofman A. G., Zaibel R., Levine A. M., and P. Yehiam, Phys. Rev. A 41 (11), 6434 (1990). [5] McKane A. J., Luckock H. C., and Bray A. J., Phys. Rev. A 41 (2) 644-656 (1990). [6] Bray A. J., McKane A. J., and Newman T. J., Phys. Rev. A 41 (2), 657-667 (1990). [7] Vigier J. P., Phys. Lett. A 270 pp. 221-231 (2000). [8] Vlad M. O., Physica A 208, pp. 167-176 (1994). [9] Osche G. R., Ann. Fond. Louis de Broglie 36 pp. 61-71 (2011). [10] Gerritsma R., Kirchmair G., Z¨ahringer F., Solano E., Blatt R. and Roos C. F., Nature 463, pp. 68-72 (2010). [11] Gouan`ere M., Spighel M., Cue N., Gaillard M. J., Genre R., Kirsch R., Poizat J. C., Remillieux J., Catillon P., and L. Roussel, Fond. Louis de Broglie 30 (1) pp. 109-114 (2005). [12] Mecklenburg M., and B. C. Regan, Phys. Rev. Lett. bf 106 229901 (2011); Erratum Phys. Rev. Lett. 106 229901(E) (2011). [13] Arza A., Arias P., and J. Gamboa, Parametric Resonance and Dark Matter Axion-Like Particles arXiv:1506.02698v1 [hep-ph]. [14] Allahverdi R., Campbell B. A., and R.H.A. David Shaw, PArametric Resonance for Complex Fields arXiv:hep-ph/9909256v1.

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[15] de Broglie L., Etude critique des bases de l’interpretation actuelle de la M´ecanique Quantique [Critical study of the bases of the current interpretation of Quantum Mechanics] (Gauthier-Villars, Paris, 1956); see also, de Broglie L., Introduction a la Nouvelle Th´eorie des Particules[Introduction to the New Theory of Particles] (Gauthier-Villars, Paris, 1961), Ch.V. [16] Bohm D., Phys. Rev. 85 166 (1952); Bohm D., Phys. Rev. 85 180 (1952). [17] Linde A., Phys. Scripta T85 pp. 168-176 (2000). [18] Hiro-Oka H., and M. Hisakazu, Phys. Rev. C 64 044902 (2001). [19] Shiqian D., Gleb M., Roland H., Huanqian L., and D. Matsukevich, Phys. Rev. Lett. 119 150404 (2017) [20] Chen et al. 2007. Quantum Computing Devices: Principles, Designs, and Analysis. New York: Chapman & Hall/CRC.

ABOUT THE EDITOR Luigi Maxmilian Caligiuri Full Professor of Physics at Italian Minister of Education, University and Scientific Research (MIUR) and General Director at Foundation of Physics Research Center (FoPRC) Foundation of Physics Research Center (FoPRC), Cosenza, Italy Email: [email protected]

Luigi Maxmilian Caligiuri is full professor of Physics and also the director of Foundation of Physics Research Center (FoPRC), an independent research organization devoted to advanced research in Physics. From 2014 to 2016 he has been the Executive Director of the Scientific Projects Divisions at International Society for Space Science (ISSS). In 2016 and 2020 he has been included in the Marquis Who’s Who in the World and, in the 2017, nominated among the top 100 scientists in the world by the Cambridge Biographical Centre. He is expert member of the International Engineering and Technology Institute (IETI), belongs to European Quantum Flagship and is invited member of CEN-CENELEC Focus Group on Quantum Technology. He has published many of scientific papers in international journals of physics and engineering and is editorial board

402

About the Editor

member of many international journals about theoretical and applied physics. His most recent interests include the development of coherent quantum field theory and its application to different fields (including fundamental physics, cosmology, biophysics, quantum computation and artificial intelligence) and tachyons physics. He is considered a world expert of Quantum Hypercomputing and Coherent Quantum Field Theory.

INDEX A absolute, 194, 226, 253, 272, 276, 277, 282, 287, 308, 309, 339, 359 absorption, 9, 21, 56, 205, 281, 327 additional dimensions (XD), 178, 181, 182, 198, 199, 200, 254, 270, 290, 295, 301, 304, 305, 306, 310, 311, 320, 321, 323, 327, 328 Advanced-retarded, 199, 205, 233, 245, 301 Aharonov-Bohm effect, 86 algebra, 32, 182, 249, 251, 266, 281, 302, 321, 325, 326 algorithm, x, xi, 184, 186, 187, 188, 189, 192, 217, 218, 219, 220, 224, 280, 281, 333, 336, 374, 375, 381 amplification, 59, 60, 64, 65 amplitude, 13, 23, 29, 41, 42, 43, 64, 84, 85, 181, 194, 245, 249, 275, 324, 396 android, 223, 224, 225, 226, 227, 228, 229, 248, 258, 260, 264 androids, viii, xiii, 223, 224, 226, 227, 264 angular momentum, 20, 258, 259, 295, 296

annihilation, 2, 29, 42, 43, 45, 48, 74, 182, 246, 260, 292, 298, 301, 319, 326, 349, 358 anticipatory, 147, 148, 202, 206, 247, 266 anti-commutativity, 321, 323, 325, 326 anti-De-Sitter (AdS), 310, 311 antiparticle, 213, 277 anti-space, 321, 323, 324, 326 array, 16, 50, 230, 234, 235, 239, 242, 243, 248, 256, 259, 261, 288, 291, 297, 299, 337 artificial intelligence (AI), 101, 102, 159, 223, 225, 227, 348, 402 atom, x, 3, 4, 204, 206, 222, 230, 256, 261, 293, 296, 300, 317, 318, 320, 322, 327, 328, 329, 344, 347

B backbone, 180, 247, 248, 249, 291, 303, 307, 317 backcloth, 235, 239, 260, 296, 297, 299, 324, 326 back-reaction, 260, 297

404 backward rate, 368 barrier, xii, 23, 24, 27, 37, 87, 94, 230, 232, 238, 241, 247, 295, 296, 306, 309, 318, 339, 340, 370, 377, 380, 382, 390 basic logic, 158, 159, 168 basis, xi, xii, 35, 69, 71, 90, 145, 158, 162, 163, 178, 180, 182, 183, 184, 186, 194, 214, 222, 229, 231, 232, 234, 235, 237, 238, 240, 241, 243, 247, 250, 251, 258, 259, 270, 272, 273, 278, 279, 282, 283, 284, 285, 290, 291, 292, 294, 295, 297, 304, 306, 311,314, 356, 390 beam, 22, 23, 41, 288, 293, 366 bifurcation, 105, 129, 138, 147, 149, 246 big bang, 229, 232, 296, 306, 376 biological, xiii, 35, 162, 215, 262, 265, 306, 331, 344, 345, 347, 349, 351, 354, 355, 356, 357, 358, 359, 365, 376, 379, 385, 386 bits, x, 82, 157, 164, 165, 169, 179, 184, 192, 211, 223, 270, 271, 281, 332, 333, 336, 347, 356 black-box, 186, 187, 188, 189 Bloch sphere, 228, 236, 249, 250, 251, 255, 256, 272, 284, 315 Bogoliubov transformation, 51 Bohmian mechanics, 178 Boolean, x, xi, 187, 192, 279, 281 Bose-Einstein condensate, 347, 348 bosons, 276, 277, 287 bounds, 89, 92, 93, 218 brain, viii, xiii, 36, 37, 101, 153, 154, 155, 156, 158, 159, 160, 161, 163, 173, 174, 175, 223, 224, 225, 226, 227, 228, 230, 236, 240, 241, 248, 265, 281, 331, 343, 344, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 361, 363, 364, 365, 366,376, 377, 378, 379, 381, 383, 384, 385, 386, 387, 388 bulk, 180, 182, 183, 184, 198, 199, 210, 212, 226, 229, 236, 247, 248, 251, 264,

Index 270, 271, 272, 273, 276, 285, 290, 291, 294, 304, 305, 306, 309, 310, 317

C Calabi-Yau, 180, 182, 183, 198, 206, 229, 234, 236, 237, 239, 240, 242, 244, 245, 247, 249, 250, 252, 254, 255, 256, 270, 280, 286, 289, 290, 291, 292, 300, 305, 306, 307, 312, 315, 317, 325, 326, 327 Cauchy-Riemann relations, 204, 206 cavity, 12, 244, 259, 261, 262, 274, 289, 292, 297, 299, 312, 318, 320, 322, 324, 325, 327, 357, 358, 361 CCNOT (controlled-controlled-NOT) gate, 146 centrioles, 345, 356, 358, 386 CERN, 210, 271, 280, 295 chaos, 326 chemical activation, 21, 86 chemical reaction, 21, 87, 95 chemical specie, 21, 98 chemo-electrodynamical, 21, 22 Cherenkov radiation, 205 Chinese, 225, 264 Chiral representation, 131 Church-Turing, 185, 217, 383 classical, xi, 22, 25, 40, 41, 42, 43, 50, 54, 55, 63, 66, 79, 84, 96, 104, 105, 111, 148, 154, 157, 159, 160, 162, 163, 164, 165, 169, 175, 184, 185, 186, 187, 188, 189, 190, 192, 193, 194, 210, 211, 219, 228, 229, 230, 232, 233, 245, 246, 248, 256, 258, 261, 271,275, 279, 282, 286, 295, 299, 300, 305, 309, 311, 333, 335, 347, 386, 390 Clifford algebra, 256, 302, 321, 326 CNOT (Controlled-NOT), 66, 67, 68, 70, 71, 72, 79, 146, 165, 166, 168, 169, 285 cognitive, 159, 223, 224, 225, 227, 228, 239, 240, 241

Index coherence, 3, 5, 15, 19, 20, 21, 35, 36, 55, 58, 97, 98, 173, 198, 206, 210, 215, 231, 236, 242, 244, 247, 264, 265, 290, 291, 292, 298, 304, 306, 307, 309, 314, 316, 317, 345, 349, 350, 351, 379, 385, 386 coherence domain (CD), 5, 6, 8, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 28, 30, 33, 54, 55, 60, 62, 63, 65, 85, 86, 87, 89, 93, 94, 95, 96, 97, 98, 99 coherent behavior, 18 coherent electromagnetic field, 13 coherent equations, 9, 52 coherent excited levels, 20 coherent fraction of water, 63, 83, 87 coherent ground state (CGS), 8, 9, 12, 16, 17, 19, 28, 42, 50, 59 coherent state, 2, 5, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 34, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 54, 55, 59, 85, 86, 87, 94, 95, 98, 161, 342, 349 coherent states, 2, 20, 39, 40, 41, 42, 43, 44, 47, 51, 54, 55, 95, 161 coherent vortex, 19, 20 coherent water, 18, 19, 36, 58 collapse, 177, 193, 219, 286, 348, 350 collective, 8, 20, 21, 49, 162, 183, 349, 376 communication, 96, 100, 215, 267, 381 commutativity, 302, 321, 323, 325, 326 commutator, 47, 49 compactification, 181, 182, 198, 212, 216, 234, 244, 260, 292, 299, 305, 306 computability, 156 computable, 155, 158, 161, 172, 175, 185, 217, 374, 382 computation, vii, x, xii, xiii, 1, 55, 57, 58, 65, 66, 67, 79, 83, 86, 89, 96, 98, 100, 102, 157, 183, 185, 216, 219, 220, 221, 250, 266, 279, 283, 328, 331, 332, 334, 338, 340, 341, 342, 343, 346, 347, 351, 352, 354, 355, 356, 357, 367, 368, 369,

405 370, 373, 374, 375,379, 382, 383, 384, 386, 387, 388 computational basis, 69, 70, 71, 72, 75, 336 computational mode, 157, 158, 160, 367, 373 computational speed, xi, xii, 34, 59, 89, 100, 332, 338, 341, 352 computational step, 52, 96, 367, 368, 369 computer, ix, xi, xiii, 96, 99, 104, 105, 121, 148, 155, 158, 159, 161, 167, 168, 171, 173, 185, 217, 218, 219, 221, 225, 226, 227, 228, 265, 271, 281, 331, 332, 333, 335, 337, 338, 339, 341, 342, 343, 347, 351, 356, 357, 365, 367, 374, 375, 377, 378, 382, 383, 388 computing, vii, xi, xii, xiii, 98, 99, 101, 102, 103, 110, 147, 148, 154, 156, 157, 168, 174, 183, 184, 188, 193, 194, 216, 217, 218, 219, 220, 225, 226, 227, 247, 252, 253, 266, 273, 279, 281, 282, 309, 329, 332, 343, 349, 357, 376, 382, 397 condensation, 9, 10, 13, 50, 52, 55, 349, 384, 386 conformational states, 344, 345, 358 conscious, 156, 157, 162, 224, 227, 228, 241, 264, 265, 349, 384, 385, 386 consciousness, 157, 162, 163, 173, 174, 175, 224, 225, 235, 237, 240, 245, 265, 266, 313, 343, 345, 348, 349, 351, 356, 357, 361, 364, 365, 376, 377, 381, 385, 387, 388 constants of motion, 104, 111 control bit, 72 control qubit, 70, 82 controlled operations, 79 controlled-U (cU) gate, 148, 260, 297, 390 cosmology, 175, 199, 206, 222, 230, 232, 234, 236, 237, 247, 248, 259, 261, 267, 294, 296, 297, 299, 301, 306, 307, 308, 322, 323, 398, 402

406

Index

coupling, xiii, 2, 3, 6, 22, 25, 29, 30, 33, 58, 59, 60, 63, 65, 73, 74, 76, 78, 82, 86, 198, 242, 256, 258, 259, 260, 261, 262, 287, 296, 297, 299, 300, 309, 318, 345, 396, 398 creation operator, 2, 42, 246 critical density, 11, 21, 59 cutoff, 61, 62, 63, 64, 361 cytoskeleton, 357

D decoherence, xi, xii, xiii, 61, 83, 92, 96, 97, 157, 161, 162, 179, 183, 215, 229, 247, 248, 263, 267, 269, 270, 273, 276, 278, 280, 285, 292, 293, 308, 309, 317, 328, 342, 350, 351, 354, 355, 386 decomposition, 48, 72, 79 degrees of freedom, xi, 96, 178, 182, 228, 247, 248, 257, 261, 270, 272, 279, 286, 289, 290, 292, 300, 301, 303, 306, 307, 308, 318, 323, 396 density, 3, 4, 8, 9, 11, 12, 18, 19, 20, 22, 59, 205, 274, 391, 392, 393, 398 depth, 23, 100 dielectric, 64, 337, 363, 364 dipole, 28, 258, 295 dipole operator, 28 Dirac complex spinors, 104 dirac equations, vii, 103, 134, 222 discrete Dirac equations, 105, 126 discrete energy spectrum, 3, 91 discrete harmonic oscillators, 104, 107, 109, 110, 120, 131, 147, 149 discrete Klein-Gordon equation, 103, 105, 120, 121, 122, 127, 128, 149 discrete Majorana quantum relativistic equations, 105 discrete quantum fluctuations, 167 dissipative quantum field theory (DQFT), 155, 158, 159, 160, 161

distribution probability, 44 donor, 28 double prisms, 24 dual-rail representation, 73

E eigenstates, 20, 30, 33, 39, 42, 48, 50, 55, 95, 272, 396 eigenvalues, 30, 31, 49, 91, 96, 333 Einstein-Podolski-Rosen (EPR), 155, 167, 168, 169, 170, 172, 177, 178, 179, 194, 197, 198, 212, 214, 215 electric field, 28, 258 electromagnetic, 1, 3, 5, 6, 8, 10, 22, 29, 36, 37, 40, 43, 62, 64, 84, 85, 99, 100, 101, 102, 221, 231, 241, 242, 287, 290, 292, 298, 301, 322, 338, 342, 345, 349, 358, 359, 360, 382, 385, 387 electromagnetic potentials, 84, 85, 102 electromagnetic waves, 37, 101 energy, xi, 1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 24, 27, 28, 29, 30, 42, 58, 59, 62, 63, 82, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 107, 111, 147, 166, 167, 183, 198, 205, 210, 211, 230, 231, 232, 234, 235, 236, 240, 241, 242, 243, 244, 245, 256, 259, 260, 275, 276, 280, 287, 290, 291, 295, 296, 298, 299, 303, 307, 310, 311, 313, 314, 315, 317, 327, 332, 338, 339, 340, 341, 342, 343, 349, 352, 353, 354, 359, 367, 368, 369, 371, 376, 380, 381, 384, 388, 389, 394, 395, 397, 398 energy absorption, 211 energy cost, 96, 332, 338, 341, 384 energy discharge, 21 energy gap, 12, 14, 15, 16, 17, 19, 93, 94, 98, 183, 275 energy levels, xi, 1, 3, 5, 58, 63, 98, 100, 299

Index energy limit, 332, 341 energy of excitation, 20 energy ratio, 352 energy spread, 93, 339 ensemble, 1, 6, 8, 50, 274, 275, 285 entangled space-time, 164, 165, 167, 170, 171, 172 entanglement, x, xii, 153, 160, 164, 165, 166, 167, 168, 169, 170, 173, 175, 179, 184, 216, 275, 292, 332, 347, 355, 385, 386, 391 EPR pair, 170, 173, 194 equilibrium state, 9, 274 evanescent fields, 2, 22, 26, 58, 60 evanescent modes, 22, 24, 34, 58, 63 evanescent photons, 57, 102, 338, 343, 349, 351, 353, 354, 356, 357, 359, 360, 384, 385 evanescent tail, 15 evanescent tunneling coupling interaction, 72 evanescent waves, 22, 23, 94, 361, 363 events, 194, 204, 206, 209, 230, 254, 293, 301, 385 evolution, x, xi, xii, 3, 5, 6, 11, 31, 32, 33, 34, 50, 52, 53, 76, 84, 85, 86, 88, 92, 95, 102, 148, 178, 181, 185, 193, 198, 213, 223, 225, 227, 228, 231, 233, 234, 236, 237, 242, 251, 256, 257, 266, 273, 278, 280, 281, 283, 284, 285, 291, 293, 307, 309, 314, 319, 384, 390 exclusion zone (EZ) - water, 18, 19

F faster-than-light (FTL), 380 fault-tolerant, 68, 71 fermions, 150, 276, 277, 287 fields, 2, 3, 6, 8, 10, 15, 22, 24, 25, 26, 27, 36, 42, 52, 54, 58, 63, 83, 84, 85, 86, 100, 101, 181, 192, 195, 235, 242, 252,

407 262, 293, 301, 303, 313, 389, 390, 397, 399, 402 floating-point operations, 271 fluctuations, 2, 3, 4, 8, 16, 21, 34, 42, 58, 60, 61, 87, 165, 167, 199, 221, 277, 279, 327 Fock space, 43, 50 forces, 2, 4, 16, 54, 244, 258, 287, 290, 295, 310, 314, 329 formal language, 154, 161, 220 forward-backward, 104, 107 four-dimensional, 311 Fourier decomposition, 2, 42 Fourier transform, 186, 189 fraction, 17, 18, 34, 60, 61, 97 frame, 201, 204, 240, 316 Fredkin gate, 285 frequency, 7, 10, 11, 12, 15, 20, 21, 22, 24, 27, 33, 52, 54, 61, 62, 64, 85, 86, 90, 95, 121, 127, 211, 239, 242, 258, 259, 260, 291, 295, 297, 298, 299, 304, 316, 317, 319, 320, 324, 325, 326, 339, 354, 361, 389, 390, 391, 395, 396, 397, 398 frustrated total internal reflection (FTIR), 23

G gap junctions, 348 gas-like state, 12 gate, 66, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 88, 89, 92, 93, 95, 96, 97, 98, 100, 104, 105, 106, 138, 144, 145, 146, 165, 166, 168, 169, 186, 235, 237, 244, 252, 273, 278, 281, 282, 285, 316, 337, 338, 339, 342, 350 gates, xi, 66, 67, 68, 70, 71, 72, 78, 88, 89, 93, 96, 97, 98, 100, 106, 144, 145, 147, 168, 169, 219, 228, 235, 240, 250, 251, 252, 256, 266, 269, 273, 281, 285, 328, 333, 338, 339, 340, 341, 352 gauge field, 3

408

Index

gauge invariance, 3, 40, 214 gauge symmetry, 198 gauge theories, 40 gauge transformations, 40 General (Theory of) Relativity, 95 generalized Lorentz transformation, 200 glassy state, 36 Gödel theorem, 348 gravitation, 216, 243, 244, 266 gravity, xiii, 153, 155, 156, 166, 181, 199, 200, 213, 216, 235, 242, 249, 271, 298, 310, 313 ground state, 6, 7, 9, 11, 12, 19, 29, 42, 59, 195, 196, 197, 204, 248, 272, 277, 318 group, 18, 24, 94, 130, 192, 210, 230, 251, 252, 278, 338, 351, 358, 370, 380, 384, 387, 401 group velocity, 24, 94 Grover, 188, 189, 218 gyromagnetic ratio, 20, 296

H Hadamard gate, 68, 71, 88, 104, 106, 145, 179, 186, 285 Hadamard matrix, 104, 105, 106, 112, 114, 131, 132, 133, 145, 149 Halting problem, 374, 375 Hamiltonian, 28, 30, 32, 39, 52, 53, 54, 55, 68, 73, 74, 93, 96, 198, 245, 258, 259, 275, 284, 285, 287, 290, 296, 311 harmonic oscillators, 104, 105, 119, 120, 150, 246, 329 hierarchy, 190, 198, 199, 206, 219, 221, 245, 247, 256, 260, 261, 262, 265, 290, 291, 292, 298, 299, 300, 303, 306, 307, 311, 312, 316, 317, 318, 319, 320, 322, 324, 326, 328, 382 Higgs mechanism, 340, 349 Hilbert space, 67, 158, 183, 251, 252, 256, 272, 279, 280, 281

holographic, 166, 170, 180, 184, 190, 191, 192, 193, 197, 206, 210, 214, 219, 221, 223, 226, 239, 240, 248, 265, 281, 311, 320, 343, 349, 356, 357, 363, 364, 365, 376, 377, 386, 387 holophote, 231, 243, 261, 262, 299, 316, 317, 318 human brain, 332, 343, 351, 352, 353, 354, 356, 357, 377, 378, 379 human cognition, 366 human consciousness, xiii, 331, 347, 349, 358, 376, 382 human memory, 365 hydrophobic pockets, 344, 347 hypercomputation, xii, 89, 374, 376, 387, 388 hypercomputer, 331, 383 hypercomputing, 34, 331, 367 hyperincursive, vii, xii, 103, 104, 105, 108, 109, 110, 120, 121, 122, 126, 127, 128, 129, 147, 148, 149, 150 hyperincursive discrete equations, 103, 104, 105

I incursive discrete equations, 104, 109, 118, 129, 130, 131, 145 inertia, 20, 35, 56, 102, 298, 395, 397 inertial back-reaction, 259, 260, 297, 298 infinity, 234, 253, 257, 262, 265, 293, 296, 299, 318, 319, 322, 326, 327, 369 information, ix, x, xi, xiii, 36, 40, 42, 85, 87, 92, 95, 99, 100, 158, 166, 172, 178, 184, 186, 188, 192, 194, 198, 200, 210, 211, 212, 215, 221, 249, 259, 273, 279, 289, 291, 292, 294, 297, 303, 306, 311, 312, 314, 324, 332, 333, 349, 350, 351, 363, 377, 378, 382, 385, 386, 387, 389, 390, 396, 397, 398

Index instantaneous, viii, xiii, 24, 177, 178, 179, 182, 183, 184, 194, 197, 198, 206, 209, 212, 213, 214 instantaneous algorithms, 178, 184, 194, 198, 212 interaction, xi, xii, xiii, 2, 3, 18, 21, 22, 25, 26, 27, 28, 30, 31, 33, 34, 36, 52, 54, 55, 57, 58, 59, 60, 63, 64, 65, 76, 82, 83, 84, 85, 86, 87, 92, 93, 97, 99, 100, 163, 181, 193, 196, 212, 213, 220, 221, 224, 227, 230, 232, 247, 258, 259, 260, 276, 277, 279, 286, 287, 289, 293, 294, 297, 303, 308, 314, 326, 329, 330, 342, 345, 348, 349, 350, 358, 363, 388, 389, 390, 395, 397, 398 interfacial water, 18, 36, 60

K Kaluza-Klein, 178, 181, 209, 216, 305 Klein-Fock-Gordon equation, 358 Klein-Gordon equation, 105, 120, 121, 127, 130, 135, 141

L Lagrangian, 3, 5, 256, 311 Lamb-shift, 3 Large-Scale Additional Dimensionality (LSXD), 180, 181, 182, 198, 210, 212, 214, 231, 232, 234, 236, 239, 240, 242, 243, 244, 247, 248, 250, 252, 254, 255, 256, 264, 280, 289, 290, 291, 292, 295, 303, 304, 305, 307, 310, 311, 312, 313, 316, 317, 321, 327 Larmor radius, 182, 296, 322 laser, 40, 41, 43, 85, 248, 259, 293, 297, 298, 303, 316, 320, 326, 356, 361, 366, 390, 395, 396

409 laser oscillated vacuum energy resonator, 259, 297, 298, 303 levels, x, xi, 3, 5, 17, 19, 20, 28, 30, 66, 92, 93, 96, 97, 100, 156, 171, 293 life principle, 223, 224, 225, 227, 228, 229, 231, 235, 237, 239, 241, 243, 244, 264 light, 22, 41, 43, 55, 86, 95, 120, 167, 172, 177, 201, 202, 213, 221, 227, 230, 237, 239, 240, 241, 244, 251, 260, 261, 299, 300, 316, 324, 337, 345, 346, 353, 354, 358, 359, 360, 366, 370, 377, 379, 382, 384, 387, 388 local spacetime, 236, 315 locality, 165, 178, 179, 183, 184, 192, 193, 212, 214, 273, 279, 291, 294 logic states, 73 logical operation, 90, 91, 97, 100, 184, 338, 339 logical qubit, 164, 165, 170, 171 long - range, 2, 86 Lorentz frames of reference, 206

M Mach-Zehnder interferometer, 288, 308 macroscopic, xii, 1, 2, 3, 8, 12, 21, 25, 30, 35, 41, 42, 47, 50, 54, 55, 85, 95, 230, 233, 239, 322, 347, 349, 351, 357, 359 macroscopic quantum state, xii, 8, 42, 47 magnetic moment, 258, 259, 296, 297, 329 magnetic permittivity, 64, 65 Majorana equations, 106, 120, 123, 131, 133, 134, 136, 137, 140, 141, 142 Majorana fermion, 276 Majorana real spinors, 104 Majorana Zero Modes (MZM), 183, 276 mass, 6, 7, 9, 10, 11, 14, 17, 24, 83, 107, 120, 200, 202, 203, 213, 246, 258, 263, 302, 326, 327, 339, 340, 354, 359, 371, 380, 389, 390, 395, 397, 398

410

Index

matter, xi, 1, 2, 3, 5, 6, 8, 10, 11, 12, 20, 25, 28, 35, 36, 39, 41, 42, 43, 50, 52, 55, 56, 97, 100, 102, 154, 160, 193, 198, 229, 231, 232, 235, 237, 241, 244, 260, 265, 270, 277, 287, 296, 298, 301, 304, 309, 310, 311, 315, 318, 319, 326, 328, 377, 382, 385, 389, 397, 398, 399 measurements, 40, 68, 154, 206, 217, 221, 287, 288, 293, 320, 329, 330 memorization, 92, 99 memory, 89, 174, 333, 338, 356, 357, 363, 364, 365, 376, 384, 385, 386, 387, 390, 391, 398 metamaterial, 83, 101, 331, 357, 360, 361, 364, 365 metamaterial (MTM), 59, 63, 64, 73, 83, 101, 331, 357, 360, 361, 364, 365 meta-rules, 168, 169, 170, 172 metastable coherent excited levels, 19 meta-theory, 170, 172 metric, 24, 165, 167, 209, 234, 245, 257, 263, 281, 291, 302, 307, 314, 380 microtuble, 331, 344, 345, 347, 349, 351, 365 microtubule (MT), 346, 348, 349, 350, 354, 356, 357, 358, 359, 361, 362, 363, 364, 365, 382 Möbius transformation, 257 modes, xiii, 2, 5, 23, 25, 37, 61, 63, 157, 158, 161, 189, 244, 247, 257, 260, 261, 276, 297, 298, 300, 305, 327, 349, 389, 396, 398 M-Theory, 179, 180, 198, 215, 229, 230, 235, 245, 247, 249, 290, 307, 316, 323

N Nambu-Goldstone bosons, 349 NAND (not-AND), 163 nano-switching, 338 near field, 27

Necker cube, 237, 291, 311, 314, 315 negative refractive index, 357, 360, 387 network, xii, 57, 59, 60, 65, 98, 99, 100, 166, 168, 169, 175, 189, 243, 343, 349, 357 neural network, 163, 243 neurons, 267, 332, 344, 345, 347, 348, 349, 357, 358, 360, 382 nilpotent, 210, 213, 231, 236, 248, 255, 291, 319, 324 no-cloning theorem, 180, 193, 211, 215, 273, 328 node, 166, 167, 172, 192, 262, 318 Noetic transform, 231, 244, 251, 253, 254 no-go theorem, 168, 170, 179 non-abelian, 276 non-algorithmic, 156, 157, 159, 161, 175 non-coherent state, 10, 11 non-computable, 158, 160, 217, 382 non-linear, 84 non-locality, 183, 379 non-phased zero-point oscillations, 42 non-physical, 224 non-relativistic, 106, 134, 135, 136, 140, 142, 144 non-separable, 165 non-thermal, 21, 86, 97 non-Turing computation, xiii, 331, 382 NOT gate, 66, 104, 144, 146

O object wave field, 363 observable, 25, 205, 287, 293, 311, 333 observer, 154, 157, 163, 166, 204, 205, 206, 230, 231, 237, 251, 314, 381, 395 occupation states, 43 oracle, 186, 187, 374, 375, 378 Orchestrated Objective Reduction (OrchOR), 156, 347, 348, 350

Index oscillations, 5, 8, 9, 16, 19, 20, 21, 64, 107, 199, 239, 245, 305, 391, 392, 393 oscillator strength, 4, 7, 12 oscillators, 3, 50, 103, 105, 111, 148, 246

P parabolic mirrors, 239, 240 parallel, 92, 97, 98, 100, 157, 169, 170, 211, 214, 226, 236, 240, 271, 289, 303, 314, 325 particles, 3, 19, 23, 24, 25, 29, 52, 85, 134, 140, 164, 170, 172, 193, 194, 205, 210, 250, 252, 258, 262, 263, 276, 277, 295, 301, 302, 309, 318, 333, 335, 339, 347, 368, 370, 372, 373, 377, 379, 388, 389, 397, 398, 399, 400 Pauli gates, 105, 106, 138, 139, 144 penetration depth, 60, 64, 340, 363 Perturbative Ground State (PGS), 8, 42, 50 phase, 3, 4, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 25, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 59, 68, 69, 70, 71, 76, 77, 78, 83, 84, 85, 86, 87, 88, 95, 98, 99, 100, 102, 144, 146, 178, 180, 191, 194, 210, 211, 212, 213, 215, 220, 245, 270, 277, 278, 280, 285, 294, 300, 309, 317, 319, 324, 327, 345, 347, 358, 390, 398 phase gate, 68, 69, 70, 71, 76, 77, 78, 146 phase interaction, 95, 98, 99, 210 phase operator, 33, 39, 41, 42, 48, 49, 50, 52, 55, 56, 85, 95 phase transition, 8, 35, 50, 56, 59, 102, 270, 398 phase-shift operator, 50 photon spatial modes, 97 photons, x, xi, 1, 4, 9, 16, 22, 24, 26, 27, 28, 34, 37, 44, 50, 58, 63, 64, 73, 76, 85, 93, 94, 100, 101, 164, 179, 189, 190, 194, 209, 235, 241, 288, 313, 327, 337, 338,

411 339, 341, 343, 345, 349, 354, 355, 356, 357, 358, 361, 365, 368, 370, 372, 377, 378, 379, 384, 386, 387, 389, 396 Planck units, 164 plane-wave expansion, 29 plasma frequency, 6 Plato, 230, 367 polarization, 261, 300 polynomial, 178, 185, 187, 188, 189, 190, 191, 194, 212, 218, 219 position, 29, 41, 45, 47, 48, 104, 107, 111, 117, 127, 145, 179, 183, 253, 257, 287, 308, 309, 310, 325, 327, 390, 394, 395, 396 Positive Operator-Value Measure (POVM), 67 probability, 4, 20, 26, 27, 32, 68, 74, 88, 181, 185, 186, 194, 247, 249, 272, 283, 286, 287, 288, 289, 293, 305, 308, 335, 390, 391, 392, 393, 395, 398 processing speed, 352 processor, 192, 226, 269, 336, 349, 352, 354, 379 propagation, 16, 23, 58, 61, 73, 95, 96, 206, 209, 239, 262, 300, 301, 314, 359, 370, 388, 390 proper mass, 359, 380 proteins, 344, 347, 348, 351, 357, 385 psychon, 230, 231, 241, 242, 243

Q QED cavity, 206, 321 QED coherence, xi, 1, 2, 34, 35, 39, 41, 55, 56, 98, 100, 102 QED coherence in matter, 2, 34, 98, 100 quantized e.m. field, 3 quantized magnetic moment, 19 quantum algorithms, viii, 87, 153, 161, 177, 178, 183, 184, 185, 187, 188, 189, 214, 273, 274, 336

412 quantum circuit, 55, 68, 71, 79, 80, 184, 273 quantum computation, vii, x, xi, xii, xiii, xiv, 2, 34, 37, 39, 42, 53, 55, 56, 57, 59, 65, 66, 68, 72, 79, 82, 83, 89, 96, 98, 99, 101, 102, 103, 104, 106, 144, 153, 155, 157, 162, 184, 217, 219, 266, 273, 274, 277, 328, 332, 334, 336, 338, 341, 342, 343, 347, 350, 354, 355, 356, 357, 361, 365, 383, 384, 402 quantum computer, x, xi, xii, xiii, 59, 61, 66, 72, 76, 96, 97, 105, 106, 144, 146, 153, 154, 155, 157, 161, 167, 168, 169, 172, 184, 185, 216, 217, 218, 219, 220, 224, 225, 227, 228, 236, 264, 265, 269, 273, 274, 281, 283, 328, 332, 333, 335, 336, 341, 342, 347, 350, 351, 383, 385 quantum computers, xii, 96, 144, 146, 153, 154, 155, 157, 161, 216, 218, 219, 236, 269, 273, 274, 281, 335, 347, 383, 385 quantum computing, 1, iii, vii, ix, x, xi, xiii, xiv, 16, 39, 52, 72, 79, 83, 96, 100, 101, 104, 153, 154, 161, 168, 177, 178, 179, 182, 183, 184, 218, 220, 224, 228, 247, 252, 253, 270, 273, 274, 278, 295, 303, 317, 332, 341, 351, 357, 384, 389, 396, 400 quantum cosmology, 318 quantum coupling, 25, 63 quantum dots, 248, 386 quantum field theory (QFT), 2, 3, 20, 39, 41, 42, 45, 49, 154, 155, 158, 160, 161, 171, 192, 211, 228, 261, 340, 385, 402 quantum Fourier transform, 186, 188 quantum gates, xi, 53, 55, 66, 71, 73, 76, 78, 82, 83, 86, 88, 89, 93, 96, 100, 104, 138, 144, 146, 184, 285, 338, 356, 361, 386 quantum gravity, vii, xii, 153, 154, 155, 156, 164, 166, 171, 174, 181, 347, 355 quantum hypercomputer, 57, 59, 102 quantum hypercomputing, viii, 58, 402

Index quantum information, ix, x, xiii, xiv, 56, 65, 86, 99, 100, 101, 156, 158, 162, 164, 166, 167, 174, 183, 184, 217, 269, 273, 275, 276, 278, 279, 282, 285, 290, 309, 383, 396, 398 quantum logic, x, xi, 154, 155, 162, 167, 168, 169, 171, 228, 269, 281, 282, 333, 337, 384 quantum metalanguage, 154, 155, 156, 158, 161, 163, 164, 171, 172, 175 quantum mind, vii, 153, 154, 155, 156, 157, 175, 388 quantum optics, 12, 41, 56 quantum phase, vii, xii, 11, 20, 39, 40, 41, 42, 45, 47, 48, 49, 53, 54, 55, 56, 83, 85, 86, 96 quantum phase transition, 11 quantum potential, 178, 195, 196, 197, 198, 206, 213, 243, 256, 290, 292, 307 quantum potential-pilot wave, 198, 292 quantum transformations, x, xi, 66 quantum tunneling, xii, 2, 22, 23, 24, 28, 96, 332, 338, 339, 348, 384, 386 quantum tunneling (of virtual photons), 22 Quantum Turing Machine(s), 185, 273 quantum vacuum, 36, 101 Quantum Zeno effect (QZE), 277, 293 Quantum Zeno paradox, 308 qubits, 73, 175, 226, 249, 270, 273, 276, 335

R radiation, 56 Raleigh-Jeans law, 211 Randall-Sundrum, 181, 182, 271 reality, 179, 221, 265, 328 recursive discrete harmonic oscillators, 149 relativistic, vii, 103, 104, 120, 149, 150, 179, 249, 252 relativistic information processing, 179

Index resonance, viii, 274, 389, 391, 393, 395, 397, 399

413 Turing paradox, 286, 293 Turing test, 225, 226

S

U

sentience, 223, 224, 225, 226, 227, 231, 238, 263 signal, 186, 275, 336 single qubit, 76 space-time, 385 spin, 273, 276, 299 stochastic, viii, 389, 391, 393, 395, 397, 399 storage, 384 string theory, 286, 306 super quantum potential, 243, 292 supercoherence, 2, 20, 21, 22, 58, 98 superluminal particle, 331, 332, 339, 343, 356, 359, 367, 368, 371, 372, 373, 374, 375, 376, 377, 378, 379, 382, 388 Superluminal Universe, 36, 101 superposition, 334

uncertainty principle, 2, 178, 179, 183, 197, 211, 229, 232, 236, 247, 248, 255, 256, 257, 262, 264, 270, 286, 287, 288, 289, 299, 303, 304, 306, 307, 308, 309, 317, 318, 319, 320, 322, 324, 326, 327, 338, 341, 367, 370, 372, 376 unconscious, 162 unified field mechanics (UFM), xiii, 35, 36, 56, 102, 148, 149, 178, 182, 184, 186, 192, 193, 195, 197, 198, 206, 210, 211, 212, 213, 214, 223, 227, 228, 229, 232, 235, 242, 253, 264, 265, 270, 271, 277, 278, 279, 280, 285, 286, 289, 291, 294, 295, 297, 298, 299, 300, 301, 302, 304, 306, 307, 310, 312, 313, 316, 318, 320, 323, 325 Unified Field Theory (UFT), xiii, 177, 214, 245, 251, 270, 297 unitarity, 178, 179, 279 unitary matrix, 149 universal quantum computing, viii, 58, 180, 215, 224, 226, 227, 247, 267, 269, 285, 295, 328 universe, 174, 175, 216, 280, 329, 366, 376, 377, 379, 381, 382, 386

T tachyonic signaling, 200 time evolution, 32, 39, 40, 42, 54, 55, 89, 283, 285, 293, 333, 397 Toffoli gate, 285 topological quantum computing, 270, 274, 276, 278 transaction, 300 transactional interpretation, 182, 205, 206, 233, 252, 255, 266, 292 transistor, 274 transitions, 36 tunneling photon, xiii, 24, 25, 33, 95, 331, 332, 336, 337, 338, 339, 340, 341, 351, 358, 359, 360, 370, 376, 384, 386 tunneling-coupling interaction, 2, 57, 58, 59 Turing machine, 185, 225

V virtual photons, 22, 25

W water coherent domains, vii, xi, xii, 2, 22, 25, 34, 53, 57, 58, 59, 65, 73, 76, 89, 100

414

Index Y

Y gate, 285

Z Z gate, 285 Zeno machine, 367 Zero-Point (ZP) fluctuations, 3