Relativity, Gravitation, Cosmology: Beyond Foundations [Reprint ed.] 1536141356, 9781536141351

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CONTEMPORARY FUNDAMENTAL PHYSICS

RELATIVITY, GRAVITATION, COSMOLOGY BEYOND FOUNDATIONS

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CONTEMPORARY FUNDAMENTAL PHYSICS VALERIY V. DVOEGLAZOV - SERIES EDITOR UNIV. DE ZACATECAS, ZACATECAS, MEXICO Some Contemporary Problems of Condensed Matter Physics Stoyan J. Vlaev and L.M. Gaggero Sager (Editors) 2001. ISBN: 1-56072-889-2 The Photon Trilochan Pradhan 2001. ISBN: 1-56072-928-7 The Problem of Electron and Superluminal Signals V.P. Oleinik 2001. ISBN: 1-56072-938-4 Process Physics: From Information Theory to Quantum Space and Matter Reginald T. Cahill 2005. ISBN: 1-59454-300-3 Logical Foundation of Theoretical Physics G. Quznetsov 2006. ISBN: 1-59454-948-6 Introduction to Physics of Elementary Particles Oleg Mikhilovich Boyarkin 2007. ISBN: 1-60021-200-X

Probabalistic Treatment of Gauge Theories G. Quznetsov 2007. ISBN: 978-1-60021-627-5 Relativity, Gravitation, and Cosmology: New Developments Valeriy V. Dvoeglazov (Editor) 2010. ISBN: 978-1-60692-333-7 Progress in Relativity, Gravitation, Cosmology Valeriy V. Dvoeglazov and A. Molgado (Editors) 2012. ISBN: 978-1-61324-811-9 Einstein and Hilbert: Dark Matter Valeriy V. Dvoeglazov (Editor) 2012. ISBN: 978-1-61324-840-9 Quantum Mechanics in Spaces of Constant Curvature V.M. Redkov and E.M. Ovsiyuk 2012. ISBN: 978-1-61470-271-9 Revised Quantum Electrodynamics Bo Lehnert 2012. ISBN: 978-1-62081-484-0

Maxwell Electrodynamics and Boson Fields in Spaces of Constant Curvature E.M. Ovsiyuk, V.V. Kisel and V.M. Red’kov 2014. ISBN: 978-1-62618-891-4

Einstein and Others: Unification Valeriy V. Dvoeglazov (Editor) 2015. ISBN: 978-1-63463-276-8 Relativity, Gravitation, Cosmology: Beyond Foundations Valeriy V. Dvoeglazov (Editor) 2019. ISBN: 978-1-53614-135-1

CONTEMPORARY FUNDAMENTAL PHYSICS

RELATIVITY, GRAVITATION, COSMOLOGY BEYOND FOUNDATIONS

VALERIY V. DVOEGLAZOV EDITOR

Copyright © 2019 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

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Library of Congress Cataloging-in-Publication Data ISBN H%RRN

Published by Nova Science Publishers, Inc. † New York

CONTENTS Editorial Introduction

ix

Chapter 1

Classical Stueckelberg-Horwitz-Piron Electrodynamics Martin Land

Chapter 2

On Algebraic Structure of Matter Spectrum V. V. Varlamov

15

Chapter 3

Tachyons in the Framework of Special Relativity Edward Kapuścik

37

Chapter 4

An Extension of the Lorentz Symmetry Concerning the Limit of Ultra-High Energies I. A. Vernigora and Yu. G. Rudoy

57

Chapter 5

Spin 1/2 Particle with Anomalous Magnetic Moment in Presence of External Magnetic Field, Exact Solutions E. M. Ovsiyuk, V. V. Kisel and V. M. Red'kov

65

Chapter 6

Cox’s Particle in Magnetic and Electric Fields on the Background of Euclidean and Lobachevsky Geometries O. V. Veko

81

1

viii

Contents

Chapter 7

Nonlinear arctan-Electrodynamics and Charged Black Holes S. I. Kruglov

99

Chapter 8

Does the Temperature of Charged Black Holes Depend on the Charge? S. I. Kruglov

113

Chapter 9

Gravitation and Quantum Mechanics B. G. Sidharth, Abhishek Das and Arka Dev Roy

119

Chapter 10

A Brief Note on the Cosmic Background Radiation B. G. Sidharth

131

Chapter 11

Neutrinos in a Supernova via the Process

135

e e  in a 331 Model  

A. G.-Rodríguez, M. A. H.-Ruíz and A. G.-Sánchez Commentary

On “Electromagnetic Potential Vectors and Spontaneous Symmetry Breaking” V. V. Dvoeglazov

141

Book Reviews

145

In Memoriam: Dr. Thomas E. Phipps, Jr.

151

The Scientific Legacy of Dr. Peter Graneau: Instantaneous Interconnection of All Things N. Graneau

159

Editor Contact Information

173

Index

175

E DITORIAL I NTRODUCTION

We continue the Book Series “Contemporary Fundamental Physics” of the Nova Science Publishers. The thematic issue “Relativity, Gravitation, Cosmology: Beyond Foundations” contains the articles related to problems of modern physics. The book includes the Editorial Introduction and 11 chapters, a comment. and several reprints. This book may be considered as the continuation of books of our Series published by us in the past years. M. Land reviews the Stueckelberg-Horwitz-Piron electrodynamics which contains an additional chronology parameter. The author explains, why Stueckelberg introduced this parameter τ . It is noticied that the ”Grandfather paradox” can still be resolved within this framework. In this sense the causality is not afected by introducing τ - parameter. The Varlamov’s model “allows one to describe the all observed spectrum of states on an equal footing, including lepton, meson and barion sectors of the matter spectrum”. As far as I remember earlier attempts have been done by Barut, Wilson [1] and successors. Varlamov refers to the basic holistic ideas of Heisenberg. Kapu´scik reviews theories with superluminal velocities (tachyons).1 The 3dimensional Lorentz transformations have been presented for both events and 3-velocity, for both bradyons and tachyons. The author clarifies some misunderstandings in the famous Feinberg article [3] and discusses the Nimtz experiments [4]. 1

The extended review can be found in Ref. [2].

x

Valeriy V. Dvoeglazov

Rudoy and Vernigora proposed the paper ”Extension of the Lorentz symmetry up to conformal one in limit of ultra-high energies” (γ ∼ 1010 ÷ 1011 ). The paper was reported at recent conferences (2015). They stay, as a whole, on logic bases of the Kirzhnits-Chechin work [5], they suppose an additional idea, which consists not only of some phenomenologic “violation” of the group of Lorentz-Poincare transitions, but from its expansion up to the well-known conformal group. Thus, they come to overcome the GZK-limit. The Redkov’s group solves some important differential equations in relativistic quantum mechanics. Namely, the Dirac equation with the Pauli term was solved in the presence of magnetic field in the first paper. The Cox nonrelativistic wave equation has been investigated in presence of the external uniform magnetic and electric field in the case of Minkowski space by O. Veko. Extension of the problem to the case of open hyperbolic Lobachevsky 3-space is also given. In both his papers S. I. Kruglov considers the non-linear electrodynamics. For instance, such a phenomenon as vacuum birefringence, coupling with gravity as well. It was shown that the electric field does not have singularity of the charged objects. Moreover, as a result, the author made a conclusion that it was impossible the quantum tunneling through event and Cauchy horizons in 1 + 1 black holes. The result is in contrast with that for the Reissner-Nordstr¨om system. B. G. Sidharth et al. got a modified form of the Einstein’s equation, the main feature being a time varying gravitational constant. In fact, he attempts to integrate Dark Energy into the Einstein’s equation. He uses the sero-point field to modify the electromagnetic tensor. His model is in agreement with experimental predictions of General Relativity. Moreover, in fact, he derived the Cosmic Microwave Background of ∼ 4 mm, that there would be the maximum intensity, from the very basic principles of the zero point energy. A. Gutierrez et al. paper may be useful for understanding of physics of supernovae. V. V. Dvoeglazov presents his earlier unpublished paper, as always, dedicated to the fundamental problems of high-energy physics. My recent research has been presented at the XI Workshop (2015) and the X and XI Schools (2014 and 2016) of the DGFM of the Sociedad Mexicana de F´ısica (organized by H. H. Hern´andez Hern´andez) And, again about the newest history as in Ref. [6]. The administration of the Zacatecas University is: I.Q. Armando Silva Ch´airez (Secretario General, UAZ, 2008-2012; Rector, UAZ, 2012-2016), Dr. Antonio Guzm´an Fern´andez (Rec-

Editorial Introduction

xi

tor, UAZ, 2016-2020), C.P. Jos´e Said Saman Zajur (Internal Control, UAZ); Mtro. Juan Carlos Gir´on Enrquez, Mtra. Mar´ıa Baalberit Murga Rodr´ıguez and Mtro.Mario Salazar Mac´ıas (Tribunal Universitario, 2012-2016); Dr. J. de J. Araiza Ibarra (Director, UAF-UAZ, 2008-2012; Responsible of the Area de Ciencias B´asicas, 2012-2016), Dr. Sinhu´e Lizandro Hinojosa Ruiz (Responsible of the Maestr´ıa, 2006-2008; Responsible of the Licenciatura, 2008-2012; Director, UAF-UAZ, 2012-2016), Dr. Felipe Rom´an Puch Ceballos (Responsible of the Programa de Maestr´ıa, 2008-2014; Responsable of the Programa de Licenciatura, 2014-2016; Director, UAF-UAZ, 2016-2020), Dr. Carlos Alberto Ortiz Gonzalez (Leader of the Cuerpo Acad´emico de Gravitaci´on y F´ısica Matem´atica); Dr. Alejandro Puga Candelas (Responsible of the Acad´emia “M´etodos Matem´aticos”). Some of them were my students in the 90s. However, the problems in the Universidad Aut´onoma de Zacatecas continued. They are the old ones. I am just remembering professors Dr. M´aximo Augusto Ag¨uero Granados, Fis. Jorge A. Huerta Ruelas, Dr. Jos´e Andr´es Matutes Aquino, Dr. Abraham Medina Ovando, Dr. Miguel Eduardo Mora Ramos, M.Sc. Mario Enrique Rodr´ıguez Garc´ıa, who left for other Universities among others in 199294.

Acknowledgments I am very grateful to the Publisher, to our authors, referees and friends.

References [1] Barut A. O., Phys. Lett. B 73, 310 (1978); Phys. Rev. Lett. 42, 1251 (1979); Wilson R., Nucl. Phys. B 68, 157 (1974). [2] Recami E., Riv. Nuovo Cim. 9, 1 (1986). [3] Feinberg G., Phys. Rev. 159, 1089 (1967). [4] Nimtz G. and Haibel A., Ann. der Phys. 11, 163 (2002); E. Recami, F. Fontana and R. Garavaglia, Int J. Mod. Phys. A 15, 2793 (2000); E. Kapu´scik and R. Orlicki, Ann. der Phys. 523, 235 (2011). [5] Kirzhnits D. A. and Chechin V. A., Ultra-high energy cosmic rays and possible generalization of the relativistic theory. Yadernaya Fizica 15, 1051 (1972).

xii

Valeriy V. Dvoeglazov

[6] Dvoeglazov V. V. and Enciso Mu˜noz A., Hadronic J. Suppl. 17 (2002) Proceedings of the Zacatecas School on Theoretical Physics; Dvoeglazov V. V., Einstein and Hilbert: Dark Matter. (Nova Science Pubs, NY, 2011); Dvoeglazov V. V., Einstein and Others: Unification. (Nova Science Pubs, NY, 2014); Relativity, Gravitation, Cosmology: Foundations (Nova Science Pubs, NY, 2016). [7] https://zacatecasonline.com.mx/noticias/universidad/26227-eligentribunal-universitario.html.

Announcement The Nova Science Publishers (NY, USA) continues the Project of “Relativity, Gravitation, Cosmology”. The following topics are of our primary interest: 1. Dilaton gravity. 2. Quantum Mechanical Phases, Neutrino and Gravity. Photon and Gravity. 3. Spin connection and 4-potential. Axion, Torsion and Notoph. 4. Curvature as a Scalar Field over the Minkowski Space. 5. Multidimensional Gravity. De Sitter Gravity. Weyl Approach. 6. Relativistic Quantum Mechanics Approach to Gravity (a la S. Weinberg). Parity Violation. 7. Non-commutative Space-time. 8. vgroup > c. 9. Quantization. Quantum Gravity. 10. Dark Matter and Energy. The papers should be submitted to Prof. V. V. Dvoeglazov ( [email protected] ). The acceptable topics of papers are indicated above, and they are not restricted by those of the first special issues. We are ready to consider other candidates for the Editorial Board. Inquiries concerning paper submissions and book requests should be sent to [email protected] and/or [email protected].

Editorial Introduction Professor Dr. Valeriy V. Dvoeglazov Editor of “Contemporary Fundamental Physics” Book Series Nova Science Publishers, NY, USA and Profesor-Investigador Titular “C” Universidad de Zacatecas Apartado Postal 636, Suc. 3 Zacatecas 98061 Zac., M´exico December 2016 - April 2017 - December 2017.

xiii

ISBN: 978-1-53614-135-1 In: Relativity, Gravitation, Cosmology c 2019 Nova Science Publishers, Inc.

Editor: Valeriy V. Dvoeglazov

Chapter 1

C LASSICAL S TUECKELBERG -H ORWITZ -P IRON E LECTRODYNAMICS Martin Land∗ Department of Computer Science, Hadassah College Jerusalem, Israel

Abstract We give a brief overview of classical Stueckelberg-Horwitz-Piron (SHP) electrodynamics, which formalizes the distinction between coordinate time (measured by laboratory clocks) and chronology (temporal ordering) by defining 4D spacetime events xµ as functions of an external evolution parameter τ . Classical spacetime events xµ (τ ), evolving as τ grows monotonically, trace out particle worldlines dynamically and induce the five U(1) gauge potentials through which events interact. The field equations are Maxwell-like but τ -dependent and permit mass exchange between particles and fields. Maxwell theory emerges as an equilibrium limit of SHP.

1.

Introduction

In proposing that we interpret antiparticles as particles traveling backward in time, Stueckelberg introduced [1] a number of closely related innovations that ∗

E-mail address: [email protected].

2

Martin Land

are most readily appreciated in classical electrodynamics. In his model, a pair annihilation process is observed when a worldline reverses its time direction at time t∗ , so that laboratory apparatus registers two events — points on the worldline — at any t < t∗ but none for t > t∗ . To achieve this behavior in a classical picture, the worldline must evolve continuously as a function xµ (τ ), for µ = 0, 1, 2, 3, where τ is some Poincar´e invariant parameter. For example, the classical equivalent of a Feynman spacetime diagram might be given by the trajectory  
1¨ 2 ˙ ˙ x(τ ) = c t0 + t0 τ − 2 t0 τ , x(τ ) (1)

where t0 , t˙0 and t¨0 are positive constants. Pair annihilation is observed when the event evolves to τ ∗ = t˙0 /t¨0 and retreats in time from t∗ = t0 + t˙20 /2t¨0 . The particle pair observed at t = t0 combines the particle event (t˙ = +t˙0 ) occurring at τ = 0 and the antiparticle event (t˙ = −t˙0 ) occurring subsequently at τ = 2τ ∗ . Although τ is often called proper time, Stueckelberg observed that it must be independent of the particle motion because pair annihilation requires that the quantity c2 ds2 (τ ) = −gµν dxµ dxν = −x˙ 2 (τ ) dτ 2

gµν = diag(−1, 1, 1, 1)

(2)

reverse sign twice as the particle crosses the spacelike region separating futureoriented evolution to past-oriented evolution. The parameter τ thus plays the role of an irreducible chronological time, independent of the space and time coordinates, and similar to the external time t in nonrelativistic Newtonian mechanics [2]. Stueckelberg recognized that the standard Maxwell field F µν (x) would not permit ds2 (τ ) to change sign and proposed a modified Lorentz force   ν ρ D µ d 2 xµ e dxρ µ dx dx µν µ x˙ = + Γνρ = F (x)gνρ + G (x) (3) Dτ dτ 2 dτ dτ M dτ

in which the vector field Gµ (x) is required to overcome conservation of x˙ 2   D 1 D x˙ µ 2 M x˙ = M x˙ µ = ex˙ µ Gµ (x) −−−− −−−→ 0 . (4) Gµ → 0 Dτ 2 Dτ

However, Stueckelberg was not satisfied that he could justify a Hamiltonian K that produces his evolution equation in flat spacetime from the unconstrained symplectic equations dxµ ∂K = x˙ µ = dτ ∂pµ

dpµ ∂K = p˙ µ = − dτ ∂xµ

(5)

3

Classical Stueckelberg-Horwitz-Piron Electrodynamics

and instead continued his program in quantum mechanics, where, as in Feynman’s spacetime diagrams, the event may tunnel probabilistically across the spacelike region. Horwitz and Piron [3] returned to these questions in constructing a canonical relativistic mechanics for the two-body problem, introducing an invariant scalar interaction of the type K=

p21 p2 + 2 + V (|x1 − x2 |) 2M1 2M2

(6)

under which x˙ 2 is a dynamical quantity. Horwitz et. al. found solutions in this framework for relativistic generalizations of the standard central force problems, including quantum mechanical potential scattering and bound states [4]. Examination of radiative transitions [5] indicates that along with F µν , a scalar interaction, which by way of (5) produces the vector field eGµ = −∂Kscalar/∂xµ proposed by Stueckelberg, is required to account for known phenomenology. The origin of Kscalar was found by Sa’ad, Horwitz, and Arshansky [6] in the gauge invariance associated with the canonical system (5) and (6). As shown by Land, Horwitz and Shnerb [7] the most general classical interaction consistent with the quantum commutation relations [xµ , xν ] = 0

m [xµ , x˙ ν ] = −i~g µν (x)

(7)

is given by Stueckelberg’s evolution equation (3) with the substitutions F µν (x) → f µν (x, τ )

Gµ (x) → f 5µ (x, τ ) .

(8)

In flat Minkowski space this system is equivalent to the Lagrangian L=

1 e M x˙ µ x˙ µ + x˙ α aα(x, τ ) 2 c

f αβ = ∂ αaβ − ∂β aα

(9)

where in analogy with the notation x0 = ct we adopt the formal designations x5 = c 5 τ

∂5 =

1 ∂τ c5

(10)

and conventions µ, ν = 0, 1, 2, 3

α, β, γ = 0, 1, 2, 3, 5

gαβ = diag(−1, 1, 1, 1, ±1)

(11)

4

Martin Land

so that the scalar interaction becomes a fifth potential a5 (x, τ ). This Lagrangian is unique up to the five τ -dependent gauge transformations aα (x, τ ) → aα (x, τ ) + ∂α Λ(x, τ )

(12)

and the associated quantum mechanics i~∂τ ψ (x, τ ) = Kψ (x, τ ) =

»

– 1 “ µ e µ” “ e ” ec5 p − a pµ − aµ − a5 ψ (x, τ ) 2M c c c

(13)

admits the additional invariance 

 ie ψ(x, τ ) → exp Λ(x, τ ) ψ(x, τ ) ~c

(14)

when taken together with (12). This enlarged gauge invariance thus provides the grounding for Stueckelberg-Horwitz-Piron (SHP) electrodynamics. From the Lagrangian (9) the event dynamics are given by the Lorentz force e Mx ¨µ = f µα (x, τ )x˙ α c

d ec5 5µ (− 12 M x˙ 2 ) = g55 f x˙ µ dτ c

(15)

equivalent to Stueckelberg’s evolution equation (3) with the τ -dependent fields (8) in flat spacetime. The second equation, which follows from the first, expresses exchange of particle mass with the field as required for classical pair processes. This effect is scaled by c5 /c which we expect to be small. The usual a priori mass-shell constraint (−M x˙ 2 = M c2 where M is fixed) would not be consistent with a canonical system satisfying (5) and (7). Removing the constraint also eliminates general reparametrization invariance, two related features of a Lagrangian that unlike (9) is homogeneous of degree one in the velocities. To write a dynamical theory for the fields, one first considers the familiar action   Z e α 1 αβ 4 Sem = d xdτ j (x, τ )aα(x, τ ) − f (x, τ )fαβ (x, τ ) (16) c2 4c which is Lorentz and gauge invariant, and contains only first-order derivatives of the field. This 5D scalar structure suggests a larger symmetry, O(4,1) for g55 = 1 or O(3,2) for g55 = −1, that breaks to O(3,1) in the presence of matter. Re-expressing the velocity-potential interaction as a current-potential integral X˙ α aα →

Z

Z “ ” 1 d4 x X˙ α (τ )δ4 x − X(τ ) aα (x, τ ) = d4 x j α(x, τ )aα (x, τ ) c

(17)

Classical Stueckelberg-Horwitz-Piron Electrodynamics

5

we find the current in the form j α(x, τ ) = cX˙ α(τ )δ 4 x − X(τ )




.

(18)

This action leads to Maxwell-like field equations, a wave equation and a Green’s function from which fields for general event trajectories may be found. But the pointlike support of the current (18) poses difficulties in describing even the simple case of low energy Coulomb scattering. The potential at an observation point (x0 , x) induced by a ‘static’ particle, an event evolving uniformly along the x0 axis as xsource = (cτ, 0), is     e |x|  a0 (x, τ ) = δ τ − t − (19) 4π|x| c with support sharply focused on the lightcone of the source event’s immediate location, a form not easily reconciled with experiment.

2.

Higher Order Field Derivative

We may remove the singularity appearing in (19) by writing the non-local action [8] Sem =

Z

d4 xdτ



e α j (x, τ )aα (x, τ ) − c2

Z

iff ds 1 h αβ f (x, τ )Φ(τ − s)fαβ (x, s) (20) λ 4c

where λ is a parameter with dimensions of time. The field interaction kernel is Z i dκ h 2 00 Φ(τ ) = δ (τ ) − (ξλ) δ (τ ) = 1 + (ξλκ)2 e−iκτ (21) 2π where

  c 2  1 5 ξ= 1+ 2 c

(22)

is chosen so that the low energy Lorentz
force agrees with Coulomb’s law. The kinetic term now includes ∂τ f αβ (x, τ ) (∂τ fαβ (x, τ )) which breaks any 5D symmetry to O(3,1). The inverse function of the interaction kernel is written Z dκ 1 e−iκτ −1 ϕ(τ ) = λΦ (τ ) = λ = e−|τ |/ξλ (23) 2 2π 1 + (ξλκ) 2ξ

6

Martin Land

which satisfies Z ds ϕ (τ − s) Φ (s) = δ(τ ) λ

Z

dτ ϕ (τ ) = 1 . λ

(24)

Varying the action (20) with respect to the potentials, and using (24) to remove Φ(τ ), we obtain equations for the local field sourced by a non-local superposition of event currents, Z e e αβ ∂β f (x, τ ) = ds ϕ (τ − s) j α (x, s) = jϕα (x, τ ) (25) c c ∂α fβγ + ∂γ fαβ + ∂β fγα = 0

(26)

where (26) is the Bianchi identity for the exact form f αβ . These are formally similar to 5D Maxwell equations and are called pre-Maxwell equations. The convolution with ϕ (τ ) in (25) smooths the sharp particle current j α (x, τ ) over a neighborhood of width λ (in time) around the exact τ -synchronization of each event. Rewriting the field equations in 4D tensor, vector and scalar components, they take the form ∂µ f 5µ = ∂ν f5µ − ∂µ f5ν +

e 5 j c ϕ 1 ∂ fµν = 0 c5 ∂τ

∂ν f µν −

1 ∂ 5µ e µ f = jϕ c5 ∂τ c

∂µ fνρ + ∂ν fρµ + ∂ρ fµν = 0

(27) which may be compared with the 3-vector form of Maxwell’s equations with f 5µ playing the role of the vector electric field and f µν playing the role of the magnetic field. The pre-Maxwell equations in Lorenz gauge lead to the wave equation   g55 2 α e β α µ ∂β ∂ a = ∂µ ∂ + 2 ∂τ a = − jϕα (x, τ ) (28) c5 c which partially preserves 5D symmetries broken by the O(3,1) symmetry of the event dynamics. A Green’s function solution to   g55 2 µ ∂µ ∂ + 2 ∂τ G(x, τ ) = −δ 4 (x) δ (τ ) (29) c5

7

Classical Stueckelberg-Horwitz-Piron Electrodynamics can be used to obtain potentials in the form Z

e α d4 x0 dτ 0 G x − x0 , τ − τ 0 jϕα x0 , τ 0 . a (x, τ ) = − c

(30)

The principal part Green’s function [9] is GP (x, τ )

1 c5 ∂ 1 δ(x2 )δ(τ ) − 2 θ(−g55 gαβ xα xβ ) p 2π 2π ∂x2 −g55 gαβ xα xβ = GM axwell + GCorrelation

(31)

where GM axwell breaks any 5D symmetry to O(3,1) while GCorrelation has support determined by

−g55 gαβ xα xβ =

` 2 ´ 8 2 2 2 2 2 2 2 < − x + c5 τ = c t − x − c5 τ > 0 : `

´ x2 − c25 τ 2 = x2 − c2 t2 − c25 τ 2 > 0

,

g55 = 1

,

g55 = −1

(32)

with symmetry and causality properties dependent on the choice of g55 . The contribution to the potential from GM axwell with the current (18) Z h i e α a (x, τ ) = ds ϕ (τ − s) X˙ α(s) δ (x − X α(s))2 2π e X˙ α (τR ) (33) = ϕ (τ − τR ) 4π (xµ − X µ (τ )) X˙ (τ ) R µ R

is found using the identity Z f (τR ) dτ f (τ ) δ [g (τ )] = 0 , |g (τR )|

(34)

where τR is the retarded time that solves g (τ ) = (x − X(τR))2 = 0

θret = θ x0 − X 0 (τR )




.

(35)

We recognize (33) as the standard Li´enard-Wiechert potential multiplied by ϕ (τ − τR ) which contains the τ -dependence of the fields and expresses the relative time synchronization between the source and a test event experiencing the potential at the spacetime point x at the chronological time τ .

8

Martin Land For the ‘static’ particle the potential at an observation point (x0 , x) becomes 0 1 „ « e |x| A a (x, τ ) = ϕ @τ − t − 4π|x| c 0

a=0

a5 (x, τ ) =

c5 0 a (x, τ ) . c

(36)

Removing the higher-order term from the action, ϕ (τ − τR ) → λδ (τ − τR ), producing the complications associated with (19). The GCorrelation contribution is smaller [10] by c5 /c and drops off as 1/ |x|2 , so it may be neglected when (33) is significant. Using (23) a slow-moving test event with x˙ 0 ∼ c will experience the Lorentz force ! c5 e−|x|/ξλc 2 1 − g55 c Mx ¨ = −e (37)
2 ∇ 4π |x| 1 + c5 c

which has the form of a Yukawa-type potential with photon mass mγ ∼ ~/ξλc2 . If the test event is an antiparticle with x˙ 0 ∼ −c then the force is modified by   c5  c5  −e2 1 − g55 → e2 1 + g55 (38) c c

and so c5 /c must be smaller than the experimental error in the measured symmetry between classical particle/particle and particle/antiparticle scattering crosssections. Thus we take ξ ' 1/2 and see that the value of λ is constrained by the experimental error in the photon mass: using mγ < 10−18 eV /c2 requires λ > 10−2 seconds.

3.

Mass Exchange

As seen in (15) particles may exchange mass with fields, precisely the effect sought by Stueckelberg in order to achieve a classical description of pair creation/annihilation. In the annihilation example of section 1, the particle must continuously lose mass as the future-oriented trajectory turns toward the spacelike region and then regain this mass as it enters the past-oriented trajectory, so that x˙ 2 (∞) = x˙ 2 (−∞). Because the action is invariant under 5D translations δxα , the mass-energy-momentum tensor   1 αγ β 1 γδ θ0αβ = fΦ fγ + fΦ fγδ g αβ (39) c 4

Classical Stueckelberg-Horwitz-Piron Electrodynamics with fΦαβ (x, τ )

=

is a conserved Noether current

Z

ds Φ(τ − s) f αβ (x, s) λ

9

(40)

e βα f jα . (41) c2 Integrating over spacetime and using the current (18), the LHS and RHS become Z Z Z 1 d αβ 5β 4 4 d x ∂α θ0 = ∂5 d x θ0 = d4 x θ05β (42) c5 dτ Z e e d4 x 2 f βα (x, τ ) jα (x, τ ) = f βα (X(τ ), τ ) X˙ α(τ ) (43) c c leading to ∂α θ0αβ =

d dτ

„Z

4

d xθ

5µ

+ c5 M X˙ µ

«

=0

» „ «– Z d 1 4 55 2 g55 d x θ0 − − M x˙ =0 dτ 2

(44)

expressing conservation of total mass-energy-momentum for the particle-field system. Although mass exchange is scaled by c5 /c, the effect may be sufficiently large to overcome various no-go theorems that assume exactly fixed rest mass. For example, if an event evolving uniformly on its mass shell enters a dense plasma, its velocity x˙ µ may acquire a stochastic perturbation [11] and a momentary but significant shift in x˙ 2 . But if particle masses are not fixed a priori, some mechanism must account for their observed values. Two approaches have been recently proposed. Although GCorrelation can be neglected relative to GM axwell , we see from (32) that when g55 = 1 it has support at timelike separations for which GM axwell vanishes. In particular, a particle may interact with a field produced by its own past trajectory. It was shown in [11] that this self-interaction vanishes identically for on-shell motion, that is, when the particle energy remains constant in a co-moving frame. But in certain cases, when the particle’s time coordinate accelerates in its rest frame, the particle produces a radiation field that interacts with its future motion, tending to damp the time acceleration and restore the particle to on-shell evolution. More generally, Horwitz [12] has modeled a particle as an ensemble of n independent spacetime events Xiµ (τ ), i = 1, 2, . . ., n defined at a given τ . He has shown that the total particle mass is determined by a chemical potential. Following collisions governed by a general class of interactions that includes pair processes, particles return to their equilibrium mass values.

10

4.

Martin Land

Plane Wave Solutions

To further compare SHP with standard Maxwell electrodynamics, we consider the Fourier transform f (x, τ ) =

1 (2π)5

Z

α

d5 k eikα x f (k) =

1 (2π)5

Z

0 +g c κτ ) 55 5

d4 k dκ ei(k·x−k0 x

f (k, κ) (45)

of the 3-vector + scalar fields ei = f 0i

bi = ijk f jk

i = f 5i

0 = f 50

(46)

for which the sourceless pre-Maxwell equations are k · e − g55 κ0 = 0

k ·  − k 0 0 = 0

k·b=0

k × e − k0 b = 0

k × b + k0 e − g55 κ = 0

k ×  − κb = 0

−κe + k0  − k0 = 0

(47)

and the 5D mass shell condition kα kα = k2 − (k0 )2 + g55 κ2 = 0

(48)

follows from the wave equation. Plane wave solutions can then be written as e = e⊥ +

g55 κ k k0

b=

1 k × e⊥ k0

 = k +

κ e⊥ k0

0 =

1 k · k k0

(49)

2 = 1. Since k µ is the energy-momentum of the field, we see where we used g55 that κ represents the mass carried by the field. In the limit κ → 0 the field is massless and (0 , ) decouples from e and b, which become transverse to the direction of propagation k. The Poynting vectors found from (39) become

θ0µ =

´ kµ ` e⊥ · e⊥ + g55 k · k k0

θ5µ =

κkµ 00 θ (k0 )2

θ55 =

κ2 θ00 (k0 )2

(50)

showing again that the field mass vanishes when κ → 0.

5.

Maxwell Theory as an Equilibrium State of SHP

The vanishing 5-divergence ∂αjϕα (x, τ ) = 0 is a consequence of the gauge symmetry (12) and follows explicitly from f αβ in (25). Horwitz [6] extended an argument by Stueckelberg, showing that under the boundary conditions jϕ5 (x, τ ) −−−−−−−→ 0 τ →±∞

f 5µ (x, τ ) −−−−−−−→ 0 τ →±∞

(51)

Classical Stueckelberg-Horwitz-Piron Electrodynamics

11

τ -integration of the pre-Maxwell equations provides Maxwell equations as 9 e ∂β f αβ (x, τ ) = jϕα (x, τ ) > > > c > > = ∂[α fβγ] = 0 > > > > > ; α ∂α j = 0

−−−−Z−−−−−−→ dτ λ

where α

J (x) =

Z

dτ α j (x, τ ) λ

F

αν

8 e > ∂ν F µν (x) = J µ (x) > > c > > > > > > e > < ∂ν F 5ν (x) = J 5 (x) c > > > > ∂[µ Fνρ] = 0 > > > > > > : ∂µ J µ (x) = 0

(x) =

Z

dτ αν f (x, τ ) . λ

(52)

(53)

This integration is called concatenation and is understood as forming a combined image at a spacetime point xµ from all events that occur at the point over τ . In this image F µν decouples from F 5µ and satisfies Maxwell’s equations. In particular, concatenation of (36) using (24) recovers the Coulomb potential and concatenation of jϕµ (x, τ ) yields the Maxwell current J µ (x) = c

Z

Z “ ” “ ” dτ ds ϕ (τ − s) X˙ µ (s)δ4 x − X(s) = c dτ X˙ µ (τ )δ4 x − X(τ ) (54)

in standard form. Similarly, concatenation of (31) Z Z 1 2 dτ GM axwell = D(x) = − δ(x ) dτ GCorrelation = 0 (55) 2π recovers the 4D Maxwell Green’s function. Integrating the fields in the Fourier representation (45) introduces a factor of δ(κ) enforcing κ = 0, restoring the fields to the zero mass shell kµ kµ + g55 κ2 = kµ kµ = 0 and recovering the standard configuration for E and B plane wave polarizations. Maxwell theory may also be recovered from SHP [10] by taking c5 /c → 0 which slows the τ -evolution to zero and freezes the microscopic system into a static equilibrium. Under this condition the homogeneous equation (26) imposes the condition c5 (∂ν f5µ − ∂µ f5ν ) + ∂τ fµν = 0 −−−−−−−→ ∂τ fµν = 0 . c5 →0

(56)

Since the τ -dependence of Li´enard-Wiechert fields (33) resides in ϕ(τ − τR ), (56) can only be satisfied by taking λ → ∞ which entails ϕ(τ ) = 1 and renders

12

Martin Land

all field components τ -independent. As under concatenation, f µν decouples from f 5µ in the pre-Maxwell equations (25), satsifies Maxwell’s equations, and the pre-Maxwell current takes the form Z Z jϕα (x, τ ) = ds ϕ (τ − s) j α (x, s) −→ ds 1 · j α (x, s) = J α (x) (57)

so that jϕµ(x, τ ) = J µ (x) behaves as the divergenceless Maxwell current. The photon mass mγ ∼ ~/ξλc2 associated with the Yukawa potential in (36) vanishes in the limit λ → ∞, recovering the Coulomb potential. Because the fields are τ -independent, the Fourier representation imposes κ = 0, recovering the plane wave structure of zero mass shell Maxwell fields.

Conclusion Classical SHP is a theory of events xµ (τ ) that induce 5-currents jϕα (x, τ ) with support in a neighborhood xµ (τ ± λ). Events interact through gauge potentials aα (x, τ ) induced by these currents, governed canonically by a covariant Hamiltonian that generates evolution of a 4D block universe defined at τ to an infinitesimally close 4D block universe defined at τ + dτ . The complete worldlines characterized by the Maxwell currents J µ (x) can only be known a posteriori by concatenation of the instantaneous event currents. The Maxwell theory of interacting particle worldlines can thus be understood as an expectation state of this microscopic event dynamics. Maxwell theory is also obtained by taking c5 → 0, producing an equilibrium limit in which the system becomes τ -independent, so that the 4D block universe remains static. Stueckelberg-Horwitz-Piron electrodynamics can be approached as an abstract gauge theory in which the U(1) gauge transformation (14) depends on the evolution parameter in the dynamical framework. But just as Maxwell sought to formalize the empirical results of Cavendish and Coulomb, SHP may be seen as including within classical electrodynamics the creation/annihilation processes observed by Anderson. For Stueckelberg, pair processes provide empirical evidence that time must be understood as two distinct physical phenomena, chronology and coordinate, and so must be formalized through independent quantities τ and (x0 , x) in a physically reasonable theory. In light of this distinction, SHP theory must be symmetric [13] under the discrete Lorentz transformations P and T , which are sufficient to describe pair processes, but need not impose τ -reversal symmetry on the first order evolution equations. In SHP,

Classical Stueckelberg-Horwitz-Piron Electrodynamics

13

fields transform tensorially (sign reversal of 0-indexed quantities) under T and energy reversal follows from E = cp0 = M c2 dt/dτ → −M c2 dt/dτ . T symmetry requires equality of the cross-sections for a pair creation process and a corresponding pair annihilation process, without changing the order of events. It may be loosely claimed that τ -reversal takes place when the laboratory record of an antiparticle trajectory x = (−|u0 |τ, uτ ) is reordered as a series of events with an increasing x0 time-stamp, but this reinterpretation is performed by the observer, not by the underlying dynamics. Grandfather paradoxes are resolved by noticing that returning to a past coordinate time x0 must take place while chronological time τ continues to increase. The occurrence of event xµ (τ ) at τ is an irreversible process unchanged by a subsequent event occurring at the same spacetime location. Closed timelike curves are similarly absent in SHP quantum field theory, thus eliminating divergent matter loops.

References [1] Stueckelberg E. C. G., 1941 Helv. Phys. Acta 14 322, Stueckelberg E. C. G., 1941 Helv. Phys. Acta 14 588. [2] Horwitz L. P., Arshansky R. I. and Elitzur A. C., 1988 Found. of Phys. 18 1159. [3] Horwitz L. P. and Piron C., 1973 Helv. Phys. Acta 48 316. [4] Horwitz L. P. and Lavie Y., 1982 Phys. Rev. D 26 819. Arshansky R. I. and Horwitz L. P., 1989 J Math Phys. 30 213. Arshansky R. I. and Horwitz L. P., 1988 Phys Lett A 131 222. Arshansky R. I. and Horwitz L. P., 1989 J Math Phys. 30 66. Arshansky R. I. and Horwitz L. P., 1989 J Math Phys. 30 380. [5] Land M. C., Arshansky R. I. and Horwitz L. P., 1994 Found. of Phys. 24 563. Land M. C. and Horwitz L. P., 1995 J. Phys. A: Math. and Gen. 28 3289. Land M. C., 2001 Found. of Phys. 31 967. [6] Saad D., Horwitz L. P. and Arshansky R. I., 1989 Found. of Phys. 19 1126. [7] Land M. C., Shnerb N. and Horwitz L. P., 1995 J. Math. Phys. 36 3263. [8] Land M. C., 2003 Found. of Phys. 33 1157.

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Martin Land

[9] Land M. C. and Horwitz L. P., 1991 Found. of Phys. 21 299. [10] Land M., 2017 (to appear) Speeds of light in Stueckelberg-Horwitz-Piron electrodynamics, J. Phys.: Conf. Ser.. [11] Land M., 2017 (to appear) Mass stability in classical StueckelbergHorwitz-Piron electrodynamics, J. Phys.: Conf. Ser.. [12] Horwitz L. P., 2017 (to appear) A statistical mechanical model for mass stability in the SHP theory, J. Phys.: Conf. Ser.. [13] Land M. C., 2005 Found. of Phys. 35 1263. [14] Land M. C. and Horwitz L. P., 1991 Found. of Phys. Lett. 4 61. [15] Land M., 2013 J. Phys.: Conf. Ser. 437 012011. [16] Feynman R. P., 1950 Phys. Rev. 80 440 Feynman R. 1948 Rev P. Mod. Phys. 20 367. [17] IARD talk. [18] Anderson C. D., 1932 Phys. Rev. 41 405. [19] Bethe H. A. and Heitler W., 1934 Proc. R. Soc. London A 146 83. [20] L¨otstedt E. et. al., 2009 New J. Phys. 11 013054. [21] M¨uller C., 2010 High-energy collision processes involving intense laser fields, EMMI Workshop. [22] Horwitz L. P. and Piron C., 1973 Helv. Phys. Acta 48 316 . Arshansky R. L., Horwitz P. and Lavie Y., 1983 Found. of Phys. 13 1167. Land M. C. and Horwitz L. P., 1991 Found. of Phys. Lett. 4 61. Land M. C. and Horwitz L. P., 1991 Found. of Phys. 21 299. Land M. C., Shnerb N. and Horwitz L. P., 1995 J. Math. Phys. 36 3263. Land M. C. and Horwitz L P 1998 Land M. C., A239 135. Land M. C., 1996 Found. of Phys. 27 19. Land M. C., 2001 Found. of Phys. 31 967. Land M. C., 2011 J. Phys.: Conf. Ser. 330 012015. Land M., 2013 J. Phys.: Conf. Ser. 437 012012.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 2

O N A LGEBRAIC S TRUCTURE OF M ATTER S PECTRUM V. V. Varlamov∗ Siberian State Industrial University, Novokuznetsk, Russia

Abstract An algebraic structure of matter spectrum is studied. It is shown that a base mathematical construction, lying in the ground of matter spectrum (introduced by Heisenberg), is a two-level Hilbert space. A twolevel structure of the Hilbert space is defined by the following pair: 1) a separable Hilbert space with operator algebras and fundamental symmetries; 2) a nonseparable (physical) Hilbert space with dynamical and gauge symmetries, that is, a space of states (energy levels) of the matter spectrum. The each state of matter spectrum is defined by a cyclic representation within Gel’fand-Naimark-Segal construction. A decomposition of the physical Hilbert space onto coherent subspaces is given. This decomposition allows one to describe the all observed spectrum of states on an equal footing, including lepton, meson and baryon sectors of the matter spectrum. Following to Heisenberg, we assume that there exist no fundamental particles, there exist fundamental symmetries. It is shown also that all the symmetries of matter spectrum are divided onto three kinds: fundamental, dynamical and gauge symmetries. ∗

E-mail address: [email protected].

16

V. V. Varlamov

Keywords: matter spectrum, Hilbert space, coherent subspaces, superposition principle, reduction principle, symmetries

1.

Introduction

As is known, one of the most important undecided problems in theoretical physics is a description of the mass spectrum of elementary particles (it is one from 30 problems in Ginzburg’s list [1]). A some progress in systematization of hadron spectra has been achieved in SU(3)- and SU(6)-theories (see, for example, [2]). In the recent past it has been attempted to describe a mass distribution of baryon octets within quark model and its extensions [3, 4]. However, a lepton sector of the particle spectrum takes no place in the framework of dynamical symmetries (SU(N )-theories). Furthermore, a gauge sector has an isolated position. Such a triple division of the particle spectrum is a distinctive feature of the standard model (SM), in which an existence of the three kinds of ‘fundamental particles’ (quarks, leptons and gauge bosons) is postulated. It is not hard to see that in SM we have a description scheme of the particle spectrum from a position of reductionism, according to which all the members of hadron sector (baryons and mesons) are constructed from the quarks, and leptons and gauge bosons are understood as fundamental particles. As it is known, an antithesis to reductionism is a holism (teaching about whole). A description scheme of particle spectrum from a position of holism (an alternative to reductionism of SM) is the Heisenberg’s approach [5, 6]. According to Heisenberg, in the ground of the all wide variety of elementary particles we have a certain substrate of energy, the outline form of which (via the fundamental symmetries) is a matter spectrum. The each level (state) of matter spectrum is defined by a representation of a group of fundamental symmetry. The each elementary particle presents itself a some energy level of this spectrum. An essential distinctive feature of such description is an absence of fundamental particles. Heisenberg claimed that a notion ‘consists of’ (the main notion of reductionism) does not work in particle physics. Applying this notion, we obtain that the each particle consists of the all known particles. For that reason among the all elementary particles we cannot to separate one as a fundamental particle [6]. In Heisenberg’s approach we have fundamental symmetries instead fundamental particles. In Heisenberg’s opinion, all known symmetries in particle physics are divided on the two categories: fundamental (primary) symmetries (such as the Lorentz group, discrete symmetries, conformal group)

On Algebraic Structure of Matter Spectrum

17

and dynamical (secondary) symmetries (such as SU(3), SU(6) and so on). In the present paper we study an algebraic formulation of the main notions of matter spectrum. The general algebraic structure of matter spectrum, defined by the two-level Hilbert space, is given in the section 2. It is shown that basic energy levels (states) of matter spectrum are constructed on the ground of cyclic representations within Gel’fand-Naimark-Segal construction. A concrete realization of the operator algebra is realized via the spinor structure associated with the each cyclic representation. Pure states (cyclic representations), defining the levels of matter spectrum, are divided with respect to a charge (an action of a pseudoautomorphism of the spinor structure) on the subsets of charged, neutral and truly neutral states. It is shown that the structure of matter spectrum is defined by a partition of physical Hilbert space (a space of states) onto coherent subspaces. At this point, a superposition principle takes place in the restricted form, that is, within the limits of coherent subspaces. None of the levels (states) of matter spectrum is separate or ‘fundamental’, all the levels present actualized (localized) states of quantum micro-objects. Notion of symmetry plays a key role. Symmetries of matter spectrum are divided on the three kinds: fundamental symmetries which participate in formation of states, dynamical and gauge symmetries which relate states with each other.

2.

General Algebraic Structure of Matter Spectrum

An algebraic formulation of quantum theory was first proposed by von Neumann [7] and (in language of C ∗ -algebras) by Segal [8]. Further, an algebraic formulation of local quantum field theory was analyzed in detail by Haag [9] and by Araki [10] (see also [11, 12]). In this section we consider a general structure (algebraic formulation) of matter spectrum, which is defined by a construction of the Hilbert space of elementary particle. In its turn, this Hilbert space has a two-level structure. On the first level we have a separable Hilbert space H∞ . In H∞ according to standard rules of local quantum phenomenology (Schr¨odinger picture) we have observables (C ∗ -algebras), states, spectra of observables and fundamental symmetries. The basic observable is the energy (Hermitian operator H), the fundamental symmetry is defined by the Lorentz group SO0 (1, 3). On the second level we have a nonseparable Hilbert space HS ⊗ HQ ⊗ H∞ , in which the main structural forming components are the

18

V. V. Varlamov

states1 (rays). State vectors in HS ⊗ HQ ⊗ H∞ are constructed from irreducible finite-dimensional representations of the group SL(2, C). These representations are defined in eigenvector subspaces HE ⊂ H∞ of the energy operator H. Thus, state vectors in HS ⊗ HQ ⊗ H∞ define spin and charge degrees of freedom of elementary particle. An elementary particle presents itself a superposition of state vectors in HS ⊗ HQ ⊗ H∞ , that is, in the case of pure entangled states we have nonseparable (nonlocal) state. Therefore, two-level structure of the Hilbert space of elementary particle is defined by the pair (H∞ , HS ⊗ HQ ⊗ H∞ ).

2.1.

Hilbert Space H∞

So, on the first level we have separable Hilbert space H∞ , that is, H∞ is a Banach space endowed with enumerable base which is dense everywhere in H∞ (any element from H∞ is represented as a limit of sequence of the elements from enumerable set). Observables play a key role on the level H∞ . An original object of consideration is C ∗ -algebra A with the unit (algebra of observables or algebra of bounded observables). Hermitian elements of this algebra are bounded observables2. A positive functional ω over A (with the norm kωk ≡ ω(1) = 1) is called a state of the algebra A. A set of the all states of A we will denote as S(A). The magnitude ω(A) at A = A∗ is understood as an average value of the observable A in the state ω. S(A) is a convex set, that is, for any two states ω1 , ω2 and λ1 , λ2 ≥ 0, λ1 + λ2 = 1, we have λ1 ω1 + λ2 ω2 ∈ S(A). The state ω is called mixed state (or statistical mixture), when ω can be represented in the form ω = λω1 +(1−λ)ω2 , where 0 < λ < 1 and ω1 , ω2 are two different states of the algebra A. States, which cannot be represented as mixed states, are called pure states (pure states are extremal points of the set S(A)). A set of the all pure states of C ∗ -algebra A we denote via P S(A). Let S be a set of such 1

Segal [13] pointed out that the fundamental object associated with a physical system may be taken either as an observable or as a state. Segal wrote: “Whether observables or states are more fundamental is somewhat parallel to the same question for chickens and eggs. Leaving aside metaphysics, either notion has certain distinctive advantages as a foundational concept, but no analytical treatment starting from the states exists as yet which is of the same order of comprehensiveness and applicability as that starting from observables. In particular, the work of Birkhoff and von Neumann (1936) and of Mackey (1957), in which the states play the fundamental role, has not yet been developed to the point where their serviceability as possible frameworks for quantum field phenomenology is apparent” [13, p. 13]. 2 Hermitian elements of C ∗ -algebra A form Jordan algebra Ah . In Ah we have linear combinations with the real coefficients. The square of the each element in Ah is defined by a symmetrical product (pseudoproduct) A ◦ B = 1/4[(A + B)2 − (A − B)2 ].

On Algebraic Structure of Matter Spectrum

19

states of the algebra A, for which the condition ω(A) ≥ 0 is fulfilled for the all ω ∈ S, that is, A is a positive element of the algebra A (A can be represented in the form A = B ∗ B). Then S is called a set of physical states of A, and the pair (A, S) is called a physical system. For the arbitrary C ∗ -algebra A a transition probability between two pure states ω1 , ω2 ∈ P S(A) is given by the Roberts-Roepstorff formula [14]: |hΦ1 | Φ2 i|2 = ω1 · ω2 = 1 − 1/4kω1 − ω2 k2 , where |Φ1 i and |Φ2 i are unit vectors of the space H∞ . At this point, ω1 · ω2 = ω2 ·ω1 and ω1 ·ω2 always belongs to [0, 1]. Correspondingly, ω1 ·ω2 = 1 exactly when ω1 = ω2 . Two pure states ω1 and ω2 are called orthogonal states if the transition probability ω1 · ω2 is equal to zero. Therefore, two subsets S1 and S2 in P S(A) are mutually orthogonal when ω1 · ω2 = 0 for the all ω1 ∈ S1 and ω2 ∈ S2 . Further, nonempty subset S ∈ P S(A) is called indecomposable set in the case when S cannot be divided on the two orthogonal subsets. Following to Haag and Kastler [12], we assume that any maximal indecomposable set is a sector. So, P S(A) is divided on the sectors, therefore, in P S(A) there exists an equivalence relation ω1 ∼ ω2 if and only if there exists an indecomposable set in P S(A) containing ω1 and ω2 . Therefore, P S(A) is divided in pairs on disjoint and mutually orthogonal sectors which coincide with equivalence classes in P S(A). One of the most important aspects in theory of C ∗ -algebras is a duality between states and representations. A relation between states and irreducible representations of operator algebras was first formulated by Segal [16]. Let π be a some representation of the algebra A in the Hilbert space H∞ , then for any non-null vector |Φi ∈ H∞ the expression ωΦ (A) =

hΦ | π(A)Φi hΦ | Φi

(1)

defines a state ωΦ (A) of the algebra A. ωΦ (A) is called a vector state associated with the representation π (ωΦ (A) corresponds to the vector |Φi). Let ρ be a density matrix in H∞ , then ωρ (A) = Tr (ρπ(A)) . Analogously, ωρ (A) is a state associated with π and ωρ (A) corresponds to density matrix ρ. The states ωρ (A) are statistical mixtures of the vector states (1).

20

V. V. Varlamov

Let Sπ be a set of all states associated with the representation π. Two representations π1 and π2 with one and the same set of associated states (that is, Sπ1 = Sπ2 ) are called phenomenologically equivalent sets (it corresponds to unitary equivalent representations). Moreover, the set P S(A) of the all pure states of C ∗ -algebra A coincides with the set of all vector states associated with the all irreducible representations of the algebra A. Further, let π be a representation of C ∗ -algebra A in H∞ and let |Φi be a cyclic vector3 of the representation π defining the state ωΦ . In accordance with Gel’fand-Naimark-Segal construction (see [15]) the each state defines a some representation of the algebra A. At this point, resulting representation is irreducible exactly when the state is pure. Close relationship between states and representations of C ∗ -algebra, based on the GNS construction, allows us to consider representations of the algebra as an effective tool for organization of states. This fact becomes more evident at the concrete realization π(A). 2.1.1.

Fundamental Symmetries

Usually (in abstract-algebraic formulation), symmetry is understood as a transformation of physical system, which does not change its structural properties. In its turn, physical system is characterized by the algebra of observables A and by the set of states S(A). Following to Heisenberg, we assume that on the level of separable Hilbert space H∞ we have fundamental (primary) symmetries. Let us define fundamental symmetry as a pair of bijections α : A → A and α0 : S(A) → S(A), satisfying to coordination condition: (α0 ω)(αA) = ω(A) for the all A ∈ A, ω ∈ S(A). A set of all symmetries of the physical system form a group with multiplication defined by a composition of bijections. The product of the two symmetries (α, α0 ) and (β, β 0 ) is a symmetry (αβ, α0 β 0 ), where (αβ)(A) ≡ α[β(A)] and (α0 β 0 )(ω) ≡ α0 [β 0 (ω)]. The group G is called a group of fundamental symmetry of C ∗ -algebra A when there exists a homomorphism g → (αg , α0g ) of the group G into the group of all symmetries of the system (A, S(A)). We assume that G is a noncompact Lie group (for example, Lorentz group, Poincar´e group or conformal group). Then the following continuity condition is fulfilled: at any physical state ω ∈ S and any fixed A ∈ A the function g → ω(αg (A)) is continuous on g. At this 3

Vector |Φi ∈ H∞ is called a cyclic vector for the representation π, if the all vectors |π(A)Φi (where A ∈ A) form a total set in H∞ , that is, such a set, for which a closing of linear envelope is dense everywhere in H∞ . π with the cyclic vector is called a cyclic representation.

On Algebraic Structure of Matter Spectrum

21

point, the group G is unitary-antiunitary realized if there exists a continuous representation g → Ug of the group G defined by unitary or antiunitary operators (that is, αg are algebraic automorphisms or antiautomorphisms) in the Hilbert space H∞ such that for the all A ∈ A, g ∈ G there is αg (A) = Ug A(∗)Ug−1 , where A(∗) is A for unitary Ug and A∗ for antiunitary Ug . 2.1.2.

Concrete Realization π(A)

In this section we will consider a concrete realization of the operator algebra A. A transition A ⇒ π(A) from A to a concrete algebra π(A) is called sometimes as ‘clothing’. So, the basic observable is energy which represented by Hermitian operator H. Let G = SO0 (1, 3) ' SL(2, C)/Z2 be the group of fune ' SL(2, C) damental symmetry, where SO0 (1, 3) is the Lorentz group. Let G be the universal covering of SO0 (1, 3). Let H be the energy operator defined on the separable Hilbert space H∞ . Then all the possible values of energy (states) are eigenvalues of the operator H. At this point, if E1 6= E2 are eigenvalues of H, and |Φ1 i and |Φ2 i are corresponding eigenvectors in the space H∞ , then hΦ1 | Φ2 i = 0. All the eigenvectors, belonging to a given eigenvalue E, form (together with the null vector) an eigenvector subspace HE of the Hilbert space H∞ . All the eigenvector subspaces HE ∈ H∞ are finite-dimensional. A dimensionality r of HE is called a multiplicity of the eigenvalue E. When r > 1 the eigenvalue E is r-fold degenerate. Further, let Xl , Yl be infinitesimal operators e of the complex envelope of the group algebra sl(2, C) for universal covering G, l = 1, 2, 3. As is known [17], the energy operator H commutes with the all e Let us conoperators in H∞ , which represent a Lie algebra of the group G. sider an arbitrary eigenvector subspace HE of the energy operator H. Since the operators Xl , Yl and H commute with the each other, then, as is known [18], for these operators we can build a common system of eigenfunctions. It means that the subspace HE is invariant with respect to operators Xl , Yl (moreover, the operators Xl , Yl can be considered only on HE ). Further, we suppose that there e defined by the operators acting in is a some local representation of the group G the space H∞ . At this point, we assume that all the representing operators commute with H. Then the each eigenvector subspace HE of the energy operator is invariant with respect to operators of complex momentum Xl , Yl . It allows us to identify subspaces HE with symmetrical spaces Sym(k,r) of interlocking representations τ k/2,r/2 of the Lorentz group. Thus, we obtain a concrete realization (‘clothing’) of the operator algebra π(A) → π(H), where π ≡ τ k/2,r/2 .

22

V. V. Varlamov

The system of interlocking representations of the Lorentz group is shown on the Figure 1 (for more details see [19, 20]). Hence it follows that the each possible value of energy (energy level) is a vector state of the form (1): ωΦ (H) =

hΦ | τ k/2,r/2 (H)Φi hΦ | π(H)Φi = , hΦ | Φi hΦ | Φi

(2)

The state ωΦ (H) is associated with the representation π ≡ τ k/2,r/2 and the each ωΦ (H) corresponds to non-null (cyclic) vector |Φi ∈ H∞ . Analogously, if ρ is the density matrix in H∞ , then
ωρ (H) = Tr ρτ k/2,r/2 (H) is a statistical mixture of the vector states (2). .. .. . .. . .. . .. . . .1 , 7 ) .. .3 , 5 ) .. .5 , 3 ) .. .7 , 1 ) .. ( ( ( ( (0,4) (1,3) (2,2) (3,1) (4,0) 2 2 2 2 2 2 2 2 • (0, 7 ) • ( 1 ,3) • (1, 5 ) • ( 3 ,2)···•··· (2, 3 ) • ( 5 ,1) • (3, 1 ) • ( 7 ,0) • · · · · 2 2 2 2 · 2 2 2 · 2 • 1 5 • • 5 5 • .• ..• ·• 3 3 ·• ···· ( 2 , 2 ) ·····(2,1) ( 2 , 2 ) (3,0) .. .. (0,3) ( 2 , 2 ) (1,2) · ·• 3 • 3 •· 1 • 5 .• . .. ..• (0, 25 ) • ( 12 ,2) ···· (1, 2 ) ( 2 ,1) ·····(2, 2 ) ( 2 ,0).. ... · .. .. .• • 3 1 ·• .• . . ·• 1 3 • .. .. .. (0,2) ···· ( 2 , 2 ) (1,1) ( 2 , 2 ) ·····(2,0) .. ... .. · . . .. . . ..•··· (0, 3 ) • ( 1 ,1) • (1, 1 ) • ( 3 ,0)···..• .. . .. .. .. . 2 2 2 ··· . .. ... .. .. ··· •2 . . .. .. ..·····(0,1) • ( 12 , 12 )• (1,0)·····..• ... .. .. .. .. .. .. . .. · • · . . . . ·..• ... ... .. ... ... .. .. . .. .. ..·····(0, 21 ) • ( 12 ,0) · · · ·· . . . . . . .. .. .. .. .. .. · .·• ··...• ... .. ... ... .. .. · .· .· .. .. .· .. .. ·····(0,0) · · · · . · · . · •· · . ·· · . ·· ·· − 27 −3− 52 −2− 23 −1− 12 0 12 1 32 2 52 3 72 Figure 1. Eigenvector subspaces HE ' Sym(k,r) of the energy operator H. The each subspace HE (level of matter spectrum) is a space of irreducible representation τ k/2,r/2 belonging to a system of interlocking representations of the Lorentz group. The first cell of spinorial chessboard of second order is marked by dotted lines. Further, in virtue of the isomorphism SL(2, C) ' Spin+ (1, 3) we will cone as a spinor group. It allows us to associate in adsider the universal covering G dition a spinor structure with the each cyclic vector |Φi ∈ H∞ (in some sense, it be a second layer in ‘clothing’ of the operator algebra). Spintensor representae ' Spin (1, 3) form a substrate of interlocking representions of the group G + tations τ k/2,r/2 of the Lorentz group realized in the spaces Sym(k,r) ⊂ S2k+r ,

23

On Algebraic Structure of Matter Spectrum

where S2k+r is a spinspace. In its turn, as it is known [21], a spinspace is a minimal left ideal of the Clifford algebra C`p,q , that is, there exists an isomorphism S2m (K) ' Ip,q = C`p,q f , where f is a primitive idempotent of C`p,q , and K = f C`p,q f is a division ring of the algebra C`p,q , m = (p + q)/2. The complex spinspace S2m (C) is a complexification C ⊗ Ip,q of the minimal left ideal Ip,q of the real subalgebra C`p,q . So, S2k+r (C) is the minimal left ideal ∗

of the complex algebra C2k ⊗ C2r ' C2(k+r) (for more details see [22, 23]). Let us define a system of basic cyclic vectors endowed with the complex spinor structure (these vectors correspond to the system of interlocking representations of the Lorentz group): | C0 , τ 0,0 (H)Φi; | C2 , τ 1/2,0 (H)Φi,

∗

| C2 , τ 0,1/2 (H)Φi;

| C2 ⊗ C2 , τ 1,0 (H)Φi,

∗

| C2 ⊗ C2 , τ 1/2,1/2 (H)Φi,

| C2 ⊗ C2 ⊗ C2 , τ 3/2,0 (H)Φi, ∗

∗

∗

∗

| C2 ⊗ C2 , τ 0,1 (H)Φi;

∗

| C2 ⊗ C2 ⊗ C2 , τ 1,1/2 (H)Φi,

∗

∗

| C2 ⊗ C2 ⊗ C2 , τ 1/2,1 (H)Φi,

∗

| C2 ⊗ C2 ⊗ C2 , τ 0,3/2 (H)Φi; .........................................................

Therefore, in accordance with GNS construction we have complex vector states of the form hΦ | C2(k+r), τ k/2,r/2 (H)Φi c ωΦ (H) = , (3) hΦ | Φi c The states ωΦ (H) are associated with the complex representations τ k/2,r/2(H) and cyclic vectors |Φi ∈ H∞ . As is known, in the Lagrangian formalism of the standard (local) quantum field theory charged particles are described by complex fields. In our case, pure states of the form (3) correspond to charged states. At this point, the sign of charge is changed under action of the pseudoautomorphism A → A of the complex spinor structure (for more details see [24, 25, 26]). Following to analogy with the Lagrangian formalism, where neutral particles are described by real fields, we introduce vector states of the form r ωΦ (H) =

hΦ | C`p,q , τ k/2,r/2(H)Φi . hΦ | Φi

(4)

The states (4) are associated with the real representations τ k/2,r/2 (H), that is, these representations are endowed with a real spinor structure, where C`p,q

24

V. V. Varlamov

is a real subalgebra of C2(k+r). States of the form (4) correspond to neutral states. Since the real spinor structure is appeared in the result of reduction C2(k+r) → C`p,q , then (as a consequence) a charge conjugation C (pseudoautomorphism A → A) for the algebras C`p,q over the real number field F = R and quaternionic division ring K ' H (the types p − q ≡ 4, 6 (mod 8)) is reduced to particle-antiparticle interchange C 0 (see [24, 25, 26]). As is known, there exist two classes of neutral particles: 1) particles which have antiparticles, such as neutrons, neutrino4 and so on; 2) particles which coincide with their antiparticles (for example, photons, π 0 -mesons and so on), that is, so-called truly r (H) with the neutral particles. The first class is described by neutral states ωΦ algebras C`p,q over the field F = R with the rings K ' H and K ' H⊕H (types p − q ≡ 4, 6 (mod 8) and p − q ≡ 5 (mod 8)). With the aim to describe the r0 second class of neutral particles we introduce truly neutral states ωΦ (H) with the algebras C`p,q over the number field F = R and real division rings K ' R and K ' R ⊕ R (types p − q ≡ 0, 2 (mod 8) and p − q ≡ 1 (mod 8)). In r0 the case of states ωΦ (H) pseudoautomorphism A → A is reduced to identical transformation (particle coincides with its antiparticle). Further, if ρ is the density matrix in H∞ , then ωρc (H), ωρr (H) and ωρr0 (H) are statistical mixtures of charged, neutral and truly neutral states. Before we proceed with a construction of physical Hilbert space, let us consider in more detail the structure of states (2)-(4). The states (2)-(4) present the levels of matter spectrum, that is, actualized (localized) states of quantum microobjects (‘elementary particles’). The each state of the form (2)-(4) possesses the following characteristics (properties): energy (mass), spin and charge (the first two layers in ‘clothing’ of the operator algebra). On this level of description a state acquires a primary meaning (in spirit of Birkhoff-von NeumannMackey intepretation), and observable characteristics (energy, spin, charge, . . .) are properties of the state. Subsequent ‘clothing’ of the operator algebra leads to introduction of new properties (characteristics) of the state. For example, discrete symmetries (space inversion P , time reversal T , charge conjugation C and their combinations) are appeared as automorphisms of spinor structure associated with the each state of the form (2)-(4) [27, 28, 29, 30] (see also recent paper on spinors transformations [31]). Further, a fractal structure of matter spectrum (third layer of ‘clothing’) is defined by the Cartan-Bott periodicity of 4

However, it should be noted that the question whether neutrinos are Dirac or Majorana particles (truly neutral fermions) is still open (the last hypothesis being preferred by particle physicists).

On Algebraic Structure of Matter Spectrum

25

spinor structure [32]. However, detailed consideration of these characteristics of the states be beyond the scope of the present paper.

2.2.

Physical Hilbert Space

A set of pure states ωΦ (H), defined according to GNS construction by the equality (2), at the execution of condition ωΦ (H) ≥ 0 forms a physical Hilbert space Hphys = HS ⊗ HQ ⊗ H∞ . It is easy to verify that axioms of addition, multiplication and scalar (inner) product are fulfilled for the vectors ωΦ (H) → |Ψi ∈ Hphys. We assume that a so-defined Hilbert space is nonseparable, that is, in general case the axiom of separability is not executed in Hphys. The space Hphys is a second member of the pair (H∞ , Hphys), which defines two-level structure of the Hilbert space of elementary particle. Therefore, Hphys describes spin and charge degrees of freedom of the particle. In accordance with the charge degrees of freedom we separate three basic subspaces in Hphys. S ± 1) Subspace of charged states H± phys = H ⊗ H ⊗ H∞ . 2) Subspace of neutral states H0phys = HS ⊗ H0 ⊗ H∞ . 3) Subspace of truly neutral states H0phys = HS ⊗ H0 ⊗ H∞ . c Basis vectors |Ψi ∈ H± phys are formed by the states ωΦ (H) (see (3)). Correspondingly, |Ψi ∈ H0phys and |Ψi ∈ H0phys are formed by the states ωρr (H) and ωρr0 (H). Following to Birkhoff-von Neumann-Mackey interpretation [33, 34], we assume that on the level of physical Hilbert space Hphys the original (primary) notion is a state (ray). Let |Ψi be a state vector in p the space Hphys, then Ψ = eiα |Ψi, where α runs all the real numbers and hΨ |Ψi = 1, is called a unit ray. Therefore, the unit ray Ψ is a totality of basis state vectors {λ |Ψi}, λ = eiα , |Ψi ∈ Hphys. As is known, the magnitudes, related with observable effects, are absolute values of a semibilinear form | hΨ1 |Ψ2 i |2 (these values do not depend on the parameters λ characterizing the ray). Thus, a ray space is a ˆ = Hphys/S 1 , that is, a projective space of one-dimensional quotient space H subspaces from Hphys. All the states of physical (quantum) system (in our case, elementary particle) are described by the unit rays. We assume that basic correspondence between physical states and elements (rays) of the space Hphys includes a superposition principle of quantum theory, that is, there exists a collection of basis states such that arbitrary states can be constructed from them

26

V. V. Varlamov

via the linear superpositions. Hence it follows a definition of elementary particle. An elementary particle (single quantum microsystem) is a superposition of state vectors in physical Hilbert space Hphys. 2.2.1.

Group Action in Hphys

We assume that one and the same quantum system can be described by the two 0 0 different ways in one and the same subspace H± phys (Hphys or Hphys ) of the space Hphys one time by the rays Ψ1 , Ψ2 , . . . and other time by the rays Ψ01 , Ψ02 , . . .. One can say that we have here a symmetry of the quantum system when one and the same physical state is described with the help of Ψ1 in the first case and with the help of Ψ01 in the second case such that probabilities of transitions are the same. Therefore, we have a mapping Tˆ between the rays Ψ1 and Ψ01 . Since only the absolute values are invariant, then the transformation Tˆ in the Hilbert space Hphys should be unitary or antiunitary. Both these possibilities are realized in the case of subspace H± phys , state vectors of which are endowed with the complex spinor structure, because the complex field has two (and only two) automorphisms preserving absolute values: an identical automorphism and complex conjugation. Also both these possibilities are realized in the subspace H0phys, since in this case state vectors are endowed with the real spinor structure with the quaternionic division ring. In the case of subspace H0phys we have only unitary transformations Tˆ, because the real spinor structure with the real division ring admits only one identical automorphism. Let |Ψ1 i, |Ψ2 i, . . . be the unit vectors chosen from the first totality of rays Ψ1 , Ψ2 , . . . and let |Ψ01 i, |Ψ02 i, . . . be the unit vectors chosen from the second totality Ψ01 , Ψ02 , . . . such that a correspondence |Ψ1 i ↔ |Ψ01 i, |Ψ2 i ↔ |Ψ02 i, . . . is unitary or antiunitary. The first collection corresponds to the states {ω}, and the second collection corresponds to transformed states {gω}. We choose the vectors |Ψ1 i ∈ Ψ1 , |Ψ2 i ∈ Ψ2 , . . . and |Ψ01 i ∈ Ψ01 , |Ψ02 i ∈ Ψ02 , . . . such that  0 Ψ1 = Tg |Ψ1 i , Ψ02 = Tg |Ψ2 i , . . . (5)

It means that if |Ψ1 i is the vector associated with the ray Ψ1 , then Tg |Ψ1 i is the vector associated with the ray Ψ01 . If there exist two operators Tg and Tg0 with the property (5), then they can be distinguished by only a constant factor. Therefore, Tgg0 = φ(g, g 0)Tg Tg0 , (6)

On Algebraic Structure of Matter Spectrum

27

where φ(g, g 0) is a phase factor. Representations of the type (6) are called ray (projective) representations. It means also that we have here a correspondence between physical states and rays of the Hilbert space Hphys. Hence it follows that the ray representation T of a topological group G is a continuous homoˆ where L(H) ˆ is a set of linear operators in the morphism T : G → L(H), ˆ projective space H endowed with a factor-topology according to the mapping ˆ → Hphys, that is, |Ψi → Ψ. However, when φ(g, g 0) 6= 1 we cannot to H apply the mathematical theory of usual group representations. With the aim to avoid this obstacle we construct a more large group E in such manner that usual representations of E give all nonequivalent ray representations (6) of the group G. This problem can be solved by the lifting of projective representations of G to usual representations of the group E. Let K be an Abelian group generated by the multiplication of nonequivalent phases φ(g, g 0) satisfying the condition φ(g, g 0)φ(gg 0, g 00) = φ(g 0 , g 00)φ(g, g 0g 00). Let us consider the pairs (φ, x), where φ ∈ K, x ∈ G, in particular, K = {(φ, e)}, G = {(e, x)}. The pairs (φ, x) form a group with multiplication law of simidirect product type: (φ1 , x1 )(φ2 , x2 ) = (φ1 φ(x1 , x2 )φ2 , x1 x2 ). The group E = {(φ, x)} is called a central extension of the group G via the group K. Vector representations of the group E contain all the ray representations of the group G. Hence it follows that a symmetry group G of physical system induces a unitary or antiunitary representation T of invertible mappings of the space HS ⊗ HQ ⊗ H∞ into itself, which is a representation of the central extension E of G. On the level of physical Hilbert space Hphys a symmetry group G is understood as one from the sequence of unitary unimodular groups: SUT (2) (isospin group), SU(3), . . ., SU(N ), . . . (groups of so-called ‘internal’ symmetries). According to Heisenberg, the groups SU(N ) define dynamical (secondary) symmetries. Thus, in conformity with the two-level structure of the Hilbert space of elementary particle (single quantum microsystem), defined by the pair (H∞ , Hphys), all the set of symmetry groups G is divided on the two classes: 1) groups of fundamental (primary) symmetries Gf which form state vectors of quantum microsystem; 2) groups of dynamical (secondary) symmetries Gd which describe approximate symmetries (transitions) between state vectors of quantum system5 . 5

According to Wigner [35], a quantum system, described by an irreducible unitary represen-

28 2.2.2.

V. V. Varlamov Reduction Principle

So, dynamical symmetries Gd relate different states (state vectors |Ψi ∈ Hphys) of quantum system. Symmetry Gd can be represented as a quantum transition between the states of quantum system (levels of matter spectrum). For example, if we take baryons, then the reaction N → P e− ν (neutron decay) can be considered as a transition N → P , the reaction Σ− → N π − as a transition Σ → N and so on (here we neglect the leptons and mesons). At this point, we assume that representing operators of the complex envelope of Lie algebra of Gd realize all possible quantum transitions of the system. For example, if Gd = SU(3), then Okubo operators Aστ transit the states (particles) of the octet into each other. It is natural to regard that operators of the group Gd or its subgroup connect allied states. The chain of nested subgroups leads to a hierarchical classification of the states. A dynamical symmetry is defined by the chain of nested Lie groups: G = G0 ⊃ G1 ⊃ G2 ⊃ . . . ⊃ Gk . A system with the given dynamical symmetry is defined by an irreducible representation P of the group G in the space Hphys. A reduction G/G1 of the representation P of the group G on its subgroup G1 leads to a decomposition of (1) P into orthogonal sum of irreducible representations Pi of the subgroup G1 : (1)

(1)

(1)

P = P1 ⊕ P2 ⊕ . . . ⊕ Pi ⊕ . . . . In its turn, a reduction G1 /G2 of the representation of the group G1 on its sub(1) group G2 leads to a decomposition of the representations Pi into irreducible tation of the Poincar´e group P, is called an elementary particle. On the other hand, in accordance with SU(3)-theory an elementary particle is described by a vector of irreducible representation of the group SU(3). For example, in a so-called ‘eightfold way’ [36] the hadrons (baryons and mesons) are represented by the vectors of eight-dimensional regular representation Sym0(1,1) of the group SU(3). Thus, we have two mutually exclusive each other interpretations of elementary particle: as a representation of the group P and as a vector of the representation of the group SU(3). This opposition vanishes if we adopt that all the ‘elementary particles’ are localized states (levels) of matter spectrum. At this point, matter spectrum is realized via Hphys , the vectors of which are defined by cyclic representations of the operator algebra (energy operator H). On this level of description the group SU(3), defined in Hphys via central extension, describes dynamical (approximate) symmetries between different states (more precisely, between states from different coherent subspaces in Hphys ).

On Algebraic Structure of Matter Spectrum

29

(2)

representations Pij of the group G2 : (1)

Pi

(2)

(2)

(2)

= Pi1 ⊕ Pi2 ⊕ . . . ⊕ Pij ⊕ . . .

and so on6 .

3.

Coherent Subspaces

As is known [38], there are unit rays which are physically unrealizable. There exist physical restrictions (superselection rules) on the execution of superposition principle (for more details see [39, 40, 41]). In 1952, Wigner, Wightman and Wick [38] showed that existence of superselection rules is related with the measurability of relative phase of the superposition. It means that a pure state cannot be realized in the form of superposition of some states, for example, there is no a pure state (coherent superposition) consisting of boson |Ψb i and fermion |Ψf i states (superselection rule on spin). However, if we define the density matrix ρ in Hphys, then a superposition |Ψb i + |Ψf i defines a mixed state. Theorem 1. Physical Hilbert space Hphys is decomposed into a direct sum of (non-null) coherent subspaces M M Hphys = H± H0phys H0phys, (7) phys where ˙ |l−l|

HQ phys

=

M

H2|s|+1 ⊗ HQ ⊗ H∞ ,

Q = {±, 0, 0}.

(8)

˙ s=−|l−l|

At this point, superposition principle takes place in the restricted form, that is, within the limits of coherent subspaces. A non-null linear combination of 6

For example, one of the basic supermultiplets of SU(3)-theory (baryon octet F1/2 ), based on the eight-dimensional regular representation Sym0(1,1) of SU(3), admits the following SU(3)/ SU(2)-reduction into isotopic multiplets of the subgroup SU(2): Sym0(1,1) = ∗

∗

Φ3 ⊕ Φ2 ⊕ Φ2 ⊕ Φ0 , where Φ3 is a triplet, Φ2 and Φ2 are doublets, Φ0 is a singlet. Analogously, for the hypermultiplets of SU(6)-theory (baryon 56-plet and meson 35-plet) there are SU(6)/ SU(3)- and SU(6)/ SU(4)-reductions, where SU(4) is a Wigner subgroup [2, 37].

30

V. V. Varlamov

vectors of pure states is a vector of pure state at the condition that all original vectors lie in one and the same coherent subspace. A superposition of vectors of pure states from different coherent subspaces defines a mixed state. Proof. An original point of the proof is a correspondence ωΦ (H) ↔ |Ψi between states of operator algebra and basis vectors of the space Hphys. As it has been shown in the section 2.1, the set of all pure states P S(A) of the operator algebra A is divided in pairs on disjoint and mutually orthogonal sectors in virtue of the equivalence relation ω1 ∼ ω2 . Sectors coincide with equivalence classes in P S(A) (in essence, sector is an algebraic counterpart of the coherent subspace). Further, we assume that a some set of vectors in Hphys, containing pure states of the algebra A, form a total set in Hphys, that is, such a set X, closing of linear envelope of X is dense everywhere in Hphys. Then X cannot be represented as a union of the two (or more) nonempty mutually orthogonal subsets. We adopt that vectors |Ψ1 i, |Ψ2 i ∈ X are related by a correspondence |Ψ1 i ∼ |Ψ2 i, if |Ψ1 i and |Ψ2 i belong to a linear envelope from X. It easy to see that the correspondence |Ψ1 i ∼ |Ψ2 i is induced by the equivalence relation ω1 ∼ ω2 from P S(A). Therefore, |Ψ1 i ∼ |Ψ2 i is an equivalence relation and equivalence classes in X form a partition of X onto mutually orthogonal systems Xν , where {ν} = N is a some index collection. Taking as Hνphys a closed linear envelope of the set Xν , we come to a sought decomposition of Hphys into a direct sum of mutually orthogonal subspaces Hνphys: Hphys =

M

Hνphys.

ν∈N

Thus, there is a one-to-one correspondence between pure states and unit rays in ∪ν Hνphys. Hence it follows a restricted form of the superselection principle (namely, within the limits of subspaces Hνphys). It is obvious that three base sub0 0 spaces H± phys , Hphys and Hphys are coherent subspaces (with respect to charge) of the original physical space Hphys. Hence it follows the formula (7). A subsequent decomposition of Hphys onto coherent subspaces is realized with respect to spin, that is, with respect to a forming component HS in HS ⊗ HQ ⊗ H∞ (the formula (8)).

Example. According to modern data, neutrino is a superposition of three mass neutrino states: electron, muon and τ -lepton neutrinos. All the three states lie in coherent subspases H2 ⊗ H0 ⊗ H∞ , which belong to spin-1/2 line of the base

On Algebraic Structure of Matter Spectrum

31

subspace H0phys (subspace of neutral states). Nevertheless, all three neutrino states of matter spectrum, belonging to one and the same coherent subspace, are differed from each other by the energy value. The difference in energy is characterized by the different arrangement of corresponded representations on the spin-1/2 line.

3.1.

Gauge Symmetries

Let G = U (1)n ≡ U (1) × . . . U (1) be a compact n-parameter Abelian group (gauge group) defined in Hphys via the central extension. An arbitrary element of this group is represented by a collection of n phase factors: g(s1 , . . ., sn ) ≡ (eiα1 , . . ., eiαn ),

0 ≤ αj < 2π.

Let us define in the space Hphys an exact unitary representation U of the group G. Gauge transformations in Hphys have the form Qn i(α1 Q1 +...+αn Qn ) 1 U(g) = sQ . 1 . . . sn ≡ e

Generators Q1 , . . ., Qn of the gauge transformations are mutually commuting self-conjugated operators with integer spectrum. Qj (j = 1, . . . , n) are called charges, which correspond to a given gauge group. Then Hphys is decomposed into a direct sum M Hphys = Hphys(q1 , . . . , qn ) (9) q1 ,...,qn ∈Z

of corresponded spectral subspaces consisting of the all vectors |Ψi such that (Qj − qj ) |Ψi = 0. At this point, an arbitrary non-null vector |Ψi ∈ Hphys defines a pure state of the algebra π(H) exactly when |Ψi is an eigenvector for the all charges. Thus, we have standard (discrete) superselection rules in Hphys, and (9) is a decomposition of Hphys into a direct sum of coherent subspaces Hphys(q1 , . . ., qn ). According to modern situation in particle physics, superselection rules can be described completely by electric Q (= Q1 ), baryon B (= Q2 ) and lepton L (= Q3 ) charges, such that a decomposition onto coherent subspaces has the form M Hphys = Hphys(q, b, `). q,b,`∈Z

32

V. V. Varlamov

The decomposition of Hphys onto coherent subspaces with respect to charge and spin is given by the formula (7). With the aim to describe all spectrum of observed states (levels of matter spectrum) we introduce 2-parameter gauge group G = U (1)2 ≡ U (1) × U (1) with respect to baryon B and lepton L charges. Then the decomposition of Hphys onto coherent subspaces takes the following form: i Mh M M 0 0 (b, `) , (10) Hphys = H± (b, `) H (b, `) H phys phys phys b,`∈Z

where ˙

HQ phys (b, `)

=

|l−l| M

H2|s|+1 ⊗ HQ (b, `) ⊗ H∞ ,

Q = {±, 0, 0}.

˙ s=−|l−l|

The decomposition (10) allows one to embrace practically the all observed spectrum of states (see Particle Data Group). First of all, matter spectrum is divided onto three sectors: lepton, meson and baryon sectors. Lepton sector includes into itself charged leptons: electron e− , muon µ− , τ − -lepton (and their antiparticles). All charged leptons belong to coherent subspaces of the form H± phys (0, `). Neutral leptons (three kinds of neutrino) belong to coherent subspace H0phys(0, `). Lepton sector includes also one truly neutral state: photon γ (subspace H0phys(0, `)). In contrast to lepton sector, meson and baryon sectors (hadron sector in the aggregate) include a wide variety of states (particles). Meson sector is divided (with respect to charge) onto three sets of coherent subspaces. At first, charged mesons (π ± (pions), K ± (kaons), ρ± , . . .) belong to coherent subspaces of the form H± phys (0, 0) with integer spin (all mesons ∗

have integer spin). Further, neutral mesons (K 0 , K 0 , . . .) belong to subspaces H0phys(0, 0) of integer spin. In turn, truly neutral mesons (π 0 , η, ϕ, ρ0 , . . .) are the states belonging to coherent subspace H0phys(0, 0). The baryon sector is divided with respect to a charge on the two sets of coherent subspaces: charged baryons (p (proton), Σ± , Ξ± , . . .) form subspaces H± phys (b, 0) with halfinteger spin (all baryons have half-integer spin); neutral baryons (n (neutron), Σ0 , Ξ0 , . . .) are the states from coherent subspaces H0phys(b, 0) of half-integer spin. Truly neutral baryons are not discovered until now. In conclusion it should be noted that an addition of gauge symmetries leads to a triple symmetry (Gf , Gd , Gg ) division of matter spectrum. Namely, fundamental symmetries Gf participate in formation of pure states (rays) of quantum

On Algebraic Structure of Matter Spectrum

33

system and coherent subspaces in Hphys, dynamical symmetries Gd describe transitions between states from different coherent subspaces, and gauge symmetries Gg relate pure states within coherent subspaces.

References [1] Ginzburg, V. L.: What problems of physics and astrophysics seems now to be especially important and interesting. Uspekhi Fiz. Nauk. 169, 419–441 (1999) [in Russian]. [2] Rumer, Yu. B., Fet, A. I.: Theory of Unitary Symmetry. Nauka, Moscow (1970) [in Russian]. [3] Guzey, V., Polyakov, M. V.: SU(3) systematization of baryons. arXiv:hepph/0512355 (2005). [4] Melde, T., Plessas, W., Sengl, B.: Quark-Model Identification of Baryon Ground and Resonant States. arXiv:0806.1454 [hep-ph] (2008). [5] Heisenberg, W.: The Nature of Elementary Particle. Phys. Today 29(3), 32–39 (1976). [6] Heisenberg, W.: Schritte u¨ ber Grenzen. [Steps across borders.] M¨unchen (1977). [7] von Neumann, J.: On an algebraic generalization of the quantum mechanical formalism (Part I). Rec. Mat. [Mat. Sbornik] 1(43), 415–484 (1936). [8] Segal, I.: Postulates for general quantum mechanics. Ann. Math. 48, 930– 948 (1947). [9] Haag, R.: The framework of quantum field theory. Nuovo Cimento. 14, Suppl.1, 131–152 (1959). [10] Araki, H.: Einf¨uhung in die axiomatishe Quantenfeldtheorie [Introduction to axiomatic quantum field theory] I, II. Lectures Notes. ETH, Z¨urich (1961). [11] Haag, R., Schroer, B.: Postulates of quantum field theory. J. Math. Phys. 3, 248–256 (1962).

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V. V. Varlamov

[12] Haag, R., Kastler, D.: An algebraic approach to field theory. J. Math. Phys. 5, 848–861 (1964). [13] Segal, I.: Mathematical Problems of Relativistic Physics. AMS, Providence (1963). [14] Roberts, J. E., Roepstorff, G.: Some Basic Concepts of Algebraic Quantum Theory. Commun. Math. Phys. 11, 321–338 (1969). [15] Bogoljubov, N. N., Logunov, A. A., Oksak, A. I., Todorov, I. T.: General Principles of Quantum Field Theory. Nauka, Moscow (1987) [in Russian]. [16] Segal, I.: Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 53, 73–88 (1947). [17] Born, M., Heisenberg, W., Jordan, P.: Zur Quantenmechanik. II [To quantum mechanics. II]. Zs. Phys. 35, 557–615 (1926). [18] Dirac, P. A. M.: The principles of quantum mechanics. Clarendon Press, Oxford (1958). [19] Varlamov, V. V.: General Solutions of Relativistic Wave Equations. Int. J. Theor. Phys. 42, 3, 583–633 (2003); arXiv:math-ph/0209036 (2002). [20] Varlamov, V. V.: General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains. Int. J. Theor. Phys. 46, 4, 741–805 (2007); arXiv:math-ph/0503058 (2005). [21] Lounesto, P.: Clifford Algebras and Spinors. Cambridge Univ. Press, Cambridge (2001). [22] Varlamov, V. V.: Spinor Structure and Internal Symmetries. Int. J. Theor. Phys. 54, 10, 3533–3576 (2015); arXiv:1409.1400 [math-ph] (2014). [23] Varlamov, V. V.: Spinor Structure and Matter Spectrum. arXiv:1602.04050 [math-ph] (2016); to appear in Int. J. Theor. Phys. [24] Varlamov, V. V.: Discrete symmetries on the spaces of quotient representations of the Lorentz group. Mathematical structures and modeling 7, 115– 128 (2001).

On Algebraic Structure of Matter Spectrum

35

[25] Varlamov, V. V.: Universal Coverings of Orthogonal Groups. Adv. Appl. Clifford Algebras. 14, 81–168 (2004); arXiv:math-ph/0405040 (2004). [26] Varlamov, V. V.: CP T groups of spinor fields in de Sitter and anti-de Sitter spaces. Adv. Appl. Clifford Algebras. 25, 487–516 (2015); arXiv: 1401.7723 [math-ph] (2014). [27] Varlamov, V. V.: Discrete Symmetries and Clifford Algebras. Int. J. Theor. Phys. 40, 4, 769–805 (2001); arXiv:math-ph/0009026 (2000). [28] Varlamov, V. V.: The CPT Group in the de Sitter Space. Annales de la Fondation Louis de Broglie. 29, 969–987 (2004); arXiv:math-ph/0406060 (2004). [29] Varlamov, V. V.: CP T groups for spinor field in de Sitter space. Phys. Lett. B 631, 187–191 (2005); arXiv:math-ph/0508050 (2005). [30] Varlamov, V. V.: CPT Groups of Higher Spin Fields. Int. J. Theor. Phys. 51, 1453–1481 (2012); arXiv: 1107.4156 [math-ph] (2011). [31] Budinich, M.: On Spinors Transformations. J. Math. Phys. 57 (2016); arXiv: 1603.02181 [math-ph] (2016). [32] Varlamov, V. V.: Cyclic structures of Cliffordian supergroups and particle representations of Spin+ (1, 3). Adv. Appl. Clifford Algebras. 24, 849–874 (2014); arXiv: 1207.6162 [math-ph] (2012). [33] Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. of Math. 37, 823–843 (1936). [34] Mackey, G. W.: Quantum mechanics and Hilbert space. Amer. Math. Monthly. 64, 45–57 (1957). [35] Wigner, E. P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939). [36] Gell-Mann, M., Ne’eman, Y.: The Eightfold Way. Benjamin, New York (1964). [37] Fet, A. I.: Symmetry group of chemical elements. Novosibirsk, Nauka (2010) [in Russian].

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[38] Wick, G. G., Wigner, E. P., Wightman, A. S.: Intrinsic Parity of Elementary Particles. Phys. Rev. 88, 101 (1952). [39] Sushko, V. N., Khoruzhy, S. S.: Vector states on algebras of observables and superselection rules. I Vector states and Hilbert space. Teor. Math. Fiz. 4, 171–195 (1970). [40] Sushko, V. N., Khoruzhy, S. S.: Vector states on algebras of observables and superselection rules. II Algebraical theory of superselection rules. Teor. Math. Fiz. 4, 341–359 (1970). [41] Khoruzhy, S. S.: On superposition principle in algebraic quantum theory. Teor. Math. Fiz. 23, 147–159 (1975).

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 3

TACHYONS IN THE F RAMEWORK OF S PECIAL R ELATIVITY Edward Kapu´scik∗ University of Lodz and Institute of Nuclear Physics PAS, Krakow, Poland

Abstract The aim of this chapter is to show that the commonly accepted fundamental paradigm of Special Relativity on the maximality of the speed of light is false. Tachyons, the superluminal objects, are allowed by Special Relativity just as much as subluminal particles are. However, for tachyons there are no rest frames and the notion of rest mass or rest energy for tachyons cannot be operationally defined. Instead of that the notion of energy in the reference frame where tachyons move with infinite speed is defined. The expressions for energy and momentum of superluminal objects are derived. The momentum vanishes for objects moving with infinite speed. This may explain the weakness of interaction of superluminal objects with other objects. Some pointers as to how to find tachyons are discussed.

1.

Introduction

The speed of light plays a distinguished role in the Special Theory of Relativity. It is generally believed that, apart from its invariant character, the speed of light ∗

Professor Emeritus

38

Edward Kapu´scik

is also the maximal speed allowed by Special Relativity. This belief is continuously repeated by many prominent scientists. For example, Stephen Hawking in his book[1] wrote “that nothing may travel faster than the speed of light”. Next, more carefully, he stated that ”any normal object is forever confined by relativity to move at speeds slower than the speed of light” without specification as to what normal objects are. Below we shall show that by “normal” objects we should understand objects which possess rest frames. The same or similar statements can be found in an endless number of textbooks and scientific articles. As a matter of fact, such a statement is commonly treated as one of the fundamental paradigms of modern science. In spite of this, in the present paper, we shall show that this paradigm is false. The correct statement should be that “no object which posseses a rest frame (in particular, no observer) can move faster than the speed of light”. But there exist objects (tachyons) which can move faster than light but can never be at rest. In this sense tachyons are not “normal” objects. In the past there were many attempts to show that superluminal speeds can be incorporated into the framework of Special Relativity [2]. Unfortunately, all of them were purely formal without sufficient physical background and in one way or another used complex numbers in the course of reasoning. In addition, the expressions for energy and momentum of tachyons derived in this way are not correct. A comprehensive review of the early history of tachyons can be found in Ref. [11]. Our discussion is rigorously based on Special Relativity and does not contain any elements of speculation. The goal of our paper is to rectify the erroneous treatment of the problem of speeds in Special Relativity. The paper has partly a pedagogical character. It is needless to say that the presented result should be incorporated into the standard manner of teaching Special Relativity.

2.

Lorentz Transformations and Velocities of Motion

The Special Theory of Relativity is based on Lorentz transformations of spacetime coordinates (~x, t) of the form [3] V~ · R~x t → t = γ t+ , c2 0

~x → ~x0 = R~x + (γ − 1)

!

(1)

~ · R~x) (V ~ t, V~ + γ V V2

(2)

Tachyons in the Framework of Special Relativity

39

where γ=

V2 1− 2 c

!−1/2

(3)

~ being the relative velocity of two observers is the famous Lorentz factor with V tightly connected with two inertial reference frames in which the spacetime coordinates (~x, t) and (x~0 , t0 ) are used (R denotes the rotation of space axes used by two observers) and c is the invariant speed of light. Clearly, due to the square root in γ the relative velocities of reference frames must respect the condition ~ 2 < c2 . V

(4)

In two-dimensional spacetime formulas (1) and (2) reduce to the more familiar ones Vx t = γ t+ 2 , c 



(5)

x = γ (x + V t) .

(6)

0

0

The restriction (4) is the basis for the customary statement that in Special Relativity no velocity can exceed the speed of light. It is however clear that ~ in (1) and (2) (or (5) such conclusion is not justified because the velocities V and (6)) refer to the velocities of reference frames but not to the velocities of motions. The correct statement should therefore be that no reference frames (or, equivalently, no observers) can move faster than light. In fact, from formulas (1) and (2) it follows that the velocities of motion of point particles transform according to the rule 0 ~ + (γ − 1) (V~ ·R~v(t)) V ~ + γV ~ R v(t) d~x(t) 0 d~ x ~2 V   ~v (t) = → ~v 0 (t ) = 0 = , ~ v(t)) dt dt γ 1 + (V ·R~ 2 c

(7)

where we follow the good habit to use lower case letters for quantities which may vary in time and capital letters for quantities which are constant in time [4]. Again, in two-dimensional spacetime we have the more familiar simpler transformation rule 0 0 v(t) + V v (t ) = . (8) 1 + V cv(t) 2

40

Edward Kapu´scik

It is seen that contrary to the transformation rule for the spacetime coordinates ~ of reference which contains the factor γ, restricting the relative velocities V frames, the transformation rules (7) or (8) for velocities do not contain any factor which may restrict the magnitude of the velocities of motion ~v (t). It is worth noting that the transformation rule (8) very often is erroneously treated as the addition rule for relativistic velocities. It should, however, be clear that this is not correct because the addition operation may be applied only to objects of the same kind while in (8) we have two kinds of velocities: the relative velocity of observers restricted by the condition (4) and the unrestricted velocity of motion of an arbitrary physical object. For a mass point which is at rest in the unprimed reference frame we have ~v (t) = 0 and it follows from (7) that in the primed reference frame the mass 0 0 ~ . The restriction (4) implies then point moves with constant velocity ~v (t ) = V that mass points for which there exist rest frames cannot move faster than light. These are therefore Hawking’s “normal” objects. The situation, however, is quite different for objects for which there is no rest frame. Such objects cannot move with velocities less than the speed of light because for such velocities using Lorentz transformation we can always find a reference frame in which the velocity is zero and this would mean that the rest reference would exist. An example of objects without a rest frame are provided by photons which in all reference frame move with the same speed c. For other objects without rest frames the velocities cannot be restricted from above because then apart from the speed of light there would exist also a second invariant velocity. The only logical way out is to admit that for objects without rest frames there exist reference frames in which their velocity is infinite. Assuming that in the unprimed reference frame the velocity |~v(t)| → ∞ from (7) ~ of the same object is we get that in the primed reference frame the velocity W given by 2 ~ = c V ~ W (9) ~2 V and from (9) it follows now that ~ 2 > c2 . W

(10)

This clearly shows that in the framework of Special Relativity the statement “that nothing may travel faster than the speed of light” is not justified. However,

Tachyons in the Framework of Special Relativity

41

we must agree that the faster than light objects, called tachyons[3], are not quite “normal” because for them there are no rest frames of reference. Instead of that, in some reference frames the tachyons may travel with infinite speed. Contrary to Galilean physics, the infinite speed of tachyons is not an invariant speed because in any other reference frame according to (9) it is finite and greater than the speed of light. Reversing the relation (9) we get c2 ~ V~ = W ~2 W

(11)

and substituting this into formulas (1) and( 2) we get a new form of the Lorentz transformations [5] ! ~ · R~x W (12) t0 = γ t + ~2 W ~x0 = R~x + (γ − 1)

2 ~ · R~x W ~ +γ c W ~ t, W ~2 ~2 W W

(13)

where γ now is given by c2 γ = 1− ~2 W

!−1/2

.

(14)

It is clear that the restriction (10) must be satisfied. In two-dimensional spacetime these formulas simplify significantly. The transformations (12) and (13) imply also that the transformation rule for the velocities of motion now takes the form 0

~v (t ) =

~ v(t) ~ +γ R~v(t) + (γ − 1) W ·R~ W ~2



γ 1+

W  ~ ·R~ W v(t) ~ 2 W

c2 ~ W ~ W2

.

(15)

Again, we may put in this formula either ~v (t) = ~0 for particles having rest frames or take the limit |~v (t)| → ∞ for superluminal objects. In both cases we get the previous results. ~ | → ∞ these transforFrom (12) and (13) it is seen that for R = I and |W mations are the identity transformation.

42

3.

Edward Kapu´scik

Energy - Momentum Four - Vectors

It is well-known that in the Special Theory of Relativity the energy E and momentum P~ of a point particle with mass M are given by

and

M c2 E=q 2 1 − ~vc2 M~v P~ = q , 2 1 − ~vc2

(16)

(17)

where ~v is the velocity of the particle. The presence of the square root in (16) and (17) implies that |~v| < c. In order to overcome the limitation for velocities of motion, Gerald Feinberg in 1967 [2] proposed to replace the mass M in (16) and (17) by an imaginary quantity iµ. This allows the conversions of the expressions for energy and momentum into µc2 E=q 2 , (18) ~ v − 1 2 c µ~v P~ = q 2 , ~ v − 1 2 c

(19)

where ~v is the tachyon velocity and µ is the so-called metamass of the tachyon. Clearly, this time |~v | > c. It must be however stressed that both expressions (18) and (19) were not derived from some first principles but just proposed ad hoc in order to allow the tachyons to move faster than light. As a consequence of these formulas G. Feinberg pointed out that the energy~ of tachyons is a spacelike four-vector since momentum four-vector (E, P) E 2 − c2 P~ 2 = −µ2 c4 < 0

(20)

c|P~ | > E.

(21)

and therefore From the Lorentz transformation for the energy-momentum four-vector of the form   ~ , E → E 0 = γ E − RP~ · V (22)

43

Tachyons in the Framework of Special Relativity ~ RP~ · V E~ V~ − γ 2 V , P~ → P~ 0 = RP~ + (γ − 1) 2 ~ c V where γ is the famous Lorentz factor given by ~2 V 1− 2 c

γ=

!−1/2

(23)

,

(24)

with V~ the relative velocity of two observers and R the rotation of space axes used by the observers, G. Feinberg pointed out that in some reference frames the energy of tachyons becomes negative because in view of (21) for some velocities ~ the expression in the parenthesis in (22) is certainly negative. V The question, however, arises: how may the evidently positive quantity, given in all reference frames by (18), become negative? The resolution of this paradox consists in the observation that the proposed quantities (18) and (19), as a matter of fact, are not components of a fourvector and therefore under Lorentz transformations they do not transform according to (22) and (23). In fact, taking into account that E 0 ≡ E(~v 0)

(25)

~ (~v 0 ) P~ 0 ≡ P

(26)

and with 0

~v =

~ v ~ ~ R~v + (γ − 1) V~·R~ V − γV 2



γ 1−

V  ~ ·R~ V v c2

(27)

from (18) and (19), after some brief calculation, we get



E 0 = γ E − RP~ · V~

and ~0



P = sign E − RP~ · V~



(28)

RP~ · V~ ~ E RP~ + (γ − 1) V − γ 2 V~ . 2 ~ c V

"

#

(29)

The reason for the appearance of the absolute value in (28) comes from the fact that in the process of calculation we meet the expression v !2 u u v · V~ t 1 − R~ = 2

c

R~v · V~ 1 − c2

.

(30)

44

Edward Kapu´scik

Clearly, for standard subluminal velocities ~v we have R~v · V~ < c2 . Therefore ~ the expression 1 − R~cv2·V is always positive and the sign of the absolute value in (30) is not needed. As a result we get the standard Lorentz transformation rules (22) and (23). But it is not the case for superluminal speeds ~v for which ~ the expression 1 − R~cv2·V may be both positive and negative. Consequently, the signs of absolute value in (30) and consequently in (28) are necessary. Unfortunately, G. Feinberg did not notice this fact. As a result we must conclude that Feinberg’s expressions (18) and (19) do not provide the energy-momentum four-vector. In order to get the correct expressions for energy and momentum of superluminal objects let us observe that the Lorentz transformation rules (22) and (23) together with (25) and (26) lead to the following set of functional equations for energy and momentum 

E

h

~)−1 R~v + γ(V



P~ 



i

~) 1 γ(V h

~ ·R~ V v ~ ~2 V − V  ~ v − V c·R~ 2

~ )V ~ γ(V



  ~ ) E(~v) − RP~ (~v) · V ~ ,  = γ(V

(31)

~ ·R~ V v ~ ~ ~ ~ 2 V − γ(V )V V   v ~ ) 1 − V~ ·R~ γ(V c2

~)−1 R~v + γ(V

i



=

i RP ~ (~v ) · V~

~ − γ(V ~ ) E(~v) V~ . (32) V 2 ~ c2 V It is not difficult to solve this set of functional equations. For subluminal objects rest frames always exist and therefore we may put ~v = 0. Then from (31) and (32) we get h

~)−1 = RP~ (~v ) + γ(V



~ E −V







~ ) E(~0) − RP~ (~0) · V ~ , = γ(V

(33)

h i ~ ~ ~ ~ ~ ) − 1 RP (0) · V V ~ − γ(V ~ ) E(0) V~ . = RP~ (~0) + γ(V (34) ~2 c2 V Requiring that momentum is an odd function of velocity, we must put P~ (~0) = ~0 and using the Einstein relation E(~0) ≡ E0 = M c2 we arrive (after changing the sign of V~ ) to the standard expressions (16) and (17) for the energy and momentum of subluminal objects. 

~ −V ~ P



Tachyons in the Framework of Special Relativity

45

For superluminal objects there are no rest frames because the velocities are restricted from below by c. The velocities are, however, not restricted from above and therefore we may always consider the limit of infinite velocities ~v in (31) and (32). In order to execute such a limit let us choose the rotation R in ~ . Then taking the limit lim λ → ∞ we get such a way that R~v = λV c2 ~ E − V ~2 V c2 − V~ ~2 V

~ P

!

!





~ ) E(∞) − RP~ (∞) · V ~ , = γ(V

(35)

h i ~ ~ ~ ) − 1 RP (∞) · V V~ − γ(V ~ ) E(∞) V~ . (36) = RP~ (∞) + γ(V ~2 c2 V

Denoting 2 ~ = − c V~ , W ~2 V

(37)

~ . Requiring that the energy we may express the right-hand side in terms of W should be an even function of velocity while the momentum an odd function of ~ (∞) = 0 and finally we get[5][6] velocity we must put P ~ ) = q E∞ , E(W 2 1 − ~c 2

(38)

W

~ (W ~ )= P

E∞

q

~ 2 1− W

c2 ~2 W

~, W

(39)

where we have introduced the notation E∞ = E(∞) for the energy of objects moving with infinite speeds. It must be stressed that according to (37) the velocity of a tachyon in a given reference frame is always determined in a nonlinear way by the relative velocity of the given reference frame with respect to the reference frame in which the tachyon moves with infinite speed. As a consequence we get the unusual transformation rule for the tachyon velocities in the following form 

~ ~ u·RW ~2 W

γ 1−

~ = W 0

1−

~2 W c2

2

+ γ 2 w~c2





1−

~ u·RW ~ ~2 W

2

~2 ~ ~ + (γ − 1) ~u · RW ~u − γ W ~u . RW ~2 c2 W

"

#

(40)

46

Edward Kapu´scik

The above formulas imply several important conclusions. First, they rigorously follow from the general principles of the special theory of relativity and therefore are well justified. Second, the energy-momentum four-vector given by (38) and (39), contrary to the Feinberg energy-momentum “four-vector”, is a time-like four-vector because 2 E 2 − c2 P~ 2 = E∞ > 0.

(41)

This fact eliminates all troubles in constructing a quantum field theory of tachyons [7]. In particular, tachyons may exist with arbitrary spins contrary to the Feinberg tachyon which can only be spinless. Third, for infinite velocities the momentum of tachyons vanishes while their energy remains finite. Fourth, from the expressions (38) and (39) it follows that the tachyon velocity is given by ~ = E P~ . W (42) P~ 2 which must be confronted with the expression for the velocity of the subluminal particles P~ ~v = (43) E which follows from (16) and (17). Eq.(42) does not support the Feinberg argumentation that in order to determine the tachyon velocity it is sufficient to measure the momentum and energy of tachyons because, for the same values of momentum and energy, applying the relation (42), we may also obtain subluminal velocities. To resolve the dilemma as to whether we have to do with subluminal or with superluminal velocities we must use the time of flight method.

4.

A Description of Tachyonic Motion

In Special Relativity motions of mass points are customarily described in terms of the notion of the proper time τ related to the coordinate time t by the relation dτ =

s

1−

~v 2 (t) dt, c2

(44)

where ~v (t) is the velocity of the considered mass point. Unfortunately, the proper time approach cannot be applied to tachyons because for superluminal

Tachyons in the Framework of Special Relativity

47

speeds the proper time becomes imaginary and consequently it cannot be used as a parameter labelling the position of the tachyon on its trajectory. In the present chapter we shall show how to describe relativistic motions using only the coordinate time without any reference to the proper time. The clue to this goal is the velocity tensor introduced in [8]. For simplicity, we restrict here the considerations to the two-dimensional spacetime only. The passage to higher dimensional spacetimes is straightforward but technically more involved. We begin with recapitulation of the basic properties of the velocity tensors. Then, we shall explicitly construct such tensors for the Galilean and Minkowskian spacetimes. In the latter case we shall consider both the subluminal and superluminal motions. Finally, we shall consider the dynamical equations of motions which directly generalize the standard Newton dynamical equation. In the formalism described in [8] the velocity tensors Vνµ (~v ) are defined as functions of the standard three-dimensional velocity ~v. In terms of these tensors the standard kinematical relation d~x = ~v dt

(45)

Vνµ (~v)dxν = 0,

(46)

is written in the covariant form as

where dxν (ν = 0, 1, 2, 3) denotes the infinitesimal displacements along the trajectory of the particle. The general construction of the velocity tensors goes as follows. First, we use the tensorial transformation rule for the matrix V with matrix elements Vνµ given by the components of the velocity tensor. This rule reads V 0 (v~0 ) = SV (~v)S −1 ,

(47)

where S is the matrix with matrix elements Sνµ fixed by the transformation rule for spacetime coordinates dx0µ = Sνµ dxν . (48) Second, we use the additional conditions for the matrix V of the form T rj V = 0

(49)

for j = 1, ...n, where n is the dimension of spacetime and T rj denote the sums of the diagonal minors of order j of the matrix V . Conditions (49) ensure that all

48

Edward Kapu´scik

eigenvalues of the matrix V are equal to zero because under the conditions (49) the characteristic equation for the eigenvalues λ reduces to the simple equation λn = 0.

(50)

The eigenvalue equation (46) provides then an unique eigenvector dxµ . Third, we assume that matrix elements of V (~v) are form-invariant functions of the velocity ~v and therefore V 0 (~v ) = V (~v ).

(51)

This relation together with (47) provides us functional equations for finding the matrix elements of V . These functional equations obviously have the form V (v~0 ) = SV (~v)S −1 .

(52)

We shall now illustrate this method on the examples of the Galilean and Einsteinian two-dimensional spacetimes. The generalization to four-dimensional spacetime is straightforward but a little bit more tedious.

5.

Examples of Velocity Tensors

The Galilean transformations of spacetime coordinates t0 = t,

x0 = x + ut

(53)

lead to the following form of the matrix S: SG (u) =

1 0 u 1

!

,

(54)

where u is the relative velocity of the observer tight to the primed reference frame with respect to the observer tight to the unprimed reference frame. Then (52) gives the functional equation −1 VG (v + u) = SG (u)VG(v)SG (u) = SG (u)VG(v)SG(−u).

(55)

Taking the unprimed reference frame as the rest frame for the particle we must put v = 0 and we get VG (u) = SG (u)VG(0)SG (−u).

(56)

Tachyons in the Framework of Special Relativity

49

Finally, renaming the velocity u as v we get VG (v) = SG (v)VG(0)SG(−v).

(57)

In the rest frame equation (46) gives two equations V00 (0)dt + V10 (0)dx = 0

(58)

V01 (0)dt + V11 (0)dx = 0.

(59)

and But in the rest frame of the particle dx = 0 for arbitrary dt and therefore V00 (0) = V01 (0) = 0.

(60)

The conditions (49) for the two-dimensional spacetime reduce to only two conditions T r V (v) = 0 (61) and det V (v) = 0.

(62)

From these conditions and (60) we get then that also V11 (0) = 0. Below we shall see that it is convenient to normalize the remaining free matrix element V10 (0) to −1. Thus we have VG (0) =

0 −1 0 0

!

(63)

and from (57) we finally get the Galilean velocity tensor in the form VG (v) =

v −1 v 2 −v

!

.

(64)

Clearly with such a velocity tensor the standard equation (45) follows from the equation (46). In the case of the Einsteinian two-dimensional spacetime from the standard Lorentz transformations t + u2 x t0 = q c 2 , 1 − uc2

50

Edward Kapu´scik x + ut x0 = q 2 1 − uc2

we have

SL (u) = q

1 u2 c2

1−

  

(65)

1 u

u c2

1

  

(66)

and from equations (52) for the Einsteinian velocity tensor VE we get the functional equation VE

v+u 1 + vu c2

!

= SL (u)VE (v)SL−1 (−u).

(67)

Again, assuming that the unprimed reference frame is the rest frame of the particle we get the relation (56) with SG (v) replaced by SL (v). In the rest frame the velocity tensor VE (0) is exactly the same as in the Galilean case because the same conditions must be satisfied as for the Galilean case. Thus finally we get 1 VE (v) = 2 1 − vc2

v −1 v 2 −v

!

.

(68)

It is easy to check that with such velocity tensor from (46) equation (45) also follows. The presented formalism may also be applied to motions with superluminal speeds. The physics of tachyons is relatively poorly developed. The main reason for that is the widespread but erroneous opinion that the existence of tachyons contradicts the main principles of Special Relativity. Such opinions are based on the unjustified statement that the speed of light is the maximal speed allowed by Special Relativity[3]. To construct the velocity tensor for tachyons we must use the Lorentz transformation between the reference frame in which the tachyon moves with an infinite speed and the reference frame in which its speed is equal to w. This transformation leads to the following form of the matrix S(w) [6] 1

1

S(w) = q 1−

c2 w2

c2 w

1 w

1

!

(69)

and to the composition law of tachyonic speeds w12 =

w 1 w 2 + c2 . w1 + w2

(70)

51

Tachyons in the Framework of Special Relativity The tachyonic velocity tensor therefore satisfies the functional equation V

w 1 w 2 + c2 w1 + w2

!

= S(w1 )V (w2 )S −1 (w1 ) = S(w1 )V (w2 )S(−w1 ).

(71)

In the limit w2 → ∞ we get V (w) = S(w)V (∞)S(−w).

(72)

In the reference frame in which the tachyon moves with an infinite speed from (46) we have V00 (∞)dt + V10 (∞)dx = 0 (73) and V01 (∞)dt + V11 (∞)dx = 0.

(74)

Tachyons with infinite speeds in any finite time pass infinite distances. Therefore, equations (73) and (74) may be satisfied only for V10 (∞) = V11 (∞) = 0.

(75)

From the traceless condition for velocity tensors we get then also that V00 (∞) = 0.

(76)

Normalizing V01 (∞) = +1 we finally get VT (∞) =

0 0 1 0

!

.

(77)

With such velocity tensor in the reference frame where tachyons move with infinite speeds the time stops along their trajectories because from (77) it follows that dt = 0. (78) With VT (∞) of the form (77) the tachyonic velocity tensor VT (w) has the form VT (w) =



1 w

1  2  1 − wc 2 1

This tensor also gives equations (45).

− w12 − w1



 .

(79)

52

6.

Edward Kapu´scik

Dynamical Equations

We shall write the dynamical equation of motion in the form ∂µ Vνµ (~v) = Iν ,

(80)

where Iν describes the influence of the environment on the moving object. It is clear that (80) is the only covariant form which generalizes the standard Newton equation d~v(t) 1 ~ = F (t), (81) dt M where M is the mass of the particle and F~ (t) is the acting force. Below, we shall elaborate the meaning of the notion of Iν and its relation to the standard ~ (t). force F For this purpose we shall apply the dynamical equation (80) using the velocity tensors derived above. We begin with the Galilean velocity tensor (64) with the time-dependent velocity v(t). Since the components of the velocity tensor depends only on the time coordinate equation (80) reduces to two equations dv(t) = I0 (t). dt

(81)

I1 (t) = 0.

(82)

and From equation (81) it is clear that the time component of the influence is related to the customary force F (t) in the form I0 (t) =

1 F (t), M

(83)

where M is the mass of the particle. For the relativistic subluminal particles with the velocity tensor (68) equation (80) gives   d  v(t)  = I0 (t) dt 1 − v2 (t) 2

(84)

c

and the corresponding equation for the space component 



d 1  = I1 (t) −  dt 1 − v2 (t) 2 c

(85)

Tachyons in the Framework of Special Relativity

53

In the simplest case of constant in time I0 (t) = I we can integrate equation (84) and solve the result with respect to v(t). In this way we get v(t) = c where

2

q

2

−1 1 + 4 (It+Γ) 2(It + Γ) c2 q = < c, 2 2(It + Γ) 1 + 1 + 4(It+Γ) c2 v0

Γ=

1−

− It0

v02 c2

(86)

(87)

and v0 is the initial velocity at time t0 . It is easy to see that v(t) is always less than the speed of light and in the limit t → ∞ we get v(t) → ±c, where the sign depends on the sign of I. This result is to be compared with the case of a standard relativistic particle moving under the influence of a constant force for which we have vrel (t) = r

F t/M 1+



Ft Mc

2 .

(88)

Here M is the mass of the particle, F is the standard nonrelativistic force and the initial condition is such that Γ = 0 (i.e. at t0 = 0 we put v0 = 0). For large values of t formulas (86) and (88) coincide for Γ = 0 provided I = F/2M . Finally, we pass to the motion of tachyons for which the velocity tensor has the form (79) and equation (80) gives d dt



d dt



w(t) w 2 (t) − c2



= I0 (t)

(89)

= I1 (t).

(90)

and −

1 w 2 (t) − c2



For a constant in time I0 (t) we get w(t) =

1+

p

where Γ=

1 + 4c2 (It + Γ)2 > c, 2(It + Γ) w0 − It0 − c2

w02

and w0 is the initial velocity at time t0 .

(91)

(92)

54

Edward Kapu´scik

7.

Some Physical Considerations

At the end of the chapter we shall try to answer the question where to find tachyons or how to produce and detect them? It is rather hopeless to expect that tachyons will be found as ordinary elementary particles. Here the history of quarks may serve as a good lesson. It is, however, highly probable that tachyons can exist as some excitations in solids because in solids we have a lot of examples of instantaneous creation of global excitations, i. e. arising instantaneously in the whole bulk of matter. Such specific quasiparticles are usually not localizable but they immediately transfer energy between the walls of the bulk matter. Such a point of view is supported by the famous experiment of Guenter Nimtz [9] in which electromagnetic signals were transmitted with superluminal velocities. The experiment, in different versions, has been repeated several times and always signals traveling faster than light were observed. The reason for that is the fact that photons in incident waves which hit the insulators, such as paraffin used by Nimtz as a barrier, one of the most important excitations are Frenkel or Mott - Wannier excitons [10]. It is always assumed that finally the excitons recombine into photons and only photons may leave the solid. However it is highly probable that the interaction of excitons with the molecular lattice leads to some Umklapp processes [10] which clearly distort the energy - momentum balance characteristic to photons and the final photons cannot be created. The energy - momentum content may quite possibly be taken away by some kind of tachyons. It is also clear that tachyons produced in solids may also be annihilated in them. Therefore, the only outgoing tachyons may come from the surface of solids. Since we do not know how directly to detect tachyons a second piece of solid is needed in which due to the reverse processes (reverse to the processes in the first piece of solid) the tachyons are annihilated with the final production of real photons which then easily are detected in the standard way after they leave the second piece of solid. The fact that in the gap between barriers the tachyons travel with superluminal velocities is the reason that final photons are detected earlier than the electromagnetic signal which all the time travels with the velocity of light.

Conclusion The main result of the present chapter consists in the elementary proof that Special Relativity based on Lorentz transformations does not exclude superluminal

Tachyons in the Framework of Special Relativity

55

speeds. As a matter of fact, Special Relativity divides the energy-momentum space into two sectors in which subluminal motions and superluminal motions take place, correspondingly. The light cone is an impenetrable barrier between these two sectors. It is rather hopeless to expect that the superluminal objects will be found in the macroscopic domain. In microphysics it was recently proposed that probably superluminal speeds may occur in solids[9]. In view of the present paper this fact does not contradict Special Relativity. It must also be stressed that superluminal speeds do not violate the existing relativistic kinematical calculations because in microphysics the basic characteristics of objects are the frequencies and wave vectors of the quantum mechanical waves which due to the de Broglie ansatz are related to energies and momenta of quantum objects and not directly to their velocities. The velocities are always determined from classical relations of energy and momentum to velocities. In this sense the velocities are auxiliary quantities which help to visualize quantum processes. It is astonishing that in the centennial history of Special Relativity the problem of superluminal speeds was not correctly treated. In view of the present paper it is necessary to change the way in which Special Relativity is taught.

References [1] Hawking S. W., A Brief History of Time, Bantam Books, 1988. [2] Feinberg G., Phys. Rev. 159, 1089 (1967). [3] Jackson J. D., Classical Electrodynamics, New York, John Willey and Sons, (1998). [4] See for example, Sears F. W., Zemansky M. W. and Young H. D., College Physics, Addison-Wesley Publishing Company, 1991. [5] Kapu´scik E., Condens. Matter arXiv:1010.5886v1[physics.gen-ph].

Phys.

13,

43102

(2010),

[6] Kapu´scik E., International Journal of Theoretical Physics 54, 4041 - 4045 (20014). [7] Twareque S., Phys. Rev. D7, 1668 (1973).

56

Edward Kapu´scik

[8] Kapu´scik E. and T. Lanczewski, Physics of Atomic Nuclei, 72, 809 - 812 (2009). [9] Kapu´scik E. and R. Orlicki, Ann. Phys.(Berlin), 523, 235 - 238 (2011). [10] Kittel C., Introduction to Solid State Physics, Willey and Sons, New York, 1966. [11] Recami E., Riv. Nuovo Cimento 9, 1 (1986).

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 4

A N E XTENSION OF THE L ORENTZ S YMMETRY C ONCERNING THE L IMIT OF U LTRA -H IGH E NERGIES ∗ I. A. Vernigora† and Yu. G. Rudoy‡ People’s Friendship University of Russia, Department of Theoretical Physics and Mechanics, Moscow, Mikluho-Maklay, Russia

Abstract In this chapter we propose a theoretical group substantiation of the original Kirzhnitz-Chechin approach. This approach calls for the primary cosmic ray’s protons to overcome the of Greisen-ZatsepinKuz’min (GZK) energy limit (approximately 50 EeV) within the framework of the usual representation of a physics nature of out-Galaxy sources. It is shown that the evident form of the multiplier, which forms the Lorentz invariant in space of energy-moments, may be stated on the basis of approximate transition from Lorentz to conform to symmetry at the values of Lorentz-factor near 1010 ÷ 1011 .

Keywords: Lorentz group, conform group, cosmic rays, Greisen-ZatsepinKuz’min (GZK) limit, ultra-high energies ∗

This subject was reported on the international conference BGL-9 (Bolyai-Gauss-Lobachevski), Minsk, 27-29 Oktober, 2015. † E-mail address: [email protected]. ‡ E-mail address: [email protected].

58

I. A. Vernigora and Yu. G. Rudoy

Introduction The common characteristic of GZK-problem is that, just more than half-acentury in the cosmic rays (CR) astrophysics in the ultra-high energy region from exa- (EeV = 1018 eV ) up to zeta- electron volt (ZeV = 1021 eV ), the Greyzen-Zatsepin-Kuz’min problem remained unresolved. This was theoretically established in the framework of standard relativistic astrophysics simultaneously, and independently, in Refs [1] and [2]. According to these works, a distribution of the energies of a proton component for the primary CR must abruptly fall in energy region of order near EGZK ≈ 5 · 1015eV because of the proton’s energy loss on photoproduction of pions in reactions like p+γ → p+π 0 or p + γ → n + π 0 (with following β-decay n → p + e− + νe ). Here γ — “soft” Infra-Red photons of relict cosmic radiation, which accords to hard gammaradiation in the rest system of the primary CR protons. However, an experimental status of GZK-limit cannot quite be determined (look at [3], [4], and also reviews [5], [6]). It reflects both insufficient event statistics in the energy region EGZK ≈ 50EeV and difficulties with their identification. But in spite of this, there are some reasons to suppose that transGZK-events are taking place which allows the actual problem of theoretical substantiation to be a possibility. The calculations carried out in [1] and [2] (see also [3]), shows that protons having the rest energy E0 ≈ 1 GeV, and in a laboratory reference – energy EGZK with Lorentz-factor γ = EGZK /E0 ≈ 5 · 1010 – must utterly lose their energy during the time τ ≈ (1.0 ÷ 1.5) · 108 years, passing the distance from a source not more than l ≈ cτ = 30 ÷ 50 M pc. From this follows that the source of protons by such high energies (“zevatron”) must be “galactic” i.e., to be inside of Local super-cluster of galactics. According to Hillas’ diagram (see [4], [5]), these sources can be “hot spots” of radiogalactics – for example M87-Virgo or M82, which localises on distances approximately 15 ÷ 25 M pc from Earth, and also the remnants of super- and hyper-novas and/or γ-splashs (see [6]). It is evident that the ground EAS (Extensive Air Shower) stream of protons registered from such “single” type inner-galactic sources must be essentially anisotrope, but the available observations contradicts this conclusion (see [4], [5]). That is why inner-galactic sources are the real candidates on the role of “zevatrons”, for example, an active nucleos of galactics, but in this case the CR-protons must pass distances not less 150 M pc. It obviously exceeds the

An Extension of the Lorentz Symmetry Concerning the Limit ...

59

proposed GZK-limit, so that a problem of overcoming this second limit arises (at least theoretically). The Kirzhnits-Chechin idea of overcoming GZK-limit. Attempts of such overcoming may be divided into two groups. One of them, named the approach of “new physics” (see, for example, [8]), considers the possibilities of innergalactic borning of isotropically distributed trans-GZK-protons in processes of some relict objects decaying. However, considerably earlier, beginning from the Kirzhnits and Chechin work [9] (see also Coleman and Glashow [11] and later works [12]–[13]), considered this and another – in our view, more realistic – possibility: staying in frameworks of “usual physics” to find the resources of its “deformation”, which as a matter of principle, assumes overcoming the GZK-limit. Unfortunately, the pioneer work [10] was first noticed and cited in foreign literature only after a quarter of a century in [12]. According to [10], [11], the same role might play a deviation from the Lorentz kinematics at such high values of Lorentz factor γ. In [10] this approach was formulated only on a half-phenomenologic level. That is why an aim of this work is a more strict group substantiation of the Kirzhnits-Chechin approach [10], and carrying out on this base, the numerical estimations of overcoming GZK-limit. The initial idea consists in the appearance of “conform” corrections on powers of γ1 (which absents for massless particles in limit of γ1 = 0 ). Expression for time of protons passing. According to [1] - [3], the time of protons passing with start energy E in a local system of reading is given by expression Z ∞ Z 2γε 1 = A dε nγ (ε) dε0 ε0 ς(ε0 ) K (ε0 ) . (1) 0 0 τ (E) εthsh /2γ εthsh Here A – dimensional constant; nγ (ε) – an equilibrium Planck’s distribution (with temperature T = 2.7 K ) of the relict radiation photons with energy ε and concentration ≈ 400 cm− 3 ; 0 6 ε0 6 2γε ; γ = EE0 – Lorentz-factor of passing from proton’s Rest Reference System to protons and photons Centre of Inertia; ς (ε0 ) – section of pions photoproduction; K (ε0 ) – non-elastic losts of proton’s energy coefficient, which has a form K (ε0 ) = (1/2){1 − ∆/S(ε0)} ; ∆ = M 2 − m2 = const > 0 ; M, m –the masses of proton and pion, m = 0.14M ; S (ε0 ) = M 2 + 2M ε0 – Lorentz-invariant; ε0thsh = m (1 + mM/2) ≈ 1.07 m – threshold energy of the pion borning, at this K (ε0 ) ≈ 0.12 . It is evident that the form of functions nγ (ε) and σ (ε0 ) (which takes places in 1) is

60

I. A. Vernigora and Yu. G. Rudoy

setting unsimply. And only the function K (ε0 ) (and, hence, variable τ − 1 (E)) may be changed by changing the Lorentz-invariant S (ε0 ).

Approaches to Deformations of a Lorentz Kinematics Specifically, this logic lies in the base of Kirzhnits-Chechin approach [10], which starts from Lorentz-(L)-invariant in energy-momentum space: IL (p; E) = E 2 − p2 = IL (0; E0 ) = E02 ;

EL (p; E0 ) = E02 + p2

where EL (p; E0 ) – Lorentz’s dispersion law. For values β ≡ v L the particle) and Lorentz-factor γ we have (at c = 1 ) : vL (p; E0) =

∂EL(p; E0) p = , ∂p EL (p; E0)


1/2

, (2) (velocity of

v L (p; E0) ≡ |vL (p; E0)| 6 1; (3)

γ(p; E0) ≡

EL (p; E0) 1 1 > 1, 2 = 1 − vL2 (p; E0), 0 6 6 1. E0 γ (p; E0) γ(p; E0) (4)

In the context of this work the limit case of massless particle (E0 = 0) is of special interest. For it: IL (p; 0) = 0 ,

EL (p; 0) = p ,

v L (p; 0) = 1 ,

1 = 0. γ(p; 0)

(5)

Phenomenological approach of Kirzhnits-Chechin [10]. So long as the Lorentz kinematics (2) - (5) reduces to GZK-limit for (1), Kirzhnits and Chechin [10] constructs beside (2) a new expression for invariant I(p; E0) – and, hence, for dispersion law E(p; E0). For this purpose the expression for IL (p; E) multiplying on the scalar factor f (p; E) in work [10]. This factor is determined on the basis of clearly phenomenological considerations. In particular, as long as, according to (2) and (3), IL (p; E0) = E 2 [1 − vL2 (p; E0)], authors propose that it is obvious (in framework of idea about “minimal” deformation) to find f (p; E) in for f (vL2 (p; E)), then, with due regard (4), is equal to f (γ 2 (p; E0)). In [10] it is supposed that in massless limit γ 2 → 1 (and, correspondingly, v L → 1) the function f (γ 2 ) has a finite limit f (∞) (in general, different from 1); evidently, such preposition

An Extension of the Lorentz Symmetry Concerning the Limit ...

61

does not influence the dispersion law of massless particles because, for them I(p; 0) = f (∞)IL (p; 0) = 0, was considered the feature (5). Finally, Kirzhnits and Chechin [10] receive for function f (γ 2 ) the expression f (γ 2 ) ≈ 1 + |α| γ 4 ,

|α| γ 4 6 1,

|α| ≈ 10−44 ,

γ ≈ γ0 ≡ 1011 .

(6)

The basic idea of this work consists of the construction of a precised expression for “deformate” factor f (γ 2 ), which we will name the conform factor and design it C(ξ 2 ), where the value of ξ ≡ γ −1 is regarded as a small parameter. For convenience of comparing the results of our work with article [10] we shall designate the function C(ξ 2 ) as g(γ 2). Staying, as a whole, in logic frameworks of the work [10], we suppose an additional idea, which consists not only from some phenomenologic “violation” the group of Lorentz-Poincare transitions, but from its expansion up to the well known conform group of Weyl-Fock. It is necessary for this to substitute the Lorentz scalar IL (p; E0), which is invariant at arbitrary values E0 and ξ, on the conform scalar IC (p; E0) = C(ξ)IL (p; 0) = 0. It is an exact invariant only in a massless case1 : E0 = 0, ξ = γ −1 = 0, when in compliance with (5) one has IC (p; 0) = C(ξ)IL (p; 0) = 0. This circumstance points exactly to the unique way of construction of the perturbation theory for C(ξ) on small parameter ξ = γ −1  1. A theoretical-group approach. The most convenient and useful instrument for such approach is the translations of conform group, here it’s sufficient to be restricted by only Myobius translations C4 (c), which are known as “special conform” translations. Under these translations the arbitrary 4-vector P passes to PC = C(P ; c)P, where matrix C depends from P, and, besides that, determines by an arbitrary 4-vector c. Translation C(P ; c) has an evident form: PC = C(P ; c)P + ∆P (P ; c),

∆P (P ; c) = C(P ; c)cP 2,

PC@ = C(P ; c)P 2 (7)

and is a nonlinear, inhomogeneous and anizotrope scale translation, and, as a result, the conform factor C(P ; c) is gave by expression C(P ; c) = |σ(P ; c)]−1 ,

1 σ(P ; c) = 1 − 2cP + c2 P 2 = c2 (P − )2 . c

(8)

1 Here suppose that the value C(0), which in fact coincides with f (∞) from [10], differes from zero; below it finds a confirmation (see at formula (9).

62

I. A. Vernigora and Yu. G. Rudoy

As it visible in (7), where, in common, C 6= 1, the Lorentz-invariance value P 2 = E02 is not always the conform invariant; an elimination is P 2 = 0 (at E0 = 0 and ξ = 0). It s physically evident that for non-zero, but small quantities of E0 and ξ  1, the conform factor C(P ; c) at ξ  1 infinitesimally differs from C(0; c) = const. Let us pass from 4-vectors of energy-momentum P (E, p) to their undegree or homogeneous analogies V (E; p) ≡ P (E, p)/E = (1, p/E). Then, in accordance with kinematics (2) – (4) the vector VL(E, p) = (1, v L(p; E0)) has  2 meaning of velociy 4-vector, besides that, VL2 ≡ γ12 , VLlight = 0, hence, the Lorentz-invariant is γ 2 VL2 ≡ 1. Numerical estimations of possible overcoming of the GZK-limit. Let’s consider further the choice of vector c in C(V, c) by taking the direction AV, where A = −aγ ∗ , a > 0 – numerical factor of order 1 ÷ 102 , γ ∗ ≡ E ∗ /E − 0  1 – characteristic value of Lorentz-factor; for example, for E0 ≈ 1GeV is expedient to receive E ∗ = EP lank ≈ 1.2 · 1019 Gev, γ ∗ ' 1019 , that’s why γ ∗ /γGZK ≈ 108 ÷ 109 . Finally, the conform factor C(V, AV ) = (1 − AV 2 )−1 6 1 receives the form g(γ 2) = (1 + aγ −2 ) ≈ 1 − aγ ∗γ −2 , (9)

which in some ways differs from the expression (6) which was received by Kirzhnits and Chechin [10]: above all, as it must be, g(γ 2) 6 1, whereas f (γ 2 ) > 1. Furthermore, g(γ 2) has a point of bending and a finite limit g(γ 2) → 1 at γ → ∞, whereas for receiving of finite value of monotonically increasing function f (γ 2 ) at γ → ∞ demands an addition of summand by form − |β| γ 6 to right part of (6). In conclusion we remark what the influence on time of protons passing (1) makes the substitution of the Lorentz-invariant SL (which includes to unelastic coefficient) on the conform invariant SC = gSL, g 6 1. Omitting everywhere 1 an argument ε, we have: KL = 2(1−ϕ) , ϕ = ∆/SL, ∆ > 0, ϕ 6 1, thus, 1 0 6 KL 6 1; otherwise, KC = 2(1−λϕ) , where λ = SL /SC = (1/g) > 1. Let’s assume that based on astrophysical considerations it’s necessary to increase the time and distance of protons passing with energy EGZK in (1/η)  > 1 times: lC /lL = τC /τL = 1/η, hence KC /KL = η 6 1 and λ = (1/ϕ) 1 − η(1 − ϕ) . Taking into account, that in accordance with (9), at EGZK one has λ ≈ 1 + a(5 · 10−3 ), we find a ≈ 2 · 102 (1 − η)[(1/ϕ) − 1]. Further, restricting by value KL = 0.12 on the threshold of pions photoproduction, we find ϕ =

An Extension of the Lorentz Symmetry Concerning the Limit ...

63

0.76, and from it a ≈ 66(1 − η). For protons passing the outgalactic distance ∼ 150 ÷ 200M pc it is sufficient to suppose η ≈ 0.25, which gives a ≈ 50.

Conclusion In this chapter we focused on the idea of the pioneer work of KirzhnitsChechin [10] regarding weak “violation” of the Lorentz invariance by its extension to the conform invariance. It allows, at reasonable values of conform parameters, to increase the distance passing by primary protons of ultra highenergy cosmic rays (UHECR) up to 150 ÷ 200M pc (or more), i.e., to overcome the GZK-limit (which corresponds to values of Lorentz-factor ∼ 1010 ÷ 1011 ), if needed for description of the future broad-scale experiments – as on working settings “Pierre Auger”, HiRes, AGASA, “Yakutsk”, also on planned setting EUSO.

References [1] Greisen K., End to the cosmic-ray spectrum? Phys. Rev. Lett., 1966. Vol. 16, No. 17, Pp. 748-750. [2] Zatsepin G. T., Kuz’min V. A., Upper level of the spectrum of the cosmic rays, JETP, 1966, Vol. 4, no. 3. Pp. 78-80. [3] Stecker F. W., Effect of photomeson production by the universal radiation field on the high-energy cosmic rays, Phys. Rev. Lett., 1968. Vol. 21, no. 14. Pp. 1016-1018. [4] Panasyuk M. I., Wanderers of the Universe or the Big Bang Echo. Fryazino: Vek 2, 2005. 267 pp. [5] Zasov A. V., Postnov K. A., General astrophysics. Fryazino: Vek 2, 2006. 496 pp. [6] Ptuskin V. S., On the Origin of galactic cosmic rays. Phys. Usp., 2007. Vol. 50, no. 5. Pp. 534-540. [7] Ivanov A. A., Knurenko S. P., Pravdin M. I., Krasilnikov A. D., Steptsov I. E., A search of extragalactic sources of cosmic rays in the ultra-high en-

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[8] Olinto A. V., Adams H. J., Dermer C. D. et al., White paper on ultra-high energy cosmic rays: http://uhecr.uchicago.edu. [9] Berezinsky V. S., Ultra-high energy cosmic rays, Nucl. Phys. Bulletin, 2000, Vol. 81. Pp. 311-322. [10] Kirzhnits D. A., Chechin V. A., Ultra-high energy cosmic rays and possible generalization of the relativistic theory, Yadernaya Phyzica, 1972. Vol. 15, no. 5. Pp. 1051-1059. [11] Coleman S., Glashow S. L., High-energy tests of Lorentz invariance. Phys. Rev. D., 1999. Vol. 59, no. 11, 116008. 14 pp. [12] Gonzalez-Mestres L., Deformed Lorentz symmetry and high-energy astrophysics (I), 2000, arXiv: physics/003080 [physics.gen-ph]. [13] Scully S. T., Stecker F. W., Lorentz invariance violation and the observed spectrum of ultrahigh energy cosmic rays, Astroparticle Physics. Vol. 31, no. 3. Pp. 220-225, arXiv: 0811.2230 [astro-ph]. [14] Jacobson T., Liberati S., Mattingly D., Astrophysical bounds on Plank suppressed Lorentz violation, Lect. Not. Phys., 2005. Vol. 669. Pp. 101130, arXiv: hep-ph/0407370.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 5

S PIN 1/2 PARTICLE WITH A NOMALOUS M AGNETIC M OMENT IN P RESENCE OF E XTERNAL M AGNETIC F IELD , E XACT S OLUTIONS E. M. Ovsiyuk1,∗, V. V. Kisel2 and V. M. Red’kov3,† 1 Mozyr State Pedagogical University, Belarus 2 Belarusian State University of Informatics and Radioelectronics, Belarus 3 B. I. Stepanov Institute of Physics, Belarus

Abstract We examine a generalize Dirac equation for spin 1/2 particle with anomalous magnetic moment in presence of the external uniform magnetic field. After separation of the variables, the problem is reduced to a 4-order ordinary differential equation, which is solved exactly with the use of the factorization method. A generalized formula for Landau energy levels is found. Solutions are expressed in terms of confluent hypergeometric functions. ∗ †

E-mail address: [email protected]. E-mail address: [email protected].

66

1.

E. M. Ovsiyuk, V. V. Kisel and V. M. Red’kov

Introduction

Commonly, only the simplest wave equations are proposed for fundamental particles of spin 0, 1/2, 1. Meanwhile, it is known that other and more complicated equations can be proposed which are based on the use of extended sets of Lorentz group representations (see [1–16]). Such generalized wave equations makes it possible to describe more complex objects which have in addition to mass, spin, electric charge other electromagnetic characteristics like polarizability or anomalous magnetic moment. These additional characteristics manifest themselves in the presence of external electromagnetic fields. In particular, within that approach Petras proposed a 20-component theory for spin 1/2 which, after excluding 16 subsidiary components, turns out to be equivalent to the Dirac particle theory modified by presence of the Pauli interaction term. In other words, this theory describes a spin 1/2 particle with anomalous magnetic moment. In this chapter, we investigate a solution of that wave equation in the presence of external uniform magnetic field. Generalized formulas for Landau energy level are derived, and corresponding wave functions are constructed. Restriction to the case of neutron (uncharged spin 1/2 particle with anomalous magnetic moment) can be performed, though no bound state arise in this case.

2.

Ordinary Dirac Equation, Separation of the Variables

We use the known representation for the uniform magnetic field: A = r, B = (0, 0, B). After translating to cylindrical coordinates we get At = 0 ,

Ar = 0 ,

Az = 0 ,

Aφ = −Br 2 /2

1 2

cB × (1)

a non-vanishing component of the electromagnetic tensor is Fφr = Br. We consider the Dirac equation in the magnetic field (1), using the tetrad formalism [18] for cylindrical coordinates xα = (t, r, φ, z): 1 0 0 0 0 1 0 0 β dS 2 = dt2 − dr 2 − r 2 dφ2 − dz 2 , e(a)(x) = . (2) 0 0 1/r 0 0 0 0 1

67

Spin 1/2 Particle with Anomalous Magnetic Moment ... Generally covariant tetrad Dirac equation [18] is   e 1 ab β c γ [ i~ ( e(c) ∂β + σ γabc ) − Ac ] − mc Ψ = 0 , 2 c

(3)

β

where γbac = −γabc = −e(b)β;α e(a)eα(c) are the Ricci rotation coefficient: Aa = eβ(a)Aβ is the tetrad components of the 4-vector Aβ ; σ ab = 1/4(γ aγ b − γ bγ a) are generators for bispinor representation of the Lorentz group. We will use the shortening notation: e/c~ =⇒ e, mc/~ =⇒ M . √ The Dirac equation takes the form (let Ψ = ϕ/ r):     i∂φ eBr 0 ∂ 1 ∂ 2 3 ∂ iγ + iγ +γ + + iγ −M ϕ = 0. (4) ∂t ∂r r 2 ∂z We search solutions in the form ˛ ˛ ˛ ˛ ϕ = e−it eimφ eikz ˛˛ ˛ ˛

f1(r) f2(r) f3(r) f4(r)

˛ ˛ ˛ ˛ ˛ » –˛ ˛ ˛ ∂ ˛ , +γ 0 + iγ 1 − γ 2 µ(r) − kγ 3 − M ˛˛ ˛ ∂r ˛ ˛ ˛ ˛

f1 f2 f3 f4

˛ ˛ ˛ ˛ ˛ = 0, ˛ ˛ ˛

where µ(r) = m/r − eBr/2; further we will use the shortening notation eB =⇒ B. When choosing Dirac matrices in the spinor basis, we find equations for the four functions fa (t, z): d + µ)f4 + ikf3 + i(f3 − M f1 ) = 0, dr d ( + µ)f2 + ikf1 − i(f1 − M f3 ) = 0, dr

(

d − µ)f3 − ikf4 + i(f4 − M f2 ) = 0, dr d ( − µ)f1 − ikf2 − i(f2 − M f4 ) = 0. dr

(

(5)

The equations are consistent with the linear constraint f3 = Af1 , f4 = Af2 , if the following condition is imposed √ M  ± 2 − M 2 − = − + M A =⇒ A = A1,2 = . (6) A M As a result, the problem is reduced to the system of two equations „

« d + µ f2 + i(k −  + M A)f1 = 0, dr

„

« d − µ f1 + i(−k −  + M A)f2 = 0. dr

(7)

68

E. M. Ovsiyuk, V. V. Kisel and V. M. Red’kov

In accordance √ with (6), we have√two types of states: ( 2 − M 2 = p ) AM =  + 2 − M 2 ,     d d + µ f2 + i (k + p) f1 = 0 , − µ f1 − i (k − p) f2 = 0 ; (8) dr dr √ √ AM =  − 2 − M 2 , ( 2 − M 2 = p )     d d + µ f2 + i (k − p) f1 = 0 , − µ f1 − i (k + p) f2 = 0 . (9) dr dr For definiteness, we follow the variant (8).

3.

Solving Equations for r-Variable

From (8) we obtain the second order equation for R1 " #   d2 f1 m B m Br 2 + 2+ − − + λ2 f1 = 0 , dr 2 r 2 r 2

(10)

where λ2 = 2 − m2 − k2 . Parameter λ2 describes the contribution of the electron transversal motion to the total energy, this part of the energy is quantized. Note that we diagonalize the operator −i

∂ Ψ=mΨ, ∂φ

(11)

which represents the third projection of the total angular momentum of the Dirac particle in cylindrical tetrad basis: ∂ Jˆ3 ΨCart = (−i + Σ3 ) ΨCart = m Ψ = m ΨCart ∂φ

(12)

therefore for m are permitted only half-integer values m = ±1/2, ±3/2, .... We turn to eq. (10) and introduce variable x = Br 2 /2, equation becomes1   d2 f1 df1 m(1 − m) 2λ2 4x 2 + 2 + − x + 1 + 2m + f1 = 0 . (13) dx dx x B 1

Without loss of generality, we assume that the parameter B positive.

69

Spin 1/2 Particle with Anomalous Magnetic Moment ...

We seek solutions in the form f1 (x) = xA e−Cx R(x). If A, C are chosen according A = m/2 , (1 − m)/2 , C = ±1/2, the equation for R reads     1 dR m λ2 d2 R − A− − R = 0, x 2 + 2A + − x dx 2 dx 2 2eB which is the confluent hypergeometric equation x Y 00 + (γ − x)Y 0 − αY = 0 ,

α =A−

m λ2 − , 2 2B

γ = 2A +

1 . 2

To obtain solutions that vanish at the origin r → 0 and infinity r → ∞, we must take positive values for A and C: ( m = +1/2, +3/2, ..., A = m/2 , C = +1/2 , A= m = −1/2, −3/2, ..., A = (1 − m)/2 . To obtain polynomials, we impose the known restriction α = −n, n = 0, 1, 2, ... This leads to the following rule for quantization of the parameter λ2 : λ2 m =A− +n . 2eB 2 Depending on the sign of the quantum number m we get two formulas for λ2 = 2 − M 2 − k2 : m>0, m 0): a=−

m , 2

(m < 0) ;

a=

m 1 + >0 2 2

(m ≥ 0) .

(34)

Conditions of terminating the hypergeometric series to polynomials α = −n (introduce the notation 2 − k2 = λ) gives quantization rule for the energy values: N = M 2 + 2B(a +

√ 1 m − + n) =⇒ λ = ( N − Γ)2 > 0 . 2 2

(35)

From (35) we find the formula for the allowed values of λ 2 − k2 =

!2 1 m M 2 + 2B(a + − + n) − Γ . 2 2

r

(36)

Depending on the sign of m, we obtain two formulas: m < 0,

m a=− , 2

2 − k2 =

r

!2 1 M 2 + 2B( − m + n) − Γ ; (37) 2

76 m ≥ 0,

E. M. Ovsiyuk, V. V. Kisel and V. M. Red’kov p 2 m 1 + , 2 − k2 = a= M 2 + 2B(1 + n) − Γ . 2 2

(38)

The particle with anomalous magnetic moment has two series of energy levels, formally differing in sign of the parameter Γ.

6.

Further Analysis of Solutions

Let us consider the function f2 (r) as the primary. Based on the ratios above it allows us to calculate the other three functions. We start from the explicit form of the function f2 , the solution of equation (32): f2 = xae−x/2 F (α, γ, x) ,

(39)

where the parameters are given by √ 1 m M 2 − Γ2 Γ λ λ α=a+ − + − − = −n, 2 2 2B B 2B

γ=

1 + 2a . (40) 2

The function f4 can be found according to the following relationship: 1 2Γ ( − k)

f4 =

„ « d2 m(m + 1) B(2m − 1) B2 2 Γ2 − M 2 − k 2 +  2 + 2 − + − r f2 ; dr r2 2 4

from where after translating to the variable x we find f4 =

2B 2Γ ( − k)

„

x

« d2 1 d λ M 2 − Γ2 m 1 m(m + 1) 1 + + − + − − − x f2 . dx2 2 dx 2B 2B 2 4 4x 4

(41)

Depending on the values of the parameter m, we have two different cases: m , 2

1 f2 = x−m/2 e−x/2 F (−n, −m + , x) , 2 √ 1 M 2 − Γ2 Γ λ λ 1 α=m+ + − − = −n, γ = −m + , 2 2B B 2B 2

A) m < 0,

a=−

2B f4 = 2Γ ( − k)

√

! √ d2 1 d m 1 Γ λ m(m + 1) x x 2 + +n− + − − − f2 , dx 2 dx 2 4 B 4x 4

λ=

r

1 M 2 + 2B( − m + n) − Γ ; 2

(42)

77

Spin 1/2 Particle with Anomalous Magnetic Moment ... B)

3 m 1 + , f2 = x(m+1)/2 e−x/2 F (−n, m + , x) , 2 2 2 √ 2 2 M −Γ Γ λ λ 3 − − = −n, γ =m+ , α=1+ 2B B 2B 2

m ≥ 0,

2B f4 = 2Γ ( − k)

a=

! √ d2 1 d m+1 1 Γ λ m(m + 1) x x 2 + +n+ + − − − f2 , dx 2 dx 2 4 B 4x 4

√

λ=

p M 2 + 2B(1 + n) − Γ .

(43)

Using the relations (42) and (43) we find the explicit expression for the function f4 : for the variant A, m < 0,

√ λ −m/2 −x/2 1 λ f4 = − x e F (−n, −m + , x) = − f2 (x) , −k 2 −k √

(44)

for the variant B, m ≥ 0,

f4 = −

√

√ λ (m+1)/2 −x/2 λ 3 x e F (−n, m + , x) = − f2 (x) . −k 2 −k

(45)

Changing in these formulas the parameter Γ on −Γ, we obtain the relations describing the second series of states. It is not difficult to calculate the explicit form of the other two functions f1 , f3 ; corresponding calculations are made which are not detailed in this chapter.

Conclusion We have examined a generalize Dirac equation for spin 1/2 particle with anomalous magnetic moment in presence of the external uniform magnetic field. After separation of the variables, the problem is reduced to a 4-order ordinary differential equation, which is solved exactly with the use of the factorization method. A generalized formula for Landau energy levels is found. Solutions are expressed in terms of confluent hypergeometric functions. Restriction to the case of neutron (uncharged spin 1/2 particle with anomalous magnetic moment) can be performed, though no bound states arise in this case.

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E. M. Ovsiyuk, V. V. Kisel and V. M. Red’kov

References [1] Fradkin E. S., To the theory of particles with higher spins. JETP. 1950. Vol. 20. P. 27–38. [2] Feinberg V. Ya., To interaction of the particlesof higher spins with electromagnetic fields. Trudy FIAN. USSR. 1955. Vol. 6. P. 269–332. [3] Petras M., A note to Bhabha’s equation for a particle with maximum spin 3/2. M. Petras Czehc. J. Phys. - 1955. - Vol. 5. . 3. - P. 418-419. [4] Ulehla I., Anomalous equations for spin 1/2 particles. JETP. 1957. Vol. 33. P. 473–477. [5] Fedorov F. I., Pletyukhov V. A., Wave equatioons withe repeated representations of the Lorentz group Intejer spin. Vesti of NASB. Ser. fiz.-mat. nauk. 1969. . 6. P. 81–88; Wave equatioons withe repeated representations of the Lorentz group Half-intejer spin. Vesti of NASB. Ser. fiz.-mat. nauk. 1970. . 3. P. 78–83; Wave equatioons withe repeated representations of the Lorentz group for spon 0 particle Vesti of NASB. Ser. fiz.-mat. nauk. 1970. 2. P. 79–85; Wave equatioons withe repeated representations of the Lorentz group for spon 1 particle Vesti of NASB. Ser. fiz.-mat. nauk. 1970. . 3. P. 84–92. [6] Capri A. Z., Nonuniqueness of the spin 1/2 equation. Phys. Rev.- 1969. Vol. 178. . 5. P. 181–1815; First order wave equations for half-oddintegral spin. Phys. Rev. 1969. Vol. 178. P. 2427–2433; Electromagnetic properties of a new spin-1/2 field. Progr. Theor. Phys. 1972. Vol. 48. P. 1364–1374. [7] Shamaly A., Capri A. Z., First-order wave equations for integral spin. Nuovo Cimento. B. 1971. Vol. 2. P. 235–253; Unified theories for massive spin 1 fields. Can. J. Phys. 1973. Vol. 51. P. 1467–1470. [8] Khalil M. A. K., Properties of a 20-component spin 1/2 relativictic wave equation. Phys. Rev. D. 1977. Vol. 15. P. 1532–1539 : Barnacle equivalence structure in relativistic wave equation. Progr. Theor. Phys. 1978. Vol. 60. P. 1559–1579; An equivalence of relativistic field equations. Nuovo Cimento. A. 1978. Vol. 45. P. 389–404; Reducible relativistic wave equations. J. Phys. A. Math. and Gen. 1979. Vol. 12. P. 649–663.

Spin 1/2 Particle with Anomalous Magnetic Moment ...

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[9] Bogus A. A., Kisel V. V., Equations with epeated representations of the Lorent group and interaction of the Pauli type. Izvestia Wuzov. Phizika. 1984. 1. P. 23–27. [10] Bogus A. A., Kisel V. V., Moroz L. G., Levchuk M. I., To desription pf scalr paticle with polarizability. Covariant methods in theoretical physics. Elementary particles physics and relativity theory. IP NAN BSSR. Minsk, 1981. P. 81–90. [11] Bogus A. A., Kisel V. V., Fedorov F. I., On interpretation of subsidiary components of wave funcrtions in electromagnetic interaction. Doklady NAN BSSR. 1984. Vol. 277. 2. P. 343–346. [12] Kisel V. V., Tokarevskaya N. G., Red’kov V. M., Petrash theory for particles with spin 1/2 in curved space-time. Preprint 737. Institute of Physics of NAS of Belarus: Minsk, 2002. [13] Bogush A. A., Kisel V. V., Tokarevskaya N. G., Red’kov V. M., Petrasha theory for particles with spin 1/2 in curved space-time. Vesti of NASB. Ser. fiz.-mat. nauk. 2002. 1. P. 63–68. [14] Kisel V. V., Tokarevskaya N. G., Red’kov V. M., Spin 1/2 particle with anomalous magnetic moment in a curved space-time, non-relativistic approximation. Pages 36-42 in: Proceeding of 11th International School & Conference Foundation & Advances in Nonlinear Science, Eds.: Kuvshinov V.I., Krylov G.G., Minsk, 2004. [15] Bogush A. A., Kisel V. V., Tokarevskaya N. G., Red’kov V. M., Petras Theory of a Spin-1/2 Particle in Electromagnetic and Gravitational Fields. 44 pages. – arXiv:hep-th/0604109. [16] Pletyukhov V. A., Red’kov V. M., Strazhev V. I., Relativistic wave equations and intrinsic degrees of freedom. – Belarusian Science: Minsk, 2015. 328 pages (in Russian).

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[17] Ovsiyuk E., Kisel V., Red’kov V., On a Dirac particle in an uniform magnetic field in 3-dimensional space of constant negative curvature. Nonlinear Dynamics and Applications. Vol. 18, P. 145–164. [18] Red’kov V. M., Fields of particles in Riemannian space and the Lorentz group. – Belarusian Science: Minsk, 2009. – 486 pages (in Russian).

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 6

C OX ’ S PARTICLE IN M AGNETIC AND E LECTRIC F IELDS ON THE B ACKGROUND OF E UCLIDEAN AND L OBACHEVSKY G EOMETRIES O. V. Veko∗ Kalinkovichi Gymnasium, Belarus

Abstract A generalized non-relativistic Schr¨odinger equation for Cox’s not point-like scalar particle with intrinsic structure has been solved exactly in presence of external uniform magnetic and electric fields in the case of Euclidean space. Extension of these problems to the case of the hyperbolic Lobachevsky 3-space is examined. Complete separation of the variables in the system of special cylindric coordinates in this curved model has been performed for cases of magnetic and electric fields. In the presence of the magnetic field, the quantum problem in radial variable has been solved exactly, wave functions and corresponding energy levels have been found; the quantum motion in z-direction is described by a 1-dimensional Schr¨odinger-like equation in an effective potential of a barrier type which turns out to be too difficult for analytical treatment. In the presence of an electric field on the background of the curved model, the situation is similar: a radial equation is solved exactly in hypergeometric ∗

E-mail address: [email protected].

82

O. V. Veko functions, an equation in z-variable can be be treated only qualitatively because of its complexity.

Keywords: Cox’s particle, generalized Schr¨odinger equation, magnetic field, electric field, Minkowski space, Lobachevsky space

1.

The Schr¨odinger Equation in the Magnetic Field

In 1982 W. Cox [1] proposed a special wave equation for a scalar particle with a larger set of tensor functions than in the usual Proca’s approach: he used the set of a scalar, 4-vector, antisymmetric and (irreducible) symmetric tensor; thus using the 20-component wave function. In this chapter we start with the generalized system Proca’s type obtained after elimination from the initial system of Cox’s equations, both second-rank tensors. Such a relativistic theory of a scalar Cox’s not point-like particle with intrinsic structure is specified in external uniform magnetic and electric fields, first in Minkowski space. Then an extension is done to arbitrary curved space-time background. In relativistic Cox’s theory, the limiting procedure to non-relativistic approximation was performed for arbitrary curved space-time model in [2], [3]; this generalized Pauli-like equation is specified in external magnetic and electric fields. ~ = 1 ~x × B ~ be directed along the axis z. Let the uniform magnetic field A 2 Recalculating the potential to cylindrical coordinates, we obtain Ar = 0, Aφ = −

Br 2 , Az = 0, Frφ = −Br . 2

(1)

The metric tensor in these coordinates and field variables are determined by dS 2 = c2 dt2 − dr2 − r2 dφ2 − dz 2 ,

B3 = −Br, B 3 = −Br−1 , Bi B i = B 2 .

(2)

We start with the Schr¨odinger wave equation for Cox’s particle in the following form [2], [3] (the mass of the particle is denoted by the letter M ) Dt Ψ =

◦ ∗ ◦ ∗ 1 ◦ ∗ (D 1 D 1 + D 2 r −2 D 2 + D 3 D 3 )Ψ, 2M

(3)

Cox’s Particle in Magnetic and Electric Fields ...

83

where e Br2 , D3 = i~∂z , c 2 ◦ ◦ ◦ 1 e Br2 , D 3 = i~∂z , D1 = i~(∂r + ), D 2 = i~∂φ + r c 2 „ « ∗ 1 1 ΓB e Br2 2 (D − ΓB D ) = i~∂ − (i~∂ + ) , D1 = 1 3 r φ 1 + Γ2 B 2 1 + Γ2 B 2 r c 2 „ « ∗ 1 1 e Br2 ( D2 + ΓB3 D 1 ) = (i~∂φ + ) + i~ΓBr∂r , D2 = 2 2 2 2 1+Γ B 1+Γ B c 2 D1 = i~∂r , D2 = i~∂φ +

∗

D3 =

(D3 + Γ2 B 3 B3 D3 ) = i~∂z ; 1 + Γ2 B 2

Below we will use the notation eB/2~c = b, ΓB = γ . We compute   ~2 1 γ 1 ◦ ∗ 2 = − ∂ + ∂ − ∂ ∂ + iγbr∂ + 2iγb , D1D1 r r φ r 2M 2M (1 + γ 2 ) r r r   1 ◦ ∗ ~2 1 2 2 2 1 (∂φ − ibr ) + γ(∂φ − ibr ) ∂r , D2D2 = − 2M 2M (1 + γ 2 ) r 2 r 2 1 ◦ ∗ ~ (1 + γ 2 )∂z2 . D3D3 = − 2M 2M (1 + γ 2 )

(4)

(5)

After using the substitution for the wave function Ψ = e−iEt/~eimφ eikz R(r),

=

2mE (1 + γ 2 ) ~2

we get the radial Schr¨odinger equation  2  d 1 d (m − br 2 )2 2 2 + + 2iγb +  − − (1 + γ )k R = 0. dr 2 r dr r2

(6)

(7)

By physical reasons, parameter γ must be imaginary: γ = −iη; so the radial equation reads   2 d 1 d (m − br 2 )2 2 2 + + 2ηb +  − − (1 − η )k R = 0. (8) dr 2 r dr r2 With the use of notation  − (1 − η 2 )k2 + 2ηb = 0 , equation (8) can be written as  2  d 1 d (m − br 2 )2 0 + − + R=0, (9) dr 2 r dr r2

84

O. V. Veko

which coincides with the differential equation arising in the usual problem of the Schr¨odinger particle in the magnetic field. Its solutions are known. Here we present only an expression for the energy spectrum (turning back from 0 to )  = 4b(n +

m+ | m | +1 ) + (1 − η 2 )k2 − 2ηb . 2

From this, after translating to ordinary units, we find   p2 1 eB m+ | m | +1 η 1 eB E= + ~ n+ − ~. 2M 1 − η2 M c 2 2 1 − η2 M c

(10)

(11)

With the notation η = ΓB, Γ∗ = Γ, ω = eB/M c, the formula for the energy levels can be written as   p2 ω~ m+ | m | +1 ω~ ΓB E= + n+ − . (12) 2M 1 − (ΓB)2 2 1 − (ΓB)2 2 Thus, the intrinsic structure of the Cox’s particle modifies the frequency of the quantum oscillator ω

2.

=⇒

ω ˜=

ω eB , ω= . 1 − Γ2 B 2 Mc

(13)

Cox’s Particle in the Magnetic Field in the Lobachevsky Space

In a special (cylindrical) coordinate system in the Lobachevsky space, analogue of the uniform magnetic field is determined by the relations [4] (we use dimensionless coordinate r obtained by dividing on the curvature radius ρ): √

−g = ρ3 sh r ch2 z,

dS 2 = c2 dt2 − ch2 z(dr 2 + sh2 rdφ2 ) + dz 2 ,

Aφ = −Bρ2 (ch r − 1), Frφ = −Bρ sh r, B B3 = −Bρ sh r, B 3 = − , Bi B i = B 2 ch−4 z. ρ sh r ch4 z

We start with the wave equation in the form [2]   ◦ ∗ ◦ ◦ ∗ 1 1 ∗ Dt Ψ = D 1 D 1 + D 2 2 D 2 + D 3 D3 Ψ, 2M ρ2 sh r

(14)

Cox’s Particle in Magnetic and Electric Fields ...

85

where e D2 = i~∂φ + Bρ2 (ch r − 1), D3 = i~∂z , c ◦ ◦ ◦ e 2 sh z ch r ), D 2 = i~∂φ + Bρ (ch r − 1), D 3 = i~(∂z + 2 ), D1 = i~(∂r + sh r c ch z » – ∗ 1 ΓBch−2 z e i~∂r − (i~∂φ + Bρ2 (ch r − 1)) , D1 = −4 2 2 shr c 1 + Γ B ch z h i ∗ 1 e 2 (i~∂φ + Bρ (ch r − 1)) + i~ΓBch−2 z sh r∂r , D2 = −4 2 2 c 1 + Γ B ch z ∗ (D3 + Γ2 B 3 B3 D3 ) = i~∂z . D3 = 1 + Γ2 B 2 ch−4 z D1 = i~∂r ,

(15)

Below the notation is used: (eBρ2 /~c) = b, ΓBch−2 z = γ(z). Then we compute ◦ ∗ 1 ~2 ch−2 z 11 g = − × D1 D1 2M ρ2 2M ρ2 (1 + γ 2 (z))   ch r ch r − 1 γ(z) 2 ∂r + ( + iγ(z)b )∂r − ∂r ∂φ + iγ(z)b , sh r sh r sh r ◦ ∗ 1 ~2 ch−2 z 22 g = − × D D 2 2 2M ρ2 2M ρ2 (1 + γ 2 (z))   1 1 2 2 [∂φ − ib(ch r − 1)] + γ(z)[∂φ − ib(ch r − 1)] sh r ∂r , sh r ◦ ∗ 1 ~2 sh z 33 g = − (∂z + 2 )∂z . D D 3 3 2 2 2M ρ 2M ρ ch z

Substitution for the wave function is Ψ = e−iEt/~eimφ Z(z)R(r) ,

=

E ~2 /2M ρ2

,

(16)

then the Schr¨odinger equation (15) gives (by physical reasons function γ(z) must be purely imaginary: iγ(z =⇒ γ(z))    ch−2 z ch r [m − b(ch r − 1)]2 2 ∂r + ∂r − + bγ(z) + 1 − γ 2 (z) sh r sh2 r    sh z  + ∂z + 2 ∂z R(r)Z(z) = 0 . (17) ch z

86

O. V. Veko

In this equation, the variables are separated: « d2 ch r d [m − b(ch r − 1)]2 + − R+ dr2 sh r dr sh2 r „ « 1 b γ(z) ch−2 z d sh z d (1 − γ 2 (z)) ch2 z +  + ( + 2 ) Z = 0. Z 1 − γ 2 (z) dz ch z dz 1 R

„

The radial equation for the function R(r) reads  2  d ch r d [m − b(ch r − 1)]2 + − + Λ R = 0, dr 2 sh r dr sh2 r and the equation for Z(z) is (remember that γ = BΓ)  2  d sh z d bγ − Λ ch2 z +2 ++ Z = 0. dz 2 ch z dz ch4 z − γ 2

3.

(18)

(19)

(20)

Analysis of the Equation in the Variable z

In equation (20), let us eliminate the first derivative term: Z=

1 f (z), ch z

U (z) = −

bγ − Λ ch2 z , ch4 z − γ 2

„

« d2 +  − 1 − U (z) f (z) = 0. dz 2

(21)

Eq. (21) can be viewed as the Schr¨odinger equation in the effective potential field U (z). The corresponding effective force is Fz = −

dU Λ ch4 z − 2bγ ch2 z + γ 2 Λ = 2 ch z sh z . dz (ch4 z − γ 2 )2

(22)

We find the points of local extremum fot the potential U (z): z = 0 and the roots of a quadratic equation Λ ch4 z − 2bγ ch2 z + γ 2 Λ = 0

=⇒

`

´ b ch2 z |1,2 = γ ± Λ

r

(

b2 − 1)γ 2 . Λ2

(23)

Below it will be shown that when considering the bound states (for motion in the variable r) we have Λ2 > b2 . This means that the square root in (23) is an imaginary number. Consequently, the point of zero force (equilibrium points) except z = 0 cannot exist. The situation is illustrated in the Fig. 1.

Cox’s Particle in Magnetic and Electric Fields ...

87

Figure 1. Effective potential U (z). After the change of the variable, ch2 z = y, the differential equation (21) reads »

d2 + dy2

„

31 1 1 + 2y 2y−1

«

– d  bγ − Λ y + + Z(y) = 0 . (24) dy 4y(y − 1) (y − γ)(y + γ)4y(y − 1)

We consider this equation near the singular points (note that the points y = 0, ±γ(| γ | 0) » „ « – d2 d 1 m2 (m − 2B)2 y(1 − y) 2 + (1 − 2y) − − 4B 2 + − Λ R = 0. dy dy 4 y 1−y

(26)

We find the behavior of R near singular points 0 and 1 (note that the point y = 1 does not belong to the physical range of the coordinate y): y −→ 0 , R(y) ∼ ya , a = ±

m ; 2

y −→ 1 , R(y) ∼ (1 − y)b , b = ±

m − 2B . 2

Making the substitution R = y a (1 − y)bF , at a = ±m/2, b = ±(m − 2B)/2, we obtain the equation R = z a (1 − z)b F,

y(1 − y)F 00 + [(2a + 1) − 2(a + b + 1)y]F 0

−[a(a + 1) + 2ab + b(b + 1) − B 2 + Λ]F = 0, which is the equation of hypergeometric type y(1 − y)F 00 + [γ − (α + β + 1)y]F 0 − αβF = 0 .

89

Cox’s Particle in Magnetic and Electric Fields ...

Thus, the solutions are constructed as follows: (bound states correspond to positive values of the parameter a and negative values of the parameter b): z = − sinh2

r , 2

z ∈ (−∞, +0] ,

r r r R = (− sinh )|m|(cosh )−|m−2B| F (α, β, γ, −sinh2 ) , 2 2 2

(27)

where α and β are defined by the following expressions |m| | m − 2B | , b=− , γ = 2a + 1 = + | m | +1 , 2 r 2 r 1 1 1 1 2 α = a + b + − B + − Λ , β = a + b + + B 2 + − Λ . (28) 2 4 2 4 a=+

A detailed study shows that here we have a finite series of bound states described by the relation m+ | m | + n + 1/2 ≤ B, n = 0, 1, . . ., NB , 2     m+ | m | m+ | m | 1 2 Λ − 1/4 = 2B + n + 1/2 − +n+ . 2 2 2 m < 2B,

(29)

In usual units the last relation can be written as: Λ−

1 1 = ρ2 Λ 0 − , 4 4

ρ2 Λ 0 −

lim Λ0 =

ρ→∞

2M P2 (E − ), 2 ~ 2M

m < 2B, m + n + 1/2 ≤

eB 2 ρ , ~c

1 eB 2 m+ | m | m+ | m | =2 ρ ( + n + 1/2) − ( + n + 1/2)2 , n = 0, 1, . . . , NB . 4 ~c 2 2

In the limit of vanishing curvature, we obtain the known result in flat space E−

P2 eB~ m+ | m | = ( + n + 1/2) . 2M Mc 2

(30)

To get the results for the opposite orientation of the magnetic field B < 0 , it is necessary to use the symmetry of the original differential equation (25): m → m0 = −m ,

B → B 0 = −B .

90

O. V. Veko From (29) it follows Λ − 1/4 = 2BN − N 2 , n = 0, 1, . . ., NB , where m+ | m | 1 ≤N = + n + 1/2 ≤ B; 2 2

therefore Λ obeys the restriction B ≤ Λ ≤ B2 +

5.

1 . 4

(31)

Cox’s Particle in the Electric Field, Minkowski Space

Schr¨odinger equation for Cox’s particle in the electric field has the form [2] „ « » – Γ2 Ei E i µ + ΓE j Dj 1 ◦ Γ2 Ej (E i Di ) + µΓEj kj Dt − c Ψ = − g D + Ψ; (32) D j k 2(1 + Γ2 Ei E i ) 2M 1 + Γ2 Ei E i

the notation is used: A0 = −eEz, Ei = (F01 , F02, F03 ) , g 11 E1 = E 1 , g 22E2 = E 2 , g 33 E3 = E 3 , ◦ i~ ∂ √ i~∂t − eA0 = Dt , i~∂k = Dk , √ −g =D k . k −g ∂x Let us use cylindric coordinates dS 2 = c2 dt2 − dr 2 − r 2 dφ2 − dz 2 ,

E3 = E, E 3 = −E, E3 E 3 = −E 2 . (33)

First, we get (let it be ΓE = γ)   Γ2 Ei E iµ + ΓE j Dj γ 2 µ + γD3 Dt − c = i~∂ + eEz + c . t 2(1 + Γ2 EiE i ) 2(1 − γ 2 ) Next, we consider the Hamiltonian    ◦ ◦ ◦ 1 1 µγ H= . D 1 D1 + D 2 2 D2 + D 3 D3 + 2M r 1 − γ2

(34)

(35)

Cox’s Particle in Magnetic and Electric Fields ...

91

In explicit form, the extended Schr¨odinger equation looks as follows (to allow for the imaginary character of γ, we make formal change iγ −→ γ)   γ M c2 γ 2 + ~c∂z Ψ = i~∂t + eEz − 2(1 + γ 2 ) 2(1 + γ 2 ) ! ∂φ2 −~2 1 (mc/~)γ 2 2 ∂r + ∂r + 2 + ∂z − ∂z Ψ. (36) 2M r r 1 + γ2 With the substitution Ψ = e−iW t/~eimφ Z(z)R(t) and the notation M 2 c2 1 = 2 , 2 ~ λ

2M W =w, ~2

2M eEz = ν , ~2

we get 1 Z(z)

„

∂z2 + ν z + w −

1 γ2 λ2 1 + γ 2

«

Z(z) +

1 R(r)

„ « 1 m2 ∂r2 + ∂r − 2 R(r) = 0. r r

(37)

After separation of the variables (w⊥ > 0 stands for the separation constant) we derive  2  d 1 d m2 + − 2 + w⊥ R(r) = 0, (38) dr 2 r dr r d2 1 γ2 ( + ν z + w 0 )Z(z) = 0, w 0 = w − w⊥ + 2 . (39) dz λ 1 + γ2 In fact, (38) and (39) coincide with the well known equations for an ordinary particle in the uniform electric field. Equation in the variable z looks as a one-dimensional Schr¨odinger equation in the potential of the form U (z) = −ν z , ν > 0: (

d2 + w 0 + ν z)Z(z) = 0. dz 2

(40)

The form of the curve U (z) says that any particle moving from the right must be 0 reflected by this barrier in vicinity of the point z0 = − wν (we assume that electric force acts in positive direction of the axis z). Solutions of the equation (40) can be expressed in Airy functions. Indeed, in (40) let us change the variable νz + w 0 = ax, (

d2 a3 + x)Z(x) = 0 ; dx2 ν 2

92

O. V. Veko

let it be (for definiteness ν > 0) a3 = −1, ν2

a = −ν 2/3 ,

x=

νz + w 0 w0 1/3 = −ν z − ; −ν 2/3 ν 2/3

then we arrive at the Airy equation  2  d − x Z(x) = 0 ; dx2

(41)

(42)

to the turning point z0 = −w 0 /ν there corresponds the value x0 = 0. Eq. (42) can be related to the Bessel equation. Indeed, let us introduce the variable ξ=

2 3/2 x , 3

x=

3 2/3 ξ , 2

(43)

then Airy equation gives (

1 d d2 + 2 − 1)Z = 0 . 3ξ dξ dξ

Applying the substitution Z = ξ 1/3 f (ξ), we arrive at the Bessel equation  2  d 1 d 1/9 + − 1 − 2 f (ξ) = 0 (44) dξ 2 ξ dξ ξ with two solutions f1 (ξ) = J+1/3 (iξ) , f2 (ξ) = J−1/3 (iξ) .

(45)

Thus, general solutions of Airy equation can be constructed as linear combinations of Z1 (x) = ξ 1/3 J+1/3 (iξ) , Z2 (x) = ξ 1/3 J−1/3 (iξ) ,

iξ = i

√ 2 ν w 0 3/2 . (z + ) 3 ν

(46)

Far on the left from the turning point we have imaginary infinity and far on the right from the turning point we have real infinity, respectively: √ √ 2 ν w 0 3/2 2 ν w0 iξ = i (z + ) ∼ +i∞ and iξ = i (z + )3/2 ∼ ∞. 3 ν 3 ν

93

Cox’s Particle in Magnetic and Electric Fields ... With the use of the known relation Jµ (y) =

(y/2)µ −iy 1 e 1F1 (µ + , 2µ + 1, 2iy) Γ(µ + 1) 2

and with the notation y = iξ, µ = +1/3, −1/3, one expresses two independent solutions of the Schr¨odinger equation as follows (iξ/2)µ ξ 1 e1 F1 (+µ + , +2µ + 1, −2ξ) , Γ(µ + 1) 2 −µ (iξ/2) 1 Z2 = ξ 1/3 J−1/3 (iξ) = ξ 1/3 eξF1 (−µ + , −2µ + 1, −2ξ) . Γ(−µ + 1) 1 2 Z1 = ξ 1/3 J+1/3 (iξ) = ξ +1/3

6.

Cox’s Particle in the Electric Field in the Lobachevsky Model

We determine the concept of generalized electric field in the special system of cylindric coordinates in the curved space as follows [3] √ dS 2 = dt2 − ch2 z(dr2 + sh2 rdφ2 ) − dz 2 , −g = sh r ch2 z, E E , E3 = − , E3 E 3 = −E 2 cosh−4 z. A0 = −Eρ tanh z, E3 = cosh2 z cosh2 z

(47)

Below we use operators (the coordinate tc/ρ −→ t is dimensionless) i

◦ ~c ~ i~/ρ ∂ √ −g =D k . ∂t − eA0 = Dt, i ∂k = Dk , √ k ρ ρ −g ∂x

We start with the extended Schr¨odinger equation [2] (let it be E 2 cosh−4 z = −γ 2 (x)) „

Dt − c

Γ2 Ei E i µ + ΓE j Dj 2(1 + Γ2 Ei E i )

«

Ψ=

„ « ◦ Γ2 Ej (E i Di ) + µΓEj 1 kj = (−g ) D + Ψ. D j k 2M ρ2 1 + Γ2 Ei E i

Allowing for relations   Γ2 Ei E iµ + ΓE j Dj Dt − c = 2(1 + Γ2 EiE i )   ~c eEρ 1 M cρ γ 2(z) 1 γ(z) i∂t + tanh z + + i∂z , ρ ~c/ρ 2 ~ 1 − γ 2(z) 2 1 − γ 2 (z)

94

O. V. Veko

and H=−

~2 2M ρ2

"

1 cosh2 z

∂r2 +

∂φ2 cosh r ∂r + sinh r sinh2 r

!

+ (∂z + 2

„ «# sh z M cρ iγ(z) ) ∂z − , ch z ~ 1 − γ 2 (z)

we get an explicit form of the Schr¨odinger equation ~c ρ ~2 − 2M ρ2

"

„

1 cosh2 z

« 1 M cρ γ 2 (z) eEρ 1 γ(z) tanh z + + i∂ Ψ= z ~c/ρ 2 ~ 1 − γ 2 (z) 2 1 − γ 2 (z) ! „ «# ∂φ2 sh z cosh r M cρ iγ(z) 2 ∂r + + (∂z + 2 ∂r + ) ∂z − Ψ. sinh r ch z ~ 1 − γ 2 (z) sinh2 r

i∂t +

In particular, note that two terms proportional to iγ(z)∂z compensate each other. Additionally, we should perform formal change iγ −→ γ: « eEρ 1 M cρ γ 2 (z) tanh z − i∂t + Ψ= ~c/ρ 2 ~ 1 + γ 2 (z) " ! ∂φ2 ~2 1 cosh r 2 − ∂r + ∂r + + 2M ρ2 cosh2 z sinh r sinh2 r „ « – sh z M cρ ∂ γ(z) M cρ γ(z) sh z )∂z − − 2 Ψ; (∂z + 2 ch z ~ ∂z 1 + γ 2 (z) ~ 1 + γ 2 (z) ch z ~c ρ

„

(48)

With the use of substitution Wρ , ~c

Ψ = exp(−iwt) eimφ R(r)Z(z), w = and notation W =w

~c M ρc 1 = 2w , ρ ~2 /2M ρ2 ~

ν=

eEρ ~2 /2M ρ2

1 1 M 2ρ2 c2 = M c2 2 = µ2 , 2 2 ~ /2M ρ ~2

,

we get cosh2 z

W + ν tanh z − "

cosh2 z (∂z + 2

sh z ch z

2 µ γ2 (z)

1+

γ 2 (z)

)∂z − µ

! ∂

R(r)Z(z) + γ(z)

∂z 1 + γ 2 (z)

!

∂r2 +

−µ

cosh r sinh r

∂r −

γ(z) 1 + γ 2 (z)

2

m2 sinh2

sh z ch z

#

r

!

R(r)Z(z) +

R(r)Z(z) = 0 .

In this equation one can separate the variables  2  d cosh r d m2 + − + Λ R = 0, dr 2 sinh r dr sinh2 r

(49)

(50)

Cox’s Particle in Magnetic and Electric Fields ...  2   d sh z d d γ(z) γ(z) sh z + 2 − µ −µ 2 + dz 2 ch z dz dz 1 + γ 2 (z) 1 + γ 2 (z) ch z  2 Λ 2 γ (z) − W + ν tanh z − µ Z=0; 1 + γ 2 (z) ch2 z

95

(51)

remember that γ(z) = γ ch−2 z. The most interesting is the equation in variable z. After elementary transformation it is reduced to the form  2 d sh z ch z sh z d −ch4 z + γ 2 + 2 − 2µγ sh z ch z − 2µγ 4 4 2 2 2 dz ch z dz (ch z + γ ) ch z + γ 2  µ2 γ 2 Λ +W + ν tanh z − 4 − Z=0. ch z + γ 2 ch2 z This final equation turns out to be very complex and it hardly can be solved analytically.

7.

Solving the Radial Equation in the Lobachewsky Space

In eq. (50) let us introduce a new variable x = (1 + cosh r)/2, x ∈ [1, +∞), so that   d2 R dR 1 m2 1 m2 x (1 − x) 2 + (1 − 2 x) − w⊥ + + R = 0 . (52) dx dx 4 x 4 1−x With the substitution R = xa (1 − x)b F,

a = ± | m | /2 , b = ± | m | /2

we obtain the hypergeometric equation x (1 − x)

d2 F dF + [2 a + 1 − (2 a + 2 b + 2) x] − [(a + b) (a + b + 1) + w⊥] F = 0 , dx2 dx

with parameters p 1 F = F (α, β, γ; x), α = a + b + − i w⊥ − 1/4, 2 p 1 1 β = a + b + + i w⊥ − 1/4, w⊥ > , γ = 2a+1. 2 4

(53)

96

O. V. Veko

We will specify solutions tending to zero at r = 0: F = u2 = F (α, β, α + β + 1 − γ, 1 − x) ;

(54)

when a and b take positive values a = + | m | /2 , b = + | m | /2 ; the complete radial function is R = xa (1 − x)b F (α, β, α + β + 1 − γ, 1 − x).

(55)

To find behavior at infinity, r → +∞, one should apply the following Kummer relationship u2 =

Γ(α + β + 1 − γ)Γ(β − α) −iπα Γ(α + β + 1 − γ)Γ(α − β) −iπβ e u3 + e u4 , (56) Γ(β + 1 − γ)Γ(β) Γ(α + 1 − γ)Γ(α)

1 u3 = (−x)−αF (α, α + 1 − γ, α + 1 − β, ) , x 1 u4 = (−x)−β F (β, β + 1 − γ, β + 1 − α, ) . x u2 = F (α, β, α + β + 1 − γ; 1 − x) . Therefore, asymptotic behavior at x → 1 (r → +∞) is given by R ≈ (−1)a+bΓ(α + β + 1 − γ) ×

„

« Γ(β − α) Γ(α − β) e−iπα (−x)a+b−α + e−iπβ (−x)a+b−β . Γ(β + 1 − γ)Γ(β) Γ(α + 1 − γ)Γ(α)

From this it follows x≈ „

er , R ≈ (−1)a+bΓ(α + β + 1 − γ)(−x)−1/2× 4

« √ √ Γ(β − α) Γ(α − β) e−iπα (−x)+i λ−1/4 + e−iπβ (−x)−i λ−1/4 . Γ(β + 1 − γ)Γ(β) Γ(α + 1 − γ)Γ(α)

Thus, constructed solutions represent standing radial waves. The factor √ e is not significant for probability interpretation dW = −g ψ ∗ ψ , and the term e−r/2 will be compensated by the factor sinh r ≈ e+r /2 entering the √ volume element dV = −g dr dz dφ. −r/2

Cox’s Particle in Magnetic and Electric Fields ...

97

Acknowledgments The author is grateful to V. V. Kisel, E. M. Ovsiyuk, and V. M. Red’kov for their advice and help.

Conclusion Assumed by Cox [1] a not point-like structure for a relativistic spin zero particle provides us with a highly modified non-relativistic Schr¨odinger equation for such a particle. This non-relativistic wave equation has been investigated in the presence of the external uniform magnetic and electric field in the case of Minkowski space. Extension of these problems to the case of open hyperbolic Lobachevsky 3-space is given.

References [1] Cox W. Higher-rank representations for zero-spin field theories. J. Phys. Math. Gen. 1982. Vol. 15, no 2. P. 627–635. [2] Ovsiyuk E. M., Spin zero Cox’s particle with an intrinsic structure: general analysis in external electromagnetic and gravitational fields. Ukr. J. Phys. 2015. Vol. 60. P. 485–496. [3] Kazmerchuk K. V., Ovsiyuk E. M., Cox’s particlele in m agnetic and electric field against the background of Euclidean and spherical geometries. Ukr. Phys. J. 2015. Vol. 60. P. 389–400. [4] Redkov V. M., Ovsiyuk E. M., Quantum Mechanics in space of constant curvature. Nova Science Publishers: New York, 2012.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy Dvoeglazov

Chapter 7

N ONLINEAR arctan-E LECTRODYNAMICS AND C HARGED B LACK H OLES S. I. Kruglov∗ Department of Chemical and Physical Sciences, University of Toronto, Mississauga, Ontario, Canada

Abstract We investigate a model of nonlinear electrodynamics with the Lagrangian density L = −(1/β) arctan(βFµν F µν /4). The phenomenon of vacuum birefringence is studied. The model of electromagnetic fields coupled with the gravitation field is considered. The black hole solution is obtained possessing the asymptotic Reissner-Nordstr¨om solution. The corrections to Reissner-Nordstr¨om solution are found.

1.

Introduction

Some models of nonlinear electrodynamics (NLE) possess finite self-energy of charged particles and do not have singularity of the electric field at the classical level [1], [2], [3], [4], [5]. Models of NLE can be considered as effective models taking into account quantum corrections. For example, nonlinear HeisenbergEuler Lagrangian [6] takes into consideration one-loop quantum corrections to ∗

E-mail address: [email protected].

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S. I. Kruglov

Maxwell’s electrodynamics. The Born-Infeld model [1] represents NLE that does not have singularity of the electric field and can be used for strong electromagnetic fields. NLE was introduced to eliminate infinite electric fields and to generalize classical electrodynamics. Maxwell’s theory may be considered as an approximation to NLE for weak fields. If the electromagnetic field strength is huge the self-interaction of photons should be taken into account and classical electrodynamics has to be modified [7]. The problems of the initial Big Bang singularity and early time inflation in cosmological models can be solved with the help of NLE. Cosmological models explore the classical Einstein equation and the classical NLE can be considered in the theory of gravity. The electromagnetic and gravitational fields in the early epoch of the universe were very strong, and therefore effects of NLE are important. Thus, high energies in early universe may produce nonlinear electromagnetic effects. Models of NLE are of interest in general relativity (GR) because they take into account processes of vacuum polarizations and can give an impact on the evolution of the early universe near the Planck era. In some inflationary models nonlinear electromagnetic fields can also mimic the dark energy. In early universe the magnetic fields can be greater than 1015 G and, therefore, nonlinear electromagnetic fields influence on spacetime. NLE can be considered as an effective electrodynamics in late epochs and such phenomenological approach [8] can mimic a material medium where electric permittivity and magnetic permeability depend on the field strength [9]. Thus, NLE can be used to create inflation in the early universe [10], [11]. To have accelerated expansion of the universe some models of NLE were considered [12]-[14]. The effects of coupling NLE to gravity can give negative pressures that result in the accelerated expansion of the universe [13]-[15]. In the Λ-Cold Dark Matter (ΛCDM) model the cosmological constant Λ drives the present cosmic acceleration and there is a similarity between the trace anomaly of NLE models and the cosmological constant [16]. The Einstein-Born-Infeld equations take into account nonlinear effects in strong electromagnetic and gravitational fields and were studied in [17]. Here we propose the NLE model coupled to gravitational field, and it depends on a dimensional constant β. We investigate Einstein-NLE model that influences on the universe evolution and does not have singularities. The energy-momentum tensor trace T may contribute to the cosmological constant [18] and, as a result, in curved spacetime nonperturbative effects of self-interacting quantum fields can mimic the cosmological constant. Classical gravity theory may be considered as an effective gravity theory

Nonlinear arctan-Electrodynamics and Charged Black Holes

101

at low energy and the Einstein-Hilbert classical action of GR have to possess the energy-momentum tensor trace anomaly [19]. In NLE the violation of the scale invariance, due to a dimensional constant β, can result in the negative pressure. The charged black hole can be described by Reissner-Nordstr¨om (RN) solution and it may be the final state of charged stars. We investigate new NLE model and study the Einstein-NLE solution that gives some corrections to the RN black hole solution. The static and spherically symmetric charged black holes with a source of NLE are considered. The solutions obtained give some modification to the RN geometry. In the weak-field limit the black hole geometry, within NLE, is converted into Einstein-Maxwell geometry. The static and spherically symmetric spacetime of black hole with the Heisenberg-Euler effective Lagrangian of QED as a source was studied in [20]. The paper is organized as follows. In Sec. 2 we formulate a new model of NLE with the dimensional parameter β. The energy-momentum tensor and its non-zero trace were obtained. We found the electric permittivity, ε, and the magnetic permeability, µ, depending on the electromagnetic fields. It was demonstrated that the scale invariance and dual invariance are broken in the model proposed. The effect of vacuum birefringence was studied in Sec. 3. It was demonstrated that the phenomenon of the vacuum birefringence takes place in the order of β 2 B04 . In Sec. 4 the model of NLE coupled with the gravitation field was studied. We found the black hole solution possessing the asymptotic Reissner-Nordstr¨om solution. The corrections to Reissner-Nordstr¨om solution are obtained. Section 5 is devoted to the conclusion. We use the units with c = h ¯ = 1.

2.

Nonlinear arctan-Electrodynamics

Let us consider NLE with the Lagrangian density L=−

1 arctan(βF ), β

(1)

where β is dimensional parameter with the dimension of (length)4 and βF is dimensionless, Fµν is the field strength and F = (1/4)Fµν F µν . The fundamental length of the model is L = β 1/4 and it goes probably from quantum gravity and it is connected with the maximum of the electric field strength. If L → 0 (β → 0) the Lagrangian density (1) converts into Maxwell’s Lagrangian density

102

S. I. Kruglov

L → −F . We obtain the symmetric energy-momentum tensor by varying the action with respect to the metric [21] T µν = H µλ F νλ − g µν L, where H µλ =

∂L µλ F µλ ∂L = F =− . ∂Fµλ ∂F 1 + (βF )2

(2)

(3)

From Eqs. (2),(3) one finds the symmetric energy-momentum tensor T µν = −

F µλ F νλ − g µν L. 1 + (βF )2

(4)

The symmetric energy-momentum tensor (4) possesses nonzero trace T ≡ Tµµ =

4 4F arctan(βF ) − . β 1 + (βF )2

(5)

When β → 0 one comes to classical electrodynamics, and trace (5) approaches to zero, T → 0. As the energy-momentum tensor trace is not zero, due to the dimensional parameter β, T = 6 0, the scale invariance is violated. One can obtain the dilatation current Dµ = xν Tµν with the divergence ∂µ Dµ = T . The electric displacement field is defined by the expression D = ∂L/∂E. From Eq. (1) we find the electric displacement field D=

E . 1 + (βF )2

(6)

Using the definition D = εE, one obtains the electric permittivity ε=

1 . 1 + (βF )2

(7)

From the relation H = −∂L/∂B we find the magnetic field H=

B , 1 + (βF )2

(8)

and from the definition B = µH the magnetic permeability is given by µ = 1/ε. It follows from Eqs. (6),(8) that D · H = ε2 E · B and D · H 6= E · B. Therefore, according to the criterion [22], the dual symmetry is broken in the

Nonlinear arctan-Electrodynamics and Charged Black Holes

103

model proposed. From the Lagrangian density (1), with the help of Eqs. (6),(8), the field equations can be represented in the form of the Maxwell equations ∇ · D = 0,

∂D − ∇ × H = 0. ∂t

(9)

Using the Bianchi identity ∂µ Feµν = 0, where Feµν is a dual tensor, we obtain the second pair of Maxwell’s equations ∇ · B = 0,

∂B + ∇ × E = 0. ∂t

(10)

The electric permittivity ε and the magnetic permeability µ depend on the electromagnetic fields E, B, and therefore, Eqs. (6), (8), (9), (10) are the nonlinear Maxwell equations.

3.

Vacuum Birefringence

There is the effect of vacuum birefringence in QED due to one-loop quantum corrections described by the Heisenberg-Euler effective Lagrangian [23], [24]. In classical Maxwell’s electrodynamics and BI electrodynamics the phenomenon of vacuum birefringence is absent but in generalized BI electrodynamics with two parameters [25] the effect of birefringence takes place. Now we investigate the possible effect of vacuum birefringence in the NLE described by Lagrangian density (1). Let us consider the superposition of the external constant and uniform magnetic induction field B0 = B0 (1, 0, 0) and the plane electromagnetic wave (e, b), e = e0 exp [−i (ωt − kz)] , b = b0 exp [−i (ωt − kz)]

(11)

which propagates in the z-direction. Let the magnetic induction field B is strong and the total electromagnetic fields are E = e, B = b + B0 . The electromagnetic wave fields are weak compared to the external magnetic induction field, e0 , b0  B0 . Then the Lagrangian density (1) is given by 



1 β β L = − arctan (B0 + b)2 − e2 . β 2 2

(12)

Defining the electric displacement field [26] di = ∂L/∂ei and the magnetic field hi = −∂L/∂bi , one can linearize equations with respect to the wave fields

104

S. I. Kruglov

e and b. We suppose that β 2 B04  1 and the electric permittivity tensor εij and the inverse magnetic permeability tensor (µ−1 )ij , up to O(e20 ), O(b20 ), become εij =

δij , 1 + (βB02 /2)2

(µ−1 )ij =

δij β 2 B02 B0i B0j −  2 , (13) 1 + (βB02 /2)2 1 + (βB02 /2)2

and we have the relations di = εij ej , bi = µij hj . From Eq. (13) one obtains the elements of the electric permittivity and magnetic permeability tensors ε11 = ε22 =

µ11



1 , 1 + (βB02 /2)2

2

1 + (βB02 /2)2 = , 1 − 3(βB02 /2)2

µ22 = 1 + (βB02 /2)2.

(14)

If the polarization of the electromagnetic wave is parallel to the external magnetic induction field, e = e0 (1, 0, 0), one obtains from the Maxwell equations the relation µ22 ε11 ω 2 = k2 . Therefore the index of refraction is given by nk =

√

µ22 ε11 = 1.

(15)

But when the polarization of the electromagnetic wave is perpendicular to the external induction magnetic field, e = e0 (0, 1, 0), we have the equality µ11 ε22 ω 2 = k2 , and the index of refraction becomes n⊥ =

√

µ11 ε22 =

s

1 + (βB02 /2)2 β 2 B04 ≈ 1 + . 2 1 − 3(βB02 /2)2

(16)

As a result, the phase velocities depend on the polarization of the electromagnetic wave, and the effect of vacuum birefringence takes place. If the polarization of the electromagnetic wave is parallel to the external magnetic field, e0 kB0 , the speed of electromagnetic wave is vk = 1/nk = c = 1. But when the polarization of the electromagnetic wave is perpendicular to the external magnetic field, e ⊥ B0 , the speed of the electromagnetic wave is given by v⊥ = 1/n⊥ < c. In accordance with the Cotton-Mouton (CM) effect [27] the difference in the indices of refraction is given by 4nCM = nk − n⊥ = kCM B02 ,

(17)

Nonlinear arctan-Electrodynamics and Charged Black Holes

105

and it is proportional to B02 . Because β 2 B04 is much less than βB02  1 the effect of vacuum birefringence is very weak, n⊥ ≈ 1, and we can neglect the vacuum birefringence. The Cotton-Mouton coefficient kCM obtained in the BMV [28], and PVLAS [29] experiments are bounded by kCM = (5.1 ± 6.2) × 10−21 T−2 kCM = (4 ± 20) × 10−23 T−2

(BMV), (PVLAS).

(18)

The value of kCM calculated within QED and taking into account loop correcQED tions is smaller than the experimental data (18) [28], kCM ≈ 4.0 × 10−24 T−2 .

4.

Nonlinear Electromagnetic Fields and Black Holes

We consider the GR action coupled with the nonlinear electromagnetic field described by the Lagrangian density (1) S=

Z

√ d4 x −g





1 R+L , 2κ2

(19)

where R is the Ricci scalar and κ−1 = MP l , MP l is the reduced Planck mass. The Einstein and electromagnetic equations, followed from Eq. (19), are 1 Rµν − gµν R = κ2 Tµν , 2 ! √ −gF µν ∂µ = 0. 1 + (βF )2

(20) (21)

Our goal is to obtain the static charged black hole solutions to Eqs. (20),(21). The spherically symmetric line element in (3 + 1)-dimensional spacetime is given by ds2 = −f (r)dt2 +

1 dr 2 + r 2 (dϑ2 + sin2 ϑdφ2 ). f (r)

(22)

We imply that the vector-potential possesses non-vanishing component A0 = h(r) and F = −[h0 (r)]2 /2 (the prime means the derivative with respect to the argument). Then the electric field is given by E = h0 (r). As a result, Eq. (21) becomes ! 4r 2 h0 (r) ∂r = 0. (23) 4 + β 2 [h0 (r)]4

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S. I. Kruglov

Eq. (23) can be integrated and we find 



4r 2 h0 (r) = Q 4 + β 2 [h0 (r)]4 ,

(24)

where Q is the constant of integration. It is convenient to introduce new dimensionless variables p 4r 2 (25) y = √ , x = βh0 (r). Q β Then from Eq. (24) we obtain the algebraic equation x4 − xy + 4 = 0.

(26)

One can find with the help of Cardano’s formulas the analytic solutions to Eq. (26), x(y). The function y(x) has a minimum at x = (4/3)1/4 (E = √ h0 (r) = [4/(3β 2)]1/4 and rmin = (4/3)3/8β 1/4 Q. Therefore, Eq. (26) possesses the real solutions if r > rmin . As a result, there is no singularity of the electric field E. It should be noted that there are two branches (solutions) of the function E(r) with physical values that decrease with r and with nonphysical values which increase with r. Therefore, we imply that E ≤ [4/(3β 2)]1/4 (x ≤ (4/3)1/4 ≈ 1.075). √ As a result, the maximum electric field is Emax = [4/(3β 2)]1/4 ≈ 1.075/ β and it decreases with r. The plot of the function y(x) is presented in Fig. 1. Now we can find from Eqs. (25),(26) the integral √ Z Z Q (3x4 − 4)dx 0 p h(r) = h (r)dr = , (27) 4β 1/4 x(x4 + 4) where x is the solution to Eq. (26) and x, as a function of r, is given by Eqs. (25),(26). The integral (27) can be calculated but it is complicated. Instead of the exact value we use the simple approximate value. At r → ∞ and physical value x → 0, one obtains the Taylor series (3x4 − 4) 2 7x7/2 √ + = − + O(x11/2). x 4 x(x4 + 4)

p

As a result, according to Eq. (27), (28) at r → 0, we have √   √ Q 7 9/2 13/2 A0 (r) = h(r) = −4 x + x + O(x ) . 18 4β 1/4

(28)

(29)

Nonlinear arctan-Electrodynamics and Charged Black Holes

107

45 40 35 30

y

25 20 15 10 5 0

0

0.5

1

1.5

2

2.5

3

3.5

x

Figure 1. The function y versus x.

From Eq. (24) for the leading term at r → ∞, we obtain the expression for the electric field as in Maxwell’s electrodynamics, E = h0 (r) → Q/r 2 . Thus, the integration constant Q means the charge. From√Eq. (29) taking into consideration only the first term, one finds, using x = βQ/r 2 , the ordinary potential A0 = −Q/r. Next terms in Eq. (29) give some corrections to Coulomb’s law.

4.1.

Asymptotic Reissner-Nordstr¨om Black Holes

The function f (r) in Eq. (22) can be obtained by the relation [30] f (r) = 1 +

k1 k2 1 + 2+ 2 r r r

Z

dr

Z



r 2 R(r)dr ,

(30)

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S. I. Kruglov

where k1 , k2 are constants of integration. One can find the Ricci scalar by the relation R = −κ2 T , T = g µν Tµν , (31) where the trace of the energy-momentum tensor is given by Eq. (5). From Eqs. (5),(31) and F = −(1/2)(h0 (r))2, one obtains the Ricci scalar 2

R=κ

"

#

  2[h0 (r)]2 4 arctan β[h0 (r)]2/2 − , β 1 + β 2 [h0 (r)]4 /4

(32)

where h0 (r) is the solution to Eq. (24). We find the approximate value for R with the help of Taylor series at r → ∞ (E = h0 (r) 0) motion of particles we use W+ , and for ingoing (pr = ∂r S0 < 0) W− . We study a trajectory of particles in the direction r from the inside to the outside of the horizons, and therefore, the W+ will be used. From Eq. (2) one finds   1 r2 1 1 = p − . (8) A(r) 2 G2 M 2 − GQ2 r − r+ r − r−

Thus, the expression (8) possesses simple poles at the horizons. If r 6= r± the integral (7) is well defined and real, but for a path going through the points r± the integral is not defined because A−1 (r± ) = ∞. We use a replacement r± → r± − iε for outgoing particles [15] to calculate the integral for crossing the horizons r± . As a result, we specify the complex contour that may be used for calculating the integral around r = r± . To normalize the probability, one should use the relation ImC=-ImW− =ImW+ [16], so that from Eq. (6) we

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S. I. Kruglov

have ImS0 = 2ImW+ . In this case there is not a reflection. We need the imaginary part of C for singular coordinates (that can be considered as boundary conditions) which are not well-defined across the horizon. One may evaluate the imaginary part of the integral (7) with the help of the relation [17] 1 = iπδ(r) + P r − iε

 

1 , r

(9)

and P (1/r) means the principal value of 1/r. Calculating the integral in Eq. (7) using Eqs. (8),(9), we obtain ImW+ = 2πGEM.

(10)

From Eqs. (1), in leading order of ¯h, one finds the tunnelling probability 4 8πGM E P = exp − ImW+ = exp − . ¯h ¯h 







(11)

From (11) and the Boltzmann expression one finds the emission temperature of the charged black hole ¯h TH = (12) 8πGM which coincides with the Hawking temperature for the Schwarzschild black hole and does not depend on the charge. This is in contrast with well-established results for the Reissner-Nordstr¨om system. Here only eternal black holes were considered, which forms gravitational background geometry and a thermal spectrum obtained is inconsistent with the energy conservation as the background is fixed. Our calculations were based on the quantum tunneling method applied for two horizons and implying the equilibrium temperature associated with the event and Cauchy horizons. Thus, the temperature is found to be independent of the black hole charge and takes the value expected when the conventional answer is evaluated for zero charge. The result obtained is an artifact of demanding equilibrium between the inner and outer horizons, which is only possible when the inner horizon ceases to exist, either because it merges with the singularity at zero charge or merges with the outer horizon at extremal charge. The outer (event) horizon of the black hole can coexist in thermal equilibrium with the gas of quantum particles at the Hawking temperature TH . There is no such possibility for the inner horizon, and the system can not be put in thermal equilibrium with Cauchy horizon. In addition, it is known that Cauchy horizons are dynamically unstable and are in the causal future of the event horizon. It was shown in

Does the Temperature of Charged Black Holes ...

117

[18] that working with the proper spatial distance in the complex plane a quarter circle contour should be used. Thus, a priori this excludes a contour going through both horizons. As a result, we make a conclusion that it is impossible the quantum tunneling through event and Cauchy horizons. Therefore, the interior and exterior horizons can not be in thermodynamic equilibrium and we 2 ) (where κ is the accept the expression T = κ+ ¯h/(2π) = ¯h(r+ − r− )/(4πr+ + surface gravity of the event horizon) for the temperature of charged black holes and the temperature of charged black holes depends on the charge.

References [1] Hawking S. W., Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199; erratum-ibid, 46 (1976) 206. [2] Hawking S. W., Black holes and thermodynamics, Phys. Rev. D 13 (1976) 191. [3] Hawking S. W., Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460. [4] Hartle J. B. and Hawking S. W., Path integral derivation of black hole radiance, Phys. Rev. D 13 (1976) 2188. [5] Gibbons G. W. and Hawking S. W., Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738. [6] Kraus P. and F. Wilczek, Selfinteraction correction to black hole radiance, Nucl. Phys. B 433 (1995) 403. [7] Kraus P. and F. Wilczek, Effect of selfinteraction on charged black hole radiance, Nucl. Phys. B 437 (1995) 231. [8] Parikh M. K. and Wilczek F., Hawking radiation as tunneling, Phys. Rev. Lett. 85 (2000) 5042. [9] Page D., Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Phys. Rev. D 13 (1976) 198; Particle emission rates from a black hole. 2. Massless particles from a rotating hole, ibid 14 (1976) 3260.

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[10] Chandrasekhar S., The Mathematical Theory of Black Holes (Oxford University Press, 1998). [11] Kerner R. and Mann R. B., Tunnelling, temperature and Taub-NUT black holes, Phys. Rev. D 73 (2006) 104010. [12] A. Yale and Mann R. B., Gravitinos tunneling from black holes, Phys. Lett. B 673 (2009) 168. [13] Kruglov S. I., Black hole emission of vector particles in (1+1) dimensions, Int. J. Mod. Phys. A 29 (2014) 1450118. [14] Kruglov S. I., Black hole radiation of spin-1 particles in (1+2) dimensions, Mod. Phys. Lett. A 29 (2014) 1450203. [15] Srinivasan K. and Padmanabhan T., Particle production and complex path analysis, Phys. Rev. D 60 (1999) 24007. [16] Kerner R. and Mann R. B., Fermions tunnelling from black holes, Class. Quant. Grav. 25 (2008) 095014. [17] Bogolyubov N. N. and Shirkov D. V., Introduction to the Theory of Quantized Fields (John Wiley and Sons Ltd., 1980). [18] Sean Stotyn, Kristin Schleich and Don Witt, Observer dependent horizon temperatures: a coordinate-free formulation of hawking radiation as tunneling, Class. Quant. Grav. 26 (2009) 065010.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 9

G RAVITATION AND Q UANTUM M ECHANICS B. G. Sidharth∗, Abhishek Das† and Arka Dev Roy‡ B. M. Birla Science Centre, Adarsh Nagar, Hyderabad, India

Abstract Einstein’s General Theory of Relativity is a purely classical theory. On the other hand the relatively recent Dark Energy model of Sidharth is rooted in Quantum Mechanics. We attempt to integrate Dark Energy into Einstein’s equation to obtain a modification which includes the time varying gravitation constant G.

1.

Introduction

Ever since Einstein announced his General Theory of Relativity, attempts were made to provide a unified description of Gravitation and Electromagnetism. One of the earliest attempts was that of Hermann Weyl, but this was dismissed by Einstein amongst others as being an ad hoc insertion of the gauge geometry [1, 2]. For the past century the efforts have continued and in the past few decades these have been incorporated in theories of Quantum Gravity like String Theory and Loop Quantum Gravity at least for quantizing gravity [3]. However it would ∗

E-mail address: [email protected]. E-mail address: [email protected]. ‡ E-mail address: [email protected]. †

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B. G. Sidharth, A. Das and A. D. Roy

not be incorrect to say that all these frantic efforts have not really lead to a satisfying picture that combines gravitation with Quantum Theory or vice versa. The author Sidharth has tried to approach from the point of view of Planck oscillations which has lead to some interesting consequences (Cf.ref.[4]). In the present paper we attempt to integrate the concept of Dark Energy with Einstein’s gravitation theory. Dark Energy is the new paradigm, leading to an accelerated expansion of the universe. In this sense Dark Energy is a throw back to the cosmological context which was introduced and subsequently retracted by Einstein himself. We will use this method to introduce Sidharth’s reinterpretation of Dark Energy into Einstein’s equations. It may be recalled that in 1997 when the ruling paradigm was that of the universe dominated by Dark Matter and therefore decelerating, Sidharth had suggested the opposite namely that there was Dark Energy in a new avatar which was causing a small cosmic acceleration [5, 6]. This subsequently came out in the observations of Perlmutter and others. We recall that in his original formulation Sidharth had used the Zero Point Field for Dark Energy. Interestingly more recently he showed that this Dark Energy would leave another footprint (apart from the acceleration of the universe), viz., a cosmic radio wave background. This was confirmed in the past few years by the ARCADE experiment of NASA [7]. We will integrate this Dark Energy or Zero Point Field into the Einstein equation in the sequel.

2.

Theory

In general relativity theory, the total action integral for a system consisting of a continuous distribution of matter as the source of the gravitational field and the gravitational field itself is given by I = IG + IM

(1)

where, IG is the action of the g-field in empty space in the absence of any field source, and IM is the action describing the interaction of the matter distribution with the g-field. Here, we introduce the action of the ZPF field or the field of quantum vacuum and observe the physical results that can be obtained. We rewrite equation (1) as I = IG + IM + I0

(2)

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121

where, I0 is the action depicting the interaction of the g-field and the ZPF. Now, the variation of the total action (2) of the g-field in the presence of matter and the inclusion of ZPF is given by δI = δIG + δIM + δI0 = 0

(3)

where, it is known that c3 δIG = − 16πG

Z

√ 1 (Rkn − gkn R)δg kn −gdΩ 2

and δIM = −

1 2c

Z

√ Tkn δg kn −gdΩ

Z

√ L0 −gdΩ

We choose I0 of the following form I0 = l l Fkn + where, L0 = − 41 Fkn l tensor Fkn is given as

1 c

jkn Akn c

l Fkn =

is the Lagrangian for the ZPF and the field

∂Akn ∂Alk ∂Anl + − ∂xl ∂xn ∂xk

Aµν being the ZPF tensor potential. Now, with the chosen I0 we have the variation as Z

√ 1 δ( −gL0 )dΩ c Now, since L0 = L0 (Akn , ∂µ Akn ) the above equation becomes δI0 =

1 δI0 = c

Z

√ √ ∂( −gL0 ) ∂( −gL0 ) [ δAkn + δ(∂µ Akn )]dΩ ∂Akn ∂(∂µ Akn )

Again, δ(∂µ Akn ) = ∂µ (δAkn ) and hence we may write δI0 =

1 c

Z

[

(4)

(5)

Z √ √ √ ∂( −gL0 ) ∂( −gL0 ) 1 ∂( −gL0 ) − ∂µ { }]δAkn dΩ + ∂µ [ δAkn ]dΩ ∂Akn ∂(∂µ Akn ) c ∂(∂µ Akn )

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B. G. Sidharth, A. Das and A. D. Roy

Next, applying Gauss’ theorem to the third term we find that it is reduced to the integral over the hypersurface that encloses the domain Ω in the given space-time. Since, the variations will be zero at the boundary, the second integration term vanishes. Therefore we have √ √ Z 1 ∂( −gL0 ) ∂( −gL0 ) δI0 = [ − ∂µ { }]δAkn dΩ (6) c ∂Akn ∂(∂µAkn ) Here, we introduce the following relation √ √ ∂( −gL0 ) ∂( −gL0 ) }]δAkn = βkn δg kn [ − ∂µ { ∂Akn ∂(∂µ Akn )

(7)

and thus we get Z

Z

1 8πG 16πG [Rkn − gkn R + 4 Tkn ]δg kn dΩ = βkn δg kn dΩ (8) 2 c c4 Now, since the variations of the metric tensor are arbitrary we get the modified gravitational field equations as 1 8πG 16πG Rkn − gkn R + 4 Tkn = βkn 2 c c4 where, βkn is a new tensor. This equation can also be written as 8πG 0 1 Rkn − gkn R = 4 Tkn 2 c

(9)

(10)

0 where, Tkn = −Tkn + 2βkn is the modified energy-momentum tensor in presence of the ZPF. Now, in terms of mixed tensor equation (9) can also be written as

1 8πG 16πG n Rnk − δkn R + 4 Tkn = β 2 c c4 k Putting k = n in this we will obtain R=

8πG 16πG T− β 4 c c4

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Using this relation back in equation (9) we get finally 8πG 1 8πG (Tkn − gkn T ) = 4 [2βkn − gkn β] (11) 4 c 2 c as an alternate form of equation (9). Again, it is known that Einstein had originally included the cosmological constant in his field equations. In presence of the cosmological constant the Einstein’s field equations can be written as Rkn +

8πG 1 (Tkn − gkn T ) = gkn Λ (12) c4 2 where, Λ is the so called cosmological constant. Equating this with equation (11) we have Rkn +

gkn Λ =

8πG [2βkn − gkn β] c4

Multiplying both sides by g mk we get Λδnm =

8πG [2βkn g mk − δnm β] c4

which finally gives 8πG [2βkn g mk − β] (13) c4 Thus (13) shows that the acceleration of the universe is due to the Dark Energy as in Sidharth’s 1997 model. From this novel relation it can be immediately concluded that the cosmological constant arises due to the field of the quantum vacuum or more precisely the ZPF. Besides, this relation can signify various other phenomena. Now, if we put Λ = 8πG c2 ρvac in equation (13) then we get Λ=

ρvac c2 = 2βkn g mk − β

(14)

This means that the new tensor βkn along with the metric tensor g mk gives rise to the vacuum energy density ρvac . More precisely, we can say that the above relation is the measure of the vacuum energy density. We now invoke

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B. G. Sidharth, A. Das and A. D. Roy

Sidharth’s fluctuational cosmology where he had found the relation [8] ρ ρvac = √ N where, N ∼ 1080 is the number of particles in the universe. Using this relation, we find that ρc2 √ = 2βkn g mk − β (15) N Now, since ρ is related to the time-time component (T00 ) of the energymomentum tensor Tkn , we can infer that the right hand side of equation (15) gives T00 . Now, if we integrate both sides in the volume V of the domain Ω, then we obtain Z Z c2 √ ρdV = [2βkn g mk − β]dV (16) N which gives Z M 1 √ = 2 [2βkn g mk − β]dV (17) c N Again, Sidharth [5] had arrived at the relation M = mN where, M is the total mass of the universe and m is the mass of the N number of particles [5]. By dint of this relation, we derive Z √ M 1 √ = m N = 2 [2βkn g mk − β]dV (18) c N which suggests that the total mass and the mass of all the elementary particles in the universe is related to the tensor βkn arising due to the ZPF. We will see various conclusions arising from the relation (13)- (18).

3.

Gravitational Potential

Now, in order to obtain the gravitational potential corresponding to Newton’s gravitational law the non-trivial equation is conventionally taken for k = n = 0

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125

in equation (11), without the extra terms on the right hand side. Here, we put k = n = 0 in equation (11) and get R00 +

8πG 1 8πG (T00 − g00 T ) = 4 [2β00 − g00 β] 4 c 2 c

(19)

where, it is known that for static non-relativistic case T00 ≈ ρc2 , T ≈ ρc2 , g00 ≈ 1 + 2φ ≈ 1 and R00 = c12 ∂α ∂ α φ. Now, differentiating equation (7) c2 partially with respect to g kn we have βkn = −

∂βkn kn δg ∂g kn

since, the left hand side of equation (7) is independent of the metric tensor g kn . Now, putting k = n = 0 in the above relation we would obtain β00 = 0

(20)

Thus, equation (19) yields ∇2 φ − 4πGρ = −

8πG β c2

(21)

or, ∇2 φ = 4πGρ −

8πG β c2

The solution of equation (21) would be of the form φ(r) = −G

Z

1 2β [ρ − 2 ]dV R c

Now, considering a uniform distribution over the volume V , we obtain GM 2G φ(r) = − + R R

Z

β dV c2

(22)

From this result it can be concluded that there is a modification of the Newtonian gravitation and this may account for the discrepancies observed in

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B. G. Sidharth, A. Das and A. D. Roy

the general relativistic theories. Again, from equation (22) we can also write G= −M R

φ(r) −

2 RC 2

R

βdV

Now, since β depends on βkn and consequently on Akn (ZPF tensor) which in turn depends on the time coordinate, we might infer that the gravitational constant G depends on time. The dependence may be small, but it is a result worth investigating. Now, we know that in vacuum the field equations lead to 8πG vac T = gkn Λ c4 kn Here, if we take k = n = 0 then we get Λ=

8πG vac 2φ T (1 + 2 ) c4 00 c

where, we have taken g00 ≈ 1 + we would have

2φ . c2

vac T00 =

Again, equating this with equation (13)

[2βkn g mk − β] (1 + 2φ ) c2

vac Using relation (22) and considering that T00 =

ρvac = c2 [

ρvac c2

(23) we have finally

2βkn g mk − β ] R β (1 − 2GM − c4G dV ) 2R c2 R c2

(24)

as an expression for the vacuum energy density.

4.

Varying G

As we have done before, taking k = n = 0 for the non-trivial solution corresponding to Newton’s gravitational law, we obtain from equation (22) M √ =− N

Z

β dV c2

(25)

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127

since, β00 = 0. Also, we have found in the previous section that G= −M R

φ(r) −

R

2 RC 2

βdV

Using equation (25) in this relation we would obtain Rφ(r) M {1 − √2 } N √ Now Sidharth had derived the relation T = N τ , where T is the age of the universe and τ ≈ 10−23 is the continuum region very nearly equal to the Compton time. Therefore, we may write G=−

G=−

G=−

Rφ(r) M {1 − 2τ T }

Rφ(r) 2τ {1 − }−1 M T

Since, T  τ we finally derive G = G0 (1 +

2τ ) T

(26)

where, G0 = − Rφ(r) is the initial constant value for the Newtonian field. M The dependence of G on time can be seen from the above relation. A similar result was also derived by the author Sidharth. The approach in this paper also accommodates such time-dependence of the gravitational constant. Now, we have shown that the inclusion of the ZPF in Einstein’s gravitation theory has reconfirmed and has brought forth results that are interesting and have far reaching consequences. Now, we know that the Hubble’s law for the expansion of the universe is given by dR = HR dt ⇒

Z

R

R0

dR = R

Z

(27) t

t0

Hdt

(28)

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B. G. Sidharth, A. Das and A. D. Roy

where, R0 is the radius of the universe, R is the expansion, t0 is age of the universe, t is time elapsed from the present epoch. From the above relation we can derive ln

R0 = H(t0 − t) R

Again, the author Sidharth used his fluctuational cosmology to derive that H=

Gm3 c ¯h2

G=

Gi t0 − t

and so, we have finally

2

(29)

where, Gi = m¯h3 c ln RR0 . From this approach also, we can see the time varying nature of G. Therefore, we have been able to show the time varying nature of G from two different viewpoints. It may be observed that time varying G cosmologies have been considered, e.g. in Brans-Dicke theory, Dirac cosmology and Hoyle-Narlikar theory [9, 10, 1, 4]. Further, Sidharth has in a series of papers, shown that a G following (26) or (29) can reproduce all results of General Relativity, like the bending of light, precession of the perihelion of Mercury right up to the shortening of the time periods of binary pulsars and so on [11, 12, 8]. It even explains the orbital curve anomaly of galactic rotation and has been shown to be at .... the same as MOND [4].

Conclusion The Zero Point Field is a Quantum Mechanical effect, intimately linked to such phenomena as Zitterbewegung. In the past, we have argued that when it is incorporated with Classical Mechanics, we get Quantum Mechanics [13, 14].We have now combined it with General Relativity which is purely classical to get a modified form of the Einstein equation, the main feature being a time varying gravitational constant, with its multifarious effects.

Gravitation and Quantum Mechanics

129

References [1] Misner C. W., Thorne K. S. and Wheeler J. A., (1973). Gravitation (Freeman W. H., San Francisco), pp.819ff. [2] Sidharth B. G. (2005). The Universe of Fluctuations (Springer, Netherlands). [3] Rovelli C. and Smolin L. (1990)., Loop Space Representation of Quantum General Relativity, Nuclear Physics B 331, 1990, pp.80-152. [4] Sidharth B. G. (2008)., The Thermodynamic Universe (World Scientific, Singapore, 2008). [5] Sidharth B. G., (1998). Int. J. of Mod. Phys. A 13, (15), pp.2599ff. [6] Sidharth B. G., (1998). International Journal of Theoretical Physics Vol.37, No.4, pp.1307–1312. [7] Sidharth B. G., (2015). The dark energy signature, Int. J. Mod. Phys. E, 24, 2015, 1550024. [8] Sidharth B. G., (2003). Cosmology and Fluctuations in Special Issue of Chaos, Solitons and Fractals Eds. Narlikar J. V. and Sidharth B. G., 16(4) 2003,pp.613-620. [9] Barrow J. D. and Parsons P., (1997). Phys.Rev.D. Vol.55, No.4, 15 February 1997, pp.1906ff. [10] Narlikar J. V., (1993). Introduction to Cosmology (Cambridge University Press, Cambridge), p.57. [11] Sidharth B. G., (2000). Nuovo Cimento, 115B (12) (2), 2000, pp.151ff. [12] Sidharth B. G., (2006). Foundations of Phys.Letts. 19(6), 2006, 611-617. [13] Sidharth B. G., (2009). Int. J. Mod. Phys. E Vol. 18, No. 9 (2009) 1863 -1869. [14] Sachidanandam S., (1983). Physics Letters Vol. 97A, No.8, 19 September 1983, pp.323–324.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Chapter 10

A B RIEF N OTE ON THE C OSMIC B ACKGROUND R ADIATION B. G. Sidharth G. P. Birla Observatory & Astronomical Research Centre B. M. Birla Science Centre, Adarsh Nagar, Hyderabad, India

Abstract In this chapter we first argue that there would be a sea of microwaves in the universe and that further these would have a peak at about 4mm corresponding to the Cosmic Microwave Background Radiation.

Introduction Our starting point is the author’s 1997 cosmology which used the ubiquitous zero point energy to deduce a model of an accelerating universe with a small cosmological constant unlike in its previous 1960s version of Zeldovich and others which lead to the famous cosmological constant problem [1]. At this time the ruling cosmological paradigm was exactly the opposite – a dark matter dominated decelerating universe [2, 3, 4]. It has also been known that the Zero Point Field leads to the Lamb shift [5, 6]. It is quite remarkable that radiations due to the Lamb shift have a frequency of around 1000 megacycles, which falls right within the microwave region to a few centimeters radiation. So there is a sea of microwave photons in the universe whose frequencies differ by

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small amounts. We shall now argue that it is this microwave sea which leads to the Cosmic Microwave Background. In the paper [7] and its updated version [8] it was shown that a collection of Photons with nearly the same energy or frequency exhibit a condensation type of behaviour. To see what this means, our starting point is the formula for the average occupation number for photons of momentum ~k for all polarizations [9]: 2 hn~k i = β¯hω (1) e −1 Let us specialize to a scenario in which all the photons have nearly the same energy so that we can write, hn~k i = hnk~0 iδ(k − k0 ),

(2)

where hn0k i is given by (1), and k ≡ |~k|. The total number of photons N , in the volume V being considered, can be obtained in the usual way, N=

V [k] (2π)3

Z

∞

dk4πk2 hnk i

(3)

0

where V is large. Inserting (2) in (3) we get, N=

2V 0 4πk 2 [Θ − 1]−1 [k], Θ ≡ β¯hω, 3 (2π)

(4)

In the above, [k] ≡ [L−1 ] is a dimensionality constant, introduced to compensate the loss of a factor k in the integral (3), owing to the δ-function in (2): That is, a volume integral in ~k space is reduced to a surface integral on the sphere [~k] = k0 , due to our constraint that all photons have nearly the same energy. We observe that, Θ = ¯hω/KT ≈ 1, since by (2), the photons have nearly the same energy ¯hω. We also introduce, v=

V 2πc 2π λ3 ,λ = = and z = N ω k v

λ being the wavelength of the radiation. We now have from (4), using (5), 0

vk 2 8π (e − 1) = 2 [k] = 0 [k] π kz

(5)

A Brief Note on the Cosmic Background Radiation

133

As λ = 2π/k, from (5), we get: z=

4λ 8π = [k] k0 (e − 1) (e − 1)

(6)

From (6) we conclude that, when λ=

e−1 = 0.4[L] 4

(7)

then, z≈1

(8)

or conversely. We are working in the cgs system, so that the units in (7) are cm. As pointed out, this could be further confirmed from other points of view. By the above argument though there may be other photons around, these nearly mono energetic photons would condense to a value around 4 millimeters wavelength which is roughly the cosmic microwave background. So of the entire microwave spectrum it is around 4mm that there would be the maximum intensity. This is the Cosmic Microwave Background.

Acknowledgments The author is grateful to the Editor for useful comments.

References [1] Weinberg S. (1979)., Phys. Rev. Lett. 43, pp.1566. [2] Sidharth B. G., (1999)., Proc. of the Eighth Marcell Grossmann Meeting on General Relativity (1997) Piran T., (ed.) (World Scientific, Singapore), pp.476–479. [3] Sidharth B. G., (1998)., Int. J. of Mod. Phys. A 13, (15), pp.2599ff. [4] Sidharth B. G., (1998)., International Journal of Theoretical Physics Vol.37, No.4, pp.1307–1312. [5] Bjorken J. D. and Drell S. D., (1964)., Relativistic Quantum Mechanics (Mc-Graw Hill, New York), pp.39.

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[6] Itzykson C. and Zuber J., (1980)., Quantum-Field Theory (Mc-Graw Hill, New York), pp.139. [7] Sidharth B. G., (2000)., Chaos, Solitons & Fractals 11, 2000, pp.14711472. [8] Sidharth B. G. and Valluri S. R., (2015)., Int.J.Th.Phys Vol.54(8), 2015, pp.2792-2797. [9] Huang K., (1975)., Statistical Mechanics (Wiley Eastern, New Delhi), pp.75ff.

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy Dvoeglazov

Chapter 11

N EUTRINOS IN A S UPERNOVA VIA THE P ROCESS e+ e− → ν ν ¯ IN A 331 M ODEL A. Guti´errez-Rodr´ıguez1 , M. A. Hern´andez-Ru´ız2 and A. Gonz´alez-S´anchez1,3 1 Unidad Acad´emica de F´ısica, Universidad Aut´onoma de Zacatecas, Zacatecas, M´exico. 2 Unidad de Ciencias Qu´ımicas, Universidad Aut´onoma de Zacatecas, Zacatecas, M´exico. 3 Departamento de Investigaci´on, UPAEP, Puebla, M´exico

Abstract We calculate the emissivity due to neutrino-pair production in e+ e− annihilation in the context of a 331 model in a way that can be used in supernova calculations. We find that the emissivity is almost independent of the mixing angle φ of the model in the allowed range for this parameter.

1.

Introduction

The detection of neutrinos from SN1987A by the Kamiokande II [1] and Irvine-Michigan-Brookhaven [2] detectors confirmed the standard model of core-collapse (type II) supernovae [3] and provided a laboratory to study the properties of neutrinos [4] and exotic particles such as axions. The collapse of stellar iron-core into a neutron star is preceded by a high-power pulse of

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A. G.-Rodr´ıguez, M. A. H.-Ru´ız and A. G.-S´anchez

neutrino emission. The energy loss rate due to neutrino emission receives contributions from both weak nuclear reactions and purely leptonic processes. However, for the large values of density and temperature which characterize the final stage of stellar evolution, the latter are largely dominant, and are mainly produced by four possible interaction mechanisms: ¯ (pair annihilation), e+ + e− → ν +ν ¯ (ν-photoproduction), γ + e± → e± + ν +ν ¯ (plasmon decay), γ ∗ → ν +ν ± ± ¯ (bremsstrahlung on nuclei). e + Z → e + Z + ν +ν Our main objective in this work is to provide suitable expressions for the emissivity of pair production of neutrinos via the process e+ e− → ν¯ ν in the context of a 331 model [5] and in a form which can be easily incorporated into realistic supernova models to evaluate the energy lost in the form of neutrinos. Pisano and Pleitez [6] proposed this model based on the gauge group SU (3)C × SU (3)L × U (1)N . This model has the interesting feature that each generation of fermions is anomalous, but that with three generations the anomalous canceled.

2.

Process e+ + e− → ν + ν¯

The amplitude of transition for the process e+ (p1 ) + e− (p2 ) → ν¯(k1 , λ1) + ν(k2 , λ2).

(1)

is given by g 2 ab ν [¯ u (k2 , λ2) γ µ (gVν − gA γ5 ) v (k1 , λ1)] 2 cos2 θW e × [¯ v (p1 ) γµ (agVe − bgA γ5 ) u (p2 )] ,

M = −

(2)

where the constant a and b depend only on the parameters of the 331 model [5]

Neutrinos in a Supernova via the Process e+ e− → ν ν¯ in a 331 ... sin φ a = cos φ − p 3 − 4 sin2 θW

and

1 − 2 sin2 θW b = cos φ + p , 3 − 4 sin2 θW

137 (3)

where φ is the mixing parameter of the 331 model [5]. We calculate the emissivity associated with neutrino pair production which is given by [7, 8, 9, 10] Qν ν¯

4 = (2π)8

Z

d3 p1 d3 p2 d3 k1 d3 k2 (E1 + E2 )F1 F2 δ(4) (p1 + p2 − k1 − k2 )|M|2 , (4) 2E1 2E2 21 22

where the quantities F1,2 = [1 + exp(Ee− ± µe− )/T ]−1 are the Fermi-Dirac distribution functions for e± , µe is the chemical potential for the electrons and T is the temperature. From the transition amplitude Eq. (2) and the formula of the emissivity Eq. (4) we obtain i

h

[1]

e 2 e Qν ν¯ = G2F a2 b2 (gVe )2 + (gA I1 , ) + 2gVe gA

(5)

where I1 explicitly is given by I1 =

1 24(2π)7

Z

  d3 p 1 d3 p 2 (E1 + E2 )F1F2 3m2e (p1 · p2 ) + 2(p1 · p2 )2 + m4e . E1 E2

(6)

In a similar way, we obtain h

[2]

i

e 2 e I2 , Qν ν¯ = G2F a2 b2 (gVe )2 + (gA ) − 2gVe gA

h

[3]

i

e 2 Qν ν¯ = G2F a2 b2 (gVe )2 − (gA ) m2e I3 ,

(7)

where

I2

1 = I1 = 24(2π)7 h

Z

d3 p1 d3 p2 (E1 + E2 )F1 F2 E1 E2 i

3m2e (p1 · p2 ) + 2(p1 · p2 )2 + m4e ,

I3 =

1 (2π)7

Z

h i d3 p1 d3 p2 (E1 + E2 )F1 F2 m2e + (p1 · p2 ) . 2E1 2E2

(8)

138

A. G.-Rodr´ıguez, M. A. H.-Ru´ız and A. G.-S´anchez

The calculation of the emissivity can be more easily performed by expressing the latest integrals in terms of the Fermi integral: G± s

=

1 m3+2s e

Z

∞

E

me /KT

2s+1

p

E 2 − m2e dE. 1 + e(E±µe )/KT

(9)

From (9), Eqs. (8) are expressed as

I1nm =

I3nm =



mn+m+8 e 3G−n G+m + 2G−n+1 G+m+1 + G−n−1 G+m−1 2 2 6(2π)5 2 2 2 2    4 G−n+1 − G−n−1 + G+m+1 − G+m−1 , 9 2 2 2 2   mn+m+6 e G−n−1 G+m−1 + G−n G+m . 2 2 (2π)5 2 2

(10) (11)

Therefore, Eqs. (5) and (7) are explicitly [1]

h

e 2 e Qν ν¯ = G2F a2 b2 (gVe )2 + (gA ) + 2gVe gA [2]

h

e 2 e Qν ν¯ = G2F a2 b2 (gVe )2 + (gA ) − 2gVe gA [3]

h

i

h

ih

ih

i

I110 + I101 ,

(12)

I210 + I201 ,

(13)

i

e 2 Qν ν¯ = G2F a2 b2 (gVe )2 − (gA ) m2e I310 + I301 .

i

(14)

Finally, the expression for the emissivity of neutrino pair production via the process e+ e− → ν¯ ν in the context of a 331 model is given by [1]

[2]

[3]

Q331 ν ν¯ (φ, β) = Qν ν¯ (φ, β) + Qν ν¯ (φ, β) + Qν ν¯ (φ, β) ,

(15)

where the dependence of the φ mixing parameter of the 331 model is contained in the constants a and b, while the dependence of the β degeneration parameter is contained in the Fermi integrals G± s.

Acknowledgments We acknowledge support from CONACyT, SNI and PROFOCIE (M´exico).

Neutrinos in a Supernova via the Process e+ e− → ν¯ ν in a 331 ...

139

Conclusion For our analysis we consider the following data: the Fermi constant GF = 1.166 × 10−5 GeV −2 , angle of Weinberg sin2 θW = 0.223 and the electron mass me = 0.51 M eV , thereby obtaining the emissivity of the 331 neutrinos Q331 ν ν¯ = Qν ν¯ (φ, β). For the mixing angle φ of the 331 model, we use the reported data of Cogollo et al. [5]:

Figure 1. The emissivity for e+ e− → ν¯ ν as a function of the mixing angle φ and the degeneration parameter β.

−3.979 × 10−3 ≤ φ ≤ 1.309 × 10−4 ,

(16)

with a 90% C.L.. In Figure 1 we show the emissivity as a function of the mixing angle φ and the degeneration parameter β. We observe that emissivity decreases when β increases, which is due to the reduction in the number of positrons available necessary to cause the collision. Also we see that the emissivity is unaffected by the φ parameter. There are other effects which may change the emissivity,

140

A. G.-Rodr´ıguez, M. A. H.-Ru´ız and A. G.-S´anchez

for example, the radiative corrections at one-loop level. In summary, we have analyzed the effects of the mixing angle φ of a 331 model on the emissivity of the neutrinos via the process e+ e− → ν¯ ν . We find that the emissivity is almost independent of the mixing angle φ of the model in the allowed range for this parameter. As expected, in the limit of vanishing φ we recover the expression for the emissivity QSM ν ν¯ (β) for the SM previously obtained in the literature [7].

References [1] Hirata K. et al., Phys. Rev. Lett. 58, 1490 (1987). [2] Bionta R. M. et al., Phys. Rev. Lett. 58, 1494 (1987). [3] Arnett W. D., Ann. Rev. Astron. Astrophys. 27, 629 (1989). [4] Schramm D. N. and Truran J. W., Phys. Rept. 189, 89 (1990). [5] Cogollo D. et al., Mod. Phys. Lett. A23, 3405 (2009). [6] Pisano F. and Pleitez V., Phys. Rev. D46, 410 (1992). [7] Yakovlev D. G., Kaminker A. D., Gnedin O. Y. and Haensel P., Phys. Rept. 354, 1 (2001). [8] Lenard A., Phys. Rev. 90, 968 (1953). [9] Guti´errez-Rodr´ıguez A. et al., Int. J. Mod. Phys. A25, 2551 (2010). [10] Hern´andez-Ru´ız M. A. et al., Eur. Phys. J. A53, 16 (2017).

In: Relativity, Gravitation, Cosmology ISBN: 978-1-53614-135-1 c 2019 Nova Science Publishers, Inc. Editor: Valeriy V. Dvoeglazov

Commentary

O N “E LECTROMAGNETIC P OTENTIAL V ECTORS AND S PONTANEOUS S YMMETRY B REAKING ” V. V. Dvoeglazov∗ UAF, Universidad Aut´onoma de Zacatecas, Zacatecas, M´exico

Abstract The appearance of terms, which are analogous to ones required for symmetry breaking, in Lagrangian of Ref. [1] is shown to be caused by gauge invariance of quantum electrodynamics (QED) and by inaccuracy of the cited author in the choice of canonical variables. These terms do not have physical significance within modern quantum electrodynamics.

PACS: 03.50.De, 04.20.Cv, 04.20.Fy, 11.10.Ef In Ref. [1, 2] the following k– space Lagrangian for electromagnetic field, interacting with the current ~j and with the charge density ρ, has been obtained: h

i

˙ ~k) + A ~ ∗ (~k) − ~k ~j(~k) − mΨ( ¯ ~k)Ψ(~k)+ Λ(~k) = iΨ+ (~k)Ψ( + i ∗

 ~kC ~ ∗ (~k)ρ(~k) | ρ(~k) | 2 1  ~ ⊥ (~k) |2 −~k2 | A ~ ⊥ (~k) |2 . (1) − + |C ~k 2 2 2~k 2

E-mail address: [email protected]; Website: http://fisica.uaz.edu.mx/˜valeri/.

142

V. V. Dvoeglazov

~ the vector potential, and C ~ =A ~˙ = ∂ A~ are The approach was used, in which A, ∂t supposed to be independent to each other. The author of cited paper considers (1) as the Lagrangian with the spontaneous-symmetry-breaking terms (fourth and fifth in the above formula). Let us mark, the approach using the additional vector variable (it is des~ in Ref. [1]), which is different from field variables, and it is ignated as C considered as independent, is not an innovation. This is just the well-known Hamiltonian canonical formalism (see e.g., [3]-[6]). In Ref. [5] the canonical~ in [1], if we conjugated variable to Aµ is defined identically with the quantity C 1 don’t take into account the inessential coefficient 4π : πi =

~ 1 ∂L 1 ∂A = . 4π ∂(∂tAi ) 4π ∂t

(2)

This canonical-conjugated quantities are due to use of the following Lagrangian: L=−

1 1 1 ∂Aµ ∂Aν Aµ,ν Aµ,ν = − Fµν F µν − 8π 16π 8π ∂xν ∂xµ

(3)

But, in the case of the x– space Lagrangian 1 L = − Fµν F µν 4

(4)

~ and C ~ =A ~˙ are not the canonical-conjugated quantities, as opthe quantities A posed to the case of classical mechanics where ~x, the coordinate, and ~x˙ , the velocity, are, in fact, the canonical quantities. It is not clear, what quantzation procedure are implied by the author of Ref [1]. In the case of canonical quantization the Lagrangian ∼ Fµν F µν does not give us π0 , which is equal to zero. In the case of Lagrange quantization it is not clear, what commutation rules ~ x), C( ~ x~0 )]. Moreover, in the Lagrange apshould be implemented, e.g., for [A(~ proach the field and the momenta of field are not considered as two independent quantities. It is also not obvious, how Pµ , the energy-momentum operator, is ~ and C ~ in the quantum case. expressed by A Let us not forget, under quantization of electromagnetic field it is impossible to use the Lorentz condition ab initio. According to Fermi [7] it exists as the condition for the state vectors only. It is necessary to choose the definite Lagrangian and the definite quantization approach. and we are able to use

On “Electromagnetic Potential Vectors and Spontaneous Symmetry ...” 143 the Lorentz condition in a weaker form only after setting up the commutation relations: ! ∂A(−) Φ = 0. (5) ∂x In the case of the Lagrangian (3) we are able to quantizate the electromag~˙ as independent. Following the netic field canonically, using the variable ~π = A techniques of [1], we then have the additional terms to the k– space Lagrangian, which do contract one of the term in (1): i ~ ~ ~ ∗ ~  1 ~ ~ ~  ~ ~ ∗ ~  ~ ~ ~  ~ ~ ∗ ~  ρ(k) kC (k) + kC(k) kC (k) − kA(k) kA (k) . ~k 2 ~k 2 (6) The total Lagrangian does not contain the symmetry breaking terms. As a result of gauge invariance of QED it is possible to use the other LaLadd = −



2

grangians differing from (4) by the supplementary term λ1 ∂A , which, on ∂x the first view, brings nothing in (18) of cited paper. However, in this case the µ i i0 and the expressions (10, canonical quantities are π0 = λ1 ( ∂A ∂xµ ) and π = F 11) between the canonical quantities in Ref. [1] are no longer kept. Moreover, it is well-known that the Lagrangian can be defined up to the total derivative only. If we implement the function ∂ µ fµ = ∂ µ (g · h)µ it is easy to select g and h in such kind that both of the symmetry-breaking terms in (18) are contracted out. ~ = 0 and ~k · C ~ = 0) In the end, it is not clear, why the Coulomb gauge (~k · A was used by the author of [1] in the formula (18), the Lagrangian, but it was not used before, e.g., in (10) and (15). 2 In conclusion, the appearance of interaction terms of the form a · | Ψ | 4 +b · | Ψ | in the QED Lagrangian is caused by gauge invariance of electrodynamics, implementing the Lorentz condition ab initio, inaccuracy of the author in the choice of the canonical-conjugated quantities. These terms are nonphysical and can be eliminated as a result of using the appropriate gauge. Consequently, they have no any physical meaning in quantum theory. I would still like to mention that investigations of interaction electromagnetic field with currents deserves serious elaboration. In Ref. [8] the equation was presented: (−iγ µ ∂µ − m)Ψ(~x) = 2 µ

= e γ Ψ(~x)

Z

¯ y )γµΨ(~y) + eγ µ Ψ(~x)Ain d~yD(~x − ~y)Ψ(~ x) µ (~

(7)

144

V. V. Dvoeglazov

(where D(~x − ~y) is the Green’s function for electromagnetic field, Ain x) is the µ (~ solution of Maxwell’s equations), which should be resolved (see also [6]).

Acknowledgments I am very grateful to Prof. A. M. Cetto, Head of the Departamento de F´ısica Te´orica, IFUNAM, for creation of excellent conditions for research. The technical help of A. Wong is greatly acknowledged. This work has been financially supported by the CONACYT (M´exico) under the contract No. 920193.

References [1] J. V. Shebalin, Nuovo Cim., 108 B, 99 (1993). [2] J. V. Shebalin, Physica D, 66, 381 (1993); Phys. Lett. A. 226, 1 (1997). [3] A. Visconti, Quantum Field Theory. Vol. I (Pergamon Press, 1969), p. 254. [4] S. N. Gupta, Quantum Electrodynamics (Gordon and Breach Sci.Publ., 1977), p. 59. [5] A. A. Sokolov et al., Quantum Electrodynamics (Mir Publisher, Moscow, Russia, 1988), in English, p. 107. Revised from the 1983 Russian edition. [6] G. K¨all´en, Quantum Electrodynamics (Springer-Verlag, 1972), p. 48. [7] E. Fermi, Rev. Mod. Phys., 4, 87 (1932). [8] Foundations of Radiation Theory and Quantum Electrodynamics. Ed. A. O. Barut (Plenum Press, 1980), p. 169.

BOOK REVIEWS* Einstein and Poincaré: The Physical Vacuum Edited by Valeri Dvoeglazov ISBN: 0-9732911-3-3 $20.00 Softcover, 184 pp. Apeiron, 2006 Review by Peter Graneau

This book gathers together 13 papers invited by the editor, Valeri Dvoeglazov, who is a professor of physics at the University of Zacatecas in Mexico. The common thread running through the papers is the discussion of an abstract medium filling all space and described by such terms as aether, physical vacuum, field, energy of radiation, cosmic background, curved space-time, and so on. There is no agreement between the authors and even the spelling of the word aether is interchangeable with ether. As one of the authors (M. C. Duffy) points out, there are so many different ethers mentioned in the physics literature that it is impossible to evaluate them all. Einstein did away with Maxwell’s elastic ether medium and is sometimes claimed to have eliminated all ethers from physics. This is denied *

Originally appeared in Infinite Energy, Volume 72, 2007, pp. 1-2. Reprinted with permission of Infinite Energy Magazine.

146

Book Reviews

by the authors in Einstein and Poincaré, who made extensive use of the electromagnetic field, the radiation energy concept, and the mathematics of empty space-time. Not all 14 authors, from 13 countries, of the 13 papers accept the special and general theories of relativity. They all rely on an ether of one form or another and agree with Einstein that a medium, like the field or spacetime, is necessary because the alternative Newtonian action-at-adistance is “spooky.” Spookiness is the scientific reason which led to Einstein’s local action to which the 14 authors of the book subscribe, regardless of whether Einstein’s relativity theory is true or false. Einstein and Poincaré is a book of conflict and largely unresolved issues. Only five of the thirteen papers profess agreement with Einsteinian relativity. Three papers are hostile to relativity with its time dilation and Lorentz contraction. The remaining five papers are uncertain as to the reality of modern physics. None of the papers examines Newtonian mechanics and Newtonian absolute space which was kind of an ether giving rise to the forces of inertia. Mach and his principle, which provide the strongest evidence for action-at-a-distance, are not mentioned. The paper which I found most interesting is written by Reginald T. Cahill, the Head of the School of Physics of Flinders University in Adelaide, Australia. He says: “General Relativity turns out to have been a major blunder.” He arrives at this conclusion from an examination of Einstein’s postulates of relativity theory. According to Cahill‒and other authors in the book‒it has been shown that the Michelson-Morley experiment proves that there exists a preferred local frame of reference. Cahill maintains a new theory of gravity is required which explains “dark matter” distributed throughout the universe and at the same time provides a new fine structure constant as a second gravitational constant. Another paper which attracted my attention is Kholmetskii’s “Energy Space-Time and the General Relativity Principle.” The author is in the physics department of Belarus State University. He points out that the equations of the special relativity theory are invariant under transformations between inertial reference frames, while in the general theory of relativity the equations are covariant. This mathematical difference, in the judgment of Kholmetskii, indicates that Einstein’s special theory of relativity is not a

Book Reviews

147

consequence of his general relativity theory. Therefore the two theories should not be compatible. Kholmetskii then proposes an experiment which tests the consistency of the two theories. Some readers will be intrigued by Yves Pierseaux’s (Free University of Brussels) paper on the dichotomy of spherical and ellipsoidal wave-fronts caused by traveling light sources, with the former having been proposed by Einstein and the latter by Poincaré. The French scientist believes in a relativistic ether because it allows him to define the state of rest. Pierseaux’s conclusion is that there exist two different relativistic theories of space-time. Two of the best known critics of modern physics have made contributions to Dvoeglazov’s work. They are Professor Selleri of the University of Bari and Professor Vigier (deceased) of the University of Paris. Selleri comments on Einstein’s clock paradox and asserts that it can be resolved with an active inertial background of space. Vigier, a former assistant of de Broglie, treats Dirac’s ether of quantum theory. Einstein and Poincaré has an historical aspect. It reveals the widespread discontent with the teaching of physics at the beginning of the twenty-first century. A major paradigm change looks ever more likely. It may still take a long time to come because, as the book proves, there is so little agreement between the critics. Both the editor and the publisher deserve praise for their courage to confront the science teaching establishment.

Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory Thomas E. Phipps, Jr. ISBN 0-9732911-4-1 $20.00 Softcover, 258 pp. Apeiron, 2006 Review by Bill Cantrell Crisp, clear and invigorating! These are words that describe the latest book from renowned dissident physicist Dr. Tom Phipps. His new book goes to battle against the illogic of modern physics by highlighting some of the mistakes and errors associated with the current formulation of

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electromagnetic field theory and its attempted rescue by special relativity theory (SRT). The avid reader of dissident material will find a treasure trove of new information on this topic, along with a detailed proposal for an experimentum crucis to decide between the validity of SRT and his alternative theory. Readers of Phipps’ first book, Heretical Verities, will also find this an indispensable addition to their dissident library. Old Physics for New makes a key contribution to dissident progress by clearly delineating the inescapable problems in the mathematical formulation of Maxwell’s Equations for moving reference frames. The book is not overly heavy on the esoteric equations so characteristic of electromagnetic field theory; but rather, the author uses mathematical description only where appropriate. The reader is treated to Phipps’ eloquent writing style in describing the many problems with mainstream theory, and a lively supposition of why science chose the paths that it did. The book makes for quite an enjoyable read, something like a detective novel, because Phipps leads the reader through his alternative neo-Hertzian theory in a clear and uncomplicated manner revealing every clue and every inescapable conclusion along the way. There are some gratifying details inside. Length invariance and Euclidian geometry reign supreme in Phipps’ theory for there is no room for space-time symmetry or length contraction. He also discusses the fact that the GPS evidence violates SRT on definitional and experimental grounds, resulting in the need to simply ignore SRT so that GPS can be designed to work correctly. An entire chapter is devoted to the topic of stellar aberration, which to the reader’s delight, does an outstanding job of explaining the history of stellar aberration, what it is, how it occurs, and why it is irreconcilable with SRT. Based on the experimental evidence, there is a need for time dilation. The author has devoted two chapters to clock rate asymmetry and “collective” time. Clock compensation is discussed in considerable detail from a pragmatic standpoint involving the actual nuts and bolts of clock compensation‒not the mind-numbing thought experiments involving trains so beloved of some in the mainstream. Dirty little experimental secrets and cold hard logic abound here. Most importantly, Phipps throws down the gauntlet to mainstream relativists with two proposals for a decisive test of his alternative neoHertzian theory versus SRT. First, he proposes a measurement of the angle

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of stellar aberration using very long baseline interferometry (VLBI), an experiment that can be performed using present-day technology. Second, he proposes an experiment for light speed measurement in a free-falling inertial system, the outcome of which could require the reformulation of the relativity principle as currently understood. This is the stuff of scientific revolutions and well worth the price for a front-row seat.

IN MEMORIAM: DR. THOMAS E. PHIPPS, JR.*

Our colleague, the brilliant Dr. Thomas Phipps, Jr., passed away on July 11, 2016 at the age of 91. During World War II, Phipps worked in P.M. Morse’s Operations Research Group in the Navy Department. Phipps obtained a Ph.D. in nuclear physics from Harvard University in 1950, with an experimental thesis on molecular beam nuclear magnetic resonance under Norman Ramsey.

*

Originally appeared in Infinite Energy, September/October 2016, Issue 129, pp. 1-3. Reprinted with permission of Infinite Energy Magazine.

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Phipps went on to twelve years in the Pentagon‒ten years in systems analysis for the Navy and two in research management for the Department of Defense. Until 1980, he worked at Navy laboratories in California and Maryland. After his retirement in 1980, Phipps opened a private physics laboratory, collaborating with his father, Thomas Phipps, Sr. (Emeritus Professor, University of Illinois). Some of the experimental work conducted at this lab led to the publication of Phipps’ masterpiece Heretical Verities: Mathematical Themes in Physical Description (1986). In an Infinite Energy review of the book (#17, 1998), Jeffery Kooistra wrote: “Lots of books get ignored that perhaps should be, but this isn’t one of them. Friends of new energy research will be delighted by the attacks Phipps brings against Establishment physics, Establishment publications, and particularly, Establishment thinking. But a glance at the copyright date will reveal that Phipps was saying all this even before the cold fusion fiasco brought to light just how little science has to do with Big Science these days.” Infinite Energy distributed Heretical Verities for Phipps in the last few years; the last copy sold a few months ago. We were considering reprinting the important book and hope to still do so. In 2006 Phipps published his second book, Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory. Bill Cantrell wrote in an Infinite Energy review (#72, 2007) that “the avid reader of dissident material will find a treasure trove of new information on this topic, along with a detailed proposal for an experimentum crucis to decide between the validity of SRT and his alternative theory.” We have recently sold our last copy of the book, but copies are still available on Amazon. Phipps published about 50 papers in mainstream physics journals, and many more in dissident physics journals (including Infinite Energy). Many of his papers are available in the Natural Philosopher’s Database: http://db.naturalphilosophy.org/member/?memberid=170&subpage=abstracts

Tom will be missed by all who knew him. Some of his friends have offered the following memorial contributions:

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— David Roscoe — What can I say about Tom Phipps that others have not already said with absolute conviction? Not much, in reality. So, I will content myself with a few recollections. Prior to 1989 (I think that was the year), I knew nothing of Tom Phipps, nor was even aware of his name. But, in that year, at a meeting of “off piste” astronomers/physicists in Paris, C. Roy Keys (who, at the time, was very active in organizing such meetings) brought to our collective attention the book Heretical Verities, of which he had several copies to be shared between those attending who were interested. To my shame, my initial reaction to Roy’s insistence that this book was very much worth a read, was skepticism along with a sigh of resignation as I agreed to at least open the front cover. I waited until my return home to Sheffield before opening that book...but, once opened, I was completely captivated at every possible level: Tom was not an ideologue; if he chose to write about some accepted theory of physics (or indeed of mathematics), it was always because the theory concerned, in some way or other, rested upon unanalyzed assumptions (I state the case mildly), which he would then proceed to unpick with forensic skill. He wrote in such a way that we, the readers, could very quickly understand the underlying problems concerned (even if they had never been apparent to us before), and using language of such vivid descriptive force that we would never forget the issues at hand. The landscape of the subjects he chose to address was (almost) without boundaries...Tom did not consider himself as “a this” or “a that”; for him, the whole of science and mathematics was fair game for his interest and forensic ability. So, I read and re-read Heretical Verities several times over the following two months after the Paris meeting and, like Neal Graneau (and I guess many others), continually dip into that lovely book, sometimes for pure and delightful entertainment but at other times, when considering some problem or other, because a little voice says to me, “I seem to remember Tom had something to say about that...” Not very long after my first acquaintance with Tom’s name, I wrote to him and we became regular correspondents (old-fashioned letters

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only, no emails then). In 1995, my youngest, William, was born and, to our great pleasure, Tom agreed to act in the role of the newborn’s Godfather (I had to assure him that his duties were minimal...the odd birthday card would be sufficient), but the real point was, the honor was all ours, that such a warm, good and great man would agree to occupy such a position in our family’s life. It was one of my wife’s great pleasures to receive personal letters from Tom, for they were so full of warmth, wisdom and wit. He was truly a letter-writer from the golden age of letter writers. Tom’s companion in his last few years, Kathleen Leahr, managed to give us sufficient notice of his last few days that we were able to set out our proper farewells in the written word in time for Tom to receive them (courtesy of express UPS) and to reply in his usual absolutely gracious way. Thank you, Tom, for gracing the lives of this family, and farewell, My Friend.

— Neal Graneau — What is sadly lacking in most practitioners in the field of modern physics is not great mathematical ability, complex machinery or fast computers, but the highly unusual combination of analysis of diverse information, uncompromising honesty and the talent to transmit the conclusions with flair, clarity and poignant allusion. I would propose that anyone who had studied the writings of Tom Phipps would agree that he was perhaps the foremost master of this latter amalgam of unusual skills. I return to his writings again and again as a soul not content with the accepted, university promoted scriptures, but one who seeks understanding from an independent sage who refused to accept paradox, sought the simplest models that explain all known facts and then communicated them with language that is as close to Shakespearian as serious science can be presented without any loss in precision. The main reason I constantly sought Tom’s opinions, both published and in correspondence, was his honesty. He had an instinct that sensed deliberate or even in most cases accidental illogic and naturally sought the cause. His campaign to bring these errors to the recognition of the

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Establishment was a hard fought battle and even with his passing remains a struggle that has inspired others who all share Tom’s hope, that however long it takes, intellectual honesty will be restored in the most sacred of sciences, physics. One of my achievements of which I am most proud was to be in a position to perform a complex experiment in my laboratory at Oxford University based on a mathematical shape independence theorem proposed by Tom in 1996 concerning the measurement of longitudinal electrodynamic forces as predicted by Andre Marie Ampère in 1822. In conjunction with Tom’s close friend in Sheffield University, David Roscoe, and my father Peter Graneau, we performed the experiment, analyzed the results and argued against a wall of disparaging referees, but eventually published a paper together in 2001 [European Physical Journal D, “An Experimental Confirmation of Longitudinal Electrodynamic Forces,” Vol. 15, Issue 1, pp. 87-97]. Tom and all of us considered this to be the most convincing demonstration of the validity of the pre-Maxwellian Instantaneous-Action-at-a-Distance (IAAAD) paradigm to date, thereby delivering another fatal blow to the modern post-Maxwellian physics of Lorentz and Einstein against which Tom railed throughout his career. As many will attest, Tom was a consummate correspondent and there will be files both in cabinets and hard disks full of lively debates over the wide range of subjects in which he was engaged. I took great pleasure and learned a tremendous amount of physics, philosophy and scientific method by eavesdropping on the letters written between Tom and my father covering the field of electrodynamics and the failures of Einstein’s theory of Special Relativity. It was a great honor when I became part of Tom’s salon in my own right many years later. Tom eventually became my most trusted sounding board when I needed to hear a truly honest opinion on a new theory or experimental concept. He was able to let me down gently or fill me with confidence more than any other colleague I have ever had and for that I will miss him tremendously. Along with a number of other physicists dotted around the world, Tom was a crucial part of a small band of adherents to the IAAAD philosophy of matter interaction. Tom’s lucid writing made clear that

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while we had a consensus on where modern physics was decisively broken, there would be no clear path regarding what to replace it with. Tom had a very gentle manner of proposing his replacement theories with strength, but not so vehemently as to leave no room for other concepts and healthy debate. His attractive style and foresight into the vagaries, strengths and weakness of human thought will ensure that anyone who reads his works will be rewarded and, without doubt, will have had their mind changed in some way. To me, this will remain Tom’s greatest gift and legacy.

— Cynthia K. Whitney — Tom Phipps came into my life along with Peter Graneau, back in the mid-1980s. What wonderful friends these individuals proved to be! Tom was possessed of a wicked wit that leavened the lives of all who knew him. Such levity was much needed by members of a community experiencing severe cognitive dissonance. We were all finding ample evidence that something was not right in Modern Physics, but at the same time we were finding that, among physicists, there was little willingness to examine the evidence objectively, which would have ultimately demanded that they become willing to consider appropriate revisions to longstanding doctrines. The big problem is this: much of the 20th century belief system is tied to Einstein’s Special Relativity Theory (SRT), and SRT is, in turn, founded in part on his Second Postulate; namely, that: the speed of light is the constant number c relative to all inertial observers. There is a clause missing from this Postulate, one that everyone before Einstein had assumed, that Einstein himself assumed, and that almost everyone after Einstein assumed, all of them without ever stating it, much less testing it. The un-written clause is: “…over the entire light propagation path…all the way back to the light source…no matter how far back that light source was…be it a distant star, a distant galaxy, or even the Big Bang creation event!” You can tell that this un-written clause is always assumed, because calculations before, during and after Einstein have always involved simple ratios like R/c, where R is the length of the light propagation path.

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If c did not have the same reference all along the propagation path, then c would not really be a constant, and the simple ratio with c in the denominator would not be appropriate. Think about it: SRT actually represents the ultimate in anthropocentrism. It is like what we had before we had Science, when Cosmology was a branch of Theology, and Man was at the center of God’s Universe, and everything else orbited around Man! Ideas alternative to Einstein’s have always been needed, and Ritz was early to offer one; namely, that the reference for c be always, not the receiver, but rather the source. This idea is not anthropocentric, but it did not work out for the first test cast: stellar aberration. And then came Sagnac. The Sagnac effect does not support either Einstein or Ritz. The Sagnac effect supports a more modest statement; namely, that the speed of light starts as c relative to the source, becomes c relative to each successive bit of matter that the light encounters, and so ends as c relative to the receiver. This modest idea works well. I look forward to discussing it one day with my now-departed friends: Tom Phipps, Peter Graneau, Jan Post, Bob Heaston and so many others.

— Greg Volk — I first contacted Dr. Phipps in 2008 after reading a paper written in 1927 by his father, Dr. Thomas E. Phipps Sr., which measured properties of hydrogen using then-new atomic techniques. I was delighted to learn that Dr. Phipps Jr. was indeed the son of the same, that he had earned a Ph.D. in physics from Harvard, that he had worked closely with Nobel Laureate Norman Ramsey, that his career connected him with several other amazing physicists, that in his early retirement conducted experiments with his father aimed at reinterpreting conventional thinking in modern physics, and that he was among the most prolific writers and critics in the dissident universe. Wow! Though I realize the NPA’s Sagnac Award has impacted little, I will always keep a place in my heart for Dr. Phipps and the other 2010 recipients, whose contributions, in my opinion, compare favorably with

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the physics Nobel Laureates of the same year. Like Avogadro, Phipps’ interpretations of relativity will, I believe, ultimately prove correct. Though he did not live to see all of his experiments conducted, he most certainly did propose specific tests that determine measurable secondorder differences between conventional thinking about light, and his own neo-Hertzian relativity. But beyond thoughts of his own, Phipps was amazingly well-read, critiquing hundreds of books and papers of other dissident authors. This contribution alone merits him the title “renaissance man.” I feel sorry for his online detractors, who never bothered to read his material, since Phipps’ credentials were so unimpeachable, his prose so lucid and delightful, his facts so clearly presented, and claims so understated. I consider it a great honor to have known him, and exchanged ideas with him. He was a great man of science in the tradition of Newton, Maxwell and, well, Sagnac.

— Brigitte Graneau — We mourn the loss of a great and independent mind. The death of Thomas Phipps leaves a void among those who pursue science unhindered by the constraints of the establishment. Dr. Phipps has encouraged many to follow their convictions and I trust that his inspiration will continue to produce progress into the unknown. My sincere sympathy goes to his family and colleagues.

THE SCIENTIFIC LEGACY OF DR. PETER GRANEAU: INSTANTANEOUS INTERCONNECTION OF ALL THINGS†

Dr. Neal Graneau Physicist and new energy pioneer Dr. Peter Graneau passed away peacefully on February 25, 2014, with his family near him in Concord, Massachusetts. He was 92 years old. Peter was born on March 13, 1921 †

Originally appeared in Infinite Energy, March/April 2014, Issue 114, pp. 10-14. Reprinted with permission of Infinite Energy Magazine.

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in Silesia, Germany, where his father was a landowner and businessman. After the war, that part of Germany was annexed by Poland and his parents were able to move to a property they owned further west. Since his studies at the University of Berlin had been interrupted by the war, he was able to move to England, become a British subject and attend the University of Nottingham, where he studied physics and was awarded the B.Sc. (First Class Honours). While at University, he met the girl he would fall in love with and eventually marry, Brigitte, who from that time until today gave unfaltering support to his scientific work. They married in 1955 and moved to London, where he joined a large industrial laboratory BICC (British Insulated Callender’s Cables) as assistant research manager while he continued to work on his Ph.D. thesis, entitled “Coupled Circuit Theory for Electromagnetic Testing.” He was eventually awarded the Ph.D. degree in 1962 after a successful resubmission and his first taste of serious scientific controversy. At this time he was also appointed as a Fellow of the Institute of Physics. At BICC (1955-1967), Peter’s aim was to bring about collaboration between industry and academia, not a common practice at that time. He was very successful in initiating many joint projects, especially in the advancement of nondestructive fault testing of standard electrical cables, and exploring novel forms of electrical energy transmission using liquid nitrogen cooling and vacuum insulation. He also enjoyed a time playing with trains while contributing to the electrification of Britain’s railway network. During this period, he produced 16 fundamental electromagnetic (EM) theory publications, based on the merger between his academic and professional work. These papers covered the mechanisms of inductance, EM induction and superconductivity. His goal was to create calculating techniques based on the pre-Maxwellian electrodynamics theories which did not require considering magnetic fields and their artificially imposed relativistic time delays. These algorithms have been demonstrated to be accurate and involve far fewer calculation elements and steps, but have still not been taken up by the power engineering or physics community as it is still not culturally acceptable that EM calculations can be performed without magnetic fields.

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Peter and Brigitte’s son, Neal, was born in London in 1963. Fourteen years later, Peter would commence a close scientific collaboration that would last until today. In the early 1960s, Peter was asked to serve on a U.S. committee under the auspices of the “Highway Beautification Act,” championed by Lady Bird Johnson. His part in the grand scheme to improve the urban and rural U.S. landscape was to develop new and efficient power transmission technologies to facilitate the economical undergrounding of the unsightly overhead electrical power corridors that blight the environment. He eventually forged a conglomerate of three Cambridge, Massachusetts institutions—Simplex Wire & Cable, Arthur D. Little and MIT—and in 1967, together with his wife and son, moved to Concord, Massachusetts to lead this project. He set up a research laboratory in the Simplex facilities and built prototype sections of cryogenically cooled vacuum insulated rigid cable, designed for high power underground AC power transmission with very low loss. He formed a close collaboration with the Vacuum Barrier Corporation, whose liquid nitrogen handling skills were vital to the potential success of the “cryocable” concept. This work was proceeding well, but came to a temporary suspension when the site of the Cambridge Simplex plant was sold to MIT in 1969, with the buildings being demolished. Undeterred by these events, together with the president of Simplex, he formed a company called Underground Power Corporation with offices in Weston, Massachusetts and at the same time established a new electrodynamics and power transmission laboratory at MIT in the Francis Bitter National Magnet Laboratory, with funding from the U.S. National Science Foundation and Department of Energy. There he built a 138 kV test cryocable section including connector, termination and switching components. He designed high voltage surge diverters which were successfully tested at the Westinghouse Waltz Mill test facility in Pennsylvania. He also explored the economics and feasibility of utilizing sodium as a low-cost cable conductor. During this period he filed more than a dozen patents. This work culminated in his first book, Underground Power Transmission: The Science, Technology and Economics of High Voltage Cables (Wiley, 1979), which covered existing technology plus his own contributions.

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As a result of a health scare in 1977, Peter deliberately sought a retreat from the high pressure commercially oriented research in which he had been devoting most of his efforts and took advantage of an opportunity to return to his true love of delving into the fundamental laws of electromagnetism that began during his Ph.D. research. With a new grant from the National Science Foundation, he was able to construct a series of rudimentary, but highly poignant experiments in his MIT laboratory that aimed to produce effects that would distinguish between the predictions of the pre-Maxwellian electrodynamics developed by scientists such as Ampère, Neumann, Weber and Kirchoff and those of the field theories of Maxwell, Lorentz and Einstein. Peter delved deeply into the history of the various theories and discovered that in many situations, the two sets of laws predicted the same effects, but were based on entirely different philosophical principles. The earlier laws were modelled on Newtonian physics, which can be generically described as “instantaneous action at a distance” (IAAAD). This notion assumes that every individual particle attracts or repels every other particle in the universe at all times. The strength of this force is proportional to the product of their masses (gravity), charges (electrostatic force) and inversely proportional to the square of the distance of their separation. In the armory of IAAAD forces, Newton quantified gravity, Coulomb did the same for the electrostatic force and Ampère deduced a similar equation for small elements of electrical conductors, which are known as current elements. Maxwell went as far as describing Ampère as “the Newton of electricity.” However, Maxwell in the 1870s was on a self-proclaimed mission to develop a field theory specifically to introduce an undetectable substance that inhabited the space between all pieces of matter with as many properties as required to explain all known experiments. For no reason, other than fashion, this new field theory approach became popular in England and within 20 years had pervaded continental Europe and the rest of the scientific world and the IAAAD era came to an end. When Einstein was a student in the last years of the 19th century, he was only taught field theory, even though this theory was known to be flawed by its lack of Galilean invariance. At that moment, it would have been entirely logical to return to the IAAAD theories which were necessarily Galilean invariant as a

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consequence of their Newtonian origin. However, history shows that physicists, like politicians, never elect to make a U-turn, and history shows that the physics community preferred to patch up the existing flawed theory to make it even more complicated. This patch was the development of Lorentz invariance and led to special relativity and ultimately general relativity. At the beginning of the 20th century, it simply was not possible to experimentally distinguish between the two sets of theories either by experiment or calculation. The necessary experiments required high current power supplies and the calculations involved digital computers, neither of which yet existed. As a consequence, logic would dictate that both theories should have been kept alive until they could be distinguished by predictions and then experiment would determine which to keep. Needless to say, this did not happen; however in 1977, Peter Graneau realized that the necessary high current power supplies were already in his lab and that his son had the skills to write the appropriate computer programs in FORTRAN on the mainframe computer at his high school, Phillips Exeter Academy. During the Christmas 1977 vacation, the two set to work to design experiments that were simple to build and easy to model and calculate predictions. The two laws to be compared were the Lorentz law (the force law attached to Maxwellian field theory) and the IAAAD Ampère’s force law (the original electrodynamic force law, empirically derived in 1822). The fundamental difference between the two is that Ampère’s law predicts that electrodynamic force can have a component in the same direction as the current flow in a current element. The Lorentz force denies the existence of such a longitudinal component. Approximately 20 experiments were performed in the MIT laboratory and all confirmed the existence of the longitudinal force component. These experiments ranged from DC currents of hundreds of amps in liquid mercury and copper circuits to much higher pulsed currents of tens of kiloamps in railgun and exploding wire configurations. Some of the effects that were analyzed, such as the location of the recoil forces in a railgun, have direct implication on technologies for which large amounts of taxpayer money is currently being spent. Unfortunately, the still unfashionable nature of IAAAD theory means that many scientists are still struggling to make

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their data fit the flawed field theory. Peter Graneau remained convinced throughout his life that someday, logic will prevail and the science and engineering community will accept new ideas rather than continuing to believe existing dogma, but this process has taken longer than he expected. The first of Peter’s publications demonstrating longitudinal electrodynamic force appeared in Nature in 1982 and a string of over 50 papers have appeared over the intervening 32 years, most of which were co-authored with his son, Neal. The work up to 1985 was captured in his second book, Ampère-Neumann Electrodynamics of Metals (Hadronic Press, 1985). A more complete summary was revealed in his third book, Newtonian Electrodynamics (World Scientific, 1996), which was coauthored with Neal and gave the full mathematical derivations and predictive capabilities of the Ampère-Neumann electrodynamics. It established a more complete definition of the Ampèrian current element and focused on the principle that if Ampère’s law predicts the force between two current elements, then Neumann’s poten tial defines the stored energy between them. This potential then controls a variety of other electrodynamic effects such as inductance, induction and effects normally ascribed to EM radiation. Most importantly this book presents a complete theory of electrodynamic force, inductance and induction that does not require the notion of a magnetic field. The extension of the theory to distant high frequency instantaneous interactions between sources and antennas (otherwise known as EM radiation) has been published in the last book co-authored by Peter and Neal, In the Grip of the Distant Universe: The Science of Inertia (World Scientific, 2006), and elsewhere. The outstanding issues of handling magnetic materials and particle beams with IAAAD theory have yet to be tackled. Peter Graneau always maintained that to have still unanswered questions will hopefully one day encourage more scientists to enter the field. In 1984, during a series of experiments with Neal in which they were experimenting with electrode configurations in salt water to drive a small model boat with the Ampère force, they accidentally discovered that if there was a spark breakdown in the water adjacent to the boat, the resulting force did not propel it forward as expected, but sent it flying to the ceiling where it shattered into many pieces. Over the next few years,

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water arc experiments were performed in semi-confined cavities and five papers were published with a series of his students quantifying the strength of these explosions and unsuccessfully attempting to predict the magnitude of this force with any form of EM or other force law. The lack of theoretical understanding brought this work to a temporary scientific hiatus. In addition, the laboratory at MIT had to be disbanded due to the expansion of the MIT Plasma Fusion Center. Between 1987-1997, Peter Graneau was also a visiting professor at the newly established Center for Electromagnetics Research at Northeastern University (Boston, Massachusetts). During some of this period, he was able to concentrate again on the foundations of IAAAD Newtonian electrodynamics. However after a few years, his thoughts returned to the anomalous water arc explosion experiments. He deduced that if the force of the explosions was larger than could be predicted by any known force law, then there must be a hidden source of energy which is released during the explosion. Experiments were all pointing to the conclusion that the most likely energy storage mechanism was the intermolecular bonding network (hydrogen bonds in water) that defines the liquid state. As a result, he set about forming an international research team to engage in the high current pulsed arc liberation of stored hydrogen bond energy from water. A large part of his experimental equipment was moved to a private laboratory in Canada which quickly set up experiments to examine the anomalous explosions. By 1996, his son, Neal, had secured funding and was able to reconfigure his laboratory in the University of Oxford, UK, to work specifically on the understanding of the energetics of water arc explosions. Until 2005, Peter enjoyed many of his “retirement” years travelling between these two institutions, designing and participating in many exciting trials of both a fundamental and engineering nature. The goal of this research was to further confirm the existence of this hidden stored energy source in water and other liquids, and find techniques to exploit it with the ultimate goal of achieving a new energy generation mechanism and source. The work proved to be challenging for several reasons: a) the difficulty of accurately measuring the energy of an explosion and b) the inefficiency of converting the kinetic energy of a low mass of high velocity water to a heavy energy harvesting device such as a turbine or

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piston. With this consortium, he published more than a dozen papers on the liberation of intermolecular hydrogen bond energy. Unfortunately the funding for both laboratories was terminated in 2006, however the nine most significant publications from this era were compiled into a soft cover compendium entitled Unlimited Renewable Solar Energy from Water, which is available from Infinite Energy. In 2006, foreign travel became more difficult for Peter and he became a technical editor of Infinite Energy and was able to contribute more broadly to monitoring and encouraging unconventional energy research as well as furthering his renewable energy interests by promoting the science of liquid bond energy liberation for the benefit of mankind. He continued this role until August 2013, during which time he wrote a dozen editorials (all of which are on the Infinite Energy website, http://www.infinite-energy.com/iemagazine/ readarticles.html). In that period, he contributed ideas and suggestions to the work which Neal was performing with new colleagues in the Technical University in Delft, Holland in which the water arc research had been extended to a continuous low power gain using the electrospray mechanism. All of this research now points to a future electricity generating technology which will produce power 24 hours a day in all locations on the earth and derives its input from the atmospheric thermal heat bath that surrounds us and is driven by the sun. During their years of joint water arc research, Peter and Neal were able to spend some of their time together on their other pet scientific project, the understanding of the force of inertia. This force, which is normally called fictitious because it does not fit neatly into the present physics paradigm dominated by the theory of special relativity, is nevertheless required to explain the bursting of rotating discs, the pain in your elbow when it presses against the car door during a high speed turn and in fact everything we physically feel. The two authors developed a theory of inertia based on the instantaneous interaction principle that they had developed during their study of IAAAD electrodynamics. It appears that the inertia that we feel on earth, and which makes all objects resist acceleration in proportion to their mass, can be due to instantaneous force interactions between every atom and every other atom in the universe. This highly thought-provoking concept

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was first conceived by Ernst Mach (1838-1916) and is often called “Mach’s Principle,” although he never delved into its mathematical consequences. Peter Graneau recognized that although it contradicts much of the philosophy of modern physics, it is consistent with all known experiments and the electromagnetic theories he was promoting and provides a vastly simpler mathematical framework with which to understand the laws of nature. The Graneau mathematization of Mach’s principle was unexpectedly accepted for publication in the most prestigious journal dedicated to promoting general relativity (General Relativity and Gravitation) in 2003. Feeling that perhaps a change of attitude was in the wind, Peter and Neal wrote another book together entitled, In the Grip of the Distant Universe: The Science of Inertia (World Scientific, 2006). It primarily described the long and tortuous history of this much misunderstood subject, presented the outlines of an IAAAD theory of the effects normally ascribed to EM radiation, demonstrated the circular reasoning behind all of the so-called experimental confirmations of special and general relativity and concluded with their IAAAD theory of inertia. One of the final conclusions presented in the published paper and the book was that if the force of inertia and the force of Newtonian gravity were one and the same thing, then in order to explain the physics we observe on earth, the universe must be accelerating away from itself. Moreover, the acceleration is related to the product of two constants, one of which is the mysterious Newtonian gravitational constant and the other is defined by the matter distribution of the entire universe which can be finite even for an infinite universe. In the last 15 years, it has become clear that the universe is actually accelerating away from itself, which runs quite contrary to the famous Big Bang theory and consequently, modern physics is now inventing yet more invisible notions such as dark energy to explain this anomaly. Peter and Neal were very proud to have discovered that the most recent cosmological measurements could be much more easily explained by a return to IAAAD Newtonian physics and first year undergraduate mathematics. However, just as in the field of electrodynamics, modern cosmologists have no stomach for a paradigm shift yet. A broad encapsulation of Peter Graneau’s research career reveals

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that he was motivated both by a love of pure fundamental physics at its most raw and unconventional edge as well as a strong sense of providing creative and entirely unexpected engineering solutions to real problems. In 2009, he received the Sagnac award from the Natural Philosophy Alliance “in recognition of a lifetime commitment to excellence in scientific pursuit, for experiments in water plasma explosions and railgun recoils, and for theoretical presentations of Amperian longitudinal forces, instantaneous Machian interactions, and the unique role of water in renewable energy.” It can be seen that many of Peter’s “pre-conventional” ideas and projects have not yet reached the marketplace due to the normal human resistance to change. High power underground cryocables and sodium cables will only be developed when the price and shortage of copper demand a change from existing power transmission methods. Similarly, only when fossil fuels and nuclear energy have become economically and environmentally impractical will there be serious demand for the required R&D investment in a renewable solar energy technology which functions 24 hours a day in all locations. Peter knew that many of his ideas were simply ahead of their time and hoped that history will eventually demonstrate their utility. Similarly, while the world of field theory or even aether-based physics continues to stumble onward by continually inventing new and invisible repair mechanisms every time it runs into a conflict with experiment, there will be only marginal interest in a return to the vastly simpler IAAAD physics that he promoted. He nevertheless maintained faith that since the brightest minds, 400 years ago, were eventually able to strip away the hideously complex epicycles of Ptolomaic astronomy and replace them with the much simpler heliocentric Copernican vision, this process could happen again. Despite the obstacles in getting his work more globally accepted, Peter Graneau was thoroughly content that he had laid much of the framework on which a new future physics could be based. This contribution has been recognized by many colleagues and is his lasting legacy. Similarly, if his stored liquid energy technology can overcome the known and yet unknown engineering obstacles that stand in its way, it may be an even more important and lasting legacy for the benefit of mankind.

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It is extraordinary that a single theme has run throughout Peter Graneau’s entire professional career and that is the concept of instantaneous interconnection between all particles across the entire universe. It seems to apply equally to electrical as well as mechanical force interactions. It runs throughout his published output of five books and over 150 published journal papers and conference presentations. He also felt so strongly that he recently created a spiritual angle to it. In his last months he expressed the view that since we are all interacting continuously with every atom in the universe, their combined motions retain a memory of us after we leave our earth bound body. He often expressed to his family that if he passed away he would be joining the distant universe, where he now resides in peace. ◆

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The New Energy Foundation Board of Directors and the staff of Infinite Energy greatly appreciate the efforts that Peter Graneau made on our behalf as a technical editor for the magazine. We have lost a true friend and co-pilot in the tireless search for new energy and new physics. We hope that readers will re-visit Peter’s writings on our website, http://www.infinite-energy.com/iemagazine/readarticles.html Peter’s family indicated that he was “proud to have become one of Concord’s authors” (Massachusetts). He will be buried in Concord’s Sleepy Hollow Cemetery, where literary greats Henry Thoreau, Nathaniel Hawthorne, Ralph Waldo Emerson and Louisa May Alcott are also buried. The World Science Database has an extensive list of Peter’s papers available: http://www.worldsci.org/people/Peter_Graneau.

Andre Koch Torres Assis It is with great sadness that I write this. Peter Graneau was a good friend of mine and very important in the beginning of my career. Around 1988 I discovered his papers dealing with Ampère’s force between current elements. This subject had essentially disappeared from the textbooks during most of the 20th century. These textbooks discuss

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only the force of Grassmann and the law of Biot-Savart. And what is nowadays called Ampère’s circuital law is not due to Ampère himself, but to Maxwell. Ampère never spoke nor worked with the concept of a magnetic field. Peter made important experiments and calculations related to Ampère’s force between current elements, bringing it once more to the forefront of modern science. I first met him in 1989 during a Conference on the Foundations of Mathematics and Physics in the 20th Century: The Renunciation of Intuition. It was held in September 1989 at the Department of Mathematics of the University of Perugia in Italy, organized by Umberto Bartocci and J. Paul Wesley. He invited me and we worked together for a year at the Center for Electromagnetics Research at Northeastern University (Boston) from October 1991 to September 1992. It was a wonderful stay. Our exchange of ideas was excellent. We published three papers together: “Kirchhoff on the Motion of Electricity in Conductors” (1994), “The Reality of Newtonian Forces of Inertia” (1995) and “Nonlocal Forces of Inertia in Cosmology” (1996). Peter was a person of strong opinions and fought hard for his beliefs. He loved not only Ampère’s electrodynamics but also action at a distance and Mach’s principle. He was in favor of the reality of Newton’s vis insita (innate force of matter) or vis inertiae (force of inertia). He also had a deep interest in the origin of inertia and believed that it did come from a gravitational interaction of the test body with the distant masses in the cosmos. One magical moment I had with Peter was a walk we took together at Walden Pond, a lake in Concord, Massachusetts, close to his home. The philosopher Henry David Thoreau lived there for two years, from 1845 to 1847, and wrote a famous book describing his experiences living there, Walden; or, Life in the Woods. During our tour on the lake we discussed physics and the origin of inertia. We had a very good insight about the influence of the distant masses in the cosmos acting on the spinning top. These real inertial forces, arising from a gravitational interaction, prevent a gyroscope from falling to the ground. We published these ideas in a joint paper in 1995 discussing the reality of Newtonian forces of inertia.

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In 1999 I published the paper “Arguments in Favor of Action at a Distance,” which appeared in the book Instantaneous Action at a Distance in Modern Physics: Pro and Contra, edited by Chubykalo, Pope and Smirnov-Rueda. I dedicated it to Peter with the following words, which are appropriate to finish my remembrances: “This paper is dedicated to Peter Graneau, the strongest advocate of action at a distance known to me. I have profited greatly from many conversations held with him.”

Thomas E. Phipps, Jr. Real contributors to physical science are few and rare in our times. The fundamentals are all supposed to be known. The purposes and effects of higher education in physics are to quell any doubts and suppress any curiosity about those fundamentals. Yet it is precisely at that level that contributions are most needed and most effective. I have been privileged to know a handful of doubters of the rule that all is well in the sub-basement of physics. Among them the Graneaus, father and son, have been outstanding examples. Now the number of this precious few is reduced by one, per- haps the foremost. Peter Graneau is no longer with us. Not only was he a leader in experimental studies that revealed important new facts about nature, he was a scholar able to read from the past of physics some of the crucial messages that fashion has by-passed and forgotten. I have in mind particularly the amazing electrodynamic insights of Ampère, a still not adequately appreciated pioneer of the subject. To have unearthed and empirically validated the existence of Ampère longitudinal forces, as Peter and Neal have done (and as Peter documented in his classic Ampère-Neumann Electrodynamics of Metals) is a far greater contribution to physical science than any number of string theories, Theories of Everything, or other unverifiable Nobelworthy tarradiddles with which the academic physicist-politicians of our era disport themselves in preference to getting down to work in the laboratory. That of course is only the barest beginning of what Peter achieved. Others will know more of his numerous further accomplishments‒I mention only what impinged directly on my own

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research. The world of physics has lost a unique asset. We shall not see his like again.

Cynthia Kolb Whitney Peter was a very courageous person, and he inspired me to be a courageous person. He knew full well that the science establishment of his day had been hampered by adherence to some indefensible ideas, and that throughout history there had been individuals with ideas that might work out better, but had been set aside. He studied in detail the ideas of André-Marie Ampère, a giant from before the time of Maxwell. I am still studying Peter’s work involving Ampère’s ideas. I believe those ideas, used in conjunction with ideas from statistical mechanics, will ultimately take us to the unification between electromagnetism and gravity. So my presently-developing paper on this subject will be dedicated to Peter.

Peter Graneau illustrates a point during his talk at the 2007 LANR Colloquium at MIT.

Infinite Energy technical editors Peter Graneau, Bill Zebuhr and Scott Chubb at the 2007 LANR Colloquium at MIT.

EDITOR CONTACT INFORMATION Valeriy Dvoeglazov UAF, Universidad Autónoma de Zacatecas, Zacatecas, México Email: [email protected] Website: http://fisica.uaz.edu.mx/˜valeri/.

INDEX A amplitude, 136, 137 annihilation, 2, 8, 12, 135, 136 anthropocentrism, 157 antiparticle, 2, 8, 12, 24

B baryon(s), 15, 16, 27, 28, 29, 31, 32, 33 bending, 62, 128 Bianchi identity, 6, 103 Big Bang, 63, 100 birefringence, viii, 99, 101, 103, 104, 105, 108, 109 black hole, viii, 99, 101, 105, 108, 109, 113, 114, 116, 117, 118 Bohr, Niels, 110 bonding, 165 boson(s), 16, 29, 115

contour, 115, 117 cosmic rays, ix, 57, 58, 63, 64 Coulomb gauge, 143

D dark energy, 129 decay, 28, 58, 136 decomposition, 15, 28, 30, 31, 32 deformation, 59, 60 differential equations, viii dimensionality, 21, 132 Dirac equation, viii, 65, 66, 67, 69, 77 dispersion, 60, 61 displacement, 102, 103 distribution, 16, 58, 59, 120, 125, 137 distribution function, 137 divergence, 10, 102 duality, 19

E C Cauchy horizon, viii, 108, 113, 115, 116, 117 CERN, 110 charge density, 141 charged black holes, 109, 113, 114, 117 classes, 19, 24, 27, 30 classical electrodynamics, 2, 12, 100, 102 classical mechanics, 142 Clifford algebra, 22 construction, 15, 17, 20, 23, 24, 25, 47, 61

early universe, 100 elaboration, 143 electric charge, 66 electric field, viii, 6, 81, 82, 90, 91, 93, 96, 97, 99, 100, 101, 105, 106, 107, 109 electricity, 166 electromagnetic, viii, 54, 66, 78, 79, 97, 99, 100, 101, 103, 104, 105, 109, 141, 142, 143, 144, 146, 148 electromagnetic fields, 66, 78, 99, 100, 101, 103, 109 electromagnetism, 162

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electron, 30, 32, 58, 68, 137, 139 elementary particle, 16, 17, 18, 24, 25, 27, 28, 54, 124 emission, 113, 114, 116, 117, 118, 136 energy, viii, ix, 5, 8, 9, 10, 12, 14, 15, 16, 17, 18, 21, 22, 24, 28, 30, 37, 38, 42, 43, 44, 45, 46, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 75, 76, 77, 81, 84, 99, 100, 101, 102, 108, 109, 116, 122, 123, 124, 126, 129, 131, 132, 136, 142, 145, 146, 159, 165, 166 energy conservation, 116 energy density, 123, 126 equilibrium, 1, 9, 11, 12, 59, 86, 113, 114, 116, 117 Euclidean space, 81 evaporation, 109, 114 evolution, 1, 2, 3, 4, 9, 11, 12, 100, 136

H hadrons, 27 Hamiltonian, 2, 12, 90, 142 Hawking radiation, 117 Hawking, Stephen, 38 Hermitian operator, 17, 21 Hilbert space, 15, 17, 18, 19, 20, 21, 24, 25, 26, 27, 29, 35 hydrogen, 165, 166 hydrogen bonds, 165

I induction, 103, 104, 109 inertia, 166 integration, 10, 11, 106, 107, 108, 122 isospin, 27

F fermions, 24, 115, 136 field theory, 13, 17, 23, 33, 46, 148, 164 force, 2, 3, 4, 5, 8, 52, 53, 86, 91, 163, 165, 166, 170 formation, 17, 32 formula, 19, 30, 31, 41, 61, 65, 75, 77, 84, 132, 137, 142, 143 foundations, 165 fractal structure, 24

G Galaxy, 57 gauge group, 31 gauge invariant, 4 gauge theory, 12 general relativity, viii, 128, 129, 133 gravitation, 99, 101, 108, 109, 119, 120, 125, 127 gravitational collapse, 117 gravitational constant, viii, 127, 128 gravitational field, 97, 100, 120, 122 gravity, viii, x, 100, 101, 109, 117, 119

L Lagrangian density, 99, 101, 103, 105 Lagrangian formalism, 23 lepton, vii, 15, 16, 30, 31, 32 Lie algebra, 21, 28 Lie group, 20, 28 light, 12, 13, 37, 38, 39, 40, 41, 42, 50, 53, 54, 55, 62, 128, 149 linear systems, 71

M magnetic field, viii, 6, 65, 66, 70, 77, 80, 81, 82, 84, 88, 89, 100, 102, 104 magnetic moment, 65, 66, 69, 76, 77, 79 magnitude, 18, 40, 165 mapping, 26, 27 mass, 1, 4, 8, 9, 10, 11, 13, 16, 24, 30, 37, 40, 42, 46, 52, 53, 66, 82, 105, 108, 114, 124, 139, 166 matrix, 19, 22, 24, 29, 47, 48, 49, 50, 61 matter, vii, 4, 13, 15, 16, 17, 22, 24, 28, 30, 31, 32, 38, 43, 54, 55, 59, 120, 121, 131 Maxwell equations, 6, 10, 11, 103, 104 Mercury, 128 mesons, 16, 24, 27, 28, 32 microwaves, 131

Index Minkowski spacetime, 108 mixing, 135, 137, 138, 139, 140 models, 99, 100, 136 momentum, 8, 9, 10, 21, 37, 38, 42, 44, 45, 46, 54, 55, 60, 62, 68, 100, 101, 102, 108, 109, 122, 124, 132, 142

N neutral, 17, 23, 24, 25, 30, 32 neutrinos, 24, 30, 135, 136, 139, 140 neutrons, 24 Newtonian physics, 162 nuclei, 136

P particle creation, 117 particle mass, 4, 9 particle physics, 16, 31 permeability, 100, 101, 102, 103, 104 permittivity, 100, 101, 102, 103, 104 phenomenology, 3, 17, 18 photons, 24, 40, 54, 58, 59, 100, 131, 132, 133 physical phenomena, 12 physics, vii, viii, 16, 31, 33, 41, 50, 55, 57, 59, 64, 79, 109, 111, 145, 156, 158, 160, 163, 166, 168, 172 pions, 32, 58, 59, 62 Poincaré, 145, 146 polarizability, 66, 79 polarization, 104 positrons, 139 principles, viii, 34, 42, 46, 50, 162 probability, 19, 96, 114, 115, 116 propagation, 10, 157 protons, 57, 58, 59, 62, 63 pulsars, 128

Q QED, 101, 103, 105, 111, 141, 143 quantization, 69, 75, 142 quantum electrodynamics, 141 quantum field theory, 13, 17, 33, 46 quantum fields, 100

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quantum gravity, 101 quantum mechanics, viii, 3, 4, 33, 34, 35, 113 quantum objects, 55 quantum theory, 17, 25, 36, 143 quarks, 16, 54 quasiparticles, 54

R radiation, 9, 58, 59, 63, 113, 114, 117, 118, 131, 132, 133, 144, 145, 146 reactions, 58, 136 reading, 59 reductionism, 16 reference frame, 37, 39, 40, 41, 43, 45, 48, 50, 51 relativity, 38, 46, 79, 100, 120, 146, 147, 149, 163 renewable energy, 166

S scalar field, 10 scalar particles, 113, 114 scattering, 3, 5, 8 solution, 6, 66, 76, 99, 101, 106, 108, 109, 115, 125, 126, 144 spacetime, 1, 2, 3, 4, 7, 9, 11, 12, 39, 40, 41, 47, 48, 49, 69, 79, 82, 100, 101, 105, 108, 114, 122, 146 special relativity, 148, 163, 166 special theory of relativity, 46 speed of light, 37, 38, 39, 40, 41, 50, 53 spin, 18, 24, 25, 29, 30, 31, 32, 65, 66, 69, 77, 78, 79, 96, 97, 115, 118 spinor fields, 34 stability, 13, 114 stars, 101 state(s), vii, 3, 12, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 35, 36, 66, 68, 75, 77, 86, 88, 89, 101, 142, 165 structure, 4, 11, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 78, 81, 82, 84, 96, 97 substitution(s), 3, 62, 83, 88, 90, 92, 94, 95 substrate, 16, 22 supernovae, viii, 135

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symmetry, viii, 4, 5, 6, 7, 8, 12, 16, 17, 20, 21, 26, 27, 28, 32, 57, 64, 89, 102, 109, 141, 142, 143 synchronization, 6, 7

T technology, 149, 166 temperature, 59, 113, 114, 116, 117, 118, 136, 137 tetrad, 66, 67, 68, 69 thermodynamic equilibrium, 113, 114, 117 thermodynamic properties, 114 thermodynamics, 117 total energy, 68 trajectory, 2, 8, 9, 12, 47, 114, 115 transformation(s), vii, 4, 12, 24, 26, 31, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 54, 94 treatment, 18, 38, 81, 88 tunneling, viii, 113, 114, 116, 117, 118 tunneling effect, 114 two-dimensional space, 39, 41, 47, 48, 49

U uniform, viii, 65, 66, 70, 77, 80, 81, 82, 84, 91, 96, 103, 125 universe, 12, 100, 120, 123, 124, 127, 128, 131, 166

V vacuum, viii, 99, 100, 101, 103, 104, 105, 108, 109, 111, 120, 123, 126, 145 variables, 65, 77, 81, 82, 86, 91, 94, 106, 141, 142 vector, 2, 3, 6, 10, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 42, 43, 44, 46, 61, 62, 67, 82, 105, 115, 118, 142 velocity, vii, 4, 9, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 60, 142

W wave vector, 55 weakness, 37, 156 WKB approximation, 114, 115