Quantum Antennas 9780367757038, 9781003163626

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Quantum Antennas

Quantum Antennas

Harish Parthasarathy Professor Electronics & Communication Engineering Netaji Subhas Institute of Technology (NSIT) New Delhi, Delhi-110078

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CRC Press

Taylor & Francis Group Boca Raton London New York

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Table of Contents Chapter 1: Basic quantum electrodynamics required for the analysis of quantum antennas 1-12 1.1 Introduction 1 1.2 The problems to be discussed 2 1.3 EM field Lagrangian density 4 1.4 Electric and magnetic fields in special relativity 4 1.5 Canonical position and momentum fields in electrodynamics 4 1.6 The matter fields in electrodynamics 5 1.7 The Dirac bracket in electrodynamics 5 1.8 Hamiltonian of the em field 6 1.9 Interaction Hamiltonian between the current field and the electromagnetic field 7 1.10 The Boson commutation relations for the creation and annihilation operator fields for the EM field in momentumspin domain 8 1.11 Electrodynamics in the Coulomb gauge 8 1.12 The Dirac second quantized field 9 1.13 The Dirac equation in an EM field, approximate solution using Perturbation theory 11 1.14 Electromagnetically perturbed Dirac current 11 Chapter 2: Effects of the gravitational field on a quantum antenna and some basic non-Abelian gauge theory 13-44 2.1 The effect of a gravitational field on photon paths 13 2.2 Interaction of gravitation with the photon field 14 2.3 Quantum description of the effect of the gravitational field on the photon propagator 15 2.4 Electrons and positrons in a mixture of the gravitational field and an EM field Quantum antennas in a background gravitational field 16 2.5 Dirac equation in a gravitational field and a quantum white noise photon field described in the Hudson-Parthasarathy formalism 17 2.6 Dirac-Yang-Mills current density for non-Abelian gauge theories 18 2.7 Dirac brackets 19 V

VI

Quantum Antennas

2.8 Harish-Chandra’s discrete series representations of SL(2,R) and its application to pattern recognition under Lorentz transformations in the plane 2.9 Estimating the shape of the antenna surface from the scattered EM field when an incident EM field induces a surface current density on the antenna that is determined by Pocklington’s integral equation obtained by setting the tangential component of the total incident plus scattered electric field on the surface to zero 2.10 Surface current density operator induced on the surface of a quantum antenna when a quantum EM field is incident on it 2.11 Summary of the second quantized Dirac field 2.12 Electron propagator computation 2.13 Quantum mechanical tunneling of a Dirac particle through the critical radius of the Schwarzchild blackhole 2.14 Supersymmetry-supersymmetric current in an antenna comprising superpartners of elementary particles

21

21 23 26 29 31 32

Chapter 3: Conducting fluids as quantum antennas 45-70 3.1 A short course in basic non-relativistic and relativistic fluid dynamics with antenna theory applications 45 3.2 Flow of a 2-D conducting fluid 54 3.3 Finite element method for solving the fluid dynamical equations 55 3.4 Elimination of pressure, incompressible fluid dynamics in terms of just a single stream function vector field with vanishing divergence 56 3.5 Fluids driven by random external force fields 57 3.6 Relativistic fluids, tensor equations 58 3.7 General relativistic fluids, special solutions 59 3.8 Galactic evolution using perturbed fluid dynamics, dispersive relations. The unperturbed metric is the Roberson-Walker metric corresponding to a homogeneous and isotropic universe 59 3.9 Magnetohydrodynamics-diffusion of the magnetic field and vorticity 60 3.10 Galactic equation using perturbed Newtonian fluids 61 3.11 Plotting the trajectories of fluid particles 61 3.12 Statistical theory of fluid turbulence, equations for the velocity field moments, the Kolmogorov-Obhukov spectrum 62

Quantum Antennas

VII

3.13 Estimating the velocity field of a fluid subject to random forcing using discrete space velocity measurements based on discretization and the Extended Kalman filter 3.14 Quantum fluid dynamics. Quantization of the fluid velocity field by the introduction of an auxiliary Lagrange multiplier field 3.15 Optimal control problems for fluid dynamics 3.16 Hydrodynamic scaling limits for simple exclusion models 3.17 Appendix: The complete fluid dynamical equations in orthogonal curvilinear coordinate systems specializing to cylindrical and spherical polar coordinates

Chapter 4: Quantum robots in motion carrying Dirac current as quantum antennas 4.1 A short course in classical and quantum robotics with antenna theory applications 4.2 A fluid of interacting robots 4.3 Disturbance observer in a robot 4.4 Robot connected to a spring mass with damping system

63

63 65 66

67

71-76 71 73 74 75

Chapter 5: Design of quantum gates using electrons, positrons and photons, quantum information theory and quantum stochastic filtering 77-98 5.1 A short course in quantum gates, quantum computation and quantum information with antenna theory applications 77 5.2 The Baker-Campbell-Hausdor formula. A, B are n x n matrices 90 5.3 Yang-Mills radiation field (an approximation) 91 5.4 Belavkin filter applied to estimating the spin of an electron in an external magnetic field. We assume that the magnetic field is B0(t)  R3 93 Chapter 6: Pattern classification for image fields in motion using Lorentz group representations 99-104 6.1 SL(2,C), SL(2,R) and image processing 99 Chapter 7: Optimization problems in classical and quantum stochastics and information with antenna design applications 7.1 A course in optimization techniques 7.2 Group theoretical techniques in optimization theory

105-120 105 113

Quantum Antennas

VIII

7.3 Feynman’s diagrammatic approach to computation of the scattering amplitudes of electrons, positrons and photons

Chapter 8: Quantum waveguides and cavity resonators 8.1 Quantum waveguides

119

121-124 121

Chapter 9: Classical and quantum filtering and control based on Hudson-Parthasarathy calculus, and filter design methods 125-164 9.1 Belavkin filter and Luc-Bouten control for electron spin estimation and quantum Fourier transformed state estimation when corrupted by quantum noise 125 9.2 General Quantum filtering and control 127 9.3 Some topics in quantum filtering theory 127 9.4 Filter design for physical applications 160 Chapter 10: Gravity interacting with waveguide quantum fields with filtering and control 165-176 10.1 Waveguides placed in the vicinity of a strong gravitational field 165 10.2 Some study projects regarding waveguides and cavity resonators in a gravitational field 167 10.3 A comparison between the EKF and Wavelet based block processing algorithms for estimating transistor parameters in an amplifier drived by the Ornstein-Uhlenbeck process 169 10.4 Computing the Haar measure on a Lie group using left invariant vector fields and left invariant one forms 170 10.5 How background em radiation affects the expansion of the universe 171 10.6 Stochastic BHJ equations in discrete and continuous time for stochastic optimal control based on instantaneous feedback 173 10.7 Quantum stochastic optimal control of the HP-Schrodinger equation 174 10.8 Bath in a superposition of coherent states interacting with a system 176 Chapter 11: Basic triangle geometry required for understanding Riemannian geometry in Einstein’s theory of gravity 177-178 11.1 Problems in mathematics and physics for school students 177 11.2 Geometry on a curved surface, study problems 178

Quantum Antennas

IX

Chapter 12: Design of gates using Abelian and non-Abelian gauge quantum field theories with performance analysis using the Hudson-Parthasarathy quantum stochastic calculus 179-192 12.1 Design of quantum gates using Feynman diagrams 179 12.2 An optimization problem in electromagnetism 181 12.3 Design of quantum gates using non-Abelian gauge theories 184 12.4 Design of quantum gates using the Hudson-Parthasarathy quantum stochastic Schrodinger equation 185 12.5 Gravitational waves in a background curved metric 185 12.6 Topics for a short course on electromagnetic field propagation at high frequencies 187 Chapter 13: Quantum gravity with photon interactions, cavity resonators with inhomogeneities, classical and quantum optimal control of fields 13.1 Quantum control of the HP-Schrodinger equation by state feedback 13.2 Some ppplications of poisson processes 13.3 A problem in optimal control 13.4 Interaction between photons and gravitons 13.5 A version of quantum optimal control 13.6 A neater formulation of the quantum optimal control problem 13.7 Calculating the approximate shift in the oscillation frequency of a cavity resonator having arbitrary cross section when the medium has a small inhomogeneity 13.8 Optimal control for partial dierential equations

191-218 191 193 197 198 203 208

210 215

Chapter 14: Quantization of cavity fields with inhomogeneous media, field dependent media parameters from BoltzmannVlasov equations for a plasma, quantum Boltzmann equation for quantum radiation pattern computation, optimal control of classical fields, applications classical nonlinear filtering 219-250 14.1 Computing the shift in the characteristic frequencies of oscillation in a cavity resonator due to gravitational effects and the effect of non-uniformity in the medium 219 14.2 Quantization of the field in a cavity resonator having non-uniform permittivity and permeability 221

Quantum Antennas

X

14.3 14.4 14.5

14.6 14.7

14.8 14.9 14.10 14.11 14.12 14.13 14.14

14.15

Problems in transmission lines and waveguides Problems in optimization theory Another approach to quantization of wave-modes in a cavity resonator having non-uniform medium based on the scalar wave equation Derivation of the general structure of the field dependent permittivity and permeability of a plasma Other approaches to calculating the permittivity and permeability of a plasma via the use of Boltzmann’s kinetic transport equation Derivation of the permittivity and permeability functions using quantum statistics Approximate discrete time nonlinear filtering for non-Gaussian process and measurement noise Quantum theory of many body systems with application to current computation in a Fermi liquid. Optimal control of gravitational, matter and em fields Calculating the modes in a cylindrical cavity resonator with a partition in the middle Summary of the algorithm for nonlinear filtering in discrete time applied to fan rotation angle estimation Classical filtering theory applied to Levy process and Gaussian measurement noise. Developing the EKF for such problems Quantum Boltzmann equation for calculating the radiation fields produced by a plasma

222 223

224 228

229 231 231 234 237 239 240

243 246

Chapter 15: Classical and quantum drone design 251-254 15.1 Project proposal on drone design for the removal of pests in a farm 251 15.2 Quantum drones based on Dirac’s relativistic wave equation 252 Chapter 16: Current in a quantum antenna 16.1 Hartree-Fock equations for obtaining the approximate current density produced by a system of interacting electrons 16.2 Controlling the current produced by a single quantum charged particle quantum antenna

255-260

255 257

Quantum Antennas

XI

Chapter 17: Photons in a gravitational field with gate design applications and image processing in electromagnetics 261-288 17.1 Some remarks on quantum blackhole physics 261 17.2 EM field pattern produced by a rotated and translated antenna with noise deblurring 263 17.3 Estimation of the 3-D rotation and translation vector of an antenna from electromagnetic field measurements 265 17.4 Mackey’s theory of induced representations applied to estimating the Poincare group element from image pairs 266 17.5 Effect of electromagnetic radiation on the expanding universe 272 17.6 Photons inside a cavity. 274 17.7 Justication of the Hartree-Fock Hamiltonian using second order quantum mechanical perturbation theory 276 17.8 Tetrad formulation of the Einstein-Maxwell field equations 277 17.9 Optimal quantum gate design in the presence of an electromagnetic field propagating in the Kerr metric 280 17.10 Maxwell’s equations in the Kerr metric in the tetrad formalism 286 Chapter 18: Quantum fluid antennas interacting with media 289-303 18.1 Quantum MHD antenna in a quantum gravitational field 289 18.2 Applications of scattering theory to quantum antennas 290 18.3 Wave function of a quantum field with applications to writing down the Schrodinger equation for the expanding universe 292 18.4 Simple exclusion process and antenna theory 294 18.5 MHD and quantum antenna theory 295 18.6 Approximate Hamiltonian formulation of the diffusion equation with applications to quantum antenna theory 297 18.7 Derivation of the damped wave equation for the electromagnetic field in a conducting media in quantum mechanics using the Lindblad formalism 298 18.8 Boson-Fermion unication in quantum stochastic calculus 302 References

304

Preface The quantum antenna consists of just electrons and positrons which satisfy the Dirac second quantized field equations. The current density of this field is obtained as a quadratic function of the Dirac field operators: J μ (x) = −eψ(x)∗ αμ ψ(s), αμ = γ 0 γ μ This current density produces a quantum electromagnetic field described by the retarded potentials  μ J (t − |r − r |, r ) 3  d r Aμ ((t, r) = 4π|r − r | Thus, Aμ (t, r) is a Bosonic field represented as a quadratic functional of Fermionic field operators. Apart from this quantum em field produced by the quantum antenna, there is in space a free photon field described by a linear superposition of creation and annihilation operators of the photon field. Since the second quantized Dirac field ψ(x) can be represented as a quadratic functional of the electron-positron creation and annihilation operator fields, it follows that the total electromagnetic field in space can be represented as a linear functional of photon creation and annihilation operators plus a quadratic functional of electron-positron creation and annihilation operators. This means that by using standard commutation rules for the photon creation and annihilation operators and anticommutation rules for electron-positron creation and annihilation operators, we can calculate the mean and mean square fluctuations of the quantum electromagnetic field produced by the antenna along with the free photon electromagnetic field in any given state, say for example, in a state in which there is a specific number of electrons and positrons with prescribed four momenta and spins and photons with specific four momenta and helicities. Or we may assume that the state of the photon component is a coherent state comprising af an infinite number of photons.

XIII

Chapter 1

Basic quantum electrodynamics required for the analysis of quantum antennas 1.1

Introduction

In this lecture, we discuss various quantum mechanical models for the current field produced in an antenna by the motion of electrons, positrons as well as non-Abelian gauge particles interacting with the gravitational field described the general relativistic metric tensor. Quantization of the electromagnetic field, second quantization of the Dirac field, computation of the current density of the Dirac field and its effect on quantum fluctuations in the radiated electromagnetic field is discussed. Other related problems like image processing algorithms on the antenna pattern are suggested. The Feynman path integral approach to computing the photon propagator in a gravitational field is proposed. This would enable us to study the effect of the gravitational field on quantum electromagnetic field fluctuations.

1

2

1.2

Quantum Antennas

The problems to be discussed

1.2.1

Quantization of electromagnetic field and Dirac field in the presence of a background gravitational field

1.2.2

Lagrangian and Hamiltonian densities for photons interacting with electrons and positrons

1.2.3

Quantization in the Coulomb and Lorentz gauges using operator theory and using the Feynman path integral

1.2.4

Electrons, positrons and photons–scattering matrix using the interaction picture

1.2.5

photon creation and annihilation operator fields and electron-positron creation and annihilation operator fields in the momentum-spin/helicity domain

1.2.6

canonical commutation relations for the photon field and canonical anticommutation relations for the electronpositron field

1.2.7

Current in the Dirac field–the second quantization picture

1.2.8

photons in the second quantized electromagnetic field

1.2.9

Interaction Lagrangian and Hamiltonians between the photon field and electron-positron field

1.2.10

Second quantization using the Boson Fock space and Fermion Fock space

1.2.11

Coherent states of the photon field

1.2.12

Hudson-Parthasarathy quantum stochastic calculus for dynamically modeling photon bath noise

1.2.13

Dirac second quantized Hamiltonian in terms of creation and annihilation operator fields of electrons and positrons

1.2.14

Quantum electromagnetic four potential produced by the Dirac second quantized current field

Statistical moments of the electromagnetic potentials and fields produced by free photons and the radiation fields produced by electron and positron current

Quantum Antennas

3

density.

1.2.15

The Feynman diagrammatic rules for calculating the S-matrix of interacting electrons, positrons and photons

1.2.16

Derivation of the Feynman rules using operator fields and using the Feynman path integral for fields

1.2.17

Dirac bracket for systems with constraints–justification

1.2.18

Dirac bracket for the canonical position and momentum fields for the electromagnetic field

1.2.19

Dirac’s equation perturbed by a quantum stochastic process

Approximate solution using time dependent perturbation theory, computation of the Dirac current after perturbation. Computation of the Dirac current moments in specific states of the electron-positron-photon fields.

1.2.20

An atom in an electromagnetic field–solution using the Glauber-Sudarshan diagonal coherent representation

1.2.21

The computation and significance of the propagators of electrons and photons

1.2.22

Non-Abelian gauge theories: The Yang-Mills theory and its second quantization based on the Feynman path integral

1.2.23

Current in non-Abelian gauge theories

1.2.24

Approximation solution for the matter and gauge fields for non-Abelian gauge theories using perturbation theory for partial differential equations

1.2.25

Maxwell’s equations in a background gravitational field–interaction between photons and gravitons

The effect of photon noise on the gravitational field—Einstein field equations for gravitation perturbed by the quantum stochastic photon noise using the Hudson-Parthasarathy quantum stochastic calculus.

4

Quantum Antennas

1.2.26

Dirac’s equation in a background gravitational field– the gravitational spinor connection

1.2.27

Second quantization of the Dirac field in a background gravitational field

1.2.28

The electroweak theory and strong interactions as examples of Yang-Mills theories

The gauge groups U (1) × SU (2) and U (1) × SU (2) × SU (3).

1.2.29

The complete picture of interaction between electrons, positrons, photons and gravitons

Influence of gravitational waves on the photon field and on the electron-positron field.

1.3

EM field Lagrangian density LEM = KFμν F μν , Fμν = Aν,μ − Aμ,ν , K = −1/16π

1.4

Electric and magnetic fields in special relativity F0r = Er , F12 = −B3 , F23 = −B1 , F31 = −B2 LEM = (1/8π)(E 2 − B 2 )

(CGS units with c = 1 are used so that 0 = 1/μ0 = 1/4π). We prefer to use another system of units which gives 0 = 1/μ0 = 1, c = 1). Then, LEM = (1/2)(E 2 − B 2 ) = (−1/4)Fμν F μν = (−1/4)(2F0r F 0r + Frs F rs )

1.5

Canonical position and momentum fields in electrodynamics Πr =

∂LEM = −F0r ∂Ar,0

5

Quantum Antennas Canonical position fields are Ar , r = 1, 2, 3 and corresponding canonical momentu fields are Πr , r = 1, 2, 3. In the Coulomb gauge, Ar,r = divA = 0 and ∇2 A0 = −J 0 . So A0 becomes a matter field. CCR’s are [Ar (t, r), Πs (t, r )] = iδsr δ 3 (r − r ) Incorrect because of the constraints Ar,r = 0. Total Lagrangian density for em field interacting with charged matter is LEM M = LEM − J μ Aμ = (−1/4)Fμν F μν − J μ Aμ Another constraint is obtained from the equations of motion ∂r

1.6

∂LEM M ∂LEM M = 0 ∂A,r ∂A0

The matter fields in electrodynamics

Note that J μ , A0 are matter fields while Ar , Πr , r = 1, 2, 3 are em field fields. This equation of motion gives F0r,r − J 0 = 0 or equivalently, ie

divΠ + J 0 Πr,r + J 0 = 0

This is a constraint equation since it does not involve time derivatives.

1.7

The Dirac bracket in electrodynamics χ1 = Ar,r , χ2 = Πr,r + J 0

constraints are χ1 = 0, χ2 = 0. To compute the Dirac bracket between Ar and Πs , we use [χ1 (t, r), χ2 (t, r )] = [Ar,r (t, r), Πs,s (t, r )] = −iδsr ∂r ∂s δ 3 (r − r ) = −i∇2 δ 3 (r − r ) Its Fourier transform is ik 2 . Inverse of the matrix   0 ik 2 −ik 2 0

6

Quantum Antennas is



0 −i/k 2

i/k 2 0



Also, [Ar (t, r), χ1 (t, r )] = 0, [Ar (t, r), χ2 (t, r )] = −iδsr ∂s δ 3 (r − r ) = −i∂r δ 3 (r − r ) [χ1 (t, r), Πs (t, r )] = [Ar,r (t, r), Πs (t, r )] = iδsr ∂r δ 3 (r − r ) = i∂s δ 3 (r − r ) [χ2 (t, r), Πs (t, r )] = [Πs,s (t, r), Πs (t, r )] = 0, The Fourier transform of the Dirac bracket of Ar (t, r) and Πs (t, r ) w.r.t r − r is therefore [Ar (t, .), Πs (t, .)](k) =    −k s 0 i/k 2 r r iδs − [0, k ] −i/k 2 0 0 = iδsr − ik r k s /k 2 Taking the inverse Fourier transform therefore gives the Dirac bracket as [Ar (t, r), Πs (t, r )] = iδsr δ 3 (r − r ) − i

1.8

∂2 (1/4π|r − r |) ∂xr ∂xs

Hamiltonian of the em field H = Πr Ar,0 − LEM M = −F0r Ar,0 + (1/4)Fμν F μν + J μ Aμ = −F0r (Ar,0 + A0,r ) + (1/4)(2F0r F 0r + Frs F rs ) + J μ Aμ +F0r A0,r

Now,

F0r A0,r = −Πr A0,r = −(Πr A0 ),r + Πr,r A0

which after neglect of a total divergence (whose spatial integral will therefore vanish) we get on using the constraints Πr,r = −J 0 that H = (1/2)(Π2 + (∇ × A)2 ) − (J r Ar ) The Hamiltonian density of the matter field alone is J 0 A0 and hence the total Hamiltonian of the field interacting with charged matter is HEM M = (1/2)(Π2 + (∇ × A)2 ) + J μ Aμ

7

Quantum Antennas In the general Lorentz gauge, the term J μ Aμ is taken as our field-charged matter interaction Hamiltonian. We write Π⊥ = Π − ∇A0 Then,

= divΠ⊥ = divΠ − ∇2 A0 = −J 0 + J 0 = 0

and further, 

Π2 d3 r = 

=



(Π2⊥ + (∇A0 )2 + 2(Π⊥ , ∇A0 ))d3 r

(Π2⊥ − A0 ∇2 A0 )d3 r =



(Π2⊥ + J 0 A0 )d3 r

where we have used the fact that   (Π⊥ , ∇A0 )d3 r = − A0 divΠ⊥ d3 r = 0 Thus, an alternative more useful expression for the Hamiltonian density after neglect of perfect spatial divergences (which do not contribute to the Hamiltonian) is given by HEM M = (1/2)(Π2⊥ + (∇ × A)2 ) + J 0 A0 − J r Ar

1.9

Interaction Hamiltonian between the current field and the electromagnetic field

 and this justifies taking HI (t) = ( J μ Aμ d3 r as our Hamiltonian of interaction of the photon field with the electron-positron field. Note that the field commutation relations are not affected if Π is replaced by Π⊥ since the difference between Π and Π⊥ is a matter field in the Coulomb gauge and matter fields commute with the position and momentum fields of the em field. Exercises: The free photon field Arf (x) satisfies the wave equation Arf = 0 whose solution can be expressed as a superposition of plane waves:  Arf (t, r) =

((2|K|)−1 er (K, σ)c(K, σ)exp(−ik.x)+(2|K|)−1 e¯r (K, σ)c(K, σ)∗ exp(ik.x))d3 K

where k = (|K|, K), k.x = kμ xμ = |K|t − K.r

8

Quantum Antennas

1.10

The Boson commutation relations for the creation and annihilation operator fields for the EM field in momentum-spin domain

A simple calculation shows that the em field Hamiltonian is  HEM = (1/2) ((A,t )2 + (∇ × A)2 )d3 r  =

|K|c(K, σ)∗ c(K, σ)d3 K

This looks like the energy of a continuous system of quantum Harmonic oscillators, two of them associated with each spatial frequency K. Further justification follows from the assumption that if the operators c(K, σ), c(K, σ)∗ are assumed to satisfy the CCR [c(K, σ), c(K  , σ  )∗ ] = δ 3 (K − K  )δσ,σ , [c(K, σ), c(K  , σ  )] = 0, [c(K, σ)∗ , c(K  , σ  )∗ ] = 0 then their dynamical behaviour described by Heisenberg’s matrix mechanics is dct (K, σ)/dt = i[HEM , c(K, σ)] = −i|K|ct (K, σ) gives that ct (K, σ) = c(K, σ)exp(−i|K|t), ct (K, σ)∗ = c(K, σ)∗ exp(i|K|t) and hence we get the correct time dependence of Arf (t, r). It can also be verified that these CCR’s give the correct CCR’s for Arf (t, r) and Πs (t, r). That we leave as an exercise.

1.11

Electrodynamics in the Coulomb gauge

Remark: We have



Π2⊥ d3 r

 =

 =

2 3

(F0r + A0,r ) d r =

(Π − ∇A0 )2 d3 r 

2

(Ar,0 ) =



(A,0 )2 d3 r

which is why we can express the em field Hamiltonian  (1/2) (Π2⊥ + (∇ × A)2 )d3 r 

as (1/2)

((A,0 )2 + (∇ × A)2 )d3 r

9

Quantum Antennas

in the Coulomb gauge. Note that the Coulomb gauge condition Arf,r = 0 implies that the field polarization vector er (K, σ) satisfies K r er = 0 and hence there are just two independent directions of photon polarization for each wave vector K. These are specified by σ = 1, 2 and may be chosen so that (er (K, 1)), (er (K, 2)), (K r ) are mutually orthogonal 3-vectors.

1.12

The Dirac second quantized field

The free Dirac equation for the electron is (iγ μ ∂μ − m)ψ(x) = 0 where γ μ are the Dirac matrices satisfying the anticommutation relations {γ μ , γ ν } = 2η μν and ψ(x) ∈ C4 . This equation on premultiplying by (iγ ν ∂ν + m) gives the Klein-Gordon equation of special relativity: ( + m2 )ψ(x) = 0 in agreement with the Einstein energy-momentum relation (E 2 − P 2 − m2 )ψ = 0, E = i∂t , P = −i∇ The solution for free Dirac waves is  ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P where p = (E, P ), E = E(P ) =

 P 2 + m2

and u(P, σ), v(P, σ) satisfy in view of the Dirac equation (γ μ pμ − m)u(P, σ) = 0, (γ μ pμ + m)v(P, σ) = 0 The second implies (γ μ pμ + m)v(−P, σ) = 0 Thus u(P, σ), σ = 1, 2 are orthogonal eigenvectors of the matrix γ μ pμ with eigenvalues m and v(−P, σ), σ = 1, 2 are orthogonal eigenvectors of the matrix γ μ pμ with eigenvalues −m. Thus, a(P, σ) annihilates an electron of momentum P energy E(P ), spin σ mass m while b(−P, σ)∗ creates a positron of momentum P energy E(P ) and spin σ and mass m.

10

Quantum Antennas The Dirac equation in an electromagnetic field is given by the standard quantum mechanical rule of replacing pμ by pμ + eAμ and is thus given by [γ μ (i∂μ + eAμ (x)) − m]ψ(x) = 0 It is easily verified from this equation that the current J μ = −eψ(x)∗ αμ ψ(x), αμ = γ 0 γ μ is conserved ie μ J,μ =0

The Dirac equation can be derived from the Lagrangian density LD = ψ(x)∗ γ 0 (γ μ (i∂μ + eAμ ) − m)ψ(x) Exercise: Verify that the space-time integral of LD is real. Exercise: Verify by applying the Legendre transformation that the Hamiltonian of the Dirac equation interacting with the electromagnetic field is HDEM = (α, P + eA) + βm − eA0 where (α, P + eA) = αr (P r + eAr ) = γ 0 γ r (P r + eAr ), β = γ 0 Exercise: Verify that if we define the Pauli spin matrices by   0 1 σ 0 = I2 , σ 1 = , 1 0 2



σ =

0 i

−i 0





3

,σ =

1 0

0 −1



and σμ = ημν σ ν so that σ0 = σ 0 , σr = −σ r , r = 1, 2, 3 and finally the Dirac matrices by  γμ =

0 σμ

σμ 0



then the desired anticommutation relations are satisfied by the γ μ and moreover α0 = I and αr are Hermitian. This guarantees that the Dirac Hamiltonian is a Hermitian operator as it should be.

11

Quantum Antennas

1.13

The Dirac equation in an EM field, approximate solution using Perturbation theory

The Dirac current perturbation in the presence of a weak em field. We assume Aμ to be of the first order of smallness. Then Write ψ(x) = ψ0 (x) + δψ(x) Then the zeroth order part ψ0 satisfies the free Dirac equation and is therefore expressible as a linear functional of a(P, σ) and b(P, σ)∗ . The first order part δψ[(x) satisfies iδψ,t (x) = [(α, −i∇) + βm]δψ(x) + e[(α, A) − A0 ]ψ0 (x) This can be solved in the four momentum domain as ˆ δψ(p) = (ip0 − (α, P ) − βm)−1 F[(e(α, A) − A0 )ψ0 ](p)

1.14

Electromagnetically perturbed Dirac current

The above formula can be used to evaluate the first order shift in the Dirac current caused by the interaction of the quantum antenna with the photon field: δJ μ = −2eRe(δψ(x)∗ αμ ψ0 (x)) where by Re, we mean ”Hermitian part”.

Chapter 2

Effects of the gravitational field on a quantum antenna and some basic non-Abelian gauge theory 2.1

The effect of a gravitational field on photon paths

On a single photon, we know from the GTR that its path bends in a gravitational field. Its paths are defined by null geodesics: gμν (x)(dxμ /dλ)(dxν /dλ) = 0, after solving the geodesic equations d2 xμ /dτ 2 + Γμαβ (dxα /dτ )(dxβ /dτ ) = 0 and then taking the limit dτ → 0. Suppose the metric does not depend on t. Then, one of the Euler-Lagrange equations is d/dλ(g0ν (r)dxν /dτ ) = 0 or equivalently, g00 (r)dt/dτ + g0m (r)dxm /dτ = K where K is a constant → ∞ as dτ → 0. This equation can be used to eliminate d/dτ from the other equations. For example, if there is a spatial coordinate say xs on which the metric does not depend. Then, we get another first integral gs0 dt/dτ + gsm dxm /dτ = K 

13

14

Quantum Antennas where K  is another infinite constant. Hence, taking the ratio gives g00 + g0m dxm /dt = K/K  = β gs0 + gsm dxm /dt where now β is a finite constant, although it is the ratio of two infinite constants. This fact can be used to determine the path of a light ray in a gravitational field.

2.2

Interaction of gravitation with the photon field

The interaction between the gravitational field and the electromagnetic field is contained in the Maxwell Lagrangian: √ LEM = (−1/4)Fμν F μν = (−1/4)g μα g νβ Fμν Fαβ −g = LEM 0 + LEM G For weak gravitational fields (as compared to the photon strength) we write gμν = ημν + hμν (x) and then g μν = ημν − hμν where hμν = ημα ηνβ hαβ Specifically,

h00 = h00 , h0r = hr0 = −h0r = hr0 , hrs = hrs

Then, the interaction Lagrangian is LEM G = Fμν (x)Fαβ (x)Cμναβ (x) √ where Cμναβ (x) is the component in (−1/4)g μν g αβ −g that is linear in hμν (x). This is computed as √ (−1/4)g μν g αβ −g ≈ (−1/4)(ημν − hμν )(ηαβ − hαβ )(1 − h/2) ≈ (−1/4)ημν ηαβ − (1/4)(ημν hαβ + ηαβ hμν + ημν ηαβ h/2) giving Cμναβ (x) = (−1/4)(ημν hαβ + ηαβ hμν + ημν ηαβ h/2) where h = ημν hμν = hμμ

15

Quantum Antennas We can more generally calculate LEM G upto quadratic orders in hμν writing the result as Cμναβ (x) = C1 (μναβρσ)hρσ (x) + C2 (μναβρσδζ)hρσ (x)hδζ (x) where now C1 (μναβρσ) and C2 (μναβρσδζ) are numerical constants. The effect of this interaction LEM G on the classical wave propagation equation of photons is easily computed using the Euler Lagrange equations: ∂μ

∂LEM 0 ∂LEM G = −∂μ ∂Aν,μ ∂Aν,μ

or equivalently using the standard General relativistic Maxwell equations √ (F μν −g),ν = 0 which approximates to (ημα − hμα )(ηνβ − hνβ )(1 − h/2)Fαβ ),ν = 0 or equivalently, upto quadratic orders in the gravitational field (ημα ηνβ Fαβ ),ν = = [(D1 (μνρσαβ)hρσ + D2 (μνρσδζαβ)hρσ hδζ )Fαβ ],ν This equation can be solved using first and second order perturbation theory with the GTR Lorentz gauge condition √ (Aμ −g),μ = 0 which approximates upto second order terms in the gravitational potentials to ημν Aν,μ = [(E1 (μαβ)hαβ + E2 (μαβρσ)hαβ hρσ )Aν ],μ The method of perturbatively solving this is to expand (1) (2) 3 Aμ = A(0) μ + Aμ + Aμ + O(|h| ) (0)

(1)

where Aμ satisfies the usual special relativistic wave equation, Aμ is the cor(2) rection to the em potentials that is a linear functional of hμν and Aμ is the correction that is a quadratic functional of hμν .

2.3

Quantum description of the effect of the gravitational field on the photon propagator

The corrected photon propagator due to a background gravitational field is given by < 0|T {Aμ (x)Aν (x )}|0 >=

16

Quantum Antennas   ≈

exp(i(SEM 0 + SEM G ))Aμ (x)Aν (x )Πμ,z dAμ (z)

2  exp(iSEM 0 )(Aμ (x)Aν (x ) + iSEM G Aμ (x)Aν (x ) − SEM G Aμ (x)Aν (x )/2)

×Πμ,z δ((Aν (z)

 −g(z)),ν )dAμ (z)

where the δ-function accounts for the Lorentz gauge. This path integral can be evaluated using the standard theory of Gaussian integrals. The δ function part can be replaced by     Πμ,z δ((Aν (z) −g(z)),ν ) = C exp(i η,μ (z)(Aμ (z) −g(z))d4 z)Πz dη(z)

2.4

Electrons and positrons in a mixture of the gravitational field and an EM field Quantum antennas in a background gravitational field

The interaction of electrons, positrons and photons in a background gravitational field gμν (x) is described by the Lagrangian density LEM + LEM G + LD + LDG + LDEM = √ (−1/4)Fμν F μν −g + Re(ψ ∗ (x)γ 0 (γ a Vaμ (x)(i∂μ + eAμ (x) + iΓμ (x)) − m)ψ(x)) where b Γμ = (−1/2)V aν (x)Vν:μ (x)J ab

where Vμa is a tetrad for the gravitational field, ie, a locally inertial frame: gμν (x) = ηab Vμa (x)Vνb (x) and J ab = (1/4)[γ a , γ b ] is the Dirac spinor Lie algebra representation of the Lorentz transformation basis elements (σ ab )μν = δμa δνb − δνa δμb Writing the Dirac spinor representation of the Lorentz group as Λ → D(Λ), we have J ab = dD(σ ab ) and like σ ab , they satisfy the standard commutation relations of the Lorentz algebra: [J ab , J cd ] = η ac J bd + η bd J ac − η ad J bc − η bc J ad

17

Quantum Antennas The gravitational connection satisfies the gauge covariance property: D(Λ(x))Vaμ (x)(∂μ + Γμ (x)))D(Λ(x))−1 = Λ(x)ba Vbμ (x)(∂μ + Γμ (x))

for any local Lorentz transformation λ(x). Here, Γμ (x) is the transformed gravitational connection: 

 Γμ (x) = (−1/2)J ab Vaν (x)Vbν:μ (x)

where



Vaμ (x) = Λab (x)Vbμ (x) This is verified by taking the local Lorentz transformation Λ(x) to be infinitesimal: Λab (x) = δba + ωba (x) where ωab (x) + ωba (x) = 0, ωab (x) = ηac ωbc (x) The resulting Lagrangian above yields the Dirac equation for electrons and positrons in a gravitational field and electromagnetic field as (γ a Vaμ (x)(i∂μ + eAμ (x) + iΓμ (x)) − m)ψ(x) = 0 This equation is both diffeomorphic and Lorentz covariant. The Dirac current due to the gravitational correction can be evaluated using this equation and hence the effect of the gravitational field on the electromagnetic field radiated by the quantum antenna comprising electrons and positrons can be determined.

2.5

Dirac equation in a gravitational field and a quantum white noise photon field described in the Hudson-Parthasarathy formalism

The Hudson-Parthasarathy theory: The Dirac equation in a gravitational field and interacting with an electromagnetic field that is modeled as quantum noise as well as the Maxwell field comprising a classical component and a quantum noise component can jointly be expressed by the following quantum stochastic differential equation: γ a Va0 (t, r)idψt (r)) + γ a Vam (t, r)i∂m ψt (r)dt +γ a Vaμ (t, r)Lβμα (t, r)ψt (r)dΛβα (t) +γ a Vaμ (t, r)iΓμ (t, r)ψt (r)dt − mψt (r)dt = 0

18

Quantum Antennas where Lβμα (t, r) are system operators, ie built out of multiplication operators and spatial differential operators. It is more natural to consider them as only multiplication operators since then e−1 Lβμα (t, r)dΛβα (t)/dt can formally be interpreted as the four vector potential Aμ (t, r) corresponding to the noisy photon bath. Here Λα β , α, β ≥ 0 are the standard Hudson-Parthasarathy noise operators satisfying the quantum Ito formula μ μ α dΛα β (t).dΛν (t) = ν dΛβ (t) α with α β being zero if either α = 0 or β = 0 and δβ otherwise. The operators β Lμα (t, r) are forced to satsify certain constraints dictated by the fact that this equation describes a unitary evolution. These constraints can easily be deduced using the quantum Ito formula.

2.6

Dirac-Yang-Mills current density for nonAbelian gauge theories

When instead of electrons and positrons, we consider antennas having other kinds of elementary particles described by non-Abelian gauge theories of the Yang-Mills type, for example the weak and strong forces, then the dynamics is described by a Lagrangian density LY M = T r(Fμν F μν ) + LM (ψ, ∇μ ψ) where the connection is given by ∇μ = ∂μ + Aμ (x) with Aμ (x) ∈ g g being the Lie algebra of the gauge group G ⊂ U (N ) and Fμν (x) ∈ g, the field tensor being defined as the curvature of the connection: Fμν = [∇μ , ∇ν ] = Aν,μ − Aμ,ν + [Aμ , Aν ] The matter field equations are [iγ μ ∇μ − m]ψ(x) = 0 which is to be interpreted as [iγ μ ∂μ ⊗ IN + iγ μ ⊗ Aμ (x) − m]ψ(x) = 0 The Dirac current has to be generalized to this non-Abelian setting and the Yang-Mills field equations have to be solved using perturbation methods since unlike the Maxwell equations, the contain nonlinear terms too. It should be

19

Quantum Antennas noted that under a local gauge transformation g(x) ∈ G, the matter field transforms as ψ(x) → g(x)ψ(x) while, the gauge field Aμ (x) has to transform to Aμ (x) so that the field equations remain invariant, ie, g(x)∇μ g(x)−1 = ∇μ This gives

Aμ (x) = g(x)Aμ (x)g(x)−1 + g(x)∂μ (g(x)−1 )

and generalizes the Lorentz gauge transformation of the electromagnetic field.

2.7

Dirac brackets

. Suppose the Lagrangian L = L(q, Q, p, P ) where (q, Q) are the position variables and (p, P ) are the momentum variables with q = (q1 , ..., qn ), p = (p1 , ..., pn ) and Q = (Q1 , ..., Qr ), P = (P1 , ..., Pr ). The constraints are Q = Q(q, p), P = P (q, p) ie, Q, P are functions of q, p. In view of these constraints, we cannot assume that {Q, u} = ∂u/∂P and {u, P } = ∂u/∂Q. So we must modify our definition of the Poisson bracket to take these constraints into account. If we write Q = Q(q, p) and P = P (q, p), and compute our Poisson brackets based on (q, p) only we find that (u, v) = T (uT,q + uT,Q Q,q + uT,P P,q )(v,p + QT,p v,Q + P,p v,P )

−(uT,p + uT,Q Q,p + uT,P P,p )(v,q + QT,q v,Q + P,qT v,P ) = uT,q v,p − uT,p v,q + T v,P +uT,q QT,p v,Q + uT,q P,p

−uT,p QT,q v,Q − uT,p P,qT v,P T T T T +v,p Q,q u,Q + v,p P,q u,P T T T T −v,q Q,p u,Q − v,q P,p u,P

+uT,Q (Q,q QT,p − Q,p QT,q )v,Q T +uT,Q (Q,q P,p − Q,p P,qT )v,P

+uT,P (P,q QT,p − P,p QT,q )v,Q T +uT,P (P,q P,p − P,p P,qT )v,P

Now, define the matrix  D = D(q, p) =

(Q, QT ) (P, QT )

(Q, P T ) (P, P T )



20

Quantum Antennas Then, we can write  (u, v) = (u, v)(q,p) + [uT,Q , uT,P ].D.

v,Q v,P



+(u, QT )v,Q + (u, P T )v,P − (v, QT )u,Q − (v, P T )u,P =   v,Q (u, v)(q,p) + [uT,Q , uT,P ].D. v,P   (Q, v) +[uT,Q , uT,P ] (P, v)   v,Q T T +[(u, Q ), (u, P )] v,P In this formula, we are assuming that the observable u, v depend explicitly on (q, p, Q, P ). If they depend only on q, p as is natural to expect since Q, P are both functions of (q, p) by virtue of the constraints, then it is natural to define   u,Q = u,P   (u, Q) D−1 (u, P ) for this is the same as saying that (u, Q) = (Q, QT )u,Q + (Q, P T )u,P and likewise for v. Note that this equation is the same as (u, Q)T = (u, QT ) = uT,Q (Q, QT ) + uT,P (P, QT ) With this definition, we get from the above, (u, v)(q,p) = (u, v) − [(u, Q) , (u, P ) ]D T

T

−1



(v, Q) (v, P )



This is called the Dirac bracket and works very well in problems involving constraints. Other related problems to be discussed: The spectral action principle in the unification of gravity with the electroweak and strong forces, ie, the fields of Leptons, quarks, photons gauge bosons and gluons starting from the heat equation. Image processing for antenna patterns: The aim is to estimate the antenna parameters like the shape of the antenna and the distribution of current on its surface from measurements of the far field electromagnetic radiation pattern. Scattering theory in quantum mechanics and its relation to antenna theory. Both of them involve solving Helmholtz equation with source.

Quantum Antennas

2.8

21

Harish-Chandra’s discrete series representations of SL(2, R) and its application to pattern recognition under Lorentz transformations in the plane

. Conclusions: In this lecture, we have proposed the analysis of quantum fluctuations in the electromagnetic field pattern produced by an antenna comprising electrons and positrons described by the second quantized Dirac field. We have also proposed the study of the Dirac field of electrons and positrons interacting with a gravitational field and the Maxwell electromagnetic field using the spinor connection of the gravitational field.

2.9

Estimating the shape of the antenna surface from the scattered EM field when an incident EM field induces a surface current density on the antenna that is determined by Pocklington’s integral equation obtained by setting the tangential component of the total incident plus scattered electric field on the surface to zero

Assume that the antenna surface is parametrized by (u, v) → R(u, v). Assume that an incident electromagnetic field Ei (ω, r), Hi (ω, r) falls on this surface. The induced surface current density is Js (u, v). The electric field (scattered) produced by this surface current density is given by  Es (ω, r) = Gm (ω, r, u, v)Jsm (u, v)dS(u, v) where Gm is a 3 × 1 complex vector valued function and the summation is over m = 1, 2. Here Jsm , m = 1, 2 are the components of the surface current density relative to the surface tangent basis R,u , R,v . This means that Js (u, v) = Js1 (u, v)R,u (u, v) + Js2 (u, v)R,v (u, v) The Kernels Gm , m = 1, 2 are determined as follows: First the scattered magnetic vector potential is  As (ω, r) = (μ/4π) Js (u, v)exp(−jk|r − R(u, v)|)dS(u, v)/|r − R(u, v)|

22

Quantum Antennas The scattered electric field is then Es (ω, r) = ∇ × (∇ × As )/jω = (∇(∇.As ) + k 2 As )/jω of if the medium is inhomogeneous and anisotropic, then (ω, r) is a 3×3 matrix and Es (ω, r) = (jω(ω, r))−1 ∇ × (∇ × As (ω, r)) Thus, we get G1 (ω, r, u, v) = (jω)−1 (μ/4π)∇×(∇(exp(−jk|r−R(u, v)|)/|r−R(u, v)|)×R,u (u, v)),

= (jω)−1 (μ/4π)[(R,u (u, v), ∇)+k 2 ](exp(−jk|r−R(u, v)|)/|r−R(u, v)|)−−−(1) and G2 (ω, r, u, v) = (jω)−1 (μ/4π)∇×(∇(exp(−jk|r −R(u, v)|)/|r−R(u, v)|)×R,v (u, v))−−−(2) The components Jsm (u, v), m = 1, 2 of the surface current density are now determined by setting the tangential components of the electric field on the antenna surface equal to zero, ie, (Es (ω, R(u, v)) + Ei (ω, R(u, v)), R,u (u, v)) = 0, (Es (ω, R(u, v)) + Ei (ω, R(u, v)), R,v (u, v)) = 0, These equations can be expressed as integral equations:   (Gm (ω, r, u , v  ), R,u (u, v))Jsm (u , v  )dS(u , v  ) m=1,2

S

  m=1,2

= (Ei (ω, R(u, v)), R,u (u, v)) − − − (3), (Gm (ω, r, u , v  ), R,v (u, v))Jsm (u , v  )dS(u , v  ) S

= (Ei (ω, R(u, v)), R,v (u, v)) − − − (4), From these equations, we can determine the surface induced current density Js (u, v) on the antenna surface and hence the scattered electromagnetic field Es (ω, r), Hs (ω, r). Now, if the surface shape is unknown, we assume it to be given in parametric form as R(u, v) =

p 

θk ψk (u, v) = R(u, v, θ)

k=1

where ψk are test/basis functions as in the method of moments and the θk s are unknown parameters to be estimated. We next observe that the Green’s functions Gm , m = 1, 2 defined by (1) and (2) can be expressed explicitly as functions of ω, r, R(u, v), R,u (u, v) and R,v (u, v). We can thus express this relation as Gm (ω, r, u, v) = Fm (ω, r, R(u, v), R,u (u, v), R,v (u, v))

23

Quantum Antennas = Fm (ω, r, u, v, θ), θ = (θ1 , ..., θp )T Thus, the integral equations (3) and (4), when solved, yield the surface current densities Jsm (u, v) as functions of (u, v, θ) and thus the scattered electric field  Es (ω, r) = Fm (ω, r, u, v, θ)Jsm (u, v, θ)|R,u (u, v, θ) × R,v (u, v, θ)|dudv is known as a function of θ. θ may then be estimated by matching this scattered electric field to given measurements and hence the shape of the object can be obtained.

2.10

Surface current density operator induced on the surface of a quantum antenna when a quantum EM field is incident on it

Problem: If the incident field is a quantum electromagnetic field built out of a superposition of creation and annihilation operators, then the surface current density Js (u, v) will also be a superposition of creation and annihilation operators. This can be seen from the equations (3) and (4) which tell us that Js (u, v) is a linear functional of Ei which is in turn a linear functional of the creation and annihilation operators. We can thus write Js (ω, u, v) =

N 

(ak χk (ω, u, v) + a∗k χ ¯k (ω, u, v))

k=1

where

[ak , a∗j ] = δkj , [ak , am ] = 0, [a∗k , a∗m ] = 0

Now suppose that the incident field is in a coherent state  ∗np 1 a∗n |φ(u) >= exp(−|u|2 /2) 1 ...ap |0 > /n1 !...np ! n1 ,...,np ≥0

where u ∈ Cp . Then the average value of the surface current density in this state is given by < φ(u)|Js (ω, u, v)|φ(u) >=  (χk (ω, u, v) < φ(u)|ak |φ(u) > +χ ¯k (ω, u, v) < φ(u)|a∗k |φ(u) >) k

=

 (χk (ω, u, v)uk + χ ¯k (ω, u, v)¯ uk ) k

Problem: Compute the higher moments of the surface current density in this coherent state: < φ(u)|Js (ω1 , u1 , v1 )⊗...⊗Js (ωm , um , vm )⊗J¯s (ω1 , u1 , v1 )⊗...⊗J¯s (ωn , un , vn )|φ(u) >

24

Quantum Antennas In order to do this after applying the appropriate commutation relations, you will need to compute things such as 1 p n1 < φ(u)|a∗m ...a∗m a1 ...anp p |φ(u) >= p 1 1 p un1 1 ...unp p u ¯m um p 1 ...¯

Problem: Compute using the formula for the moments of the surface current density in a coherent state of the incident field, the moments of the scattered em potential and field in the far field zone:  r.R(u, v))dS(u, v) Es (ω, r) = (μexp(−jkr)/4πr) Js (ω, u, v)exp(jkˆ Now consider the Dirac second quantized current density field J μ (x) = −eψ(x)∗ αμ ψ(x) We write



ψ(x) =

(u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P

Express J μ (ω, r), ie the current density in the frequency domain using the product theorem for Fourier transforms:  μ ˆ  , r)∗ αμ ψ(ω + ω  , r)dω  J (ω, r) = (−e/2π) ψ(ω R

Now express this Fourier transformed current density in terms of the creation and annihilation operators of the electrons and positrons:  ˆ r) = ψ(ω,

[u(P, σ)a(P, σ)δ(ω+E(P ))exp(iP.r)+v(P, σ)b(P, σ)∗ δ(ω−E(P ))exp(−iP.r)]d3 P

The final expression for J μ (ω, r) will be of the form  J μ (ω, r) = [K1μ (ω, P, P  , σ, σ  , r)a(P, σ)∗ a(P  , σ  )+ K2μ (ω, P, P  , σ, σ  , r)a(P  , σ  )∗ b(P, σ) +K3μ (ω, P, P  , σ, σ  , r)b(P, σ)a(P  , σ  )∗ +K4μ (ω, P, P  , σ, σ  , r)b(P, σ)b(P  , σ  )∗ ]d3 P d3 P  As mentioned in the introduction, apart from the electromagnetic field generated by this current field, there is a free photon electromagnetic field built out of a linear superposition of photon creation and annihilation operators c(K, s). So the total electromagnetic four potential will be of the form  μ A (ω, r) = (μ/4π) J μ (ω, r )exp(−jω|r − r |/c)d3 r /|r − r |+

25

Quantum Antennas 

(c(P, s)F(μ ω, P, s, r) + c(P, s)∗ F¯ μ (ω, P, s, r))d3 P  =

[K1μ (ω, P, P  , σ, σ  , r )a(P, σ)∗ a(P  , σ  )+

K2μ (ω, P, P  , σ, σ  , r )a(P  , σ  )∗ b(P, σ) +K3μ (ω, P, P  , σ, σ  , r )b(P, σ)a(P  , σ  )∗ +K4μ (ω, P, P  , σ, σ  , r )b(P, σ)b(P  , σ  )∗ ]G(ω, r − r )d3 P d3 P  d3 r  + (c(P, s)F(μ ω, P, s, r) + c(P, s)∗ F¯ μ (ω, P, s, r))d3 P where G(ω, r) = (μ.exp(−jK|r|)/4π|r|), K = ω, c = 1 Equivalently, writing Lμr (ω, P, P  , σ, σ  , r) =



G(ω, r − r )Krμ (ω, P, P  , σ, σ  , r )d3 r

we have the following final expression for the total quantum electromagnetic four potential: Aμ (ω, r) =  = [Lμ1 (ω, P, P  , σ, σ  , r)a(P, σ)∗ a(P  , σ  )+ Lμ2 (ω, P, P  , σ, σ  , r)a(P  , σ  )∗ b(P, σ) +Lμ3 (ω, P, P  , σ, σ  , r)b(P, σ)a(P  , σ  )∗ +Lμ4 (ω, P, P  , σ, σ  , r)b(P, σ)b(P  , σ  )∗ ]d3 P d3 P   + (c(P, s)F(μ ω, P, s, r) + c(P, s)∗ F¯ μ (ω, P, s, r))d3 P We now introduce the following notations: |p1e , σ1e , ..., pne , σne , p1p , σ1p , ..., pmp , σmp > stands for the state in which there is are n electrons of four momenta pke with corresponding spins σke , k = 1, 2, ..., n and m positrons of four momenta pkp with corresponding spins σkp , k = 1, 2, ..., m. Further we know that (See reference [1]) a(P, σ)|pe , σe >= δ(σ, σe )δ 3 (P − Pe )|0 > a(P, σe )∗ |0 >= |pe , σe > and likewise for positrons. More generally, we have a(Pe , σe )|p1e , σ1e , ..., pne , σne , p1p , σ1p , ..., pmp , σmp >

26

Quantum Antennas =

n 

(−1)k−1 δ(σe , σke )δ 3 (Pe , Pke )|p1e , σ1e , ..., pˆke , σ ˆke ,

k=1

..., pne , σne , p1p , σ1p , ..., pmp , σmp > where a hat above pke , σke indicates that these variables do not occur and a(Pe , σe )∗ |p1e , σ1e , ..., pne , σne , p1p , σ1p , ..., pmp , σmp > = |pe , σe , p1e , σ1e , ..., pˆke , σ ˆke , ..., pne , σne , p1p , σ1p , ..., pmp , σmp > and likewise for positrons. We also have the anticommutation rules: {a(P, σ), a(P  , σ  )∗ } = δ 3 (P − P  )δ(σ, σ  ) {b(P, σ), b(P  , σ  )∗ } = δ 3 (P − P  )δ(σ, σ  ) and all the other anticommutation relations are zero. These anticommutation rules and the action of creation and annihilaition operators can in principle be realized in a mathematically rigorous way using the Fermion Fock space (Reference [4]).

2.11

Summary of the second quantized Dirac field

Deriving the anticommutation rules for the Dirac field operators. The free Dirac equation is (iγ μ ∂μ − m)ψ(x) = 0 or equivalently in terms of the anticommuting Hermitian matrices β = γ 0 , αr = γ 0 γ r , 1 ≤ r ≤ 3 we have i∂0 ψ = ((α, −i∇) + βm)ψ The solution for ψ(x) is a representation as a superposition of plane waves:  ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P where

p0 = E(P ) =

 m2 + P 2 > 0

and in the second quantized picture, a(P, σ), b(P, σ) are operators in a Fermion Fock space and likewise their adjoints a(P, σ)∗ , b(P, σ)∗ . Note that E(−P ) = E(P ) For the above wave field to satisfy the Dirac equation, we must evidently have the purely algebraic relations (γ μ pμ − m)u(P, σ) = 0, (γ μ pμ + m)v(P, σ) = 0, σ = 1, 2

27

Quantum Antennas or equivalently, ((α, P ) + βm)u(P, σ) = E(P )u(P, σ), σ = 1, 2 ((α, P ) + βm)v(−P, σ) = −E(P )v(−P, σ), σ = 1, 2 Since the eigenvectors of a Hermitian matrix can be chosen to be an orthonormal basis for the underlying vector space and since the eigenvectors of a Hermitian matrix corresponding to distinct eigenvalues are orthogonal, we have that u(P, σ)∗ v(P, σ  ) = 0, σ, σ  = 1, 2 and we can ensure that u(P, σ)∗ u(P, σ  ) = 0, σ = σ  − − − (a) v(P, σ)∗ v(P, σ  ) = 0, σ = σ  − − − (b) The second quantized Hamiltonian of the Dirac field is given by  HD = ψ(x)∗ ((α, −i∇) + βm)ψ(x)d3 r where

x = (t, r) = (t, x1 , x2 , x3 )

We observe that p.x = E(P )t − P.r and ((α, −i∇) + βm)u(P, σ)exp(iP.r) = exp(iP.r)((α, P ) + βm)u(P, σ), = E(P )u(P, σ).exp(iP.r), ((α, −i∇) + βm)v(P, σ)exp(−iP.r) = exp(−iP.r)(−(α, P ) + βm)v(P, σ) = −E(P )v(P, σ).exp(−iP.r) and hence, the Hamiltonian at time t = 0 is given by  HD = (u(P, σ)a(P, σ)exp(iP.r) + v(P, σ)b(P, σ)∗ exp(−iP.r))∗ .E(P  )(u(P  , σ  )a(P  , σ  )exp(iP  .r) − v(P  , σ  )b(P  , σ  )exp(−iP  .r))d3 P d3 P  d3 r  = E(P  )[u(P, σ)∗ u(P  , σ  )a(P, σ)∗ a(P  , σ  )exp(i(P  − P ).r)+ v(P, σ)∗ u(P  , σ  )b(P, σ)a(P  , σ  )exp(i(P  + P ).r) −u(P, σ)∗ v(P  , σ  )a(P, σ)∗ b(P  , σ  )∗ exp(−i(P  + P ).r) −v(P, σ)∗ v(P  , σ  )b(P, σ)b(P  , σ  )∗ exp(i(P − P  ).r)]d3 P d3 P  d3 r  3 E(P )[u(P, σ)∗ u(P, σ)a(P, σ)∗ a(P, σ)−v(P, σ)∗ v(P, σ)b(P, σ)b(P, σ)∗ ]d3 P = (2π)

28

Quantum Antennas where we have used the identity  exp(i(P − P  ).r)d3 r = (2π)3 δ 3 (P − P  ) and the orthogonality relations (a) and (b). We now wish to decide the normalizations of u and v, ie, to evaluate u(P, σ)∗ u(P, σ) and v(P, σ)∗ v(P, σ). These are evaluated as follows. First note that the Lagrangian density of the free Dirac field is ∗ LD (ψ, ψ ∗ , ψ,μ , ψ,μ )= ψ(x)∗ γ 0 (iγ μ ∂μ − m)ψ(x) It follows that the momentum conjugate to ψ is π=

∂LD = iψ ∗ ∂ψ,0

Thus, the canonical anticommutation rules {ψl (t, r), πm (t, r)} = iδlm δ 3 (r − r ) give us

{ψl (t, r), ψm (t, r )∗ } = δlm δ 3 (r − r ) − − − (c)

So we must have  [ul (P, σ)um (P  , σ  )∗ {a(P, σ), a(P  , σ  )∗ }exp(i(P.r − P  .r )) +ul (P, σ)vm (P  , σ  )∗ {a(P, σ), b(P  , σ  )}exp(i(P.r + P  .r )) +vl (P, σ)um (P  , σ  )∗ {b(P, σ)∗ , a(P  , σ  )∗ }exp(−i(P.r + P  .r )) +vl (P, σ)vm (P  , σ  )∗ {b(P, σ)∗ , b(P  , σ  )}exp(−i(P.r − P  .r ))]d3 P d3 P  = δlm δ 3 (r − r ) − − − (d) This forces us to introduce the anti-commutation relations {a(P, σ), a(P  , σ  )∗ } = δσ,σ δ 3 (P − P  ) {b(P, σ), b(P  , σ  )∗ } = δσ,σ δ 3 (P − P  ) and all the other anti-commutators vanish, along with the normalizations  (ul (P, σ)um (P, σ)∗ + vl (−P, σ)vm (−P, σ)∗ ) = δlm σ=1,2

We note that Π(P ) =

 σ=1,2

u(P, σ)u(P, σ)∗T

29

Quantum Antennas is the orthogonal projection onto the space of eigenvectors of the momentum space free Dirac Hamiltonian corresponding to the energy eigenvalue E(P ) while I − Π(P ) =



v(−P, σ)v(−P, σ)∗T

σ=1,2

is the orthogonal projection onto the space of eigenvectors of the free Dirac Hamiltonian in the momentum space corresponding to the energy eigenvalue −E(P ).

2.12

Electron propagator computation

The electron propagator is defined as Slm (x|x ) =< 0|T (ψl (x)ψm (x )∗ )|0 > where T denotes the time ordering operator. In other words, with θ denoting the Heavy-side step function, we have Slm (t, r|t , r ) = θ(t − t ) < 0|ψl (t, r)ψm (t , r )∗ |0 > −θ(t − t) < 0|ψm (t , r )ψl (t, r)|0 > We shall evaluate this propagator using two methods. First, the purely operator theoretic method combined with properties of Dirac eigenvectors and second the differential equation method combined with the operator theoretic method. The first method: We note that    ∗ ψl (t, r)ψm (t , r ) = ( [ul (P, σ)a(P, σ)exp(−ip.x)+vl (P, σ)b(P, σ)∗ exp(ip.x)]d3 P )  ×

(um (P  , σ  )∗ a(P  , σ  )∗ exp(ip .x )+vm (P  , σ  )∗ b(P  , σ  )exp(−ip .x ))d3 P d3 P 

so that 

< 0|ψl (t, r)ψm (t , r )∗ |0 >= [ul (P, σ)um (P  , σ  )∗ exp(−ip.x + ip .x ) < 0|a(P, σ)a(P  , σ  )∗ |0 > d3 P d3 P   =

ul (P, σ)um (P  , σ  )∗ exp(−ip.x + p .x )δ 3 (P − P  )δσ,σ d3 P d3 P  

=

ul (P, σ)um (P, σ)∗ exp(−iE(P )(t − t ) − iP.(r − r ))d3 P  =

Πlm (P )exp(−i(E(P )(t − t ) + P.(r − r )))d3 P

30

Quantum Antennas

Likewise, < 0|ψm (t , r )∗ ψl (t, r)|0 >  = vm (P  , σ  )∗ vl (P, σ) < 0|b(P  , σ  )b(P, σ)∗ |0 > exp(−ip .x +ip.x)d3 P d3 P  

vl (P, σ)vm (P, σ)∗ exp(i(E(P )(t − t ) − P.(r − r )))d3 P

=  =

(δlm − Πlm (P ))exp(i(E(P )(t − t ) + P.(r − r )))d3 P

Thus, our final expression for the electron propagator is    Slm (x|x ) = θ(t − t ) Πlm (P )exp(−i(E(P )(t − t ) + P.(r − r )))d3 P −θ(t − t)



(δlm − Πlm (P ))exp(i(E(P )(t − t ) + P.(r − r )))d3 P

We can thus change the integration variable P to −P and express this propagator in matrix form as S(x|x ) = θ(t−t )



Π(−P )exp(−ip.(x−x ))d3 P −θ(t −t)



(I−Π(−P ))exp(ip.(x−x ))d3 P

where the integration is over the mass shell, ie p0 = E(P ). The second method: We write Slm (x|x ) = Slm (t, r|t , r ) = θ(t − t ) < 0|ψl (t, r)ψm (t , r )∗ |0 > −θ(t − t) < 0|ψm (t , r )∗ ψl (t, r)|0 > Thus,

(iγ μ ∂μ − m)S(x|x ) = iγ 0 ∂0 S(x|x ) + (iγ r ∂r − m)S(x|x )

= iγ 0 δ(t − t ) < 0|ψl (t, r)ψm (t, r )∗ |0 > +iγ 0 δ(t − t ) < 0|ψm (t, r )∗ ψl (t, r)|0 > +θ(t − t ) < 0|((iγ μ ∂μ − m)ψ(t, r))l ψm (t , r )∗ |0 > −θ(t − t) < 0|ψm (t , r )∗ ((iγ μ ∂μ − m)ψ(t, r))l |0 > = iγ 0 δ(t t ) < 0|{ψl (t, r), ψm (t, r )∗ }|0 >= −iγ 0 δ(t − t )δlm δ 3 (r − r ) = iγ 0 δ 4 (x − x ) where we have used the fact that ψ(t, r) satisfies the free Dirac equation. Taking the four dimensional space-time Fourier transform followed by Fourier inversion then gives us   −4 iγ 0 (iγ μ pμ − m − i)−1 exp(ip.(x − x ))d4 p S(x|x ) = (2π) The reason for including a term i with  → 0+ is for ensuring regularity of the propagator as t − t → ±∞. We shall elaborate on this using contour integrals.

31

Quantum Antennas

2.13

Quantum mechanical tunneling of a Dirac particle through the critical radius of the Schwarzchild blackhole

. It is well known [Steven Weinberg, Gravitation and Cosmology-Principles and applications of the general theory of relativity] that Dirac’s relativistic equation for a particle of mass m in curved space-time with metric gμν (x) is given by [γ a Vaμ (x)(i∂μ + iΓμ (x)) − m]ψ(x) = 0 where Vaμ (x) is a tetrad for the metric gμν , ie, η ab Vaμ (x)vbν (x) = g μν (x) and Γμ (x) is the spinor connection of the gravitational field. It is calculated as Γμ (x) = (−1/2)J ab Vaν (x)Vbν:μ (x) where J ab = (1/4)[γ a , γ b ] are the standard Lie algebra generators of the Dirac spinor representation of the Lorentz group. It is easily shown that this Dirac equation is invariant under local Lorentz transformations. Now let gμν be the metric of a Schwarzchild blackhole having mass M expressed in the cartesian system, ie, in the coordinate system x, y, z, t where x = r.cos(φ)sin(θ), y = r.sin(φ)sin(θ), z = r.cos(θ) We determine a tetrad Vaμ (x) corresponding to this metric and the corresponding spinor connection Γμ (x). Solve the time dependent Dirac equation with this metric and assuming that at time t = 0, the Dirac wave function ψ(t, r) is zero for |r| > 2GM/c2 , find the probability density of the particle at time t > 0 at points r with |r| > 2GM/c2 . This probability density is simply ψ(t, r)∗ ψ(t, r). More generally, determine the Dirac four current density J μ (t, r) = ψ(t, r)∗ γ 0 γ μ ψ(t, r) for |r| > 2GM/c2 . This problem illustrates the fact that although classical general relativity mechanics prevents a particle in escaping through the event horizon of the blackhole in finite coordinate time, quantum general relativity yields a small tunnelling probability for the particle through the event horizon.

32

2.14

Quantum Antennas

Supersymmetry–supersymmetric current in an antenna comprising superpartners of elementary particles

Abstract: Supersymmetry provides a nice mathematical basis for the unification of various field theories. Many kinds of fields like the scalar Klein-Gordon field with Higgs potential, the ]Dirac Fermionic field, the Maxwell electromagnetic field, the non-Abelian matter and gauge field and the gravitational field appear in supersymmetry as components of a big superfield that is a polynomial in Majorana Fermionic variables. Apart from these fields, in supersymmetry, we are forced to introduce other component fields like the gaugino field and the gravitino field which are called the superpartners of the corresponding gauge field and the gravitational metric tensor field and also other auxiliary fields. The introduction of superpartners compels us to postulate that every particle has a superpartner. More precisely, a Boson has a superpartner that is a Fermion and vice versa. The importance of supersymmetry in the context of antennas comes from the fact that when one writes down the supersymmetric Lagrangian of a Chiral superfield, then this Lagrangian contains both the kinematic part of the Lagrangian of the Dirac field and the Lagrangian of the scalar KG field along with some other auxiliary fields. When one sets the variational derivative of this Lagrangian density w.r.t. the auxiliary fields to zero, then the resulting Lagrangian density contains the Dirac Lagrangian with a mass dependent on the scalar field through a superpotential. Thus, the current density of the Dirac field depends on the scalar KG field and the corresponding radiation produced by such a current will therefore also be dependent upon the scalar field. There is another aspect to supersymmetric effect in antennas. This is based on the notion of supercurrent. Noether’s theorem states that when the Lagrangian density is invariant under a Lie group of transformations of the field, then there is an associated conserved current. The supersymmetric action integral is invariant under the transformations of the component fields defined by the Salam Strathdhee supersymmetry generators and hence, we can associate a Noether current density to this. This Noether current density associated with transformations of the component fields under a supersymmetric transformation of the superfield will have a four divergence equal to the change in the Lagrangian density under an infinitesimal supersymmetric transformation and this guarantees that the action integral is invariant under supersymmetric transformations. This relation is satisfied provided that we assume that the component superfields satisfy the Euler-Lagrange equations of motion for the given supersymmetric Lagrangian density. This current density will not be conserved since it is only the action integral that is invariant under supersymmetric transformations and not the Lagrangian density. The four divergence of this Noether current density gives us the infinitesimal change in the Lagrangian density under supersymmetric transformation. On the other hand, the Lagrangian density itself being a function of the component fields, undergoes a transformation under supersymmetry and this infinitesimal change is a four divergence of another current. Therefore, the

33

Quantum Antennas

difference between the four divergences of the two currents, one the Noether current and two the current arising from the change in the Lagrangian under supersymmetry must be zero. Thus, we get a conservation law. This law is the conservation of the supercurrent and we would like to evaluate the radiation fields produced by such super currents. There is another aspect to supersymmetry when one considers gauge fields and their superpartners the gaugino fields. The electromagnetic field is a U (1) gauge field. On the other hand, there exist non-Abelian gauge fields which arise in the weak and strong nuclear interactions. These non-Abelian gauge fields are messenger fields which communicate the nuclear forces just as the electromagnetic field is an Abelian gauge field which communicates the electromagnetic forces between charges. In supersymmetry we talk of invariance under supersymmetric gauge transformations. Just as the total Lagrangian density of the matter and gauge field is invariant under gauge transfromations in the Dirac theory and its generalized non-Abelia Yang-Mills theory, so also we must look for a Lagrangian density built solely out of the gauge fields and the gaugino fields that is invariant non only under supersymmetric gauge transformations but also under general supersymmetry transformations. Such a Lagrangian density does exist (Reference [1]) and it contains apart from the Lagrangian of the Maxwell field and the Lagrangian of the Yang-Mills non-Abelian gauge field, the Lagrangian of the gaugino and auxiliary fields. When we talk of radiation by matter in quantum antennas as mentioned earlier, we should apart from taking the matter current as the Dirac current of electrons and positrons and the corresponding radiation field as the Maxwell field, also talk about taking the matter current coming from other components of the superfield like the scalar KG field and the corresponding radiation fields being not only the Maxwell field, but the other gauge fields, gaugino fields and auxiliary fields. The relation between the matter fields and the gauge, gaugino and auxiliary fields will come from the total Lagrangian of the matter fields and gauge, gaugino and auxiliary fields. Such a Lagrangian can be derived from a superfield and it respects both supersymmetry invariance and supersymmetric gauge invariance (Reference III). Majorana Fermion:



where e = iσ 2 = Then, we have ∗





θ =  =



ζ −eζ ∗

θ=

0 −e

0 −1

ζ∗ −eζ e 0

1 0 

 θ



34

Quantum Antennas

We assume that the components of the Majorana Fermion θ = (θa , a = 0, 1, 2, 3) mutually anticommute: θa θb + θb θa = 0 

Define = Note that

e 0

0 e



σ 2T = −σ 2 , eT = −e,

∗ stands for complex conjugate, not for conjugate transpose. For conjugate transpose, we shall use the notation a → aH . Also define the Dirac gamma matrices:   0 σμ γμ = σμ 0 where



0 1 1 0   0 −i , σ 2 = −σ2 = i 0   1 0 σ 3 = −σ3 = 0 −1

σ 0 = σ0 = I, σ 1 = −σ1 =

 ,

are the usual Pauli spin matrices. We have σ μH = σ μ , σ 0T = σ 0 , σ 1T = σ 1 , σ 2T = −σ 2 , σ 3T = σ 3 σ 2 σ 0 σ 2 = σ 0 , σ 2 σ 1 σ 2 = iσ 2 σ 3 = −σ 1 = σ1 , σ 2 σ 2 σ 2 = σ 2 = −σ2 , σ 2 σ 3 σ 2 = iσ 1 σ 2 = −σ 3 = σ3 where we have used the standard relations σ 1 σ 2 = −σ 2 σ 1 = iσ 3 , σ 2 σ 3 = −σ 3 σ 2 = iσ 1 , σ 3 σ 1 = −σ 1 σ 3 = iσ 2 along with σ μ2 = I, μ = 0, 1, 2, 3 We also have the conjugation relations: σ 2 σ 1∗ σ 2 = σ1 , σ 2 σ 2∗ σ 2 = σ2 , σ 2 σ 3∗ σ 2 = σ3

35

Quantum Antennas Thus,

σ 2 σ μ∗ σ 2 = σμ

and hence eσ μ∗ e = −σμ where we use

σ 0∗ = σ 0 , σ 1∗ = σ 1 , σ 2∗ = −σ 2 , σ 3∗ = σ 3

We also have the fact that , γ5 , γ μ are six linearly independent 4 × 4 ansisymmetric matrices and hence form a basis for the vector space of all 4 × 4 antisymmetric matrices over the complex field. Here, we define   I 0 5 0 1 2 3 γ =γ γ γ γ = 0 −I 

Note that 5

γ =

e 0

0 −e



Exercise: Verify that the Dirac gamma matrices satisfy the Clifford algebra anticommutation relations with bilinear form ((η μν )) = diag[1, −, 1−, 1 − 1, ]: γ μ γ ν + γ ν γ μ = 2η μν Now let M be any symmetric 4 × 4 matrix (bosonic). Then θT M θ = 0 since θT M θ =



Mab θa θb =



Mba θa θb =



Mba (−θb θa ) = −θT M θ

a,b

Thus, any quadratic function of θ can be expressed as θT M θ where M is an antisymmetric matrix. It follows that since γ 5 , , γ μ , μ = 0, 1, 2, 3 form a basis for the six-dimensional complex vector space of 4 × 4 antisymmetric matrices, that any quadratic form in θ can be expressed as a linear combination of the six quadratic forms θT θ, θT γ 5 θ, θT γ μ θ, μ = 0, 1, 2, 3. In particular, we can write θθT = c1 .θT θ + c2 .γ 5 θT γ 5 θ + c3 γμ θT γ μ θ Note that the rhs is an antisymmetric matrix since the Fermionic variables θa anticommute. To verify this identity and also calculate the scalar coefficients c1 , c2 , c3 , we multiply both sides by , γ 5  and γ ν and take traces. Then we get using 2 = −I4 = (γ 5 )2 θT θ = −4c1 θT θ

36

Quantum Antennas θT γ 5 θ = −4c2 θT γ 5 θ so that c1 = c2 = −1/4 and finally, θT γ ν θ = c3 .T r(γ ν γμ )θT γ μ θ = −c3 T r(γ ν γμ )θT γ μ θ = −4c3 δμν θT γ μ θ = −4c3 θT γ ν θ so that c3 = −1/4 Note that we are making use of the identities θT θ.θT γ 5 θ = 0, θT θ.θT γ μ θ = 0, θT γ 5 θ.θT γ μ θ = 0 Remark: It is easy to verify that γ μT  = −γ μ and hence γ μ  = −γ μT Also,

γ 0∗ = γ 0 , γ 1∗ = γ 1 , γ 2∗ = −γ 2 , γ 3∗ = γ 3 , γ 0T = γ 0 , γ 1T = −γ 1 , γ 2T = γ 2 , γ 3T = −γ 3

and hence

γ 0H = γ 0 , γ 1H = −γ 1 , γ 2H = −γ 2 , γ 3H = −γ 3

or equivalently, γ μH = γμ Supersymmetry generators: Note that (γ 5 )2 = 2 = −I4 , (γ 5 )T = −γ 5 , γ 5T = γ 5 , T = −, γ 5  = γ 5 Now,

L = γ 5 ∂/∂θ + γ μ θ∂/∂xμ ¯ = −γ 5 L = ∂/∂θ − γ 5 γ μ θ∂/∂xμ L

or equivalently,

¯ T = LT γ 5  L

37

Quantum Antennas Define

θ¯ = −γ 5 θ

or equivalently,

θ¯T = θT γ 5 

Now, we’ve seen that γ μ  = −γ μT and hence,

γ 5 γ μ γ 5  = γ μT 

(since γ 5 anti-commutes with the γ μ s and its square is I. Note that γ 5T = γ 5 . Thus, γ 5 γ μ = γ μT γ 5  and so

¯ = ∂/∂θ + γ μT γ 5 θ∂/∂xμ L μ ¯ = −γ 5 ∂/∂ θ¯ − γ μT θ∂/∂x

Anticommutation relations between the supersymmetry generators: We use the fundamental identities {∂/∂θa , θb } = δab , {θa , θb } = 0, {∂/∂θa , ∂/∂θb } = 0 Then,

¯ b } = {(γ 5 )ac ∂/∂θc , (γ μT γ 5 )bd θd ∂/∂xμ } {La , L +{(γ μ )ac θc ∂/∂xμ , ∂/∂θb } = (γ 5 )ac (γ μT γ 5 )bd δcd ∂/∂xμ + (γ μ )ac δbc ∂/∂xμ = [(γ 5 )ac (γ μT γ 5 )bc + (γ μ )ab ]∂/∂xμ = ((γ 5 (γ μT γ 5 )T ) + γ μ )ab ∂/∂xμ μ = 2γab ∂/∂xμ

Also,

μ d θ ∂/∂xμ } {La , Lb } = {(γ 5 )ac ∂/∂θc , γbd μ c +{γac θ ∂/∂xμ , (γ 5 )bd ∂/∂θd } μ d μ = [(γ 5 )ac γbd δc + γac (γ 5 )bd δdc ]∂/∂xmu

= [γ 5 γ μT + γ μ (γ 5 )T ]ab ∂/∂xμ = 0 since γ μT  = −γ μ implies implies

γ 5 γ μT γ 5  = −γ 5 γ μ γ 5 = γ μ γ 5 γ μT = −γ μ γ 5  = γ μ (γ 5 )T

38

Quantum Antennas These equations immediately imply that ¯a, L ¯b} = 0 {L Superfields: A superfield is a function S(x, θ) of x = (xμ ) and θ = (θa ). Expanding S in powers of θ with coefficients being functions of x alone, we find that all the coefficients of the fifth and higher powers of θ may be taken as zero since any fifth and higher degree monomial in the θ is zero. Thus, we may expand a general superfield as S[x, θ] = C(x) + θT ω(x) + θT θM (x) +θT γ 5 θN (x) + θT γ μ θVμ (x) + θT θθT γ 5 (λ(x) + aγ.∂ω(x)) +(θT θ)2 (D(x) + bC(x)) where λ(x), ω(x) ∈ C4 are spinorial. Let α be a Majorana spinor variable. An infinitesimal supersymmetry transformation is defined as the operator αT γ 5 L = α ¯ T L = δL We wish to examine how the bosonic components C, ω, M, N, Vμ , λ, D transform under a supersymmetry transformation. The constants a, b will be chosen appropriately. We have δLC(x) = αT γ μ θC,μ (x) LθT ω(x) = −γ 5 ω(x) + γ μ θθT ω,μ (x) = −γ 5 ω(x) + (−1/4)γ μ (θT θ + θT γ 5 θγ 5  +θT γ ν θγν )ω,μ (x) LθT θM (x) = 2γ 5 .θM (x) + γ μ θθT θM,μ (x) = −2γ 5 θM (x) + γ μ θθT θM,μ (x) so that

δLθT θM (x) = αT γ 5 LθT θM (x) = −2αT θM (x) + αT γ 5 γ μ θθT θM,μ (x) = 2θT αM (x) − θT θθT γ μT γ 5 αM,μ (x) = 2θT αM (x) + θT θθT γ 5 γ μ αM,μ (x)

Next,

LθT γ 5 θN (x) = 2γ 5 γ 5 θN (x) +γ μ θθT γ 5 θN,μ (x) = −2θN (x) + θT γ 5 θγ μ θN,μ (x)

and hence

δLθT γ 5 θN (x) =

39

Quantum Antennas −2αT γ 5 θN (x) + θT γ 5 θαT γ 5 γ μ θN,μ (x) = 2θT γ 5 αN (x) − θT γ 5 θαT γ μT γ 5 θN,μ (x) = 2θT γ 5 αN (x) + θT γ 5 θθT γ 5 γ μ αN,μ (x) Next, LθT γ μ θVμ (x) = 2γ 5 γ μ θVμ (x) + γ ν θθT γ μ θVμ,ν (x) = −2γ 5 γ μ θVμ (x) + γ ν θθT γ μ θVμ,ν (x) so δLθT γ μ θVμ (x) = −2αT γ μ θVμ (x) + θT γ μ θαT γ 5 γ ν θVμ,ν (x) = 2θT γ μ αVμ (x) − θT γ μ θαT γ νT γ 5 θVμ,ν (x) = 2θT γ μ αVμ (x) + θT γ μ θθT γ 5 γ ν αVμ,ν (x)

To simplify these formulas further, we make use of the fact that if M is any antisymmetric skew symmetric 4 × 4 matrix, then θT M θθT β can be expressed as θT θθT δ since there are only four linearly independent monmomials of the third degree in θ. We shall now determine δ ∈ C4 in terms of M and β ∈ C4 . To do so, we first consider a matrix M of the form   0 M12 M= T −M12 0 Then θT M θθT β = T T T M12 θ2:3 (θ0:1 β0:1 + θ2:3 β2:3 ) 2θ0:1 T T θ0:1 θ0:1 M12 θ2:3 = 2β0:1 T T T −2β2:3 θ2:3 θ2:3 M12 θ0:1 T = 2θ0 θ1 β0:1 eM12 θ2:3 T T −2θ2 θ3 β2:3 eM12 θ0:1

Further, θT θθT δ = T T T T eθ0:1 + θ2:3 eθ2:3 )(θ0:1 δ0:1 + θ2:3 δ2:3 ) (θ0:1 T T T T = θ0:1 eθ0:1 θ2:3 δ2:3 + θ2:3 eθ2:3 θ0:1 δ0:1 T T = 2θ0 θ1 θ2:3 δ2:3 + 2θ2 θ3 θ0:1 δ0:1

Comparing the two expressions gives us T eβ0:1 δ0:1 = M12 eβ2:3 , δ2:3 = −M12

40

Quantum Antennas Thus,

 δ=  =  =

δ0:1 δ2:3

0 T −M12 e

0 T −M12

M12 0

 =

M12 e 0  e 0

 β 0 e

 β

= M β In particular, θT γ μ θθT β = θT θθT γ μ β = −θT θθT γ μT β Now let M be a 4 × 4 antisymmetric matrix of the form   0 M11 M= 0 M22 where T T = −M11 , M22 = −M22 M11

Then, θT M θθT β = T T T T (θ0:1 M11 θ0:1 + θ2:3 M22 θ2:3 )(θ0:1 β0:1 + θ2:3 β2:3 ) T T = (2(M11 )12 θ0 θ1 + 2(M22 )12 θ2 θ3 )(θ0:1 β0:1 + θ2:3 β2:3 ) T = 2(M11 )12 θ0 θ1 θ2:3 β2:3 T +2(M22 )12 θ2 θ3 θ0:1 β0:1

Comparing this to T T θT θθT δ = 2θ0 θ1 θ2:3 δ2:3 + 2θ2 θ3 θ0:1 δ0:1

gives us δ0:1 = (M22 )12 β0:1 , δ2:3 = (M11 )12 β2:3 

and hence δ=

(M22 )12 I2 0

0 (M11 )12 I2

 β

In particular if M = γ 5 , M11 = e, M22 = −e and we get θT γ 5 θθT β = θT θθT δ 

where δ=

−I2 0

0 I2



β = −γ 5 β

41

Quantum Antennas In short, we summarize our discussion as follows: δLθT γ μ θVμ (x) = = 2θT γ μ αVμ (x) + θT γ μ θθT γ 5 γ ν αVμ,ν (x) = 2θT γ μ αVμ (x) −θT θθT γ μT γ 5 γ ν αVμ,ν (x) = 2θT γ μ αVμ (x) − θT θθT γ 5 γ μ γ ν αVμ,ν (x) = 2θT γ μ αVμ (x) −θT θθT γ 5 ([γ μ , γ ν ]fνμ (x) + γ.∂(γ.V ))α and δLθT θθT M (x) = δLθT θM (x) = = 2θT αM (x) + θT θθT γ 5 γ.∂M (x)α, δLθT γ 5 θN (x) = = 2θT γ 5 αN (x)+ −θT θθT (γ.∂N )α Further, LθT θθT (λ + aγ.∂ω) =

From these equations we get the following formulas for the change in the components of the superfield under an infinitesimal supersymmetry transformation: δC(x) = −αT γ 5 γ 5 ω(x) = −αT ω(x) δω(x) = 2αM (x) + 2γ 5 αN (x) + 2γ.V (x)α or equivalently,

δω(x) = 2(M (x) + γ 5 N (x) + γ.V (x))α

Likewise, LθT θθT β(x) = −2γ 5 θθT β(x) + θT θγ 5 β +θT θγ μ θθT β,μ = (1/2)(γ 5 θT θβ(x) + γ 5 θT γ 5 θγ 5 β(x) +γ 5 θT γ ν θγν β(x)) +θT θγ 5 β(x) − (1/4)(θT θ)2 γ μ β,μ (x)

42

Quantum Antennas It follows therefore that

δL(θT θθT γ 5 β(x)) = αT γ 5 L(θT θθT γ 5 β(x)) =

−(1/2)(αT θT θγ 5 β(x) + αT θT γ 5 θβ(x) +αT θT γ ν θγν γ 5 β(x)) −θT θαT γ 5 β(x) + (1/4)(θT θ)2 αT γ 5 γ μ γ 5 β,μ (x) = (1/2)(θT θ(γ 5 α)T β(x) + θT γ 5 θ(α)T β(x) +θT γ μ θ(γ 5 γμ α)T β(x)) +θT θ(γ 5 α)T β(x) + (1/4)(θT θ)2 (γ μ α)T β,μ (x) By setting β = γ + a.γ.∂ω, we therefore get δM (x) = (3/2)(γ 5 α)T β(x) + (1/4)(γ 5 α)T γ.∂ω, δN (x) = (1/2)(α)T β(x) + (1/4)(α)T γ.∂ω, δVμ (x) = (1/2)(γ 5 γμ α)T β(x) + (1/4)αT γ 5 γ ν γμ ω,ν = (1/2)αT γ 5 γμ β(x) + (1/4)αT γ 5 γ ν γμ ω,ν = (1/2)αT γ 5 γμ (λ + a.γ.∂ω)+ (1/4)αT γ 5 ({γ ν , γμ } − γμ γ ν )ω,ν = (1/2)αT γ 5 γμ λ+ +(a/2)αT γ 5 γμ γ.∂ω +(1/4)αT γ 5 (2δμν ω,ν − γμ γ.∂ω) = (1/2)αT γ 5 γμ λ+ +(a/2)αT γ 5 γμ γ.∂ω +(1/4)αT γ 5 (2ω,μ − γμ γ.∂ω) Taking a = 1/2 gives us δVμ (x) = (1/2)αT γ 5 γμ λ + (1/2)αT γ 5 ω,μ It follows that δfμν = δVν,μ − δVμ,ν = (1/2)αT γ 5 (γν λ,μ (x) − γμ λ,ν (x)) We now compute δλ(x) and δD(x). We have with β(x) = λ(x) + a.γ.∂ω(x), θT θθT γ 5 δβ(x) = αT γ 5 ∂θ (θT θ)2 (bC(x) + D(x)) −θT θγ 5 γ.∂M (x)α,

43

Quantum Antennas −θT θθT (γ.∂N )α −θT θθT γ 5 ([γ μ , γ ν ]fνμ (x) + γ.∂(γ.V ))α = −2γ 5 (θT θ)(bC(x) + D(x)) −θT θγ 5 γ.∂M (x)α, −θT θθT (γ.∂N )α −θT θθT γ 5 ([γ μ , γ ν ]fνμ (x) + γ.∂(γ.V ))α We deduce that

γ 5 δ(λ(x) + a.γ.∂ω(x)) = = −2γ 5 (bC(x) + D(x)) −γ 5 γ.∂M (x)α −(γ.∂N )α 5

−γ ([γ , γ ν ]fνμ (x) + γ.∂(γ.V ))α μ

Chapter 3

Conducting fluids as quantum antennas 3.1

3.1.1

A short course in basic non-relativistic and relativistic fluid dynamics with antenna theory applications The basic physical quantities of a fluid

Density, pressure, velocity field, momentum density, mass flux, momentum flux, energy density, entropy density, enthalpy density, temperature field. Density ρ(t, r), r = (x, y, z) is the mass per unit volume in a fluid at the point r at time t. If M (t, r, δV ) is the amount of fluid mass at time t within a volume δV that surrounds r, then ρ(t, r) = lim|δV |→0

3.1.2

M (t, r, δV ) |δV |

Parameters of a fluid

Viscosity, conductivity, density if the fluid is incompressible, parameters on which the basic physical quantities can depend and which can be estimated from the equations of motion.

3.1.3

Ordinary time derivative and material derivative of a physical quantity associated to a fluid

q is a physical quantity whose density is Q(t, r). The rate of change of Q at the fixed point r in the fluid as different fluid particles enter this point and leave this point is given by ∂Q(t,r) . Its rate of change along the trajectory of a fixed ∂t

45

46

Quantum Antennas

fluid particle which is at r at time t and at r + dr at time t + dt is given by DQ/Dt = limδt→0

Q(t + dt, r + dr) − Q(t, r) dt

∂Q(t, r) + (dr/dt, ∇r )Q(t, r) ∂t ∂Q(t, r) + (v(t, r), ∇r )Q(t, r) = ∂t This is called the material derivative of Q. =

3.1.4

The mass conservation equation/equation of continuity

The general conservation equation for the density of any physical quantity taking into account the rate of its generation per unit volume.

3.1.5

The momentum equation starting from Newton’s second law of motion

. Eulerian equations for a non-viscous fluid. Momentum equation taking into account the stress tensor of a fluid, special case of diagonal stress tensor-the pressure term.

3.1.6

The momentum equation derived from conservation of momentum flux

The momentum flux tensor is T ij = ρv i v j and the momentum density or equivalently the mass flux is T 0i = T i0 = ρv i Let σ ij denote the stress tensor due to pressure and viscous forces. Then we have the obvious momentum conservation law/second law of motion, ij T,0i0 + T,jij = σ,j

or in the dyad notation, (ρv i ),0 + divT = divσ This is the differential form of the integral form    d T i0 d3 r + T ij nj dS = σ ij nj dS dt V ∂V ∂V

47

Quantum Antennas where (nj ) is the unit normal to the surface ∂V that bounds the volume V . For a fluid, we can express the stress tensor as i σ ij = η(v,j + v,ij ) + χ(divv)δ ij − pδ ij

where η, χ are coefficients of viscosity and p is the pressure. Then, the above momentum conservation law gives (ρv i ),0 + (ρv i v j ),j = η∇2 v i + (η + χ)(divv),i − p,i and on combining this with the equation of continuity ρ,0 + (ρv i ),i = 0 we get the Navier-Stokes equation i i ρv,0 + ρv j v,j = η∇2 v i + (η + χ)(divv),i − p,i

or in vector form ρv,0 + ρ(v, ∇)v = −∇p + η∇2 v + (η + χ)∇(divv)

3.1.7

Viscosity, the strain rate, shear and bulk viscosity, the Navier-Stokes equation for a viscous fluid, Expression of the Navier-Stokes equation in different orthogonal curvilinear coordinate systems

3.1.8

The fluid dynamical equations taking viscous and thermal effects into account, thermal conductivity, the energy equation for a fluid based on the first and second laws of thermodynamics

3.1.9

Vorticity, the vorticity equation, irrotational flows, the velocity potential, Bernoulli equation

3.1.10

Incompressible flows, the stream function, Bernoulli equation along a streamline for rotational flows

Here we have the following situation: divv = 0, which follows from the equation of continuity with ρ = constt. and the NavierStokes equation (v, ∇)v + v,t = −∇p/ρ + ν∇2 v

48

Quantum Antennas

where ν = η/ρ The incompressibility condition implies the existence of a stream function field ψ(t, r) ∈ R3 such that v =∇×ψ Also, we may assume that divψ = 0, for replacing ψ by ψ + ∇f , we do not alter v = ∇ × ψ and we can choose f so that 0 = div(ψ + ∇f ) = divψ + ∇2 f ie f (t, r) = (4π)−1



|r − r |−1 divψ(t, r )d3 r

The Navier-Stokes equation can be expressed as Ω × v + ∇v 2 /2 + v,t = −∇p/ρ + ν∇2 v − ∇V where Ω=∇×v is the vorticity and V (t, r) is the potential of the external force, like for example, gravity. We note that this equation is still valid if the fluid is not incompressible. Further, in the incompressible case, it follows by taking the scalar product with a unit vector field ˆl parallel to the velocity field that (1/2)

∂v 2 ∂p ∂V + v,t .ˆl + ρ−1 + ν ˆl.∇2 v − =0 ∂l ∂l ∂l

In the special case when apart from being incompressible, the fluid is in a steady state, ie, v,t = 0 and is also non-viscous, ie, η = 0, it follows from the above that ∂ 2 (v /2 + p/ρ + V ) = 0 ∂l and hence the quantity p/ρ + v 2 /2 + V is a constant along each streamline. This is a slightly restricted form of Bernoulli’s principle. If apart from being incompressible, the fluid is irrotational so that Ω = 0 and hence v = −∇φ where φ is the velocity potential, then we get ∇(p/ρ + v 2 /2 − φ,t V + η∇2 φ) = 0 This is valid even if the fluid is viscous and not in steady state. But the incompressiblity condition gives ∇2 φ = divv = 0 and hence we get the following version of the Bernoulli principle for incompressible irrotational vlows that are not necessarily in steady state: p/ρ + v 2 /2 − φ,t + V = constt.

49

Quantum Antennas

Actually, the constant on the rhs can depend on time but not on space and writing this constant as c(t), we can redefine our velocity potential as φ(t, r) − t c(s)ds without affecting the velocity field to get 0 p/ρ + v 2 /2 − φ,t + V = 0 at all points in the fluid and at all times. This is the most general form of the Bernoulli equation.

3.1.11

Some examples of fluid flows

[a] Derivation of Stokes’ law for the damping force on a sphere moving inside a viscous fluid. Exercise: Assume that a sphere of radius R is placed in a viscous fluid having a constant velocity field v0 zˆ. After placing the sphere, the velocity field of the ˆ The boundary conditions on this r + vθ (r, θ)θ. fluid changes to v(r, θ) = vr (r, θ)ˆ modified velocity field is that limr→∞ v(r, θ) = v0 zˆ and vr = 0 when r = R. Assuming the fluid field to be irrotational, we can derive it from a velocity potential φ(r, θ) as v = ∇φ = φ,r rˆ + r−1 φ,θ θˆ The incompressibility condition ∇.v = 0 becomes ∇2 φ = 0 and hence in view of the azimuthal symmetry of φ, it can be expanded using the Legendre polynomials as  φ(r, θ) = v0 rcos(θ) + c(l)r−l−1 Pl (cos(θ)), r ≥ R l≥0

Note that as r → ∞, we must have ∇φ → v0 zˆ which means that φ → v0 z = v0 rcos(θ) which is why we have omitted the terms rl Pl (cos(θ)), l ≥ 2 in the Legendre series expansion. Application of the boundary conditions gives that φ,r (R, θ) = 0 (ie the normal component of the velocity field on the surface of the sphere vanishes) (v, ∇)v = −∇p/ρ + (η/ρ)∇2 v becomes

Ω × v + ∇v 2 /2 + ∇p/ρ = (η/ρ)∇2 v

where Ω = ∇ × v = 0 and ∇2 v = ∇∇2 φ = 0 so that this equation can be satisfies by satisfying ∇2 (v 2 /2 + p/ρ) = 0 or equivalently,

p = −ρv 2 /2 + C

50

Quantum Antennas where C is a constant. This determines the pressure field once the velocity field is known. Now applying the above boundary condition φ,r (R, θ) = 0 gives us c(0) = 0, v0 − 2c(1)R−3 = 0, c(l) = 0, l ≥ 2 Thus,

c(1) = v0 R3 /2

Thus, the complete solution for the velocity potential is given by φ = v0 rcos(θ) + v0 R3 cos(θ)/2r2 = v0 cos(θ)(r + R3 /2r2 ) Hence the non-vanishing components of the velocity field in the spherical polar coordinate system are given by vr (r, θ) = φ,r (r, θ) = v0 cos(θ)(1 − R3 /r3 )r ≥ R vθ (r, θ) = r−1 φ,θ (r, θ) = −v0 sin(θ)(1 + R3 /2r3 ), r ≥ R Now using this velocity field, we can compute the spherical polar components of the viscous stress tensor, ie, the force per unit area on the surface of the sphere at each point defined as σnm = σij ni mj where n, m are unit vectors and the summation convention is adopted. In particular, after integrating over the surface area of the sphere, we are left with only the z component of the viscous force on the sphere, namely,  fz = (σzx nx + σzy ny + σzz nz )dS S

where σab = η(va,b + vb,a ) in cartesian coordinates and the components of the stress tensor are evaluated at r = R, θ, φ, dS = sin(θ)dθdφ and nx = cos(φ)sin(θ), ny = sin(φ)sin(θ), nz = cos(θ) Perform this computation and hence prove Stokes’ formula fz = 6πηRv0 Remark: Suppose we impose the boundary condition that vθ = 0 when r = R rather than vr = 0 when r = R. This would imply that φ(R, θ) = constt. and we would get a different solution, namely, φ(r, θ) = v0 cos(θ)(r − R2 /r)

51

Quantum Antennas

This boundary condition implies that the surface of the sphere is an equipotential surface. However, it is natural to believe that the normal component of the velocity of the fluid on the sphere surface is zero for otherwise, a locally turbulent flow would result. [b] Steady flow from a conical jet. The velocity field in spherical polar coordinates has the form v = vr (r, θ)ˆ r + vθ (r, θ)θˆ and we have ˆ (v, ∇)v = vr (vr,r rˆ + vθ,r θ) ˆ +r−1 vθ (vr,θ rˆ + vθ,θ θ) +r−1 vθ (vr θˆ − vθ rˆ) where we use r, rˆ,θ = θˆ θˆ,θ = −ˆ rˆ,r = 0, θˆ,r = 0 ∇2 v = v,rr + 2v,r /r + r−2 v,θθ + r−2 cos(θ)v,θ Now observe that ˆ v,r = vr,r rˆ + vθ,r θ, ˆ v,rr = vr,rr rˆ + vθ,rr θ, v,θ = vr,θ rˆ + vθ,θ θˆ +vr θˆ − vθ rˆ = (vr,θ − vθ )ˆ r + (vθ,θ + vr )θˆ v,θθ = (vr,θθ − 2vθ,θ − vr )ˆ r +(vr,θ − vθ − vθ,θθ + vr )θˆ Also, for the pressure p(r, θ), we have ∇p = p,r rˆ + r−1 p,θ θˆ Further,

divv = r−2 (r2 vr ),r + (r.sin(θ))−1 (sin(θ)vθ ),θ

Exercise:Substitute these expressions into the Navier-Stokes and incompressibility equations (v, ∇)v = −∇p/ρ + ν∇2 v, divv = 0 to derive three partial differential equations in r, θ for the three functions vr , vθ , p. Derive numerical algorithms for solving these.

52

Quantum Antennas Remark:Using the incompressibility equation, we can replace the two functions vr , vθ by a single function ψ(r, θ) so that r2 sin(θ)vr = ψ,θ , r.sin(θ)vθ = −ψ,r Then we get two pde’s for two functions ψ, p of (r, θ). [c] Flow in a cylindrical cup after stirring. Here in terms of cylindrical coordinates (ρ, φ, z), by cylindrical symmetry, we introduce the velocity field v = vρ (t, ρ, z)ˆ ρ + vφ (t, ρ, z)φˆ + vz (t, ρ, z)ˆ z Then note that ∇2 v = (ρ−1 ∂ρ + ∂ρ2 + ρ−2 ∂φ2 + ∂z2 )(vρ ρˆ + vφ φˆ + vz zˆ) = ρˆ(ρ−1 vρ,ρ + vρ,ρρ + vρ,zz − ρ−2 vρ ) ˆ −1 vφ,ρ + vφ,ρρ + vφ,zz − ρ−2 vφ ) +φ(ρ +ˆ z (ρ−1 vz,ρ + vz,ρρ + vz,zz ) Also,

(v, ∇)v = (vρ ∂ρ + ρ−1 vφ ∂φ + vz ∂z)(vρ ρˆ + vφ φˆ + vz zˆ) = ρˆ(vρ vρ,ρ − ρ−1 vφ2 + vz vρ,z ) ˆ ρ vφ,ρ + ρ−1 vφ vρ + vz vφ,z ) +φ(v +ˆ z (vρ vz,ρ + vz vz,z )

Now substitute these expressions into the Navier-Stokes equations (v, ∇)v + v,t = −∇p/ρ0 + ν∇2 v − gˆ z where p = p(t, ρ, z) so that ∇p = p,ρ ρˆ + p,z zˆ to obtain three pde’s for the four functions vρ , vφ , vz , p. The fourth equation is provided by the incompressibility condition: ρ−1 (ρvρ ),ρ + vz,z = 0 This equation implies that the two functions vρ , vz can be replaced by a single function ψ(t, ρ, z) using ρvρ = ψ,z , ρvz = −ψ,ρ Doing so, we end up with three pde’s for the three functions ψ, p, vφ of (t, ρ, z).

53

Quantum Antennas

3.1.12

Two dimensional incompressible, irrotational flows described using the velocity potential and the stream function, solution with different boundary conditions using analytic functions of a complex variable, Cauchy-Riemann equations and proof of the orthogonality of the streamlines and constant velocity potential lines, formulation of incompressible and irrotational flows as solutions to the 2-D Laplace equation, sources and sinks

Exercise: Show by integrating r−1 = (ρ2 + z 2 )−1 w.r.t. z over R and by using the fact that ∇2 r−1 = −4πδ(x)δ(y)δ(z) that

∇2 log(ρ) = 2πδ(x)δ(y)

Now consider a two dimensional incompressible flow. incompressibility implies the existence of a stream function ψ(t, x, y) such that v = ∇ψ × zˆ or equivalently, vx (t, x, y) = ψ,y , vy (t, x, y) = −ψ,x For the vorticity, we have z Ω = ∇ × v = −∇2 ψˆ and hence, we get on taking the curl of the Navier-Stokes equation, ∇ × (Ω × v) + Ω,t = ν∇2 Ω + g where g(t, x, y) = ∇ × f (t, x, y)/ρ with f as the external force per unit volume (in the xy plane). We can express this equation as ∇2 ψ,t + (ˆ z , ∇ × (∇2 ψˆ z × (∇ψ × zˆ))) = ν(∇2 )2 ψ − g Explicitly, show that this evaluates to z , ∇2 ∇ψ × ∇ψ) − ν(∇2 )2 ψ − g ∇2 ψ,t + (ˆ or equivalently,

∇2 ψ,t + ψ,y ∇2 ψ,x − ψ,x ∇2 ψ,y −ν(∇2 )2 ψ + g = 0

54

Quantum Antennas

Deduce that if K is the linear operator acting on functions on R2 with kernel K(x − x , y − y  ) = 4πlog((x − x )2 + (y − y  )2 ) then ψ,t (t, x, y) + K.(ψ,y ∇2 ψ,x − ψ,x ∇2 ψ,y )(t, x, y) − ν∇2 ψ(t, x, y) + g(t, x, y) = 0 Now introduce a perturbation parameter δ into the nonlinear term in the above equation and show that by expanding the stream function in a perturbation series as  δ n ψn (t, x, y) ψ(t, x, y) = n≥0

that

ψ0,t − ν∇2 ψ0 + g = 0, ψn,t − ν∇2 ψn +

n−1 

K.(ψk,y ∇2 ψn−k,x − ψk,x ∇2 ψn−k,y ) = 0, n ≥ 1

k=0

3.2

Flow of a 2-D conducting fluid

Now consider a 2-D conducting fluid in an electric field having the form x + Ey (t, x, y)ˆ y E = Ex (t, x, y)ˆ and a magnetic field of the form z B = Bz (t, x, y)ˆ Write down the planar equations of motion (v, ∇v) + v,t = −∇p/ρ + (σ/ρ)(E + v × B) + ν∇2 v in terms of ψ(t, x, y), Ex (t, x, y), Ey (t, x, y), Bz (t, x, y) along with the Maxwell equations Ex,x + Ey,y = 0 Ey,x − Ex,y = −Bz,t , −Bz,x = Ey,t + σ(Ey − vx Bz ), Bz,y = Ex,t + σ(Ex + vy Bz ) The first Maxwell equation implies the existence of a scalar field χ(t, x, y) such that Ex = χ,y , Ey = −χ,x Now show using the above that we have exactly three equations for the three functions ψ, χ, Bz . Note that p has been eliminated by taking the curl of the z and Navier-Stokes equation, ie, we use Ω = −∇2 ψˆ ∇ × (Ω × v) + Ω,t = −ν∇2 Ω + ∇ × (E + v × B)

55

Quantum Antennas Note that ∇ × (E + v × B) = zˆ(−Bz,t − (vx Bz ),x − (vy Bz ),y ) = −ˆ z (Bz,t + vx Bz,x + vy Bz,y ) with vx = ψ,y , vy = −ψ,x 2 −1

Explain using the K = (∇ ) kernel, how we get dynamical equations for ψ, χ, Bz that are first order in time.

3.2.1

Boundary conditions for non-viscous and viscous conducting fluids

3.2.2

The Reynold number, onset of turbulence, dimensional analysis of the conducting fluid equations

3.3

Finite element method for solving the fluid dynamical equations

Consider for example, the flow of a fluid in one dimension. Its velocity field is v(t, x) ∈ R and its density field is ρ(t, x) ∈ R. Let the equation of state be p = p(ρ). Then, the equations of motion and, equation of continuity can be expressed as ρ(v,t + vv,x ) = −p (ρ)ρ,x + ηv,xx + f (t, x) (ρv),x + ρ,t = 0, where f is the external force per unit volume. We can assume that in the unperturbed state ρ and v are constants with f = 0 and write the perturbed values of these as ρ0 + δρ(t, x) and v0 + δv(t, x) respectively. Then expanding upto second degree terms and discretizing the space and time indices results in second order Volterra difference equation for δρ and δv. Keeping this in mind, we study finite register effects in the simulation of vector valued two dimensional second degree Volterra difference equations of the form Y [n, m] =

p 

A1 [k, l]Y [n−k, m−l]+

k,m=1

+

p 

B1 [k, l]X[n−k, m−l]+

k,l=0

p 

A2 [k, l, r, s](Y [n−k, m−l]⊗Y [n−r, m−s])

k,l,r,s=1

p 

B2 [k, l, r, s](X[n−k, m−l]⊗X[n−r, m−s])

k,l,r,s=0

The effects of truncation and rounding in the implementation of this difference equation on a digital computer is to modify this difference equation to Y [n, m] =

p  k,l=1

A1 [k, l]Y [n−k, m−l]+

p  k,l,r,s=1

A2 [k, l, r, s](Y [n−k, m−l]⊗Y [n−r, m−s])

56

+

Quantum Antennas p 

p 

B1 [k, l]X[n−k, m−l]+

k,l=0

B2 [k, l, r, s](X[n−k, m−l]⊗X[n−r, m−s])

k,l,r,s=0

+



E1,kl [n, m] +

k,l





A2 [k, l, r, s]E2,klrs [n, m] +

k,l,r,s

E4,kl [n, m] +

k,l





E3,klrs [n, m]+

klrs

B2 [k, l, r, d]E5,klrs [n, m] +

klrs



E6,klrs [n, m]

klsr

where Ej,kl [n, m], Ej,klrs [n, m] are truncation and rounding error processes assumed to have independent components uniformly distributed over [0, Δ] or [−Δ/2, Δ/2] according as truncation or rounding is used where Δ = 2−b with b denoting the number of bits in each register. Writing Y [n, m] = Yx [n, m] + Ye [n, m] where Yx is the signal part of the output that has a larger amplitude than the noise part Ye [n, m] of the output, we get on equating the respectively the signal part and first order noise parts, p 

Yx [n, m] =

A1 [k, l]Y [n−k, m−l]+

k,l=1

+

p 

A2 [k, l, r, s](Y [n−k, m−l]⊗Y [n−r, m−s])

k,l,r,s=1

B1 [k, l]X[n−k, m−l]+

k,l=0



p 

p 

B2 [k, l, r, s](X[n−k, m−l]⊗X[n−r, m−s])

k,l,r,s=0

Ye [n, m] = A2 [k, l, r, s](Yx [n−k, m−l]⊗Ye [n−r, m−s]+Ye [n−k, m−l]⊗Yx [n−r, m−s])+

klrs

+



E1,kl [n, m] +

k,l

 k,l



A2 [k, l, r, s]E2,klrs [n, m] +

k,l,r,s

E4,kl [n, m] +





E3,klrs [n, m]+

klrs

B2 [k, l, r, d]E5,klrs [n, m] +

klrs



E6,klrs [n, m]

klsr

The second equation can be used to derive a difference equation for the output noise autocorrelation function.

3.4

Elimination of pressure, incompressible fluid dynamics in terms of just a single stream function vector field with vanishing divergence divv = 0

57

Quantum Antennas This is the incompressibility equation. Thsube Navier-Stokes equation is v,t + (v, ∇)v = −∇p/ρ + ν∇2 v + f where f is the external force per unit mass. This equation can be rearranged as v,t + Ω × v + (1/2)∇(v 2 ) = −∇p/ρ + ν∇2 v + f where Ω=∇×v is the fluid vorticity. The incompressibility condition implies v =∇×ψ where we may assume divψ = 0 in view of the fact that ψ may be replaced by ψ + ∇η where η is any scalar field without affecting v. Hence, we get Ω = −∇ψ and our N.S. equation on taking the curl gives us a nonlinear pde for the stream function vector field ψ: Ω,t + ∇ × (Ω × v) = ν∇2 ω + g, g = ∇ × f or

∇2 ψ,t + ∇ × ((∇2 ψ) × (∇ × ψ)) = ν(∇2 )2 ψ − g

which can be inverted using the Green’s function G(r) = −1/4π|r| for ∇2 :  2 G(r − r )η(r )d3 r = η(r) ∇ We get

3.5

ψ,t + G.(∇ × ((∇2 ψ) × (∇ × ψ)) = −ν∇2 ψ − G.g

Fluids driven by random external force fields

If in the above equation f is a random field, then so is g(t, r) and assuming g to be a zero mean Gaussian field with autocorrelation Rg (t, r|t , r ) =< g(t, r).g(t , r ) >, our aim is to calculate all the moments of the stream function field < ψ(t1 , r1 ) ⊗ ...ψ(tn , rn ) > from which all the statistical moments of the velocity field can be evaluated. This is achieved by solving for ψ using perturbation theory: ψ,t + δ.G.∇ × ((∇2 ψ) × (∇ × ψ)) = ν(∇2 )ψ − G.g

58

Quantum Antennas ψ = ψ0 + δ.ψ1 + δ 2 .ψ2 + ... so that equating coefficients of δ n , n = 0, 1, ... successively gives us ψ0,t − ν∇2 ψ0 + G.g = 0 ψn+1,t − ν∇2 ψn+1 = G.∇ × ((∇2 ψn ) × (∇ × ψn )), n = 0, 1, ... so that if K(t, r) denotes the Green’s function of the diffusion equation, ie, K,t (t, r) − ν∇2 K(t, r) = δ 3 (r) then we have



ψ0 (t, r) = −K.G.g(t, r) = −

K(t − t , r − r )G.g(t , r )d3 r d3 r ,

ψn+1 (t, r) = ν.K.G.∇ × ((∇2 ψn ) × (∇ × ψn ))(t, r), n = 1, 2, ... The convergence of this perturbation series needs to be investigated.

3.6

Relativistic fluids, tensor equations

Let T μν denote the energy-momentum tensor of the fluid field without taking viscous and thermal effects into account and let ΔT μν denote the contribution to the energy-momentum tensor due to viscous and thermal effects. Then the basic special relativistic fluid equations are obtained by energy-momentum conservation: (T μν + ΔT μν ),ν = f μ where f μ is the externally applied four force. Let v μ denote the four velocity √ μ dx /dτ with dτ = dt 1 − u2 where u = (ur )3r=1 with ur = dxr /dt = v r dτ /dt. Then T μν = (ρ + p)v μ v ν − pη μν We leave it as an exercise to write down the above energy-momentum conservation equations and separate it out into the momentum conservation part and the mass conservation part. A nice account of how to calculate ΔT μν using particle number conservation (nv μ ),μ = 0 and the first and second laws of thermodynamics T dS = dU + pdV, dS ≥ 0 in the form T dσ = d(ρ/n) + pd(1/n) where σ is the entropy per particle and n is the number of particles per unit volume is given by Steven Weinberg, ”Gravitation and Cosmology, Principles and applications of the general theory of relativity”, Wiley.

59

Quantum Antennas

3.7

General relativistic fluids, special solutions

For radially moving matter fields, the energy-momentum tensor in the coordinate system (t, r, θ, φ) has the form T 00 = (ρ + p)v 02 − pg 00 , T 11 = (ρ + p)v 12 − pg 11 , T 22 = −pg 22 , T 33 = −pg 33 , T 01 = T 10 = (ρ + p)v 0 v 1 where

g00 v 02 + g11 v 12 = 1

ie,

−1/2

v 0 = g00

(1 − g11 (v 1 )2 )1/2

We are assuming a radially symmetric metric, ie, a metric of the form dτ 2 = A(t, r)dt2 − B(t, r)dr2 − r2 (dθ2 + sin2 (θ)dφ2 ) The Einstein field equations are Rμν − (1/2)Rgμν = −8πGTμν ρ, p are assumed to be functions of (t, r) only. The only non-trivial Einstein field equations are corresponding to the indices (0, 0), (0, 1), (1, 1), (2, 2). These are four equations for the four functions A, B, ρ, v 1 . Here, v 1 is assumed to be a function of (t, r) only. Note that the (3, 3) component of the Einstein field equation is the same as the (2, 2) component since R33 = R22 sin2 (θ), T33 = T22 sin2 (θ), T22 = −pg 22 = p/r2 Problem: When in addition, there is an EM field having radial symmetry, then what are the Einstein-Maxwell field equations ?

3.8

Galactic evolution using perturbed fluid dynamics, dispersive relations. The unperturbed metric is the Robertson-Walker metric corresponding to a homogeneous and isotropic universe g00 = 1, g11 = −f (r)S 2 (t), g22 = −S 2 (t)r2 , f (r) = 1/(1 − kr2 )

Let δgμν denote the perturbed metric tensor due to inhomogeneities. We may perturb our coordinate system slightly to ensure that δg0μ = 0. Then, the number of non-trivial perturbed metric tensor components are six, ie, δgrs , 1 ≤ r ≤ s ≤ 3. These satisfy the perturbed Einstein field equations with the perturbed energy-momentum tensor: δRμν = −8πG(δTμν − δΔTμν − (1/2)δ(T + ΔT )gμν − (1/2)(T + ΔT )δgμν )

60

Quantum Antennas The number of these equations is ten. The functions to be solved for are δgrs , δv r and δρ, namely ten in number. Note that the pressure is determined from the density using the equation of state. If we include the electromagnetic field, by regarding quadratic components of the EM field as being of first order, then we must add to the rhs of the above perturbed Einstein field equation the energymomentum tensor of the EM field and in addition, take into account the Maxwell equations √ (F μν −g),ν = 0 where we treat Aμ as the fundamental EM four potentials which are of (1/2)th order of smallness with raising and lowering of indices including computation of the metric determinant g being done using the unperturbed Robertson-Walker metric. These give us extra four equations for the EM four potential. We note that the perturbed Einstein field equations imply the perturbed fluid equations [δT μν + δΔT μν + S μν ]:ν = 0 where the covariant derivatives are evaluated using the unperturbed metric.

3.9

Magnetohydrodynamics–diffusion of the mag-netic field and vorticity

The nonrelavitistic MHD equations are v,t + (v, ∇)v = −∇p/ρ + ν∇2 v + (σ/ρ)(J × B) J = σ(E + v × B) along with the incompressibility condition divv = 0, and the Maxwell equations divE = ρq /, divB = 0, curlE = −B,t , curlB = μJ + μE,t where ρq,t = −divJ We can write  B = curlA, E = −∇Φ − A,t , A(t, r) = (μ/4π)  Φ(t, r) = (1/4π)

J(t − |r − r |/c, r )d3 r /|r − r |,

ρq (t − |r − r |/c, r )d3 r /|r − r |

61

Quantum Antennas Thus,



B = (μ/4π)

J(t−|r−r |/c, r )(r −r)d3 r /|r−r |3  +(μ/4π) J,t (t−|r−r |/c, r )(r−r )/|r−r |2 )d3 r

If the EM field perturbation with respect to zero electric field and constant magnetic field are weak, then we may assume these to be of the first order of smallness say δE, δB and the fluid velocity field as a small perturbation of a constant velocity field V . Thus, v(t, r) = V + δv(t, r), B(t, r) = B0 + δB(t, r), E(t, r) = δE(t, r) Then, upto linear orders, we have taking into account density fluctuations, δv,t + (V, ∇)δv = −∇δp/ρ + (p/ρ2 )∇δρ + ν∇2 δv+ (σ/rho)[(δE + V × δB) + δv × B0 ] × B0 − (σ/ρ2 )(V × B0 )δρ +(σ/ρ)(V × B0 ) × δB, divδB = 0, curlδE = −μδB,t , curlδB = μδJ + μδE,t where δJ = σ(δE + V × δB + δV × B0 )

3.10

Galactic equation using perturbed Newtonian fluids

According to Hubble’s law, the unperturbed velocity field is of the form V (t, r) = H(t)r. Let δv(t, r) denote its perturbation. Let δρ denote the density perturbation. Then, the Navier-Stokes and matter conservation equations are δv,t (t, r) + H(t)(r, ∇)δv(t, r) + H(t)δv(t, r) = −p (ρ)∇δρ(t, r) + ν∇2 δv(t, r) ρdivδv(t, r) + H(t)r.∇ρ + δρ,t (t, r) = 0

3.11

Plotting the trajectories of fluid particles

If v(t, r) is the velocity field, then the trajectories of a fluid particle are obtained by solving the ode’s dr(t)/dt = v(t, r(t)) and we can approximate this by a difference equation r(t + Δ) = r(t) + Δv(t, r(t)) + Δ2 (∂/∂t + (v(t, r(t)), ∇))v(t, r(t))/2!+ +... + (Δn+1 /(n + 1)!)(∂/∂t + (v(t, r(t)), ∇))n v(t, r(t)) + O(Δn+2 ) In this way, the trajectories of the fluid particles can be plotted.

62

Quantum Antennas

3.12

Statistical theory of fluid turbulence, equations for the velocity field moments, the Kolmogorov-Obhukov spectrum

In terms of components, the incompressible steady state fluid equations are vk,k (r) = 0, vk (r)vi,k (r) = −p,i (r)/ρ + νvi,kk (r) with the summation convention being adopted. The velocity field and the pressure field are assumed to depend only on the spatial location r = (x, y, z) and not on time. We assume homogeneous turbulence, ie, < vi (r)vj (r ) >= Bij (r − r ) < vi (r)vj (r )vk (r ) >= Cijk (r − r , r − r ) etc. Eliminating the pressure gives us (vk vi,k ),j − (vk vj,k ),i = ν(vi,jkk − vj,ikk ) Alternately assuming < vi (r)p(r ) >= Ai (r − r ), < vi (r)vj (r )p(r ) >= Aij (r − r , r − r ) etc, we obtain from the incompressibility equation, < vk,k (r)vj (r ) >= 0 or Bkj,k (r) = 0, < vk,k (r)vi (r )vj (r ) >= 0 or

Ckij,k (r, r ) = 0

and from the Navier-Stokes equations, < vk (r)vi,k (r) >= − < p,i (r) > /ρ + ν < vi,kk (r) > Assuming zero mean pressure and zero mean velocity, this gives Bik,k (0) = 0 Also, < vi,k (r)vk (r)vj (r ) >= − < p,i (r)vj (r ) > /ρ + ν < vi,kk (r)vj (r ) > or

Cikj,k (0, r − r ) − Aj,i (r − r) − νBij,kk (r − r ) = 0

63

Quantum Antennas

We can now obtain more concrete results by assuming that the velocity correlation Bij (r − r ) can be expressed in the isotropic form as Bij (r − r ) =< vi (r)vj (r ) >= P (|r − r |)ni nj + Q(|r − r |)δij where (ni ) is the unit vector along r − r , ie, n = (ni ) = (r − r )/|r − r | and further assume that Ai (r − r ) =< vi (r)p(r ) >= S(|r − r |)ni and Cijk (r, r ) = 0. Exercise: Derive the equations satisfied by the functions P (|r|), Q(|r|), S(|r|).

3.13

Estimating the velocity field of a fluid subject to random forcing using discrete space velocity measurements based on discretization and the Extended Kalman filter

3.14

Quantum fluid dynamics. Quantization of the fluid velocity field by the introduction of an auxiliary Lagrange multiplier field

The fluid equations are (v, ∇)v + v,t = −∇p/ρ + ν∇2 v + f − − − (1) where f is the external force field. This is to be supplemented with the incompressibility condition divv = 0 − − − (2) Using Lagrange multiplier fields λ(t, r) ∈ R3 and μ(t, r) ∈ R, the Lagrangian density from which the fluid equations are derived is L(μ, λ, v, p) = λT ((v, ∇)v + v,t + ∇p/ρ − ν∇2 v − f ) − μdivv − − − (3) or equivalently, using the Einstein summation convention after integrating by parts, L = λi vj vi,j + λi vi,t + λi p,i /ρ + νλi,j vi,j − λi fi − μvj,j The state equations are the constraint equations: ∂L ∂L = 0, =0 ∂λ ∂μ

64

Quantum Antennas

and these are (1) and (2). The costate equations ∂t

∂L ∂L ∂L + ∂j = ∂vi,t ∂vi,j ∂vi

give λi,t + (λi vj ),j + νλi,jj − μ,i − λj vj,i = 0 This can be expressed in vector notation as λ,t + λ.divv + (v, ∇)λ + ν∇2 λ − ∇μ − ∇v ((λ, v)) = 0 where the differential operator ∇v acts only on v. Finally, the costate equation ∂i

∂L =0 ∂p,i

gives divλ = λi,i = 0 In order to quantize this, we must find out the Hamiltonian using the Legendre transformation: ∂L = λi πi = ∂vi,t So the Hamiltonian density is H = πi vi,t − L = −πi vj vi,j − πi p,i /ρ − νπi,j vi,j + πi fi + μvj,j The Hamilton equations are πi,t = − vi,t =

∂L δH ∂H =− + ∂j , δvi ∂vi ∂vi,j ∂H δH ∂H = − ∂j , δπi ∂πi ∂πi,j ∂H =0 ∂μ

We leave it to the reader to check that these yield the correct equations of motion. Let πλi denote the momentum density conjugate to λi and πμ the momentum density conjugate to μ. We note that πi is the momentum density conjugate to vi . Also denote by πp the momentum density conjugate to p. Then our constraint equations are πμ = 0, πλi = 0, πp = 0 We also have constraint equations χ = vi,i = 0, Mi = πi − λi = 0

65

Quantum Antennas

We then form the Dirac bracket taking these constraints into account. For example, some of the standard Poisson brackets are [vi (t, r), χ(t, r )] = 0, [χ(t, r), πi (t, r )] = i∂i δ 3 (r − r ) [χ(t, r), Mi (t, r )] = [vj,j (t, r), πi (t, r )] = i∂i δ 3 (r − r ) etc. These formulas can be used to form the Dirac bracket for quantization Reference: Steven Weinberg, ”The quantum theory of fields”, vol.1, Cambridge University Press.

3.15

Optimal control problems for fluid dynam-ics

The problem is to match the fluid velocity field v(t, r) to a given/desired velocity field vd (t, r) over the time interval [0, T ] and space region B ⊂ R3 . Assuming the fluid to be incompressible, we can write divv = 0. We also assume the desired velocity field vd to be incompressible, ie, divvd = 0. This means that we can derive v, vd from stream fields ψ, ψd using v = curlψ, vd = curlψd We may assume without loss of generality that divψ = divψd = 0 and then the problem of matching the velocity fields amounts to matching the stream vector fields ψ(t, r) and ψd (t, r). Taking the curl of the Navier-Stokes equation gives ∇2 ψ,t + ∇ × (∇2 ψ × (curlψ)) = ν(∇2 )2 ψ − g(t, r) where g = curlf with f begin the control force field. We now regard g as the control force field and it should satisfy the constraint divg = 0. Thus, the objective is to minimize  S(g, ψ, λ, μ) =  ψ(t, r) − ψd (t, r) 2 dtd3 r [0,T ]×B

 −

[0,T ]×B

λ(t, r)T (∇2 ψ,t (t, r)+∇×(∇2 ψ(t, r)×(curlψ(t, r))) −ν(∇2 )2 ψ(t, r)−g(t, r))dtd3 r  −

μ(t, r)divg(t, r)dtd3 r [0,T ]×B

We leave it as an exercise to carry out the variation of S and derive the input and costate equations.

66

Quantum Antennas

3.16

Hydrodynamic scaling limits for simple exclusion models

The lattice over which the exclusion process runs is ZdN . The exclusion process is ηt : ZdN → {0, 1}. ηt (x) is one or zero according as the site x ∈ ZdN is occupied or not by a particle. If site x is occupied while the site y is not at time t, ie ηt (x) = 1, ηt (y) = 0, then there is a probability p(x, y)dt of the particle at x jumping to y, otherwise not. Thus, we can define a family of independent Poisson processes {Nt (x, y) : t ≥ 0}, x, y ∈ ZdN having rates λp(x, y)  with y∈Zd p(x, y) = 1 and the exclusion process ηt then satisfies the stochastic N differential equation  dηt (x) = (−ηt (x)(1 − ηt (y))dNt (x, y) + ηt (y)(1 − ηt (x))dNt (y, x)) y:y =x

The process ηt is Markov and has the infinitesimal generator L where for f : d {0, 1}ZN → R, we have  η(x)(1 − η(y))p(x, y)(f (η (x,y) ) − f (η)) Lf (η) = λ x =y

where

η (x,y) : ZdN → {0, 1}

is the map defined by η (x,y) (z) equals η(z) if z = x, z = y, equals η(x) if z = y and η(y) if z = x. In other words, η (x,y) is the state that interchanges the state of the sites x and y and keeps the state of the other sites the same. We write ρ(t, x/N ) = c(N )E(ηt (x)), x ∈ ZdN where c(N ) is a normalization constant. Then, in the limit N → ∞, ρ(t, .) converges to a function on the d-dimensional torus Td = [0, 1]d and we call this function ρ(t, θ), θ ∈ Td as the limiting density of the exclusion process. Varadhan and other researchers have derived nonlinear partial differential equations for ρ(t, θ) like the Burger’s equation by imposing various conditions on the transition probabilities p(x, y) using Large deviation theory (Reference:The Collected papers of S.R.S.Varadhan, Hindustan Book Agency). The basic idea is to start with a smooth function J : Td → R and consider  J(θ)ρ(t, θ)dd θ [0,1]d

as the limit of



J(x/N )ρ(t, x/N )N −d

x∈Zd N

as N → ∞. We can easily using the sde for ηt write down  dρ(t, x/N )/dt = c(N )λE [(ηt (y)(1 − ηt (x))p(y, x) − ηt (x)(1 − ηt (y))p(x, y)] y:y =x

67

Quantum Antennas

3.17

Appendix: The complete fluid dynamical equations in orthogonal curvilinear coordinate systems specializing to cylindrical and spherical polar coordinates

Let (q1 , q2 , q3 ) be an orthogonal curvilinear coordinate system. Let σab be the stress tensor of pressure and viscosity in the cartesian system and σ ˜ij in the curvilinear system. Then, we have σ ˜ij = σab eia ejb where (ei1 , ei2 , ei3 ) is the unit vector along the qi direction expressed w.r.t. the cartesian basis. Thus, (eia , a = 1, 2, 3) = ∇qi /|∇qi | = Hi−1 ( with Hi = |

∂xa 3 ) ∂qi a=1

3  ∂r ∂xa 2 |= ( ) ∂qi ∂qi a=1

The total pressure and viscous force per unit volume along the ei direction is given by s˜i = σab,b eia = (σab eia ),b − σab eia,b Now we note that since the matrix ((eia )) is orthogonal, we have ˜ij eia ejb σab = σ and hence we can write ˜ij ejb σab eia = σ and so (σab eia ),b = (˜ σij ejb ),b = σ ˜ij,k qk,b ejb + σ ˜ij ejb,b =σ ˜ij,k (∇qk , ej ) + σ ˜ij ejb,b Now, Also,

(∇qk , ej ) = |∇qk |(ek , ej ) = Hk−1 δkj ejb,b = divej = (H1 H2 H3 )−1 (jrs)(Hr Hs ),j

by the formula for divergence in an orthogonal curvilinear coordinate system. Here, j, r, s denote curvilinear indices while a, b, c denote Cartesian indices. Thus, (σab eia ),b =

68

Quantum Antennas Hk−1 σ ˜ik,k + (H1 H2 H3 )−1 σ ˜ij (jrs)(Hr Hs ),j Further, ˜rs era esb eia,b σab eia,b = σ Now, esb eia,b = (es , ∇)eia = Hs−1 eia,s = Hs−1

∂eia ∂qs

on using the formula for the gradient in an orthogonal curvilinear coordinate system. Now, era eia,s is zero if r = a and if r = a, we can evaluate it as follows. For example, consider e2a e2a,1 = 0 because e2a e2a = 1 and hence e2a e2a,1 = 0 which means that e2,1 is orthogonal to e2 and hence in the linear span of e1 and e3 . We can thus write e2,1 = c1 e1 + c3 e3 with c1 = (e2,1 , e1 ), c3 = (e2,1 , e3 ) Now, (e2,1 , e1 ) = −(e2 , e1,1 ) Remark: From the formula for the curl in an orthogonal curvilinear coordinate system, we know that curl(ek /Hk ) = 0, k = 1, 2, 3 Also from the formula for the divergence in an orthogonal curvilinear system, we know that div(e1 /H2 H3 ) = div(e2 /H3 H1 ) = div(e3 /H1 H2 ) = 0 These equations give us divek and curlek in the curvilinear system in terms of the Hj s and their partial derivatives w.r.t the q  s. We now have (e1,1 , e2 ) = −(e1 , e2,1 ) (e1 , e2,1 ) = (e1 , (H2−1 r,2 ),1 ) = H2−1 (e1 , r,21 ) = H2−1 H1−1 (r,1 , r,21 ) = (2H1 H2 )−1 (r,1 , r,1 ),2 = (2H1 H2 )−1 (H12 ),2 = H1,2 /H2 (e1,3 , e2 ) = −(e1 , e2,3 ) = −H1−1 (r,1 , (H2−1 r,2 ),3 ) = −(H1 H2 )−1 (r,1 , r,23 ) = (H1 H2 )−1 (r,12 , r,3 )

69

Quantum Antennas We now observe the following: (r,1 , r,2 ) = 0 (r,13 , r,2 ) + (r,1 , r,23 ) = 0 (r,1 , r,3 ) = 0 (r,12 , r,3 ) + (r,1 , r,23 ) = 0 (r,2 , r,3 ) = 0 (r,12 , r,3 ) + (r,2 , r,13 ) = 0 From these, we deduce that (r,1 , r,23 ) = 0 and likewise, (r,12 , r3 ) = 0, (r,13 , r,2 ) = 0 Thus, we get (e1,3 , e2 ) = 0, (e1,2 , e3 ) = 0, (e3,2 , e1 ) = 0, (e2,1 , e3 ) = 0 ie (er,s , em ) = 0

if r, s, m are all distinct. Thus, we can calculate er,s as linear combinations of  e1 , e2 , e3 and hence evaluate s˜i in terms of the σ ˜ij s, the Hj s and their partial  derivatives w.r.t the qj s. We leave this as an exercise to the reader. Finally, write down the Navier Stokes equations in the form ρ((v, ∇)v + v,t ) = s˜i ei + f we need to express (v, ∇)v in the orthogonal curvilinear system and if we use the explicit form of s˜i ei as −∇p + η∇2 v, then we must also express ∇2 v in the orthogonal curvilinear system. The gradient of p part is easy. It is ∇p = Hi−1 p,i ei The other term is

∇2 v = ∇2 (vi ei ) =

(H1 H2 H3 )−1 ((H2 H3 (vi ei ),1 /H1 ),1 +(H3 H1 (vi ei ),2 /H2 ),2 +(H1 H2 (vi ei ),3 /H3 ),3 ) Since we have already calculated ei,j , 1 ≤ i, j ≤ 3 as linear combinations of the ej s, we leave it as an exercise to evaluate the curvilinear components of ∇2 v in terms of the partial derivatives of the curvilinear components vi of v w.r.t. the qj s.

Chapter 4

Quantum robots in motion carrying Dirac current as quantum antennas 4.1 4.1.1

A short course in classical and quantum robotics with antenna theory applications The Lagrangian for a rigid body

Let B ⊂ R3 denote the volume of the rigid body at time t = 0 and R(t)(B) the same at time t. Thus, R(t) ∈ SO(3). The kinetic energy of the body at time t is given by  |R (t)r|2 d3 r = (1/2)T r(R (t)JR (t)T ) K(t) = (ρ/2) B



where J =ρ

rrT d3 r

B

is the moment of inertia matrix of the rigid body. Taking into account the gravitational potential and external torques, we can write down the Lagrangian as L = K(t) − V (t) where V (t) = mgdR33 (t) + τ (t)T (φ(t), θ(t), ψ(t))T where R(t) = Rz (φ(t))Rx (θ(t))Rz (ψ(t) with φ, θ, ψ being the Euler angles.

71

72

4.1.2

Quantum Antennas

The Hamiltonian for a rigid body

Study project: [1] Express the velocity of each point of a rigid body in terms of the time derivative of the rotation matrix applied to the initial position of the point and hence integrate its norm square over the entire rigid body to obtain the kinetic energy of the rigid body as a quadratic function of the time derivative of the rotation matrix with the quadratic function being determined in terms of the moment of inertia matrix of the body. [2] Express the gravitational potential of the rigid body as a linear function of the rotation matrix and hence determine the total Lagrangian of the body in terms of the instantaneous rotation matrix and its time derivative. [3] By using the well known formula for the differential of the exponential map in Lie group theory, express the Lagrangian of the rigid body in terms of the standard three Lie algebra coordinates and their time derivatives. [4] Perform the Legendre transformation on this Lagrangian to obtain the Hamiltonian of the rigid body in terms of the Lie algebra coordinates and their corresponding canonical momenta.

4.1.3

The Lagrangian and Hamiltonians of a d-link robot with 3-D rigid body links

Study project: Generalize the results of the previous subsection to a robot consisting of d three dimensional rigid links with each link pivoted at the top of the previous link. Describe the state of each link at a given time in terms of a rotation matrix and hence obtain the kinetic energy of the robot as a quadratic function of the time derivatives of d rotation matrices. Likewise, express the total potential energy the robot as a linear function of the d rotation matrices and set up the Lagrangian and Hamiltonian in terms of the Lie algebra coordinates of each rotation matrix using the differential of the exponential map.

4.1.4

The equations of motion of a d-link robot subject to gravitation, external forces and torques

Starting with the result of the previous subsection, add to the Lagrangian additional terms coming from the torques at each pivot provided by motors attached the pivot and hence set up the Euler-Lagrange equations of motion in Lie algebra coordinates. If an external force acts on each link causing the robot to acquire a translational velocity at its base, then describe the contribution of these forces to the Lagrangian in terms of the coordinates and velocities of the pivot of the first link.

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Quantum Antennas

4.1.5

Stochastic differential equations for a d-link robot in the presence of noise in the machine torques and noise in the human hand operator force

Add White Gaussian noise terms to the torque in the equations of motion of the previous subsection and express the resulting equations of motion as a system of 3d coupled stochastic differential equation by noting that White Gaussian noise times dt is the differential of Brownian motion.

4.1.6

Master-slave robots acting in teleoperation with feedback, a stochastic analysis

Describe the motion of two d link robots with the torque applied to the second one having an error feedback component coming from the difference between the Lie algebra coordinates and their velocites of the two robots. Explain using Lyapunov energy theory how these feedback torques can be used to achieve trajectory tracking.

4.2

A fluid of interacting robots

If the lattice site x ∈ Zd is occupied by a robot at time t, we put ηt (x) = 1, otherwise ηt (x) = 0. The transition from site x to site y is accord with a Poisson process Nt (x, y) having rate p(x, y) and the transition takes place at time t iff ηt (x) = 1 and ηt (y) = 0. Thus, the process ηt : Zd → {0, 1} satisfies the sde dηt (x) =



ηt (y)(1 − ηt (x))dNt (y, x) − ηt (x)(1 − ηt (y))dNt (x, y)

y =x

Let X denote the space of all maps η : Zd → {0, 1}, ie, X = {0, 1}Z

d

The generator of the Markov process ηt described by the above sde is therefore given by  Lf (η) = x = yp(x, y)η(x)(1 − η(y))(f (η (x,y) ) − f (η)) To describe fluid dynamics using this model, we introduce an empirical density ρ˜t by the equation   J(y/N )˜ ρt (y/N ) = N −d J(y/N )ηt (y) N −d y∈Zd N

y∈Zd N

where J : [0, 1]d → R

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Quantum Antennas 

is any function. We expect that as N → ∞, this will converge to [0,1]d J(θ)ρt (θ)dθ where ρt (θ) satisfies an appropriate pde which is a nonlinear version of the standard heat equation. Note that

ρ˜t (y/N ) = ηt (y) so we get   d(N −d J(y/N )ρt (y/N )) = N −d J(y/N )(ηt (x)(1−ηt (y))dNt (x, y) x =y

y∈Zd N

−ηt (y)(1−ηt (x))dNt (y, x)) so taking expectations on both sides, and denoting ρt (x/N ) = E[˜ ρt (x/N )] = E[ηt (x)] = P r(ηt (x) = 1) we get d/dt(N −d 



J(y/N )ρt (y/N )) =

y∈Zd N

J(y/N ) − J(x/N ))p(x, y)E[ηt (x)(1 − ηt (y))]

x=y

4.3

Disturbance observer in a robot

The dynamics of the robot is M (q)q  + N (q, q  ) = τ (t) + d(t) where q(t) ∈ Rd and M (q) ∈ Rd×d with M (q) > 0 for all q ∈ [0, 2π)d . d(t) is the disturbance to be estimated. Define ˆ = z(t) + p(q  (t)) d(t) where Then,

ˆ z  = L(q, q  )(N (q, q  ) − τ − d) ˆ dd/dt = z  + p (q  )q  ˆ + p (q  )M (q)−1 (τ + d − N )) = L(q, q  )(N − τ − d)

Assume that Then, we get

p (q  )M (q)−1 = L(q, q  ) ˆ ˆ dd/dt = L(q, q  )(d − d)

75

Quantum Antennas Thus, if L(q, q  ) is a positive definite matrix for all q, q  , then we can expect that ˆ → 0 as t → ∞. In the special case when L(q, q  ) = CM (q)−1 where d(t) − d(t) C is a constant matrix, our disturbance observer reduces to ˆ dˆ = z + p(q  ) z  = C(N (q, q  ) − τ − d), We then require that

p (q  ) = C

or equivalently,

p(q  ) = Cq 

which means that this disturbance observer is simply ˆ dˆ = z + Cq  z  = C(N (q, q  ) − τ − d), and this results in

ˆ ˆ dd/dt = CM (q)−1 (d − d)

so we can expect convergence of d − dˆ to zero provided that for all q, all the eigenvalues of the matrix CM (q)−1 have negative real parts.

4.4

Robot connected to a spring mass with damping system

. The original robot dynamical state equations are X  (t) = ψ(t, X(t)) + G(t, X(t))τ (t) 

where X(t) = [q(t)T , q T (t)]T ∈ R2d are the d-link robot angles and angular velocities. τ (t) is the external torque coming from the motors at the joints of the robot as well as from the environment. The end-effector position of the robot is η(X(t)) ∈ R3 and this is connected to a spring mass system defined by a dynamical equation x (t) = v(t), v  (t) = −γv(t) − kx(t) + fe (t) − frob (t) where x(t) = η(X(t)) = η(q(t)), fe (t) is the external force acting on the spring mass system coming from the environment and −frob (t) is the back reaction of the robot end effector acting on the spring mass system. Thus, frob (t) = fe (t)−γx (t)−kx(t)−x (t) = fe (t)−γη  (q(t))q  (t)−kη(q(t))−(η  (q(t))q(t)) = fe (t)−χ(X(t), X  (t)) is the net external force acting on the end effector of the robot and if J(X(t)) = J(q(t)) denotes the Jacobian matrix of the transformation q(t) → x(t) from

76

Quantum Antennas the robot angles to the end-effector positions, then the net torque acting on the robot from its end effector is by D-Alembert’s principle of virtual work, given by J(X(t))T frob (t) and thus, the robot dynamical equation when we take the computed torque with trajectory tracking error feedback torque uc (t) into account, is given by −1 ˆ (K(t)(Xd (t) X  (t) = ψ(t, X(t))+G(t, X(t))(J(X(t))T frob (t)+G(t, X(t))  ˆ ˆ −X(t))+X d (t)−ψ(t, X(t)))

or equivalently, X  (t) = ψ(t, X(t)) + G(t, X(t))J(X(t))T (fe (t) − χ(η(Xd (t), Xd (t))) −1 ˆ ˆ ˆ +G(t, X(t))G(t, X(t)) (K(t)(Xd (t) − X(t)) + Xd (t) − ψ(t, X(t)))

Here, the computed torque is −1 ˆ ˆ ˆ (K(t)(Xd (t) − X(t)) + Xd (t) − ψ(t, X(t))) τc (t) = G(t, X(t))

If we wish to eliminate the external force fe (t), ie, we regard this force as a disturbance to the robot dynamics, then we should subtract off its estimate fˆe (t) from this dynamics thereby resulting in the modified dynamics X  (t) = ψ(t, X(t)) + G(t, X(t))J(X(t))T (fe (t) − fˆe (t)) −1 ˆ ˆ ˆ +G(t, X(t))G(t, X(t)) (K(t)(Xd (t) − X(t)) + Xd (t) − ψ(t, X(t)) ˆ It should be noted that X(t) is the observer output, ie, an estimate of the state X(t) based on measurement data collected upto time t: {z(s) : s ≤ t}, where

dz(t) = h(t, X(t))dt + σv dV (t) is the measurement model. The observer dynamics is a generalized version of the EKF: ˆ ˆ ˆ dX(t) = ψ(t, X(t)) + L(t)(dz(t) − h(t, X(t))dt) where L(t) is the output error feedback coefficient matrix. When tracking is ˆ good and the observer is also good, we have X(t) ≈ X(t) ≈ Xd (t), fe (t) ≈ fˆe (t),   in which case we get approximately, X (t) = Xd (t) or equivalently, X(t) ≈ Xd (t) which demonstrates self-consistency. In practice, the external force/disturbance fe (t) will be functions of some unknown parameters θ(t) like the amplitude, frequency and phase of sinusoids and then we would represent it as fe (t, θ(t)) ˆ ˆ would be a part and its estimate as fe (t, θ(t)) where the parameter estimates θ(t) of an extended state vector estimate. It should be noted that in general, process noise is present in the robot state dynamics and hence the correct dynamical equation for in terms of computed torque and trajectory tracking error feedback without subtracting out the estimate of the external forces would be given by X  (t) = ψ(t, X(t)) + G(t, X(t))J(X(t))T (fe (t) − χ(η(Xd (t), Xd (t))) −1  ˆ ˆ ˆ +G(t, X(t))G(t, X(t)) (K(t)(Xd (t)−X(t))+X d (t)−ψ(t, X(t)))+G(t, X(t))W (t)

where W (t) = σdB(t)/dt with B(.) being vector valued Brownian motion. More precisely, this equation should be multiplied by dt and expressed in the form of an Ito stochastic differential equation.

Chapter 5

Design of quantum gates using electrons, positrons and photons, quantum information theory and quantum stochastic filtering 5.1

A short course in quantum gates, quantum computation and quantum information with antenna theory applications

5.1.1

one qubit quantum state and quantum gate: Examples including NOT gate, Phase gate, Hadamard gate

5.1.2

Multiple qubit quantum gates: CNOT gate, other controlled unitary gates, swap gate, quantum Fourier transform gate

5.1.3

Information/Von-Neumann entropy of a mixed quantum state

5.1.4

Examples of entropy computation in quantum systems. Entropy computation for noisy quantum evolutions

Assume that system 1 evolves according to the dynamics ρ (t) = −i[H1 , ρ(t)] + θ1 (ρ(t))

77

78

Quantum Antennas

and system 2 evolves according to σ  (t) = −i[H2 , σ(t)] + θ2 (σ(t)) where θ1 , θ2 are Lindblad maps: θ1 (ρ) = (−1/2)

N 

(L∗k Lk ρ + ρL∗k Lk − 2Lk ρL∗k )

k=1

θ2 (σ) = (−1/2)

M 

(Pk∗ Pk σ + σPk∗ Pk − 2Pk σPk∗ )

k=1

Then calculate the rate of change of the relative entropy between the two states at time t: H(ρ(t), σ(t)) = T r(ρ(t)(log(ρ(t)) − log(σ(t)))) hint: Use the formula for the differential of the exponential map in the form: d exp(Z(t)) = exp(Z(t))g(ad(Z(t))−1 (Z  (t)) dt where g(z) =

z 1 − exp(−z)

Equivalently, Z  (t) = g(ad(Z(t))(exp(−Z(t)).

d exp(Z(t))) dt

Take ρ(t) = exp(Z(t)) and deduce that d log(ρ(t)) = g(ad(log(ρ(t))))(ρ(t)−1 ρ (t)) dt Now note that g(z) =

 1 =1+ c[k]z k 2 1 − z/2! + z /3! + ... k≥1

Note that ρ(t) has eigenvalues in the range [0, 1]. So log(ρ(t)) has all eigenvalues in the range (−∞, 0]. Hence, ad(log(ρ(t)) has eigenvalues in the range R = (−∞, ∞). So in order that we be able to substitute ad(log(ρ(t))) for z in the above equation, we require that the Taylor series for g(z) be convergent for all z ∈ R which may not be the case. Note that T r(ρ (t)) = 0 and that for k ≥ 1, T r(ρ(t)ad(log(ρ(t))k (ρ(t)−1 ρ (t))) = T r(ad(log(ρ(t)))k (ρ (t))) = 0 since the commutator of two operators has trace zero (provided that the operators are bounded). Thus, we get T r(ρ(t)

d log(ρ(t))) = 0 dt

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Quantum Antennas and so

d T r(ρ(t).log(ρ(t))) = T r(ρ (t).log(ρ(t))) dt = T r(T (ρ(t)).log(ρ(t)))

where T (X) = −i[H1 , X] + θ1 (X) Now, T r([H1 , ρ].log(ρ)) = T r([H1 .log(ρ), ρ]) = 0 so

5.1.5

d T r(ρ(t).log(ρ(t))) = T r(θ1 (ρ(t)).log(ρ(t))) dt

Entropy of a quantum antenna interacting with a photon bath

Antenna in a quantum electromagnetic field: The antenna consists of N electrons and N positrons which start in a given pure state with prescribed momenta and spins for these particles. This system evolves under the interaction with a photon bath according to the interaction Hamiltonian   HI (t) = J μ (x)Aμ (x)d3 r = −e ψ(t, r)∗ αμ ψ(t, r)Aμ (t, r)d3 r where ψ(t, r) =

N 

ak fk (t, r) + b∗k gk (t, r)

k=1

with ak , bk denoting respectively the annihilation operator of the k th electron and the k th positron, Aμ (t, r) =

p 

¯ k (t, r) ck hk (t, r) + c∗k h

k=1

with ck denoting the annihilation operator of the k th photon in the bath. We write the initial state of the antenna and photon bath as |ψi >= |e1 , ..., eN , p1 , ..., pN , φ(u) > where ei is a one or a zero according as the ith electron is present or not and pi is a one or a zero according as the ith positron is present or not while u = (u1 , ..., up )T ∈ Cp defines the coherent state of the photon bath. Specifically,  |φ(u) >= un c∗n |0 > /n! n

80

Quantum Antennas

where

∗np 1 c∗n = c∗n 1 ...cp

The action of the operators ak , a∗k , bk , b∗k , ck , c∗k on |ψi > is as follows: ak |e1 , ..., eN , p1 , ..., pN , φ(u) >= δ( 1 − ek ))|e1 , ..., 1 − ek ..., eN , p1 , ...pN , φ(u) > a∗k |e1 , ..., eN , p1 , ..., pN , φ(u) >= δ(ek )|e1 , ..., 1 − ek , ...eN , p1 , ..., pN , φ(u) > and likewise for ak , b∗k , ck |e1 , ..., eN , p1 , ..., pN , φ(u) >= uk |e1 , ..., eN , p1 , ..., pN , φ(u) > The Hamiltonian of the system of electrons and positrons inside the quantum antenna is N  HA = (a∗k ak − b∗k bk ) k=1

where the canonical anticommutation rules (CAR) are satisfied: {ak , a∗j } = δkj , {bk , b∗j } = δkj , {ak , aj } = 0, {bk , bj } = 0, {ak , bj } = 0, {ak , b∗j } = 0, {a∗k , a∗j } = 0, {bk , b∗j } = 0, {a∗k , bj } = 0 and the canonical commutation rules (CCR) are satisfied by the ck , c∗k : [ck , c∗j ] = δkj , [ck , cj ] = 0, [c∗k , c∗j ] = 0 Exercise: Express HI (t) as a cubic functional of {ck , c∗k }, {a∗k , bk }, {ak , b∗k } and hence design a perturbation theoretic technique for calculating the state at time t in the interaction picture:  t HI (s)ds)}|ψi > |ψ(t) >= T {exp(−i 0

Hence, calculate the state of the antenna at time t as ρA (t) = T rB (|ψ(t) >< ψ(t)|) where T rB denotes partial trace over the photon state. Also calculate the entropy of the antenna at time t. Finally, using the expression J μ (t, r) = −eψ(t, r)∗ αμ ψ(t, r) (in the interaction representation, the observables ψ(t, r) evolve according to the unperturbed Hamiltonian of the electrons and positrons) calculate the far field electromagnetic field radiated out by the antenna and the moments of the associated Poynting vector in the state |ψ(t) > or equivalently, in the state ρA (t). We can express the evolution of ρA (t) in terms of the Lindblad operators obtained by tracing out over the photon bath and hence evaluate the rate of its d T r(ρ(t).log(ρ(t))). entropy change − dt

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Quantum Antennas

5.1.6

Entangled quantum states and the inequalities of John Bell:The impossibility of constructing a hidden variable theory in quantum mechanics

Let X1 , X2 , X3 be three classical Bernoulli random variables on a given probability space, ie, they assume only the values ±1. Then, it is easy to see that X1 (X2 − X3 ) ≤ 1 − X2 X3 and hence taking expectations, E(X1 X2 ) − E(X1 X3 ) ≤ 1 − E(X2 X3 ) Interchanging X2 , X3 gives us therefore |E(X1 X2 ) − E(X1 X3 )| ≤ 1 − E(X2 X3 ) This is called Bell’s inequality. It is violated by quantum observables taking values ±1 only. For example, let σk , k = 1, 2, 3 be the Pauli spin matrices. Their eigenvalues are ±1. Now consider the mixed state ρ = I2 /2 in the Hilbert space C2 . Let a, b, c be unit vectors in R3 and define the observables 3  X1 = (a, σ) = ak σk , X2 = (b, σ), X3 = (c, σ) k=1

Then X1 , X2 , X3 all have eigenvalues only ±1 and are therefore quantum Bernoulli random variables. Further, their correlations in the state ρ are r(k, j) = T r(ρXk Xj ) and we find that r(1, 2) = r(2, 1) = (a, b), r(2, 3) = r(3, 2) = (b, c), r(3, 1) = r(1, 3) = (a, c) Now define θkj by r(k, j) = cos(θkj ) Then if Bell’s inequality is to be satisfied by these quantum correlations, we must have |cos(θ12 ) − cos(θ13 )| ≤ 1 − cos(θ23 ) Note that θ12 is the angle between the vectors a, b, θ23 is the angle between b, c and finally θ13 is the angle between a, c. Show that unit vectors a, b, c can be chosen so that this inequality is violated.

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Quantum Antennas

5.1.7

Communication using entangled states : Quantum teleportation and superdense coding

5.1.8

A property of quantum entropy

If A, B are two systems and |eA i >, i = 1, 2, ..., n are orthonormal vectors in A’s  Hilbert space while |eB i >, i = 1, 2, ..., n are orthonormal vectors in B s Hilbert space, then consider the following pure state in the tensor product of A s and B  s Hilbert space: n  √ B p(i)|eA |ψ >= i ⊗ ei > where p(i) ≥ 0,



i=1

p(i) = 1. Show that < ψ|ψ >= 1

and that A s state is ρA = T rB (|ψ >< ψ|) =



A p(i)|eA i >< ei |

i

while B  s state is ρB = T rA (|ψ >< ψ|) =



B p(i)|eB i >< ei |

i

Deduce that S(ρA ) = S(ρB ) where S(.) denotes Von-Neumann entropy. If the joint system of A and B evolves according to the Hamiltonian H = HA + HB + VAB where HA acts in A s Hilbert state only, HB acts in B  s Hilbert space only and VAB acts in the tensor product of the two spaces, then write down the evolution of the above state |ψ > in the interaction picture assuming that |eA i > ., i = 1, 2, ..., n are eigenstates of HA and |eB i >, i = 1, 2, ..., n are eigenstates of HB .

5.1.9

Entanglement assisted quantum communication

Evaluation of the maximum rate of information transmission when the transmitter and receiver share an entangled state. [a] Let |ea >, a = 1, 2, ..., N be an onb for Alice’s Hilbert space and |fa >, a = 1, 2, ..., N an onb for Bob’s Hilbert space. Alice and Bob share the maximally entangled state |Φ >= N −1/2

N  a=1

|ea ⊗ fa >

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Quantum Antennas Alice appends the state |ψ >=

N 

C(a)|ea >

a=1

Now Alice appends this state to her share in the entangled state |Φ > that she shares with Bob. The resulting state of Alice and Bob is  |χ >= N −1/2 |ψ > |ea > |fa > a

Now Alice applies a unitary operator W to her share of this state with W defined by the N 2 × N 2 matrix W (a, b|c, d) defined by W (|ea > |eb >) =



W (c, d|a, b)|c > |d >

c,d

Then the resulting state of Alice and Bob becomes  (W ⊗ IB )|χ >= N −1/2 W (|ψ > |ea >)|fa > and since W (|ψ > |ea >) =



C(b)W (|eb > |ea >) =

b

=



W (c, d|a, b)|ec > |ed >

b,c,d

so that the resulting state of Alice and Bob can be written as (W ⊗ IB )|χ >= N −1/2



C(a)W (c, d|a, b)|ec > |ed > |fa >

abcd

Now suppose that Alice applies the measurement {|ea > |eb >: 1 ≤ a, b ≤ N } to her share of this state (ie the projections |ea > |eb >< ea | < eb | = |ea >< ea | ⊗ |eb >< eb | : 1 ≤ a, b ≤ N ). Then if |ec > |ed > her outcome, the state of Bob after applying the collapse postulate becomes  |η(c, d) >= N −1/2 C(a)W (c, d|a, b)|fa > a,b

Alice reports via classical communication, her  measurement outcome (c, d) to Bob and hence Bob knows the numbers C(a) b W (c, d|a, b), a = 1, 2, ..., N from which he gets to know the state |ψ > ie the numbers {C(a):a=1,2,..., N}that Alice wanted to transmit to him.

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Quantum Antennas

[b] Other descriptions of super-dense coding and quantum teleportation. Let |ea >, a = 1, 2, ..., d be an onb for Cd . Assume that Alice and Bob share the epr state d  ΦAB = d−1/2 |ea >A |ea >B a=a

For 0 ≤ α ≤ d − 1, let α=

r−1 

α[k]2k , α[k] = 0, 1

k=0

be its binary expansion. Here, we are assuming that d = 2r . Let X, Y, Z denote the 2 × 2 Pauli spin matrices and define α[k] α[k] X(α) = ⊗r−1 , Y (α) = ⊗r−1 , k=0 X k=0 Y α[k] Z(α) = ⊗r−1 k=0 Z

Then we have for a, b, c, d = 0, 1 that  < m|Z c X d X a Z b |m >= 2δ[a − d]δ[b − c] m=0,1

In fact if a = d or b = c, then Z c X d X a Z b is either proportional to X, Y or Z. The diagonal entries of X and Z are zero while the sum of the diagonal entries of Y is zero. On the other hand, if a = d and b = c, then Z c X d X a Z b = I2 and the sum of its diagonal entries is 2. This proves the claim. It follows from this identity that d−1  < j|Z(β  )X(α )X(α)Z(β)|j >= j=0

Πr−1 k=0



< j[k]|Z β



[k]



X α [k] X α[k] Z β[k] |j[k] >=

j[k]=0,1   2r Πr−1 k=0 δ[α[k] − α [k]]δ[β[k] − β [k]]

= dδ[α − α ]δ[β − β  ] (We denote |ea > by |a − 1 > so that a = 0, 1, ..., d − 1). It follows from this observation that the vectors |Φα,β AB >= (X(α)Z(β) ⊗ Id )|ΦAB >= d−1/2

d−1 

(X(α)Z(β)|j >A )|j >B , α, β = 0, 1, ..., d − 1

j=0 2

forms an orthonormal basis for Cd ⊗ Cd = Cd . Note that if T is a linear transformation in Cd and if T t denotes its transpose in the basis |j >, j = 0, 1, ..., d − 1, then d−1  j=0

(T |j >A )|j >B =

d−1  j=0

|j >A T t |j >B

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Quantum Antennas

Now suppose Alice wishes to trasnmit a pair of numbers (μ, ν), μ, nu = 0, 1, ..., d− 1 to Bob. She Applies the gate X(μ)Z(ν) to her share of q-dits in the shared state ΦAB , so that the total state of Alice and Bob becomes −1/2 |Φμ,ν AB >= d

d−1 

(X(α)Z(β)|j >A )|j >B

j=0

and then transmits her share of the state (log2 (d) qubits) to Bob. Thus, Bob has the state |Φμ,ν AB > and Bob makes a measurement with respect to the Bell basis defined above {|Φα,β AB >, α, β = 0, 1, ..., d − 1} and his measurement outcome is (μ, ν). Note that the PVM operators of Bob’s measurement are |Φα,β AB >< Φα,β |, α, β = 0, 1, ..., d − 1. Thus, by sharing log d qubits with Bob, Alice can 2 AB transmit 2log2 (d) classical bits to Bob by transmitting only log2 (d) qubits. This fact can be expressed by a resource inequality: [qq, log2 (d)] + [q → q, log2 (d)] ≥ [c → c, 2log2 (d)] or more concisely as [qq] + [q → q] ≥ [2c → 2c]

5.1.10

Quantum neural networks (qnn), an example

The input states for training purpose are |ea >, a = 1, 2, ..., N and the corresponding desired output probability distributions are PY (a, k), k = 1, 2, ..., N, a = 1, 2, ..., N , ie PY (a, k) is the desired probability of obtaining the output |fk > when the input is |ea >. Here, |ea >, a = 1, 2, ..., N is an onb for the input Hilbert space Hi while |fk >, k = 1, 2, ..., N is an onb for the output Hilbert space Ho . The qnn is a unitary matrix W of size N × N mapping Hi onto Ho . We parametrize W by p Hermitian matrices X1 , ..., Xp , so that W = W (θ1 , ..., θp ) = W (θ) = exp(i

p 

θk X k )

k=1

The ”weights” θ = (θ1 , ..., θp )T are to be selected so that | < fk |W (θ)|ea > |2 ≈ PY (a, k), a, k = 1, 2, ..., N This means that the weights θ have to be trained so that the ”error energy” E(θ) =

N 

| < fk |W (θ)|ea > |2 − PY (a, k)|2

k,a=1

is a minimum. Such a training can be achieved for example using the gradient search algorithm. We can also talk about adaptive qnn’s in, for example, the

86

Quantum Antennas

following way. Assume that the desired quantum system has Hamiltonian H(t) so that the Schrodinger evolution is |ψ  (t) >= −iH(t)|ψ(t) >, t ≥ 0 H(t) is to be estimated on a real time basis. Let us approximate the corresponding unitary evolution operator  t H(s)ds)} U (t) = T {exp(−i 0

by W (θ(t)). Then θ(t) is to be varied with time so that W (θ(t))|ψ(0) > follows |ψ(t) >. This can be achieved for example, using the gradient algorithm: θ(t + Δ) = θ(t) − μ∇θ  |ψ(t) > −W (θ(t))|ψ(0) >2 where μ is an adaptation constant.

5.1.11

Design of quantum gates using the interaction of electrons,positrons, photons and gravitons

The metric tensor of the gravitational field is gμν (x) = ημν + hμν (x) where ημν is the Minkowski metric of flat space-time and hμν (x) is a small perturbation of flat space-time. The linearized Einstein field equations after choosing an appropriate coordinate system (harmonic coordinates) reduce to hμν (x) = 0 the solution to which is given by a plane wave expansion  hμν (x) = (d(K, σ)eμν (K, σ)exp(−ik.x) + d(K, σ)∗ e¯μν (K, σ)exp(ik.x))d3 K where k = (|K|, K), k.x = kμ xμ = |K|t − K.r. The sum runs over σ taking five values −2, −1, 0, 1, 2. This follows from the constraints imposed by the four coordinate conditions hμν,μ − h,ν /2 = 0 required to obtain the wave equation from the linearized version of the Einstein field equations. These constraints lead to the condition that there are only five linearly independent linear combinations of the eμν (K) s for a given value of the three vector K. For example, we can take a wave propagating along the z direction and write down these conditions to deduce this fact. This also means that gravitons have spin two. Now, we can write down the energy density of the gravitational field enclosed inside a finite volume. For doing so, we have to first

87

Quantum Antennas

write down the energy-momentum tensor of the gravitational field. That comes from Einstein field equations in the presence of matter and radiation. Suppose matter and em radiation have total energy momentum T μν . The Einstein field equations are then Rμν − (1/2)Rg μν = −8πGT μν and the covariant divergence of both sides vanishes by virtue of the Bianchi identity: (Rμν − (1/2)Rg μν ):ν = 0 Now define the Einstein tensor Gμν = Rμν − (1/2)Rg μν and express it as Gμν = Gμν(1) + Gμν(2) where Gμν(1) is linear in the hμν and its first and second order partial derivatives w.r.t to space-time. Then, Gμν(2) = Gμν − Gμν(1) . It is easy to show that the ordinary four divergence of Gμν(1) vanishes: Gμν(1) =0 ,ν and therefore, we get the conservation law (T μν + Gμν(2) /8πG),ν = 0 which means that τ μν = Gμν(2) /8πG must be interpreted as the pseudo-tensor of the gravitational field for this interpretation would guarantee that the total energy and momentum of matter, em radiation and gravitation is conserved. Now we can evaluate the energy density τ 00 of the gravitational field upto quadratic field  00 3 orders in the hμν and hence the total energy of the gravitational τ d r upto quadratic orders in the coefficients d(K, σ), d(K, σ)∗ . This would then ensure that upto second order, the Hamiltonian of the gravitational field is that of an ensemble of harmonic oscillators. The result of this calculation will yield the Hamiltonian of the gravitational field in the form  HG = f (K, σ)d(K < σ)∗ d(K, σ)d3 K with the Bosonic commutation relations [d(K, σ), d(K  , σ  )∗ ] = δ 3 (K − K  )δ(σ, σ  ) After discretizing this, we can write HG =

NG  k=1

f1 [k]d[k]∗ d[k]

88

Quantum Antennas Now the total unperturbed Hamiltonian of the electron-positron field (Dirac field) is given by ND  E[k](a[k]∗ a[k] − b[k]∗ b[k]) HD = k=1

and the unperturbed Hamiltonian of the photon field (Maxwell field)is given by HM =

M 

f2 [k]c[k]∗ c[k]

k=1

The interaction Hamiltonian between the Dirac field and the gravitational field is obtained using the spinor connection for the gravitational field. It is quadratic in the Dirac field but highly nonlinear in the gravitational field. It is represented by   HDG (t) = −e Vaμ (x)Re(ψ(x)∗ αa Γμ (x)ψ(x)) −g(x)d3 r where αa = γ 0 γ a and Vaμ (x) is the tetrad of the gravitational metric gμν (x). Γμ (x) = (−1/2)V ν )a(x)vbν:μ (x)J ab is the spinor connection of the gravitational field where J ab = (1/4)[γ a , γ b ] are the Lie algebra generators of the Dirac spinor representation of the Lorentz group. We can express HDG as  HDG (t) = (F1rs (t, d[m], d[m]∗ , m = 1, 2, ..., NG )a[r]∗ a[s]+ r,s

+F2rs (t, d[m], d[m]∗ , m = 1, 2, ..., NG )b[r]b[s]∗ +F3rs (t, d[m], d[m]∗ , m = 1, 2, ..., NG )a[r]∗ b[s]∗ )+cc. where cc denotes the adjoint of the previous terms. The interaction Hamiltonian between the Dirac field and the photon field is simpler. It is given by  HDEM (t) = −e Vaμ (x)ψ(x)∗ αa ψ(x)Aμ (x)d3 r We write Vaμ (x) = δaμ + Uaμ (x) and then express HDEM (t) = HDEM 0 (t) + HDEM G (t) where HDEM 0 (t) does not involve the gravitational tetrad Vaμ , ie, Vaμ is replaced by δaμ while HDEM G (t) involves all the three fields Uaμ , ψ(x), Aμ (x). HDEM (t) can therefore be represented using the quantum Maxwell field representation  Aμ (x) = (c[k]χkμ (x) + c[k]∗ χ ¯kμ (x)) k

Thus, we can write HDEM 0 (t) =

 kml

(g1 (t)a[k]∗ a[m]c[l]+g2 (t)b[k]b[m]∗ c[l]+g3 (t)a[k]∗ b[m]∗ c[l]+g4 (t)b[k]a[m]c[l])+cc.

89

Quantum Antennas HDEM G (t) =



(L1 (t, d[m], d[m]∗ ,

kml

m = 1, 2, ..., NG ))a[k]∗ a[m]c[l]+L2 (t, d[m], d[m]∗ , m = 1, 2, ..., NG )b[k]b Finally, we describe the interaction energy between the Maxwell photon field and the gravitational field. The corresponding Lagrangian density is proportional √ to Fμν F μν −g. This is quadratic in the EM potentials but highly nonlinear in the metric of space-time. If we subtract out the part from this not involving the metric perturbations hμν , we get the Lagrangian density of the free electromagnetic field, ie, em field in flat space-time. Thus, the interaction Hamiltonian between the Maxwell field and the gravitational field can be expressed as HGEM (t) =



(G1kr (t, d[m], d[m]∗ , m = 1, 2, ..., NG )c[k]c[r]+G2kr (t, d[m], d[m]∗ ,

k,r

m = 1, 2, ..., NG )c[k]∗ c[r])+

The total Hamiltonian is H(t) = HG + HD + HEM + HDG (t) + HDEM 0 (t) + HDEM G (t) It should be noted that in the above expression for HG , we’ve taken only quadratic terms in the d[m], d[m]∗ into account. To be more accurate, we have to express the energy density of the gravitational field τ 00 as a power series in d[m], d[m]∗ , ie, HG would be replaced by HG0 + HG1 where HG0 is the HG above, ie, quadratic in d[m], d[m]∗ while HG1 contains cubic and higher powers of d[m], d[m]∗ .

5.1.12

The notion of measurement in the quantum theory

Let ρ be a mixed state in a Hilbert space H and let |ψ > be a pure state in another Hilbert space K. Then ρ ⊗ |ψ >< ψ| is a mixed state in H ⊗ K. Let U be a unitary operator in this tensor product space and consider the state T (ρ) = T rK (U (ρ ⊗ |ψ >< ψ|)U ∗ ) We can write U=

N 

Vm ⊗ Wm

m=1

where Vm , Wm are linear operators respectively in the spaces H and K. Then, we find that N  Vm ρVn jT r(Wm |ψ >< ψ|Wn∗ ) T (ρ) = m,n=1

=

N  m,n=1

< ψ|Wn∗ Wm |ψ > Vm ρVn∗

90

Quantum Antennas

Now ((< ψ|Wn∗ Wm |ψ >)) is clearly an N × N positive definite complex matrix and hence, it has an eigendecompostion N 

λ[r]|er >< er |, λ[r] ≥ 0, < er |es >= δrs

r=1

or equivalently in terms of components, < ψ|Wn∗ Wm |ψ >=

N 

λ[r]¯ er [n]er [m]

r=1

From this, we get T (ρ) =

N 

λ[r]¯ er [n]er [m]Vm ρVn∗

m,n,r=1

=

N 

Er ρEr∗

r=1

where Er is a linear operator in H defined by N   er [m]Vm Er = λ[r] m=1

Clearly, the condition T r(ρ) = 1 implies T r(T (ρ)) = 1 and hence N 

Er∗ Er = IH

r=1

5.2

The Baker-Campbell-Hausdorff formula. A, B are n x n matrices

The aim is to define the matrix C(t) by exp(C(t)) = exp(tA).exp(tB) or equivalently, C(t) = log(exp(tA).exp(tB)) and obtain a Taylor expansion for C(t) with matrix coefficients. We note that by the formula for the differential of the exponential map, d (exp(tA).exp(tB)) = exp(tA)(A+B)exp(tB) dt = exp(C(t))((I−exp(−ad(C(t))/ad(C(t)))(C  (t))

91

Quantum Antennas or equivalently, exp(−t.ad(B))(A) + B = exp(−tB)(A + B).exp(tB) = g(ad(C(t))−1 (C  (t)) where g(z) = Thus,

z 1 − exp(−z)

C  (t) = g(ad(C(t))(exp(−t.ad(B))(A) + B) − − − (1)

By writing C(t) =



C m tm

m≥0

and substituting this into the differential equation (1), we can successively determine the coefficients Cm , m = 0, 1, .... Note that C0 = I. We will of course require the Taylor expansion of g(z).

5.3

Yang-Mills radiation field (an approxima-tion)

τa , a = 1, 2, ..., N are the Hermitian generators of the gauge group G ⊂ U (N ) and C(abc) are the associated structure constants: [τa , τb ] = C(abc)iτc The gauge field is Aμ (x) = Aaμ (x)τa and the covariant derivative is ∇μ = ∂μ + ieAμ The field tensor Fμν (x) is defined by the curvature of this connection: ieieFμν = [∇μ , ∇ν ] = ie(Aν,μ − Aμ,ν ) − e2 [Aμ , Aν ] = [ie(Aaν,μ − Aaμ,ν ) − e2 C(bca)Abμ Acν ]iτa so writing a τa Fμν = Fμν

we get a = Aaν,μ − Aaμ,ν − eC(bca)Abμ Acν Fμν

The field equations are derived from the action principle  a F aμν d4 x = 0 δ Fμν

92

Quantum Antennas This gives aμν F,μ + eC(acb)Acν F bμν = 0

or equivalently, a,ν b c ,ν Aa,ν ν,μ − Aμ,ν − eC(bca)(Aμ Aν )

+e2 C(acb)C(pqb)Acν Apμ Aqν = 0 This contains linear, quadratic and cubic terms in the gauge potentials Aaμ (x). We impose a Lorentz gauge condition Aa,μ μ =0 or equivalently, Aaμ ,μ = 0 Then, the field equations simplify to c b,ν Aa,ν μ,ν − eC(bca)Aν Aμ

+e2 C(acb)C(pqb)Acν Apμ Aqν = 0 − − − (1) We solve this approximately upto O(e2 ) using perturbation series taking e as the perturbation parameter. Specifically, + eA(a)(1) + e2 Aa(2) + ... Aaμ = Aa(0) μ μ μ Substituting this into (1) and equating coefficients of e0 , e1 , e2 respectively gives Aa(0) μ (x) = 0, − − −(2) Aa(1) μ (x) = b(0),ν −C(bca)Ac(0) ν Aμ q(0) +e2 C(acb)C(pqb)Acν(0) Ap(0) = 0 − − − (3) μ Aν

= Aa(2) μ b(1),ν b(0),ν + Ac(1) C(bca)(Ac(0) ν Aμ ν Aμ q(0) −C(acb)C(pqb)Acν(0) Ap(0) μ Aν

93

Quantum Antennas

5.4

Belavkin filter applied to estimating the spin of an electron in an external magnetic field. We assume that the magnetic field is B0 (t) ∈ R3

The spin magnetic moment of the electron is μ = gehσ/8πm = aσ say, where σ = (σ1 , σ2 , σ3 ) are the three Pauli spin matrices and we define σ0 = I 2 The HP Schrodinger equation is written down taking the system Hilbert space as h = C2 and the noise bath space as Γs (L2 (R+ )) It is given by dU (t) = (−(iH(t) + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) where H(t), P, L1 , L2 , S ∈ C2×2 = B(h), H(t) = (μ, B0 (t)) = a(σ, B0 (t)) The star unital homomorphism associated with U (t) is given by jt (X) = U (t)∗ XU (t) where X ∈ h ⊗ Γs (L2 (R+ )). The Belavkin filter measurement is taken as a mixture of quantum Brownian motion and the quantum Poisson process: Yo (t) = U (t)∗ Yi (t)U (t), Yi (t) = A(t) + A(t)∗ + cΛ(t) = B(t) + cΛ(t) where B(t) = A(t) + A(t)∗ is classical Brownian motion. The quantum Ito table is dA.dA = 0, dA.dA∗ = dt, dA.dΛ = dA, dA∗ dA = 0, dA∗ dA∗ = 0, dA∗ dΛ = 0, dΛ.dA = 0, dΛ.dA∗ = dA∗ , dΛ.dΛ = dΛ

94

Quantum Antennas Using this table, we compute (dYi (t))2 = dt + c.dB + c2 dΛ, and in general, for n ≥ 1 we write (dYi (t))n = a[n]dt + b[n]dB(t) + d[n]dΛ(t) Then, we get (dYi (t))n+1 = (dB + cdΛ)(b[n]dB + d[n]dΛ) = b[n]dt + cd[n]dΛ + d[n]dA + cb[n]dA∗ = a[n + 1]dt + b[n + 1]dB + d[n + 1]dΛ from which we infer the recursions a[n + 1] = b[n], b[n + 1] = d[n] = cb[n], d[n + 1] = cd[n], n ≥ 1 with the initial conditions a[1] = 0, b[1] = 1, d[1] = c The solution to this recursion is b[n] = cn−1 , d[n] = cn , a[n + 1] = cn−1 , n ≥ 1 Thus, for n ≥ 2, we get (dYi (t))n = cn−2 dt + cn−1 dB(t) + cn dΛ(t) = cn−2 dt + cn−1 (dB + cdΛ) = cn−2 dt + cn−1 dYo , n ≥ 2 We now have for a system observable X, djt (X) = jt (θ0t (X))dt + jt (θ1 (X))dA(t) + jt (θ2 (X))dA(t)∗ + jt (θ3 (X))dΛ(t) where

θ0t (X) = i[H(t), X] + L∗2 XL2 − P ∗ X − XP θ1 (X) = L∗2 X + XL1 + L∗2 XS, θ2 (X) = L∗1 X + XL2 + S ∗ XL2 , θ3 (X) = S ∗ XS + S ∗ X + XS

Also,

dYo (t) = dYi (t) + dU (t)∗ dYi (t)U (t) + U (t)∗ dYi (t)dU (t) dA.dYi = dt + cdA, dA∗ .dYi = cdA∗ , dΛ.dYi = dA∗ + cdΛ, dYi dA = cdA, dYi dA∗ = dt + cdA∗ , dYi dΛ = dA + cdΛ

We therefore deduce that dYo (t) = dYi (t) + jt (cL∗1 + cL2 + S ∗ )dA∗ + jt (cL∗2 + cL1 + S)dA

95

Quantum Antennas +cjt (S ∗ + S)dΛ = jt (cL∗2 + cL1 + S + 1)dA + jt (cL∗1 + cL2 + S ∗ + 1)dA∗ + cjt (S ∗ + S + 1)dΛ Problem: Calculate (dYo (t))n, n = 2, 3, ... using recursion formulas. hint: Let (dYo (t))n = jt (P0 [n])dt+jt (P1 [n])dA(t)+jt (P2 [n])dA(t)∗ +jt (P3 [n])dΛ(t), n ≥ 1 where Pk [n], k = 0, 1, 2, 3 are 2 × 2 matrices, ie, system matrices. Then using the Homomorphism property of jt and quantum Ito’s formula, we deduce that jt (P0 [n + 1])dt + jt (P1 [n + 1])dA(t) + jt (P2 [n + 1])dA(t)∗ + jt (P3 [n + 1])dΛ(t) = (dYo (t))n+1 = (jt (cL∗2 + cL1 + S + 1)dA + jt (cL∗1 + cL2 + S ∗ + 1)dA∗ + cjt (S + S ∗ + 1)dΛ) ×(jt (P0 [n])dt + jt (P1 [n])dA(t) + jt (P2 [n])dA(t)∗ + jt (P3 [n])dΛ(t)) = jt ((cL∗2 +cL1 +S+1)P2 [n])dt+jt ((cL∗2 +cL1 +S ∗ +1)P3 [n])dA +jt (c(S+S ∗ +1)P2 [n])dA∗ +jt (c(S + S ∗ + 1)P3 [n])dΛ and therefore P0 [n + 1] = (cL∗2 + cL1 + S + 1)P2 [n], P1 [n + 1] = (cL∗2 + cL1 + S ∗ + 1)P3 [n], P2 [n + 1] = c(S + S ∗ + 1)P2 [n], P3 [n + 1] = c(S + S ∗ + 1)P3 [n] The initial conditions are P0 [1] = 0, P1 [1] = cL∗2 +cL1 +S+1, P2 [1] = cL∗1 +cL2 +S ∗ +1, P3 [1] = c(S+S ∗ +1) The solution to the above recursion is therefore P3 [n] = (c(S + S ∗ + 1))n , P1 [n + 1] = (cL∗2 + cL1 + S ∗ + 1)(c(S + S ∗ + 1))n , P2 [n] = (c(S + S ∗ + 1))n−1 (cL∗1 + cL2 + S ∗ + 1), P0 [n + 1] = (cL∗2 + cL1 + S + 1)(c(S + S ∗ + 1))n−1 (cL∗1 + cL2 + S ∗ + 1), n ≥ 1 We can solve for dA, dA∗ , dΛ in terms of dt, dYo , (dYo )2 and (dYo )3 with coefficients being jt of some system operators. Specifically, we consider the three linear equations (dYo )n − jt (Po [n])dt = jt (P1 [n])dA + jt (P2 [n])dA∗ + jt (P3 [n])dΛ, n = 1, 2, 3

and use jt (X)−1 jt (Y ) = jt (X −1 Y ) and analogous relations to solve for dA, dA∗ , dΛ as linear combinations of dt, (dYo )n , n = 1, 2, 3 with coefficients being jt of some functions of the system operators Pk [n], k = 0, 1, 2, 3, n = 1, 2, 3. The generalized Belavkin filtering equation can be expressed as  dπt (X) = Ft (X)dt + Gkt (X)(dYo (t))k k≥1

96

Quantum Antennas

where Ft (X), Gkt (X) ∈ ηt = σ(Yo (s) : s ≤ t) We note that by definition, πt (X) = E[jt (X)|ηt ] and hence (the orthogonality principle) E[(jt (X) − πt (X))Ct ] = 0, Ct ∈ ηt Defining Ct ∈ ηt by dCt =



fk (t)(dYo (t))k Ct , C0 = 1

k≥1

we get by applying quantum Ito’s formula to the above equation and using the arbitrariness of the complex functions fk (t) that E(djt (X) − dπt (X)|ηt ) = 0 − − − (1), E[(djt (X)−dπt (X))(dYo (t))k |ηt ]+E[(jt (X)−πt (X))(dYo (t))k |ηt ] = 0, j ≥ 1−−−(2) Thus, assuming that expectations are in the state |f ⊗ φ(u) > where |φ(u) >= exp(−|u|2 /2)|e(u) > with |e(u) > the standard exponential vector and f ∈ C2 , |f | = 1, we get ¯(t)θ2 (X) + |u(t)|2 θ3 (X)) = πt (θ0t (X) + u(t)θ1 (X) + u  Gkt (X)πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]) − − − (3) Ft (X) + k≥1

and E(jt (θ1 (X))dA(t)(dYo (t))k |ηt ) + E(jt (θ2 (X))dA(t)∗ (dYo (t))k |ηt ) +E(jt (θ3 (X))dΛ(t)(dYo (t))k |ηt ) −



¯(t)P2 [k] + |u(t)|2 P3 [k])dt −Ft (X)πt (P0 [k] + u(t)P1 [k] + u Gmt (X)πt (P0 [k + m] + u(t)P1 [k + m] + u ¯(t)P2 [k + m] + |u(t)|2 P3 [k + m])dt

m≥1

+πt (X(P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]))dt ¯(t)P2 [k] + |u(t)|2 P3 [k])dt = 0 − − − (4a) −πt (X)πt (P0 [k] + u(t)P1 [k] + u Now, dA(t).(dYo (t))k = jt (P2 [k])dt + jt (P3 [k])dA(t) so that E(jt (θ1 (X))dA(t).(dYo (t))k |ηt ) = πt (θ1 (X)(P2 [k]) + u(t)P3 [k]))dt

97

Quantum Antennas dA(t)∗ (dYo (t))k = 0 dΛ(t)(dYo (t))k = jt (P2 [k])dA(t)∗ + jt (P3 [k])dΛ(t) and hence u(t)P2 [k]) + |u(t)|2 P3 [k]))dt E(jt (θ3 (X))dΛ(t)(dYo (t))k |ηt ) = πt (θ3 (X)(¯ Hence, (4a) can be expressed as πt (θ1 (X)(P2 [k] + u(t)P3 [k])) +πt (θ3 (X)(¯ u(t)P2 [k] + |u(t)|2 P3 [k])) +πt (X(P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k])) −πt (X)πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]) = +



Ft (X)πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]) Gmt (X)πt (P0 [k+m]+u(t)P1 [k+m]+¯ u(t)P2 [k+m]+|u(t)|2 P3 [k+m]), k ≥ 1−−−(4b)

m≥1

(3) and (4b) are to be solved for Ft (X), Gkt (X), k ≥ 1 giving the desired generalized Belavkin filter. Eliminating Ft (X) using (3), we can express the filter as dπt (X) =

+



πt (θ0t (X) + u(t)θ1 (X) + u ¯(t)θ2 (X) + |u(t)|2 θ3 (X))dt Gkt (X)((dYo (t))k − πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k])dt)

k≥1

where Gkt (X), k ≥ 1 satisfy πt (θ1 (X)(P2 [k] + u(t)P3 [k])) + πt (θ3 (X)(¯ u(t)P2 [k] + |u(t)|2 P3 [k])) +πt (X(P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k])) −πt (X)πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]) = (πt (θ0t (X)+u(t)θ1 (X)+¯ u(t)θ2 (X)+|u(t)|2 θ3 (X)) −



Gmt (X)πt (P0 [m]+u(t)P1 [m]+¯ u(t)P2 [m]+|u(t)|2 P3 [m]))

m≥1

+



¯(t)P2 [k] + |u(t)|2 P3 [k]) ×πt (P0 [k] + u(t)P1 [k] + u Gmt (X)πt (P0 [k+m]+u(t)P1 [k+m]+¯ u(t)P2 [k+m]+|u(t)|2 P3 [k+m]), k ≥ 1−−−(4b)

m≥1

or equivalently, πt (θ1 (X)(P2 [k] + u(t)P3 [k])) + πt (θ3 (X)(¯ u(t)P2 [k] + |u(t)|2 P3 [k])) +πt (X(P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k])) −πt (X)πt (P0 [k] + u(t)P1 [k] + u ¯(t)P2 [k] + |u(t)|2 P3 [k]) =

98

Quantum Antennas

(πt (θ0t (X)+u(t)θ1 (X)+¯ u(t)θ2 (X)+|u(t)|2 θ3 (X)).πt (P0 [k]+u(t)P1 [k]+¯ u(t)P2 [k]+|u(t)|2 P3 [k])

+



Gmt (X)[πt (P0 [k + m] + u(t)P1 [k + m] + u ¯(t)P2 [k + m] + |u(t)|2 P3 [k + m])

m≥1

−πt (P0 [m]+u(t)P1 [m]+¯ u(t)P2 [m]+|u(t)|2 P3 [m])πt (P0 [k]+u(t)P1 [k]+¯ u(t)P2 [k]+|u(t)|2 P3 [k])]−−−(5)

Chapter 6

Pattern classification for image fields in motion using Lorentz group representations 6.1

SL(2,C), SL(2,R) and image processing

For dealing with images having rapid time variations, special relativity must be taken into account while dealing with measurements of the images in different inertial frames. Specifically, if (t, x, y, z) is a coordinate system in the frame K and the image field in this coordinate system is f (t, x, y, z) = f (t, r), then in a frame K  that is moving relative to K with a uniform velocity after rotation, the coordinates (t , r ) are related to the coordinates (t, r) in K by a Lorentz transformation (t , r )T = L(t, r)T L will be a proper orthochronous Lorentz transformation. Writing this as xr = Lrs xs , r = 0, 1, 2, 3 with summation over the repeated index s = 0, 1, 2, 3 being implied where x0 = t, x1 = x, x2 = y, x3 = z and likewise for xr , r = 0, 1, 2, 3, we have 







x02 − x12 − x22 − x32 = x20 − x21 − x22 − x23 and hence we deduce that L200 − L210 − L220 − L230 = 1 so that L00 = ±

1 + L210 + L220 + L230

99

100

Quantum Antennas

In particular, |L00 | ≥ 1 T

Note that since L is also a Lorentz transformation, we have L200 − L201 − L202 − L203 = 1 and hence L00 = ±

1 + L201 + L202 + L203

If L00 ≥ 1, then we say that L is an orthochronous Lorentz transformation. In this case, suppose x20 − x21 − x22 − x23 > 0 and x0 > 0. Then, x0 = L00 x0 + L01 x1 + L02 x2 + L03 x3

1 + L201 + L202 + L203 x0 + L01 x1 + L02 x2 + L03 x3 ≥ 1 + L201 + L202 + L203 x0 − L201 + L202 + L203 x21 + x22 + x23 ≥ L201 + L202 + L203 (x0 − x21 + x22 + x23 ) > 0 =

This property characterizes orthchronocity of a Lorentz transformation. A Lorentz transformation has the property that det L = ±1 If detL = 1, we say that L is proper. All proper orthochronous Lorentz transformations form a group and this group is generated by proper rotations and boosts of the spacetime manifold. This group is denoted by G0 = {L : L ∈ G : detL = 1, L00 ≥ 1} where G is the group of all Lorentz transformations, ie all L such that LT ηL = η where η = diag[1, −1, −1, −1] is the Minkowski metric. G0 is the connected component of G containing the identity transformation. SL(2, C) is the covering group of G0 . In fact for any L ∈ G0 , there exists a unique pair ±A with A ∈ SL(2, C) such that AΦ(t, r)A∗ = Φ(L(t, r)) 

where Φ(t, r) =

t+z x + iy

x − iy t−z



We see therefore that G0 is isomorphic to the group SL(2, C)/{±I}. If we restrict to SL(2, R), then we obtain all Lorentz transformations in the space (t, x, 0, z), ie, rotations and boosts in the x − z plane. Thus, we can use the characters of the discrete series to do pattern recognition for planar image field moving at relativistic speeds. Exercise: Let A ∈ SL(2, R). Assume that L is the Lorentz transformation defined by Φ(L(t, r)) = AΦ(t, r)A∗ Show that L fixes y. In fact this is obvious since y occurs with a factor of i in Φ(t, r) while t, x, z appear with factors ±1. Hence if A is a real SL(2, C) matrix, it would not touch the imaginary components ±iy in Φ(t, r).

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Quantum Antennas

The discrete series for SL(2, R): Let H, X, Y be the standard generators of SL(2, C), ie,   1 0 H= , 0 −1   0 1 X= , 0 0   0 0 Y = 1 0 These satisfy the commutation relations [H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H The irreducible representations of SL(2, C) are obtained as follows. Suppose first that there is a highest weight vector v0 in this representation π. Then, X.v0 = 0, H.v0 = λ0 v0 where by X.v0 , we mean π(X).v0 etc with π denoting the irreducible representation of the Lie algebra ∫ l(2, R). Then the commutation relations imply on defining vs = Y s v0 , s = 0, 1, 2, ... that H.vs = (λ0 − 2s)vs , Y vs = vs+1 , s ≥ 0 and since the representation is irreducible, we must have Xvs = α(s)vs−1 for some scalars α(s). We find that α(s)vs = α(s)Y vs−1 = Y Xvs = [Y, X]vs +XY vs = −Hvs +Xvs+1 = −(λ0 −2s)vs +α(s+1)vs or α(s) = α(s + 1) − (λ0 − 2s) With the initial condition α(0) = 0 we get α(s) =

s−1 

(λ0 − 2k) = λ0 s − s(s − 1) = s(λ0 − s + 1)

k=0

We thus have an infinite dimensional representation of SL(2, R) spanned by the vectors {vs : s ≥ 0} with the Lie algebra actions Hvs = (λ0 − 2s)vs , Xvs = s(λ0 − s + 1)vs−1 , Y vs = vs+1 , s ≥ 0

102

Quantum Antennas

Suppose that for some s0 = 1, 2, ..., we have λ0 − s0 + 1 = 0, ie, λ0 = s0 − 1. Then, span{vs : s ≥ s0 } would be an invariant subspace for the representation and hence the representation would not be irreducible. This means that the representation is irreducible iff λ0 is not a non-negative integer. Suppose λ0 = −m for some m ≥ 1. Then we get an irreducible representation of SL(2, R) and this representation is in the discrete series. Likewise, we can start with a lowest weight vector and generate another infinite dimensional irreducible representation in the discrete series. Now define the sl(2, C) elements H  = i(X − Y ), X  = (H − i(X + Y ))/2, Y  = (H + i(X + Y ))/2 Then, [H  , X  ] = i[X − Y, H]/2 + [X, Y ] = −iX − iY + H = 2X  , [H  , Y  ] = i[X − Y, H]/2 − [X, Y ] = −iX − iY − 2H = −2Y  [X  , Y  ] = i[H, X + Y ]/2 = i(X − Y ) = H  Hence, {H  , X  , Y  } satisfy the same commutation relations as {H, X, Y } and therefore the Lie algebra spanned by {H  , X  , Y  } is isomorphic to sl(2, C). More precisely, the map {H, X, Y } → {H  , X  , Y  } is a Lie algebra automorphism which is easily proved to be an inner automorphism. It follows that if π is the same irreducible representation of sl(2, C) corresponding to λ0 = −m as defined above, then the character of the corresponding Lie group representation evaluated at exp(tH  ) is the same as its character evaluated at exp(tH). Thus, the character of this representation evaluated at exp(θ(X − Y )) is the same as the character of this representation evaluated at exp(−iθH). Now as seen above, the character of the above discrete series representation for λ0 = −m evaluated at exp(tH) is given by χm (exp(tH)) =



exp(−t(m + 2s)) =

s≥0

=

exp(−mt) 1 − exp(−2t)

exp(−(m − 1)t) exp(t) − exp(−t)

and hence, it follows by the above argument, that the character of this same representation evaluated at u(θ) = exp(θ(X − Y )) is given by χm (u(θ)) =

exp(i(m − 1)θ) exp(−iθ) − exp(iθ)

Remark: R.H and R.(X − Y ) are two non-conjugate Cartan subalgebras of sl(2, R). The first generates a noncompact subgroup L of SL(2, R) consisting

103

Quantum Antennas 

 a 0 , a > 0 while the second generates a compact subgroup 0 1/a B = SO(2) of SL(2, R) consisting of matrices   cos(θ) −sin(θ) u(θ) = , θ ∈ [0, 2π) sin(θ) cos(θ) of matrices

We have the singular value decomposition of any g ∈ SL(2, R): g = u(θ1 )a(t).u(θ2 ) where



cos(θ) −sin(θ) sin(θ) cos(θ)   t 0 e a(t) = exp(tH) = 0 e−t

u(θ) = exp(θ(X − Y )) =

 ,

The Iwasawa decomposition of G = SL(2, R) is G = KM AN where K = SO(2), M = {±I}, A = exp(R.H) and N = exp(R.X). This is essentially the QR decomposition obtained via a Gram-Schmidt orthonormalization of the columns of G. We Image processing using discrete series characters: Given any g ∈ G = SL(2, R) (or equivalently, a proper orthochronous Lorentz transformation in the xz plane), we have the basic result that g is conjugate in G = SL(2, R) either to an element of the elliptic Cartan subgroup B = exp(R.(X − Y )) or to an element of the hyperbolic Cartan subgroup L = exp(R.H). Let f1 , f2 be two functions on SL(2, R) and let χm denote the character of the above discrete series representation. Then, we consider  Fm (f1 , f2 ) = f1 (g)f2 (h)χm (gh−1 )dgdh G×G

We have that



f1 (xg)f2 (xh)χm (gh−1 )dgdh

Fm (f1 ox, f2 ox) = G×G

 =

f1 (g)f2 (h)χm (x−1 gh−1 x)dgdh =



f1 (g)f2 (h)χm (gh−1 )dgdh = Fm (f1 , f2 )

for all x ∈ G. Thus, (f1 , f2 ) → Fm (f1 , f2 ) is a G-invariant function defined on all image pairs where each image is a function on G = SL(2, R).

Chapter 7

Optimization problems in classical and quantum stochastics and information with antenna design applications 7.1 7.1.1

A course in optimization techniques Linear optimization problems using least squares methods

The orthogonal projection theorem in a finite dimensional Hilbert space.

7.1.2

Minimum mean squares estimation

Conditional expectation of L1 random variables using the Radon-Nikodym derivative; Conditional expectation using the orthogonal projection theorem for L2 (P ) random variables and the density of L2 (P ) in L1 (P ). These are applications of the orthogonal projection theorem in an infinite dimensional Hilbert space.

105

106

7.1.3

Quantum Antennas

Orthogonal projection theorem in infinite dimensional Hilbert spaces using the Apollonius theoremexistence and uniqueness theorems and properties of the orthogonal projection operator

Study project: Read about the proof of the existence and uniqueness of the best approximant of a vector from a Hilbert space in a closed subspace of it or more generally in a closed convex subset of it. The existence proof is based on constructing a sequence in the subspace/convex subset whose distance from the given vector converges to the minimum distance and then using Apollonius’s theorem to show that this sequence is Cauchy and hence converges. Likewise the uniqueness proof is based on assuming two best approximants to the given vector and then using Apollonius’ theorem to show that these two are equal. Linearity of the orthogonal projection map that takes any given vector to its best approximant in the closed subspace is established using the orthogonality priniciple which states that the approximation error is orthogonal to the subspace. This fact is established by noting that the approximation error norm square is a minimum for all vectors in the subspace. This minimum condition is expressed by setting to zero the derivative of the error norm square w.r.t a one parameter family of curves in the subspace passing through the best approximant. This orthogonality principle is also used to establish the self-adjointness of the orthogonal projection operator.

7.1.4

Orthogonal projection theorem for closed convex subsets of an infinite dimensional Hilbert space. Proofs of existence and uniqueness

Study project:The Apollonius theorem works even if the closed subspace is replaced by a closed convex subset since it involves only three vector in the subset, the first two being any two vectors in the sequence whose distance from the given vector converges to the minimum and the third being the average value of these two vectors which by convexity of the subset, is also an element of the convex set. However now we cannot talk about linearity or selfadjointness of the orthogonality operator.

7.1.5

The variational derivatives of a function defined on infinite dimensional Banach and Hilbert spaces. The Frechet and Gateaux derivatives

7.1.6

The variational principle of Lagrange on spaces of twice differentiable functions

Study project: The Euler-Lagrange equations for functions of one variable and for functions of several variables. Application to classical mechanics and classical field theory. Learn about Noether’s theorem which states that if the La-

107

Quantum Antennas

grangian is invariant under a Lie group of transformations, then we can derive a conservation law both in mechanics and in classical field theory.

7.1.7

The Euler-Lagrange variational principle combined with the Feynman path integral (sum over histories) to verify the transition from quantum mechanics to classical mechanics in the limit h → 0 via the method of stationary phase

Study project:The Feynman path integral (FPI) between two space-time points is the sum of a phase factor over different paths in time passing between the two points. The phase factor equals the action integral along the path divided by Planck’s constant. This path integral solves the Schrodinger equation ie it is precisely the Schrodinger evolution kernel as first pointed out by Richard Feynman in his PhD thesis. The contribution to the FPI from each path is interpreted as being the quantum mechanical amplitude for the particle to go from the first point to the second in the given time interval. Thus, the FPI can be used as the starting point for formulating quantum mechanics. Moreover, in this formulation, the analogy between classical and quantum mechanics becomes apparent at once. Indeed, when Planck’s constant goes to zero, then the phase becomes infinite and hence rapidly oscillates in the vicinity of any trajectory between the two points except the classical trajectory causing phase cancellations in the FPI around any trajectory except the classical one. The action functional and hence the phase is stationary around the classical trajectory by the Euler-Lagrange theorem and hence no phase cancellations occur around this trajectory. This means that only the contribution from the classical trajectory contributes to the quantum mechanical transition amplitude in the limit as Planck’s constant converges to zero or more precisely when the ratio of the action integral to Planck’s constant becomes very large as is the case for macroscopic bodies.

7.1.8

Large deviation theory as an exercise in optimization

The theorems of Sanov, Cramer, Gartner-Ellis, Bryc and Varadhan. Calculating the rate function for various sequences of random variables and random processes. Schilder’s rate function for Brownian motion. The variational principle of Varadhan, for evaluating lim→0 E[exp(φ(Z )/)] when Z ,  → 0 satisfies the LDP with a known rate function I(z).

108

7.1.9

Quantum Antennas

Nonlinear filtering theory as an optimization problem

Calculating p(xt |Yt ) on a real time basis as a solution to a stochastic pde driven by the measurement process z(t) with Yt = {z(s) : s ≤ t}. Calculating ˆ φ(t|t) = E(φ(x(t)|Yt ) as the optimal mmse of φ(x(t)) given measurements upto time t. Approximations leading to the EKF. Deriving the EKF using optimization w.r.t the Kalman gain matrix both in discrete and in continuous time.

7.1.10

Variational principles applied to the Brachistochrone problem

(curve having minimum time of descent in uniform and non-uniform gravitational fields) and applied to the catenary problem of determining the shape of the hanging chain problem with applications to transmission line theory. [10] Entropy maximization in physics for deriving the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics.

7.1.11

Optimization in filter design problems

Designing a rational transfer function that has minimum Lp distance from a given transfer function.

7.1.12

Channel capacity calculation as an optimization problem in cc and cq Shannon problems

In Cq problems, the problem is to maximize H(

 x∈A

p(x)ρ(x)) −



p(x)H(ρ(x))

x∈A

w.r.t (p(x), ρ(x)), x ∈ A where p(.) is a probability distribution on A and ρ(x) is a density matrix in a Hilbert space H for each x ∈ A.

7.1.13

Another exercise in quantum channel capacity calculation as an optimization problem

p ∗ ρi is the input state, p ρo ∗= k=1 Ek ρi Ek is the output state of a quantum noisy channel. Here, k=1 Ek Ek = I. If A is a finite alphabet and ρi : A → S(H) is a mapping from A into the space of density operators in a Hilbert  space H, then q then choose a POVM {Mk : 1 ≤ k ≤ q} in H, ie, 0 ≤ Mk ≤ I, k=1 Mk = I

109

Quantum Antennas

so that the output probability distribution q(k|x) given that the input source alphabet is x ∈ A is given by q(k|x) = T r(ρo (x)Mk ), ρo (x) =

p 

Ek ρi (x)Ek∗

k=1

Calculate the capacity of this classical channel q(.|.) and then maximize this capacity over all POVM’s {Mk }.

7.1.14

Optimization in fluid dynamics

construction of the optimal stirring forces for the fluid velocity field to match an given velocity field with applications to optical flow problems. The fluid equations are (v, ∇)v + v,t = −∇p/ρ + ν∇2 v + f (t, r) and divv(t, r) = 0 Thus, v = ∇ × ψ, divψ = 0 and ∇ × v = Ω = −∇2 ψ Taking the curl of the Navier-Stokes equation gives us a pde for ψ, ie, with ∇ × f = g, we have ∇ × (∇2 ψ × (∇ × ψ)) + ∇2 ψ,t = ν(∇2 )2 ψ + g or equivalently, with Δ = ∇2 , ψ,t = νΔψ − Δ−1 (∇ × (Δψ × (∇ × ψ)) + Δ−1 g − − − (a) Now design the forcing function g such that divg = 0 and subject to the equation of motion constraint (a), 

L(ψ(t, r), ψd (t, r), g(t, r))d3 rdt

is a minimum where ψd (t, r) is the desired stream function vector field.

110

7.1.15

Quantum Antennas

Optimization in electromagnetics

Design of an antenna to match a given radiation pattern. Suppose Pd (ω, rˆ) is the desired power flow per unit solid angle at a large distance r from the origin. Let J(ω, r) denote the current density of the source. The far field magnetic vector potential is  A(ω, r) = (μ/4πr)exp(−jkr) J(ω, r )exp(jkˆ r.r )d3 r S

where S stands for the source region (centred around the origin). The far field magnetic field is  H(ω, r) = ∇ × A(ω, r) = (exp(−jkr))/4πr)(−jk)ˆ r× J(ω, r )exp(jkˆ r.r )d3 r S

and hence the power flow per unit solid angle in the radiation zone is given by P (ω, rˆ) = (η/2)r2 |H(ω, r)|2 = 2



J(ω, r )exp(jkˆ r.r )d3 r |2

(η/16π )| S

Let σ(ω, r) denote the conductivity of the source. Then, the power dissipated in the source at frequency ω is given by  (2σ(ω, r )−1 |J(ω, r )|2 d3 r S

The optimization problem is to choose the current field J(ω, r ), r ∈ S such that for a given power dissipation at frequency ω, P (ω, rˆ) is as close as possible to Pd (ω, rˆ) for all rˆ ∈ S 2 .

7.1.16

Quantum gate design using optimization techniques. Discussion of the gravitational search algorithm (GSA)

7.1.17

Solving nonlinear least squares problems using perturbation theory

Minimizing E(θ) = y −

p  k=1

w.r.t θ ∈ R . p

Ak (θ⊗k ) 2

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Quantum Antennas

7.1.18

Bellman-Hamilton-Jacobi dynamic programming for solving deterministic and stochastic optimal control problems

The state is a Markov process X(t) ∈ Rd whose generator K depends on the input u(t) ∈ Rp which is to be controlled so that  T L(X(t), u(t), t)dt + λT Eψ(X(T )) S[X, u, λ] = E 0

is extremized. This problem amounts to minimizing the average cost over the fuel input u(.) for a fixed time duration T subject to the constraint that at time T , the average value of some function of the state is known. This is therefore a minimum fuel problem. The fuel input u(t) at time t is allowed to depend only on the state X(t) at time t, ie u(t) = f (t, X(t)) where f is a non-random function. Alternately, we could also regard the time duration T as a variable over which the average cost is to be minimized. For example, if L = 1, this becomes a minimum time problem.

7.1.19

Optimization problems for state diffusion problems

The Hudson-Parthasarathy noisy Schrodinger equation reads dU (t) = (−(iH + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) where H, P, L1 , L2 , S are system operators satisfying certain relations that guarantee U (t) to be unitary in h ⊗ Γs (L2 (R+ )). We choose a pure state |f ⊗ φ(u) > in this space where |φ(u) >= exp(−|u|2 /2)|e(u) > is coherent state of the bath and define the pure state |ψ(t) >= U (t)|f ⊗ φ(u) >∈ h ⊗ Γs (L2 (R+ )) Then |ψ(t) > satisfies d|ψ(t) >= (−(iH+P )|ψ(t) > +u(t)(L1 −L2 )|ψ(t) >)dt+L2 dB(t)|ψ(t) > +S(dA∗ dA/dt)|ψ(t) > where

B(t) = A(t) + A(t)∗

is classical Brownian motion. We have used the well known relation dΛ = dA∗ dA/dt which follows from < e(u)|dA∗ dA/dt|e(v) >=< dAe(u)|dAe(v) > /dt =u ¯(t)v(t)dt < e(u)|e(v) >=< e(u)|dΛ|e(v) > We now observe that (dA∗ dA/dt)|ψ(t) >= U (t)(dA∗ dA/dt)|f ⊗ φ(u) >=

112

Quantum Antennas u(t)U (t)dA∗ |f ⊗ φ(u) >= u(t)U (t)(dB − dA)|f ⊗ φ(u) > = u(t)dB(t)|ψ(t) > −u(t)2 dt|ψ(t) >

Thus, the HP equation when applied to the above pure state results in a ”state diffusion” (in the sense of Gisin and Percival): d|ψ(t) >= (−(iH+P )|ψ(t) > +u(t)(L1 −L2 )|ψ(t) >)dt+L2 dB(t)|ψ(t) > +S(u(t)dB(t)−u2 (t)dt)|ψ(t) > = (−(iH + P ) + u(t)L − u2 (t)S)dt + (L2 + u(t)S)dB(t))|ψ(t) > It should be noted that this is a classical stochastic differential equation and the state vector |ψ(t) > can now be regarded as a classical random process adapted to the classical Brownian motion B(.) and taking values in the system Hilbert space h. It should also be noted that in this interpretation, if < ., . >=< ., . >h is the inner product in h, then < ψ(t)|ψ(t) >= 1, rather < ψ(t)|ψ(t) > is a random process adapted to the Brownian motion B(.) and the condition < ψ(t)|ψ(t) >h⊗Γs (L2 (R+ )) = 1 translates to E[< ψ(t)|ψ(t) >] = 1 or equivalently,

 < ψ(t)|ψ(t) > dP (B) = 1

In other words, averaging w.r.t. the state of the bath is equivalent to averaging w.r.t. the probability law of the classical Brownian motion B(.). If |ψ(t) > is to be regarded as the system state at time t, then it must be normalized w.r.t the system inner product. The system state at time t then becomes |χ(t) >=< ψ(t)|ψ(t) >−1/2 |ψ(t) >= |ψ(t) > /  ψ(t)  In fact, this is the state to which the system state collapses after making a measurement of the bath over time (t, ∞) without noting the outcome. We seek now to determine using the classical Ito formula, the dynamics of |χ(t) >. We first write down the dynamics of |ψ(t) > as d|ψ(t) >= (A1 (t)dt + A2 (t)dB(t))|ψ(t) > where Then

A1 (t) = −(iH + P ) + u(t)L − u2 (t)S, A2 (t) = L2 + u(t)S d( ψ(t) −1 ) = d < ψ(t)|ψ(t) >−1/2 =

(−1/2)  ψ(t) −3 (d < ψ(t)|ψ(t) >) + (3/8)  ψ(t) −5 (d < ψ(t)|ψ(t) >)2 and d < ψ(t)|ψ(t) >= 2Re(< ψ(t)|dψ(t) >)+ < dψ(t)|dψ(t) > = 2Re(< ψ(t)|A1 |ψ(t) >)dt + 2Re(< ψ(t)|A2 |ψ(t) >)dB(t)

113

Quantum Antennas + < ψ(t)|A∗2 A2 |ψ(t) > dt (d < ψ(t)|ψ(t) >)2 = 4(Re(< ψ(t)|A2 |ψ(t) >))2 dt So, d|χ(t) >= d( ψ(t) −1 )|ψ(t) > +  ψ(t) −1 d|ψ(t) > +d( ψ(t) −1 ).d|ψ(t) > Now the above equations imply d( ψ(t) −1 ) = (−1/2)  ψ(t) −3 (d < ψ(t)|ψ(t) >) + (3/8)  ψ(t) −5 (d < ψ(t)|ψ(t) >)2

= (−1/2)  ψ(t) −3 [2Re(< ψ(t)|A1 |ψ(t) >)dt + 2Re(< ψ(t)|A2 |ψ(t) >)dB(t) + < ψ(t)|A∗2 A2 |ψ(t) > dt] + (3/2)  ψ(t) −5 (Re(< ψ(t)|A2 |ψ(t) >))2 dt Thus, d|χ(t) >= = (−1/2)  ψ(t) −3 [2Re(< ψ(t)|A1 |ψ(t) >)dt + 2Re(< ψ(t)|A2 |ψ(t) >)dB(t) + < ψ(t)|A∗2 A2 |ψ(t) > dt]|ψ(t) > +(3/2)  ψ(t) −5 (Re(< ψ(t)|A2 |ψ(t) >))2 dt|ψ(t) >

+  ψ(t) −1 (A1 |ψ(t) > dt + A2 |ψ(t) > dB(t)) −  ψ(t) −3 Re(< ψ(t)|A2 |ψ(t) >)A2 |ψ(t) > dt = dt[−Re(< χ(t)|A1 |χ(t) > −(1/2) < χ(t)|A∗2 A2 |χ(t) > +(3/2)(Re(< χ(t)|A2 |χ(t) >))2 +A1 ]|χ(t) > +dB(t)[−Re(< χ(t)|A2 |χ(t) >) + A2 ]|χ(t) > Thus, after making a measurement on the environment over the time interval (t, ∞), the state of the system collapses to the normalized state |χ(t) > which satisfies a nonlinear diffusion equation.

7.2 7.2.1

Group theoretical techniques in optimization theory Statement of the basic problems

A group G acts on a manifold M. The image field is a map f1 : M → R. The image field f after transformation by the group element g ∈ G and after having been corrupted by a noise field w(x), x ∈ M, ie w : M → L2 (Ω, F, P ) is given by f2 (x) = f1 (g −1 x) + w(x), x ∈ M

114

Quantum Antennas

The aim is to estimate the group transformation element g from measurements of f1 and f2 and to obtain the mean square error in the estimation error. More generally, we may be given a whole set of image field pairs (fa1 , fa2 ), a = 1, 2, ..., K on M such that for a fixed g ∈ G, we have fa2 (x) = fa1 (g −1 x) + wa (x), a = 1, 2, ..., K Then from measurements of these pairs, g must be estimated. The least squares method of doing this according to gˆ = argminh∈G

K   a=1

M

wa (x)|fa2 (x) − fa1 (h−1 x)|2 dμ(x)

where μ is a G-invariant measure on M and wa is a non-negative weight function on M involves a search and is computationally very expensive. Techniques based on group representation theory considerably simplify the problem to a linear problem. For example suppose we decompose L2 (M, μ) =

∞

Hk

k=1

as an orthogonal direct sum, where Hk is a Hilbert subspace of L2 (M, μ) and is invariant and irreducible under G, ie, f ∈ Hk implies f oh−1 ∈ Hk for all h ∈ G, k = 1, 2, ... and further, Hk contains no proper G-invariant subspace. Let πk denote the corresponding representation. Specifically, choose an onb {ekm : m = 1, 2, ..., dk } for Hk and let ekm (h−1 x) =

dk 

[πk (h)]m m ekm (x)

m =1

Since μ is a G-invariant measure on M, it follows that the representation πk : G → GL( C, dk ) is in fact a unitary representation, ie πk : G → U (C, dk ). In fact, we get from the above that δm,n =< ekm , ekn >=< ekm oh−1 , ekn oh−1 >=  [¯ πk (h)]m m [πk (h)]n n < ekm , ekn > m ,n

=



[¯ πk (h)]m m [πk (h)]n n δm ,n

m ,n

=

 [¯ πk (h)]m m [πk (h)]m n = [πk (h)∗ πk (h)]mn m

which proves the unitarity of πk (h). Now the equation f2 (x) = f1 (g −1 x) + w(x)

115

Quantum Antennas implies

< ekm , f2 >=< ekm , f1 og −1 > + < ekm , w >=  [¯ πk (g −1 )]m m < ekm , f1 > + < ekm , w > < ekm og, f1 > + < ekm , w >= m

 [πk (g)]mm < ekm , f1 > + < ekm , w > = m k or equivalently, denoting by f1 [k] the dk × 1 complex vector ((< ekm , f1 >))dm=1 and likewise for f2 [k] and w[k], we can express the above equation in matrix form as f2 [k] = πk (g)f1 [k] + w[k]

and in the case when we have several such image pairs, f2a [k] = πk (g)f1a [k] + wa [k] so that for each k we can estimate the matrix πk (g) by minimizing Ek (X) =

K 

wa  f2a [k] − Xf1a [k] 2

a=1

w.r.t X and by doing so for several such k, a good estimate of g may be obtained.

7.2.2

Some aspects of the root space decomposition of a semisimple Lie algebra

Use Weyl’ character formula for compact semisimple groups to construct invariants for image pairs and thereby do pattern classification/pattern recognition. Let G be a compact complex semisimple group and let g denote its Lie algebra. Let h be a Cartan subalgebra of g, note that since the group is complex, all Cartan subalgebras are conjugate to each other. Corresponding to this Cartan subalgebra, let

g=h⊕ gα α∈Δ

be its root space decomposition. Here, Δ is the set of all roots of g. Let Δ+ denote the set of positive roots. Then, the root space decomposition can also be expressed as

g=h⊕ gα ⊕ g−α α∈Δ+

Note that dimgα = 1, α ∈ Δ. Let λ be a dominant integral weight, ie, λ ∈ ∗ and λ(Hα ) is a positive integer for every simple root α. Note that corresponding to the set Δ of roots, we have a set P ⊂ Δ+ of roots such that given any α ∈ Δ+ , there exist integers n(α, β) ≥ 0, β ∈ P such that  n(α, β)β α= β∈P

116

Quantum Antennas

and further there does not exist any proper subset of P with this property. In other words, every root is either a positive integer linear combination of the simple roots P or a negative integer linear combination of the simple roots P . Define  ρ = (1/2) α α∈Δ+

In the above notation, Hα is a normalized version of [Xα , X−α ] where Xα ∈ gα with α ∈ Δ+ and the normalization is carried out in such a way so that {Hα , Xα , X−α } are the standard generators of a Lie algebra isomorphic to sl(2, C), ie, [Hα , Xα ] = 2Xα , [Hα , X−α ] = −2X−α , [Xα , X−α ] = Hα This means that α(Hα ) = 2 Corresponding to the dominant integral weight λ, there is an irreducible representation πλ of g whose character χλ on the Cartan subgroup T = exp(h) is given by  (s)exp(s(ρ + λ))  χλ (t) = s∈W s∈W (s)exp(sρ) where W is the Weyl group of h, ie, if NT denotes the normalizer of h in G, ie the set of all g ∈ G such that Ad(g)(h) ⊂ h then the Weyl group W is NT /T . Note that T = exp(h) is the centralizer of h in G, ie t ∈ T iff Ad(t)(H) = H∀H ∈ h. The Weyl group acts on h through the adjoint action. More precisely, for each s ∈ W , there is an element g ∈ NT such that Ad(g)(H) = s.H, H ∈ h and conversely, given any g ∈ NT we can find a unique element s ∈ W such that Ad(g)(H) = s.H, H ∈ h For each simple root α, ie, α ∈ P , there is an element sα ∈ W such that sα is a reflection about the plane {H ∈ h : α(H) = 0}. Equivalently, sα .H = H − α(H)Hα Any element in W is expressible as a product of the sα , α ∈ P . We note that α(sα .H) = α(H) − 2α(H) = −α(H) and further,

s2α .H = sα .H − α(H)sα .Hα

117

Quantum Antennas = H − α(H)Hα − α(H)(Hα − 2Hα ) =H ie,

s2α = 1

which confirms the fact that sα is a reflection. sα , α ∈ P are therefore called simple reflections. If s ∈ W is expressible as a product of an even number of simple reflections, then we set (s) = −1, otherwise (s) = 1. Note that the Weyl group, by duality acts on h∗ , ie, if s ∈ W and μ ∈ h∗ , then (s.μ)(H) = μ(s−1 .H) It follows that if α ∈ P , then (sα .λ)(H) = λ(sα .H) = λ(H − α(H).Hα ) = λ(H) − λ(Hα )α(H) = λ− < λ, α > α(H) ie sα .λ − λ− < λ, α > α where < λ, α >= λ(Hα ) for any λ ∈ h∗ . Note that here λ is a dominant integral weight and hence < λ, α > is a non-negative integer for all α ∈ P . Then, it follows that since any s ∈ W is a product of simple reflections and since if α, β ∈ P , then < α, β > is a negative integer, ie, α(Hβ ) is a negative integer, we have s.λ = λ −

p 

n(λ, α)α

α∈P

where n(λ, α) are non-negative integers.

7.2.3

Weyl’s integration formula and Weyl’s character formula

Let G be a compact semisimple group and T a Cartan subgroup. Consider the mapping ψ : G/T × T → G defined by ψ(g.T, h) = Ad(g)(h) = ghg −1 Note that this map is well defined since T is an Abelian subgroup. We compute the differential of this map: for Z ∈ g infinitesimal and H ∈ t infinitesimal with t being the Lie algebra of T (T is a maximal torus in G), we have g(1 + Z)h(1 + H)(1 − Z)g −1 = (g + gZ)(h + hH)(g −1 − Zg −1 )

118

Quantum Antennas = ghg −1 + ghHg −1 + gZhg −1 − ghZg −1 = ghg −1 + ghg −1 Ad(g)(H) + ghg −1 Ad(g)Ad(h−1 )(Z) − ghg −1 Ad(g)(Z)

It follows from this expression that the differential of ψ at (gT, h is given by dψ(gT,h) (Z, H) = Ad(g)(H + Ad(h)−1 (Z) − Z) Choosing successively Z to be the root vectors Xα , α ∈ Δ and H to run over a basis for h, we easily see that the Jacobian determinant of the map ψ at (gT, h) is given by Πα∈Δ (exp(α(log(h)) − 1) = |Πα∈Δ+ (exp(α(log(h))/2) − exp(−α(log(h))/2))|2 = |Δ(h)|2 where Δ(h) = Πα∈Δ+ (exp(α(log(h))/2) − exp(−α(log(h))/2)) It follows that  f (g)dg = |W |

−1

G



f (xhx−1 )|Δ(h)|2 dxdh G/T ×T

where |W | is the number of elements in the Weyl group. We note that χλ as defined above when restricted to the torus T is a finite Fourier series with integer coefficients. This is one of the requirements for any character on T because T is an Abelian group and hence since every representation of T is a direct sum of irreducible representations of T , it follows by taking the trace that every character on T is an integer linear combination of irreducible characters and any irreducible character on T is simply of the form tn1 1 ...tnl l where l = dimT , t1 , ..., tl are complex numbers of unit magnitude and n1 , ..., nl are non-negative integers. Note that by the definition of a torus, there exist linearly independent elements H1 , ..., Hl ∈ h such that exp(i2πHk ) = 1, k = 1, 2, ..., l and that every element t ∈ T can be expressed as t = exp(2πi(n1 θ1 H1 + .. + nl θl Hl )) where n1 , ..., nl are non-negative integers and θ1 , ..., θk ∈ [0, 1). We can thus define tj = exp(2πiθj Hj ), identify it with the complex number exp(2πiθj ) and note that t can then be identified with tn1 1 ...tnl l . We now note that if s1 , s2 ∈ W are such that s1 .(λ + ρ) = s2 .(λ + ρ), then λ + ρ = s.(λ + ρ), s = s−1 1 ss and we can write λ − s.λ =

 α∈P

n(α)α, n(α) ∈ Z+

119

Quantum Antennas On the other hand, ρ − s.ρ = (1/2)



α − s.α ≥ 0

α∈Δ+

Thus, we must necessarily have s = 1, ie, s1 = s 2 . It follows that each weight s.(λ + ρ), s ∈ W appears exactly once in the sum s∈W (s)exp(s.(λ + ρ)) from which, we deduce on using the fact that s.λ assumes only integer values on Hα , α ∈ P and that the same is true for ρ that   | (s)exp(s.(λ + ρ)(log(h)))|2 dh T

s∈W

= |W | Thus,



|χλ (g)|2 dg = |W |−1 G

 G/T ×T

|χλ (xhx−1 )|2 |Δ(h)|2 dxdh

=1 since

χλ (xhx−1 ) = χλ (h) = Δ(h)−1



(s).exp(s.(λ + ρ))

s∈W

Thus, the function χλ on G defined by (a) χλ is a class function and  (s).exp(s.(λ + ρ)(log(h)))  ,h ∈ T χλ (h) = s∈W s∈W (s)exp(s.ρ(log(h))) satisfies all of the following properties: [a] χ  λ restricted to T is a finite Fourier series with integer coefficients. [b] G |χλ |2 dg = 1 [c] χλ (s.h) = χλ (h), h ∈ T, s ∈ W These three conditions guarantee that χλ is an irreducible character of G whenever λ is a dominant integral weight.

7.2.4

7.3

Irreducible representations of SU (2) and hence by Weyl’s unitarian trick of SL(2,C) using [a] Wigner’s bivariate polynomials and [b] The Lie algebraic method

Feynman’s diagrammatic approach to computation of the scattering amplitudes of electrons, positrons and photons

[a] Compton scattering: An electron of four momentum p and spin σ denoted by a straight line absorbs a photon of four momentum k and helicity s, represented by a wavy line, then travels along a straight line to a point where it has

120

Quantum Antennas

a momentum p + k  , it emits a photon of momentum k  and helicity s at this point and then acquires a final momentum of p and a spin σ  . The scattering amplitude for this process is to be written down. By conservation of four momentum, p + k = p + k  and by the diagrammatic technique, this amplitude is proportional to ¯(p , σ  )γ ν S(p + k)γ μ u(p, σ)eμ (k, s) eν (k  , s )∗ u where

S(p) = (γ.p − m − i)−1

is the electron propagator, eμ (k, s) is the standard photon wave function in the momentum domain when it corresponds to an external em four potential, u(p, σ) is the electron Dirac wave function in the momentum domain. By u ¯, we mean u∗T γ 0 . [b] The electron self-energy: An electron of four momentum p and spin σ propagates through space, it then emits a photon of four momentum k, thereby acquiring a four momentum of p − k, it then propagates further and absorbs the emitted photon after which it acquires back its momentum p. The photon is thus represented by a wavy line which starts at some point on the electron line and ends at some further point. According to the standard Feynman rule, the correction to the electron propagator caused by this photon emission is given by upto a proportionality constant by  Σ(p) = Dμν (k)γ ν S(p − k)γ μ d4 k where

ημν k 2 − i is the photon propagator. Thus the correction to the electron propagator can be expressed upto a proportionality constant as  Σ(p) = (k 2 − i)−1 γμ S(p − k)γ μ d4 k Dμν (k) =

The corrected electron propagator due to such self energy terms is obtained by putting several such loops in the path of the electron and summing up all these diagrams. The resulting corrected electron propagator is thus ST (p) = S(p) + S(p)Σ(p)S(p) + S(p)Σ(p)S(p)Σ(p)S(p) + ... = (S(p)−1 − Σ(p))−1 = (γ.p − m − Σ(p))−1

Chapter 8

Quantum waveguides and cavity resonators 8.1

Quantum waveguides

Consider a waveguide of arbitrary cross section with axis parallel to the z axis. Introduce an orthogonal curvilinear coordinate system (q1 , q2 ) in the xy plane so that q1 = 0 is the boundary of the guide. Let Hk , k = 1, 2 denote the Lame’s coefficients. Denote by −γ the differential operator ∂/∂z and by jω the operator ∂/∂t. The Maxwell curl equations give  E⊥ (t, x, y, z) = ((−γ/h2n )∇⊥ (Ez [n, t, z]un (q))−(jωμ/kn2 )∇⊥ Hz [n, t.z]vn (q)׈ z) n

Ez (t, x, y, z) =



Ez [n, t, z]un (q)

n

H⊥ =

 (−γ/kn2 )∇⊥ (Hz [n, t, z]vn (q)) + (jω/h2n )(∇⊥ Ez [n, t, z]un (q)) × zˆ) n

Hz (t, x, y, z) =



Hz [n, t, z]vn (q)

n

where Ez [n, t, z], Hz [n, t, z] satisfy the eigen-equations (∂z2 + h2n − μ∂t2 )Ez [n, t, z] = 0, (∂z2 + kn2 − μ∂t2 )Hz [n, t, z] = 0 Note that γ 2 stands for the operator ∂z2 and ∂t2 is the operator −ω 2 . The different eigenvalues h2n and kn2 for the Ez and Hz components arise because of the different boundary conditions satisfied by these. These eigenvalues are determined by the solution to the boundary value problems (∇2⊥ + h2n )un (q) = 0, un (q1 = 0, q2 ) = 0,

121

122

Quantum Antennas (∇2⊥ + kn2 )vn (q) = 0, ∂vn (q1 = 0, q2 )/∂q1 = 0

These boundary conditions correspond to the fact that the tangential components of the electric field and the normal component of the magnetic field vanishes on the boundary. We may assume that {un } and {vn } are respectively orthonormal bases for L2 (S) with the Dirichlet and Neumann boundary conditions where S is the cross section of the guide. Note that q1 is directed along the normal to the guide wall while q2 is along the tangent to the guide wall at the boundary. We also note that the area measure on S is dS = dxdy = H1 H2 dq1 dq2 . We next calculate the Lagrangian density of the fields within the guide the density taken w.r.t. the z variable. It is given by L(Ez [n, t, z], Hz [n, t, z], Ez,z [n, t, z], Hz,z [n, t, z], Ez,t [n, t, z], Hz,t [n, t, z]) =



[(/2)((Ez (t, x, y, z))2 +|E⊥ (t, x, y, z)|2 )−(μ/2)((Hz (t, x, y, z))2 S

+|H⊥ (t, x, y, z)|2 )]dS The various terms are evaluated as follows:   Ez (t, x, y, z)2 dS = Ez [n, t, z]2 S



n

Hz (t, x, y, z)2 dS = S





Hz [n, t, z]2

n

 2 2 −4 2 |E⊥ (t, x, y, z)|2 dS = (c[n]h−4 n Ez,z [n, t, z] + d[n]μ kn Hz,t [n, t, z] ) S

n



where

|∇⊥ un (q)|2 dS, d[n] =

c[n] = S



|∇⊥ vn (q)|2 dS S

where we have used the fact that  (∇⊥ un (q), ∇⊥ vm (q) × zˆ)dS = S



(un,1 (q)vm,2 (q) − un,2 (q)vm,1 (q))dq1 dq2 = 0 S

using integration by parts and the fact that un (q) and vm,1 (q) vanish on the boundary of S, ie when q1 = 0. Likewise,  |H⊥ (t, x, y, z)|2 dS = 

S 2 (d[n]kn−4 Hz,z [n, t, z]2 + c[n]2 h−4 n Ez,t [n, t, z] )

n

The Lagrangian density of the waveguide confined electromagnetic field is therefore L(Ez [n, t, z], Hz [n, t, z], Ez,z [n, t, z], Hz,z [n, t, z], Ez,t [n, t, z], Hz,t [n, t, z], n = 1, 2, ...) =

123

Quantum Antennas 

+



(/2)Ez [n, t, z]2 − (μ/2)Hz [n, t, z]2 )

n 2 2 −4 2 (/2)(c[n]h−4 n Ez,z [n, t, z] + d[n]μ kn Hz,t [n, t, z] )

n

−



2 (μ/2)(d[n]kn−4 Hz,z [n, t, z]2 + c[n]2 h−4 n Ez,t [n, t, z] )

n

To quantize this confined em field, we assume that the position fields are Ez [n, z, t], Hz [n, z, t], n = 1, 2, ... and the corresponding conjugate momentum fields are ∂L = πE [n, z, t] = ∂Ez,t [n, z, t] −μ2 c[n]h−4 n Ez,t [n, z, t], πH [n, z, t] =

∂L = ∂Hz [n, z, t]

μ2 d[n]kn−4 Hz,t [n, z, t] and hence by applying the Legendre transformation, the Hamiltonian density is H(Ez [n, z, t], Hz [n, z, t], πE [n, z, t], πH [n, z, t], n = 1, 2, ...] = 

(πE [n, z, t]Ez [n, z, t] + πH [n, z, t]H[n, z, t]) − L =

n

 ((−h4n /2μ2 c[n])πE [n, z, t]2 + (kn4 /2μ2 d[n])πH [n, z, t]2 ) n

+



(μ/2)Hz [n, t, z]2 − (/2)Ez [n, t, z]2 )

n

−



2 (/2)(c[n]h−4 n Ez,z [n, t, z] )

n

+



(μ/2)(d[n]kn−4 Hz,z [n, t, z]2 )

n

This Hamiltonian density of the confined em field is therefore the sum of Hamiltonians of a discrete conutably infinite set of Harmonic oscillators some of which have negative masses. More precisely since the z coordinate variable is also involved, this Hamiltonian density is the sum of Hamiltonian densities of a countably infinite set of one dimensional strings, some of which have negative linear mass densities. The magnetic field modes have positive mass densities while the electric field modes have negative mass densities. We therefore rearrange these as H=  [(kn4 /2μ2 d[n])πH [n, z, t]2 ) + (μ/2)Hz [n, t, z]2 + (μ/2)(d[n]kn−4 Hz,z [n, t, z]2 ) n

124

Quantum Antennas −[

 2 (h4n /2μ2 c[n])πE [n, z, t]2 + (/2)Ez [n, t, z]2 + (/2)c[n]h−4 n Ez,z [n, t, z] ] n

The canonical commutation reltions are [Ez [n, z, t], πE [n , z  , t]] = iδn,n δ(z − z  ), [Hz [n, z, t], πH [n , z  , t]] = iδn,n δ(z − z  ) Hence, we can formally set up the Schrodinger equation for the wave function of the confined field within a waveguide of length d as 

d

( 0

H(Ez [n, z], Hz [n, z], −iδ/δEz [n, z], −iδ/δHz [n, z], n = 1, 2, ...)dz)ψt (Ez [m, z], Hz [m, z], m = 1, 2, ...) = i∂t ψt (Ez [m, ξ], Hz [m, ξ], m = 1, 2, ...0 ≤ ξ ≤ d)

Chapter 9

Classical and quantum filtering and control based on Hudson-Parthasarathy calculus, and filter design methods 9.1

Belavkin filter and Luc-Bouten control for electron spin estimation and quantum Fourier transformed state estimation when corrupted by quantum noise

QFT is realized by Schrodinger evolution for a fixed time duration under a designed system Hamiltonian but while implementing the evolution, it gets corrupted by quantum noise; the resulting state evolving according to the noisy HP -Schrodinger equation has to be estimated as well as the Lindblad noise appearing the HP equation has to be reduced. Belavkin quantum filter for estimating the spin of an electron as an example in quantum computation. The spin of an electron along any given direction can have only two eigenvalues ±1/2. We may represent such a particle at time t as c1 (t)|1 > +c0 (t)|0 > where c1 (t), c0 (t) are complex random processes with the constraint |c1 (t)|2 + |c0 (t)|2 = 1, t ≥ 0 where in the state |1 > the spin component along the given direction has eigenvalue +1/2 and in the state |0 >, it has the eigenvalue −1/2. This is a one qubit state in the language of quantum computation and we may wish to transmit this state through a one

125

126

Quantum Antennas

qubit quantum channel. We may on the other hand allow this state to evolve according to a noisy Schrodinger equation, take non-demolition measurements on the noisy bath passed through the system and from these measurements, estimate the evolved spin using the Belavkin filter. In such a case, the estimated state of the electron at time t is a 2 × 2 positive definite matrix with unit trace measurable w.r.t the Abelian Von-Neumann algebra of measurements upto time t. Such an estimated density can also be viewed as a 2 × 2 random matrix with some probability distribution when the Bath state is fixed, for example when the bath is in a coherent state. When the spin of the electron interacts with a real valued nonrandom time varying magnetic field B(t), its interaction energy is H(t) = ge(σ, B(t))/4m where g ≈ 2 and σ is the triplet of the Pauli spin matrices. In addition, when the electron interacts with a noisy bath modeled in the Hudson-Parthasarathy formalism as a Boson Fock space Γs (L2 (R+ )), its dynamics in described by the Hudson-Parthasarathy noisy Schrodinger evolution equation dU (t) = (−i(H(t) + P )dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t) where A(t), A(t)∗ , Λ(t) are operators in the Boson Fock space and P, L1 , L2 , S are like H(t) system space operators, ie 2 × 2 complex matrices. Let X be a system space observable, ie, a 2 × 2 Hermitian matrix like for example the spin (σ, n ˆ )/2 along a fixed direction n ˆ The Hudson-Parthasarathy noisy Schrodinger equation can be regarded as a time varying single qubit quantum channel which transforms the initial state of the electron |ψs (0) >= c1 (0)|1 > +c0 (0)|0 > into a mixed state ρs (t) at time t defined by ρs (t) = T r2 (U (t)(ρs (0) ⊗ |φ(u) >< φ(u)|)U (t)∗ ) where ρs (0) = |ψs (0) >< ψs (0)| and |φ(u) >= exp(−  u 2 /2)|e(u) > is the state of the bath, ie, the bath is in a coherent state. After transmission of the system state over the noisy HP channel, the Belavkin filter at the receiver end, decodes the system state based on non-demolition measurements. Thus, this whole setup can be regarded as a prototype single qubit quantum communication system with a receiver which operates on a real time basis. We may consider more general cases like for example suppose H is an N × N Hermitian matrix such that U0 (T ) = exp(−iT H) is the quantum Fourier transform matrix. Here N = 2r and we N −1 wish the system state at time 0 specified as |ψs (0) >= k=0 ck (0)|k > to be quantum Fourier transformed at time T to the state U0 (T )|ψs (0) >=

N −1 

N −1/2 exp(−i2πkn/N )ck (0)|n >

k,m=0

However owing to quantum noise coming from the bath some error is introduced and the Belavkin filter tries to estimate this state and if we follow this up

Quantum Antennas

127

with the Luc-Bouten algorithm of quantum control, we can remove part of the HP noise and obtain thereby a more reliable estimate of the quantum Fourier transformed state.

9.2

General Quantum filtering and control

(Ref:Naman Garg, Ph.D thesis, NSIT). In this chapter, we propose to solve three kinds of related problems related to quantum filtering and control. The first deals with designing standard finite matrix based MATLAB simulation algorithms for the Hudson-Parthasarathy noisy Schrodinger equation which is today accepted as the standard technique for describing the unitary evolution of system and noisy bath with the system getting noisy inputs from the bath in the form of the three kinds of fundamental quantum noise processes:Creation, annihilation and conservation processes. The second deals with simulating the well known Belavkin quantum filter for obtaining estimates of system states observables on a real time basis with the measurements being non-demolition measurements on the bath noise that passes through the system where the system evolves according to the noisy Hudson-Parthasarathy-Schrodinger equation. Our basic method of simulating these quantum stochastic differential equations is based on the action of noise operators on coherent vectors and constructing truncated orthonormal bases using coherent vectors for the bath space. The Belavkin filter is the quantum non-commutative generalization of the classical Kushner-Kallianpur nonlinear filter and can be used to estimate quantum processes like the spin of the electron interacting with a magnetic field. The third kind of problem we deal with concerns design and implementation of real time quantum control algorithms using a sequence of control unitaries acting on the Belavkin filtered state to obtain to lessen the amount of Lindblad noise in the Hudson-Parthasarathy evolved state as well as to track a given state trajectory. These control algorithms have their roots in Luc-Bouten’s famous thesis on filtering and control in quantum optics. The final problem dealt in this work involves computing the evolution of the Von-Neumann entropy of, (a) the Hudson-Parthasarathy noisy Schrodinger system state after tracing out over the bath variables, (b) the Belavkin filtered state and (c) the state after applying the control algorithm. These entropy computations involve Lie algebraic techniques and are important in physics for validating the second law of thermodynamics on monotonic entropy increase and also in modern quantum communication involving evaluating the amount of information transmitted through a noisy quantum channel.

9.3

Some topics in quantum filtering theory

[i] Historical survey of classical and quantum filtering theory [ii] Mathematical details of quantum filtering theory. [a] Schrodinger equation for state evolution in noiseless quantum dynamics.

128

Quantum Antennas

The Schrodinger equation for the state vector |ψ(t) > taking values in a Hilbert space H is given by id/dt|ψ(t) >= H(t)|ψ(t) > where H(t) is a Hermitian operator. This guarantees unitarity of the evolution, ie, < ψ(t)|ψ(t) >=< ψ(0)|ψ(0) >, t ≥ 0 which implies conservation of probabilities. For example, if we use the position representation PAM Dirac), then the wave function would be < r|ψ(t) >= ψ(t, r), r ∈ R3 and the above equation would read as  |ψ(t, r)|2 d3 r = 1, t ≥ 0 provided that



|ψ(0, r)|2 d3 r = 1

This means that the particle cannot with positive probability escape away to ∞ in a bound state. Schrodinger discovered the form of the Hamiltonian operator H(t) for a particle moving in R3 with a kinetic energy of p2 /2m and a potential energy of V (t, r). The total energy is then p2 /2m + V = E. According to Planck’s quantum hypothesis, the frequency of a wave associated with this energy is ω = 2πE/h where h is Planck’s constant and according to DeBroglie’s matter wave duality principle, the wave-vector associated to such a wave is k = 2πp/h and the associated plane wave to this particle should be ψ(t, r) = A.exp(−i(ωt − k.r)) After doing this, Schrodinger observed that for such a plane wave, 2πih∂ψ(t, r)/∂t = (hω/2π)ψ(t, r) = Eψ(t, r) and (−ih/2π)∇ψ(t, r) = (hk/2π)ψ(t, r) = pψ(t, r) Schrodinger postulated based on this intuitive notion, that even when the wave function is square integrable, E should be regarded as an energy operator 2πih∂/∂t and p as a momentum operator (−ih/2π)∇. Then, Newton’s relation between energy and momentum given by E − p2 /2m − V = 0 should be interpreted as a wave equation (E − p2 /2m − V )ψ(t, r) = 0 or equivalently, (ih/2π)∂ψ(t, r)/∂t − (h2 /8π 2 m)∇2 ψ(t, r) − V (t, r)ψ(t, r) = 0

129

Quantum Antennas

which is precisely the Schrodinger wave equation. Schrodinger’s genius was to state that this wave equation is valid for all nonrelativistic quantum phenomena even when its solution ψ(t, r) does not represent a plane wave. Schrodinger postulated that the bound states of atoms and molecules should correspond only to square integrable solutions and it became clear to him that such solutions can be expressed by separating the space and time variables as ψ(t, r) =

∞ 

exp(−iEn t)ψn (r)

n=1

where the ”energy eigenvalues” En are real numbers satisfying the eigenvalue problem, namely the stationary Schrodinger equation: En ψn (r) − (h2 /8π 2 m)∇2 ψn (r) − V (r)ψn (r) = 0 in the special case when the potential V does not depend explicitly on time. The eigenfunctions ψn (r) should be square integrable and this condition causes the energy eigenvalue spectrum of bound states to be discrete. He explicitly thus determined the eigenfunctions and energy spectrum of the Hydrogen atom and the quantum harmonic oscillator corresponding to potentials V = −e2 /r and V = Kr2 /2. Later on Dirac recognized that we can also have unbounded state solutions as in scattering theory in which an projectile arrives from ∞, interacts with a repulsive potential and gets scattered to ∞. Such states are not normalizable and are characterized by the energy spectrum being continuous. It was Max Born who suggested that for bound states |ψ(t, r)|2 should be interpreted as the probability density of the particle being in a unit volume around r at time t after normalizing the wave function to have a unit integral for its modulus square. It should be mentioned here that the time dependent Schrodinger equation stated above can be cast in form of a unitary operator evolution equation : |ψ(t) >= U (t)|ψ(0) >, iU  (t) = H(t)U (t) where U (t) is a unitary operator in the Hilbert space of states in which the Hamiltonian operator H(t) is defined. If the energy operator H(t) = H is time independent, then its formal solution is U (t) = exp(−itH) where the exponential of an unbounded operator must be defined via its resolvent (Kato): exp(−itH) = limn→∞ (I + itH/n)−n rather than as limn→∞ (I − itH/n)n the reason being that the resolvent of an unbounded operator is bounded over the resolvent set while the positive integer powers of an unbounded operator

130

Quantum Antennas

will generally have smaller and smaller domains. In the case when H(t) is time dependent as in quantum electrodynamics, we have to express the solutionU (t) as a Dyson series   n (−i) H(t1 )...H(tn )dt1 ...dtn U (t) = I + n≥1

0 be the vacuum state, ie, the state in which all the operators a(n)∗ a(n) have zero eigenvalue and for complex numbers u[n] ∈ C, n = 1, 2, ..., consider the state  |f (u) >= Πn u[n]m(n) a(n)∗m(n) |0 > /Πn m(n)! n

=



Πn u[n]m(n) ⊗n |m(n) > /Πn

 m(n)!

n

Then an easy computation shows that < f (u)|f (v) >= exp(< u, v >) where < u, v >=



u ¯[n]v[n]

n,j

We now choose an onb {ψn (r) = (ψn1 (r), ..., ψnd (r))T : n = 1, 2, ...} for L2 (R) ⊗ Cd and define  u[n]ψn (r) ∈ L2 (R) ⊗ Cd u(r) = n

Then, the map |e(u) >] → |f (u) > defines a Hilbert space isomorphism. In fact, this mapping is uniquely identified by the mapping  √ u⊗N → N !Πn u[n]m(n) Πn a(n)∗m(n) |0 > /Πn m(n)! m(1)+m(2)+...=N

=



√

N !u[1]m(1) u[2]m(2) ...|m(1), m(2), ... > /

 m(1)!m(2)!...

m(1)+...+..=N

which preserves inner products. The exponential vectors |e(u) >, u ∈ H span a dense linear manifold of Γs (H) and hence the action of operators on exponential vectors is sufficient to determine their action on their domain in the entire Boson Fock spaces. Keeping this in mind, for u ∈ H, we define a(u)|e(v) >=< u, v > |e(v) >,

134

Quantum Antennas

and then it is easy to verify using < e(u), e(v) >= exp(< u, v >) that the adjoint of a(u) is given by a(u)∗ |e(v) >=

d |e(v + tu) > |t=0 dt

One then also easily verifies by considering the action on exponential vectors that [a(u), a(v)∗ ] =< u, v > I, [a(u), a(v)] = 0, [a(u)∗ , a(v)∗ ] = 0, u, v ∈ H In other words, the operator fields a(u), a(u)∗ , u ∈ H define an algebra of harmonic oscillators. More precisely, choosing an orthonormal basis |ei >, i = 1, 2, ... of H, we can write for u ∈ H,   u ¯i ai , a(u)∗ = ui a∗i , ui =< ei , u > a(u) = i

i

and then (ai , a∗i ), i = 1, 2, ... define a countably infinite sequence of one dimensional quantum harmonic oscillators in the sense that the canonical commutation relations [ai , a∗j ] = δij , [ai , aj ] = 0, [a∗i , a∗j ] = 0 are satisfied. As in conventional quantum mechanics, we can also define a conservation/number operator Λi by Λi = a∗i ai , i = 1, 2, ... and then, we can obtain an onb for Γs (H) as |n1 , n2 , ... > where Λi |n1 , n2 , ... >= ni |n1 , n2 , ... > and ai |n1 , n2 , ... >=

√

ni |n1 , ..., ni−1 , ni − 1, ni+1 , ... > √ >= ni + 1|n1 , n2 , ... >

a∗i |n1 , n2 , ...

It is then easily seen that the exponential vector can be represented by   un1 1 un2 2 ...|n1 , n2 , ... > / n1 !n2 !... |e(u) >= n1 ,n2 ,...

and that the above actions of ai , a∗i on |n1 , n2 , ... > are equivalent to a(v)|e(u >=< v, u > |e(u) >, a(v)∗ |e(u) >=

 i

vi

∂ d |e(u) >= |e(u+tv) > |t=0 ∂ui dt

135

Quantum Antennas or equivalently, ai |e(u) >= ui |e(u) >, a∗i |e(u) >=

∂ d |e(u) >= |e(u + tei ) > |t=0 ∂ui dt

An easy calculation using the above identities shows further that ¯i vi < e(u)|e(v) > < e(u)|Λi |e(v) >= u Based on this idea, Hudson and Parthasarathy [1984] introduced apart from the creation and annihilation operator fields a(u), a(u)∗ , u ∈ H, a conservation operator field λ(Q) where Q is a Hermitian operator in H by the rule exp(itλ(Q))|e(u) >= |e(exp(itQ)u) >, t ∈ R or equivalently, iλ(Q)|e(u) >=

d |e(exp(itQ)u > |t=0 dt

It is easy to see then that < e(v)|λ(Q)|e(u) >=< v|Q|u >< e(v)|e(u) > and in particular if Q = |ei >< ei |, then < v|Q|u >= v¯i ui so that λ(|ei >< ei |) = Λi More generally, if Q has the spectral representation  Q= |ei > qi < ei | i

then λ(Q) =

 i

qi Λi =



qi a∗i ai

i

Most of this formalism was well known even before the seminal paper of Hudson and Parthasarathy [1984]. The crucial observation of Hudson and Parthasarathy was to introduce quantum processes, ie creation, annihilation and conservation processes by bringing time into the picture in the form of a continuous unfolding of the tensor product appearing in Boson Fock space. Without going into the mathematics, the idea here is to consider the indicator function χ[0,t] in L2 (R+ ) and then choose an orthonormal basis |fi >, i = 1, 2, ..., d for Cd so that χ[0,t] |fi >∈ H = L2 (R+ ) ⊗ Cd

136

Quantum Antennas

Then Hudson and Parthasarathy define Ai (t) = a(χ[0,t] |fi >), Ai (t)∗ = a(χ[0,t] |fi >)∗ , Λi (t) = λ(χ[0,t] |fi >< fi |) where i = 1, 2, ..., d and where χ[0,t] |fi >< fi | = χ[0,t] ⊗|fi >< fi | is a Hermitian operator in H with χ[0,t] acting as a multiplication. The action of these operators on the exponential vectors is then easily deduced to be given by  t ui (s)ds)|e(u) > − − −(a), Ai (t)|e(u) >= ( 0

< e(v)|Ai (t)∗ |e(u) >= (  < e(v)|Λi (t)|e(u) >= (



t 0

v¯i (s)ds) < e(v)|e(u) > − − −(b),

t

v¯i (s)ui (s)ds) < e(v)|e(u) > − − −(c)

0

and more generally,  < e(v)|λ(Qt )|e(u) >= (

t

< v(s)|Q|u(s) > ds) < e(v)|e(u) > 0

where Qt = χ[0,t] Q 2

and u, v ∈ H = L (R+ ) ⊗ C is specified in coordinate form as d

u(t) = [u1 (t), ..., ud (t)]T , t ≥ 0, uk (.) ∈ L2 (R+ ) Since in quantum electrodynamics, the annihilation operator fields are proportional to the magnetic vector potential in the spatial Fourier domain and since the electric field in the spatial frequency domain is also proportional to the magnetic vector potential provided that we adopt the Coulomb gauge (divA = 0) which means that the electric scalar potential A0 becomes a matter field and not any combination of the field part, we can interpret the equation (a) as giving the total complex amplitude, ie, amplitude and phase of the photons in the exponential/coherent state |e(u) > upto time t. Likewise, the equation (c) determines the number of accumulated photons in the ith a coherent state upto time t:  t < e(u)|Λi (t)|e(u) >= ( |ui (t)|2 dt) < e(u)|e(u) > 0

The equation (b) tells us the total complex amplitude of the photons in the ith mode that have been absorbed into the coherent state |e(v) >. The processes Ai (t), Ai (t)∗ , Λi (t), i = 1, 2, ..., d are known as the fundamental noise processes in the Hudson-Parthasarathy calculus. They are not random functions, they are simply families of linear operators in the Boson Fock space, but when we restrict our bath space to be in a given state like a coherent state, then these

137

Quantum Antennas

families of operators display statistics like the classical Brownian motion and Poisson processes as special commutative cases. The important point to be noted is that these operator valued processes are non-commutative unlike the classical case and hence can exhibit more general kinds of statistical behaviour than those displayed by classical stochastic processes. The non-commutativity of these processes is precisely the fact that the most general GKSL equations can be dilated into unitary evolutions using quantum noisy Schrodinger equations, ie, the Hudson-Parthasarathy noisy Schrodinger equation. This is not possible using only classical stochastic noise. The reason behind this is the quantum Ito formula discussed next which can be regarded as an Ito formula taking into account Heisenberg’s uncertainty principle arising from the non-commutativity of the operator valued processes. [d] Quantum Ito’s formula and the quantum noisy HP-Schrodinger equation. If z1 , z2 ∈ Cd , then χ0,t] zk ∈ H = L2 (R+ ) ⊗ Cd , k = 1, 2 and it is not hard to show that da(χ[0,t] z1 ).da(χ[[0,t] z2 )∗ =< z1 , z2 > dt and on the other hand all the other products of these two differentials are zero, ie, o(dt). To verify the above formula, we observe that the Harmonic oscillator commutation relations imply [a(χ[0,t] z1 ), a(χ[0,s] z2 )∗ ] =< χ[0,t] , χ[0,s] >< z1 , z2 > = min(t, s) < z1 , z2 >= (sθ(t − s) + tθ(s − t)) < z1 , z2 > and thus taking the differentials w.r.t t and using dθ(t − s) = δ(t − s)dt and xδ(x) = 0 gives us [da(χ[0,t] z1 ), a(χ[0,s] z2 )∗ ] = ((s − t)δ(t − s) + θ(s − t)dt) < z1 , z2 > = θ(s − t)dt < z1 , z2 > Now take the differential on both sides w.r.t s and use the identity δ(s − t)ds|s=t dt = dt to get the result [da(χ[0,t] z1 ), da(χ[0,t] z2 )∗ ] =< z1 , z2 > dt Now it is clear that

da(χ[0,t] z2 )∗ .da(χ[0,t] z1 ) = 0

as follows by taking the matrix element of the lhs w.r.t |e(u) > and |e(v) >: < e(u)|da(χ[0,t] z2 )∗ da(χ[0,t] z1 )|e(v) >= < da(χ[0,t] z2 )e(u)|da(χ[0,t] z1 )e(v) >= < z2 , u(t) >∗ dt. < z1 , v(t) > dt. < e(u), e(v) >= O(dt2 ) In other words, we have da(χ[0,t] z2 )∗ da(χ[0,t] z1 ) = 0

138

Quantum Antennas

We have thus proved the required quantum Ito formula (choosing z1 = fi , z2 = fj ∈ Cd ) dAi (t).dAj (t)∗ = δij dt and all the other products of these two differentials vanish. It should be noted that in deducing this quantum Ito formula, we have made use of Bosonic commutation relations and this is why one often says that quantum Ito formula can be traced to the Heisenberg uncertainty principle. The other quantum Ito formula are obtained as follows. First observe that < e(v)|dA∗i (t)dAj (t)|e(u) >= uj (t)¯ vi (t)dt2 < e(v)|e(u) > and since < e(v)|dλ(Qt )|e(u) >= d < e(v)|λ(Qt )|e(u) >  t =< e(v)|e(u) > d < v(s)|Q|u(s) > ds 0

=< e(v)|e(u) >< v(t)|Q|u(t) > dt it follows that In particular,

dAi (t)∗ .dAj (t)/dt = dλ(χ[0,t] ||fi >< fj |) dAi (t)∗ .dAi (t)/dt = dΛi (t)

and hence, application of the quantum Ito formula derived earlier gives us dAj (t)dΛi (t) = δij dAi (t), dΛi (t).dAj (t) = δij dAi (t)∗ , dΛi (t).dΛj (t) = δij dΛi (t) These are known as the quantum Ito formulae and are fundamental in studying the effect of noise on quantum systems as well as in formulating a quantum filtering theory based on non-demolition measurements. The processes Bi (t) = Ai (t)+Ai (t)∗ have all the properties of classical Brownian motion in the vacuum coherent state. First, they commute with each other: [Bi (t), Bj (s)] = 0∀t, s, i, j second, for f (t) = (fi (t)) ∈ L2 (R+ ) ⊗ Cd , < e(u)|exp(

N 

intT0 fi (t)dBi (t))|e(u) >=

i=1

< e(u)|exp(a(f χ[0,T ] ) + a(f χ[0,T ] )∗ )|e(u) >= [e] Matrix elements of noise operators in Boson Fock space relative to coherent vectors.

139

Quantum Antennas Given an HP qsde of the form  (Li dAi (t) + Mi dAi (t)∗ + Si dΛi (t))U (t) dU (t) = (−(iH + P )dt i

we can simulate this approximately as follows: We first choose a set u1 , ..., uN of linearly independent vectors in H = L2 (R+ ) ⊗ Cd and then apply the GramSchmidt orthonormalization to the corresponding exponential vectors |e(uk ) > , k = 1, 2, ..., N . Denote the resulting orthonormal vectors by |ξk >, k = 1, 2, ..., N . Thus, |ξk >=

k 

c(k, m)|e(um ) >, 1 ≤ k ≤ N,

m=1

with inverse |e(uk ) >=

k 

d(k, m)|ξm >, 1 ≤ k ≤ N

m=1

and < ξk |ξm >= δkm Since the exponential vectors span a dense linear manifold of the Boson Fock space [KRP book], it follows that if N is large, the vectors |ξk >, k = 1, 2, ...., N will span almost completely the Boson Fock space. Choosing then an orthonormal basis |ηk >, k = 1, 2, ..., p for the system Hilbert space h, we get an orthonormal set |ηr ⊗ ξk >, 1 ≤ k ≤ N, 1 ≤ r ≤ p in h ⊗ Γs (H)=system space⊗ bath space. We can then take matrix elements on both sides of the above HP equation relative to this basis and derive a sequence of linear deterministic equations for the truncated matrix elements of the unitary evolution operator U (t). The advantage of using orthonormal sets for computing matrix elements is due to the composition law: < ηr ⊗ ξk |AB|ηs ⊗ ξm >≈ p N  

< ηr ⊗ ξk |A|ηl ⊗ ξq >< ηl ⊗ ξq |B|ηs ⊗ ξm >

q=1 l=1

This equation is exact iff A, B are operators defined in the truncated system ⊗ noise space. It should be noted that this method can be combined with the fact that if L is a system operator, then LdAk (t)U (t) = (L ⊗ I)U (t)dAk (t) = (L ⊗ I)U (t) ⊗ dAk (t) and likewise with Ak (t) replaced by Ak (t)∗ or with Λk (t), the reason being that Ak (t), Ak (t)∗ , Λk (t) act in Γs (L2 ([0, t] ⊗ Cd ) while dAk (t), dAk (t)∗ , dΛk (t) act in Γs (L2 (t, t + dt]) and we have the Hilbert space isomorphism Γs (H1 ⊕ H2 ) = Γs (H1 ) ⊗ Γs (H2 )

140

Quantum Antennas

as follows by choosing wk , vk ∈ Hk , k = 1, 2 and noting that < e(w1 ⊕ v1 )|e(w2 ⊕ v2 ) >= exp(< w1 ⊕ v1 |w2 ⊕ v2 >) = exp(< w1 |w2 > + < v1 |v2 >) =< e(w1 )|e(w2 ) >< e(v1 )|e(v2 ) > =< e(w1 ) ⊗ e(v1 )|e(w2 ) ⊗ e(v2 ) > In other words for u, v ∈ H with < u|v >= 0, we can under this isomorphism identify |e(u + v) > with |e(u) ⊗ e(v) >. In view of this, we have the Hilbert space isomorphism Γs (χ[0,t+dt] H) = Γs (χ[0,t] H) ⊗ Γs (χ(t,t+dt] H) To see why dAk (t), dAk (t)∗ , dΛk (t) act in Γs (χ(t,t+dt] H) , we use the isomorphism identity just established in the form e(χ[0,T ] u) = e(χ[0,t] u) ⊗ e(χ(t,T ] , 0 < t < T in the sense that  < e(χ[0,T ] u)|e(vχ[0,T ] ) >= exp( 

t

= exp(

T

< u(s)|v(s) > ds) 0

 < u(s)|v(s) > ds).exp(

0

T

< u(s)|v(s) >)

t

=< e(uχ[0,t] ) ⊗ e(uχ(t,T ] )|e(v(χ[0,t] ) ⊗ e(vχ(t,T ] ) > We can thus write for s < t, 

t

< e(v)|Ak (t) − Ak (s)|e(u) >=< e(v)|

uk (τ )dτ |e(u) > s

 =(

t

uk (τ )dτ ) < e(v)|e(u) > s

on the one hand and on the other hand, |e(u) >= |e(χ[0,s] )u) ⊗ e(χs,t] ) ⊗ e(χ(t,∞) ) > with < e(χ(s,t] v)|(Ak (t) − Ak (s)|e(χ(s,t] )u >  t uk (τ )dτ ) < e(χ(s,t] v)|e(χ(s,t] u) > =( s

which is in view of the above isomorphism, the same as the previous equation since the isomorphism implies |e(u) >= |e(χ(0,s] u) ⊗ e(χ(s,t] u) ⊗ e(χ(t,∞) ) >

141

Quantum Antennas and further, if I is any interval in R non-overlapping with (s, t], then  uk (τ )dτ ) < e(χI )v)|e(χI .u) >= 0 < e(χI .v)|Ak (t) − Ak (s)|e(χI .u) >= ( I∩(s,t]

It should be noted that in view of these discussions, the isomorphism means that < e(v)|Ak (t) − Ak (s)|e(u) >= < e(vχ(0,s] ) ⊗ e(v.χ(s,t] ) ⊗ e(v.χ(t,∞) )|Ak (t) − Ak (s)| .e(uχ(0,s] ) ⊗ e(u.χ(s,t] ) ⊗ e(u.χ(t,∞) ) > =< e(v.χ(0,s] )|e(uχ(0,s] ) >< e(v.χ(s,t] )|Ak (t) −Ak (s)|e(u.χ(s,t] ) > . < e(v.χ(t,∞)) |e(u.χ(t,∞) ) > and likewise for the other noise operators Ak (t)∗ and Λk (t). This is the reason why we can say that if M (t) is any one of these noise operators, then for s < t, we can say that M (t) − M (s) acts in the Hilbert space Γs (χ(s,t] H) when we consider the isomorphism Γs (H) = Γs (χ(0,s] H ⊕ χ(s,t] H ⊕ χ(t,∞) H) = Γs (χ(0,s] H) ⊗ Γs (χ(s,t] H) ⊗ Γs (χ(t,∞) H) It is precisely this property of the noise operators, ie, quantum independent increment property that makes the HP equation easy to simulate using matrix elements w.r.t. bath states built out of finite linear combinations of exponential/coherent vectors. For example, the matrix elements of the terms LdA(t)U (t), M dA(t)∗ U (t), SdΛ(t)U (t) can be evaluated by noting that L, M, S act in h, the system Hilbert space, U (t) acts in h ⊗ Γs (L2 [0, t] ⊗ Cd ) = h ⊗ Γs (χ[0,t] H) while dA(t), d(t)∗ , dΛ(t) act in Γs (χ(t,t+dt] H): Let dM denote any one of dAk , dA∗k , dΛ. Then, < ηr ⊗ ξk |LdM (t)U (t)|ηs ⊗ ξl >=  < ηr ⊗ξk |L|ηq ⊗ξm >< ηq ⊗ξm |U (t)|ηq ⊗ξm >< ηq ⊗ξm |dM (t)|ηs ⊗ξl > m,q,m ,q 

=



< ηr |L|ηq > δkm < ηq ⊗ ξm |U (t)|ηq ⊗ ξm > δq s < ξm |dM (t)|ξl >

m,q,m ,q 

=



< ηr |L|ηq >< ηq ⊗ ξk |U (t)|ηs ⊗ ξm >< ξm |dM (t)|ξl >

q,m

and if dM (t) = dAk (t), we get  < ξm |dM (t)|ξl >= c¯(m, a)c(l, b) < e(ua )|dAk (t)|e(ub ) >= a,b

142

Quantum Antennas =



c¯(m, a)c(l, b)ubk (t) < e(ua )|e(ub ) > dt

a,b

while if dM (t) = dAk (t)∗ , then  < ξm |dM (t)|ξl >= c¯(m, a)c(l, b) < e(ua )|dAk (t)∗ |e(ub ) >= a,b

=



c¯(m, a)c(l, b)¯ uak (t) < e(ua )|e(ub ) > dt

a,b

and finally if dM (t) = dΛk (t), then  < ξm |dM (t)|ξl >= c¯(m, a)c(l, b) < e(ua )|dΛk (t)|e(ub ) >= a,b

=



c¯(m, a)c(l, b)¯ uak (t)ubk (t) < e(ua )|e(ub ) > dt

a,b

These formulas enable simulation of the HP qsde in the form of deterministic difference equations after time has been discretized. [f] The GKSL equation–a derivation based on the HP-Schrodinger equation. The HP equation is a dilated version of the GKSL equation. This means that the GKSL equation does not describe unitary evolution of the system. In fact, it describes the evolution of only the mixed system state in the presence of a noisy bath. It does not describe the evolution of a pure system state. If the system state is initially pure, then under interaction with the bath, it becomes mixed after sometime. This leads us to suspect that the system Hilbert space can be enlarged, ie, dilated to include the bath Hilbert space in such a way that the overall evolution of system and bath is described by a unitary evolution operator which when applied to an initally pure state on the system and bath and then averaged out over the bath via a partial trace will yield the GKSL. The answer to this is provided by the HP-noisy Schrodinger equation. To see this, we start with the HP equation  (Lk dAk + Mk dA∗k + Sk dΛk )U (t) dU (t) = (−(iH + P )dt + k

and take a system observable X. Define jt (X) = U (t)∗ XU (t) = U (t)∗ (X ⊗ I)U (t) Then jt : B(h) → B(h ⊗ Γs (χ[0,t] H) is a ∗ unital homomorphism, ie, jt (c1 X + c2 Y ) = c1 jt (X) + c2 jt (Y ), jt (XY ) = jt (X)jt (Y ), jt (X ∗ ) = jt (X)∗ Note that the system operators H, P, Lk , Mk , Sk are chosen so that quantum Ito’s formula ensures that U (t) is unitary for all t. Another application of quantum Ito’s formula yields  djt (X) = jt (θ0 (X))dt+ (jt (θ1k (X))dAk (t)+jt (θ2k (X)dAk (t)∗ jt (θ3k (X))dΛk (t) k

143

Quantum Antennas

where the θ0 , θ1k , θ2k , θ3k are linear maps from B(h) to itself expressible in term of the system operators H, P, Lk , Mk , Sk , k = 1, 2, ..., d. Now, assume that the initial state of the system ⊗ bath is ρ(0) = ρs (0) ⊗ |φ(u) >< φ(u)|, |φ(u) >= exp(−  u 2 /2)|e(u) > Then after time t, the state of the system will be ρs (t) = T r2 (U (t)ρ(0)U (t)∗ ) Thus, if X is a system observable, we have that its average at time t is given by T r(ρs (t)X) = T r(U (t)ρ(0)U (t)∗ (X ⊗ I)) = T r(ρ(0)U (t)∗ (X ⊗ I)U (t)) = T r(ρ(0)jt (X)) and its differential is given by dt.T r(ρs (t)X) = T r(ρ(0)djt (X)) = T r(ρ(0)jt (θ0 (X)))dt+



T r(ρ(0)jt (θ1k (X))dAk (t))+

k

T r(ρ(0)jt (θ2k (X))dAk (t)∗ )+T r(jt (θ3k (X))dΛk (t))

Now, it easily follows that T r(ρ(0)jt (θ0 (X)) = T r(ρs (t)θ0 (X)), and from the quantum independent increment property of the fundamental noise processes, T r(ρ(0)jt (θ1k (X))dAk (t)) = T r((ρs (0) ⊗ dAk (t)|φ(u) >< φ(u)|)jt (θ1k (X))) = uk (t)dt.T r((ρs (0) ⊗ |φ(u) >< φ(u)|)jt (θ1k (X))) = uk (t)dt.T r(ρ(0)jt (θ1k (X))) = uk (t)dt.T r(ρs (t)θ1k (X)), T r(ρ(0)jt (θ2k (X))dAk (t)∗ ) = T r((ρs (0) ⊗ (|φ(u) >< φ(u)|dAk (t)∗ ).jt (θ2k (X))) =u ¯k (t)dt.T r((ρs (0) ⊗ |φ(u) >< φ(u)|)jt (θ2k (X))) ¯k (t)dt.T r(ρs (t)θ2k (X)), =u ¯k (t).dt.T r(ρ(0)jt (θ2k (X))) = u T r(ρ(0)jt (θ3k (X))dΛk (t)) = T r(ρs (0) ⊗ dΛk (t)|φ(u) < φ(u)|.jt (θ3k (X))) = dt−1 T r(ρs (0) ⊗ dAk (t)∗ dAk (t)|φ(u) >< φ(u)|.jt (θ3k (X))) = uk (t).T r(ρs (0) ⊗ |φ(u) >< φ(u)|dAk (t)∗ .jt (θ3k (X))) = |uk (t)|2 dtT r(ρ(0)jt (θ3k (X)) = |uk (t)|2 dtT r(ρs (0)θ3k (X))

144

Quantum Antennas

If now θ maps B(h) into itself, then we have for each state ρs in h, a unique operator θ∗ (ρs ) in h with the property T r(θ∗ (ρs )X) = T r(ρs θ(X)) for all X ∈ B(h). The operator θ∗ that maps states in h into B(h) is called the dual map of θ. For example, if  θ(X) = Lk XMk k

then, θ∗ (ρs ) =



M k ρs L k

k

Combining all these relations and using the arbitrariness of the system operator X results finally in the master equation/GKSL equation in its most generalized form:  ∗ ∗ [uk (t)θ1k (ρs (t)) + u ¯k (t)θ2k (ρs (t)) + |uk (t)|2 θ3k (ρs (t))] ρs (t) = θ0∗ (ρs (t)) + k

It is easily shown that all the commonly used master equations in the theory of open quantum systems are special cases of this. That such a general master equation arises from a dilation to a unitary evolution using fundamental quantum noise processes is perhaps one of the pinnacle achievements of the Hudson-Parthasarathy theory. [g1] The need for non-demolition measurements from the standpoint of Heisenberg’s uncertainty principle for constructing quantum conditional expectations.

9.3.1

Non-demolition measurements on the bath space passed through the system

For obtaining a filtering theory for quantum noisy systems, in which the state at time t is jt (X), we need to calculate its conditional expectation given measurements upto time t. This means that if Y (s), s ≤ t is the set of measurements upto time t, then we must be in a position to define the conditional expectation πt (X) = E(jt (X)|Y (s), s ≤ t) Now in the quantum theory just as the system state at time t jt (X) is an operator in the system⊗ bath space, the measurements Y (.) must also be operators in this space. To define the above conditional expectation when the system ⊗ bath is in a given state and observables evolve with time, (ie, the Heisenberg picture of quantum mechanics), we must give meaning to the joint probability distribution of (jt (X), Y (s), s ≤ t) and this is possible in view of the Heisenberg uncertainty principle iff all the observables jt (X), Y (s), s ≤ t commute with each other for

145

Quantum Antennas

each t ≥ 0. In other words, the measurement algebra ηt = σ(Y (s) : s ≤ t) must be Abelian for each t and further [Y (s), jt (X)] = 0, t ≥ s, ie, the measurements must commute with the future state so that on making a measurement, the future values of the state do not get disturbed. Whenever such measurements exist, we say that they follow the non-demolition property. An example of such measurements was first constructed by V.P.Belavkin [] in a series of path-breaking papers and refined in a mathematically rigorous way by John Gough and Koestler []. The idea of constructing non-demolition measurements proposed by Belavkin was to first define an input measurement process Yi (t) as a linear combination of the fundamental noise processes Ai (t), Ai (t)∗ , Λi (t), i = 1, 2, ..., d, ie, d  (c[i]Ai (t) + c¯[i]Ai (t)∗ + d[i]Λi (t)) Yi (t) = i=1

and then pass this input process through the HP noisy system to get the output measurements as Yo (t) = U (t)∗ Yi (t)U (t) = U (t)∗ (I ⊗ Yi (t))U (t) Belavkin observed that for such an output process, one has Yo (t) = U (T )∗ Yi (t)U (T ), T ≥ t The trick in proving this is to use the quantum Ito formulas combined with the fact that U (T ) is unitary and that the unitarity of U (T ) is solely dependent on the system operators appearing in the HP equation which therefore commute with the input noise process values Yi (t). Specifically, the differential of U (T )∗ Yi (t)U (T ) w.r.t T for T > t is given by dT (U (T )∗ Yi (t)U (T )) = dU (T )∗ Yi (t)U (T )+U (T )∗ Yi (t)dU (T )+dU (T )∗ Yi (t)dU (T ) = 0

which follows by explicitly substituting for dU (T ) and dU (T )∗ from the HP equation. This can equivalently be seen by noting that the HP equation can be expressed as  dU (T ) = Ej dMj (T )U (T ) j

where the processes Mj (t) s consist of t, Ak (t), Ak (t)∗ and Λk (t) while the Ej s are system operators. It is clear from the quantum independent increment property of the fundamental noise processes that the dMj (T )’s commute with all system operators as well as with Yi (t) since T > t. Thus, the above equation can also be expressed in the form dT (U (T )∗ Yi (t)U (T )) = U (T )∗ (



(Ej∗ dMj (T )∗

j

+Ej dMj (T ))+



Ej∗ Ek dMj (T )∗ dMk (T ))Yi (t)U (T ) j,k

and by the unitarity of U (T ) and the quantum Ito formula applied to d(U (T )∗ U (T )) = 0, we have already the result

 j

(Ej∗ dMj (T )∗ + Ej dMj (T )) +

 j,k

Ej∗ Ek dMj (T )∗ dMk (T ) = 0

146

Quantum Antennas

thus establishing dT (U (T )∗ Yi (t)U (T )) = 0, T > t and hence U (T )∗ Yi (t)U (T ) = U (t)∗ Yi (t)U (t), T > t It is clear that since [dYi (t), dYi (s)] = 0, t = s (the independent increment property of the fundamental processes) we have that [Yi (t), Yi (s)] = 0, t = s and hence Yi (t), t ≥ 0 forms an Abelian family of operators. Hence from the unitarity of U (T ), U (T )∗ Yi (t)U (T ), t ≤ T also forms an Abelian family for each T > 0. Further since the operators Yi (t), t ≥ 0 act in the bath space, they all commute with any system operator X and hence again by the unitarity of U (T ), the family of operators U (T )∗ Yi (t)U (T ), t ≤ T commute with the operator U (T )∗ XU (T ) for any T > 0. Combined with the above observation, we obtain the result that the family of operatorsYo (t), t ≤ T commutes with U (T )∗ XU (T ) = jT (X) and this completes the proof of the non-demolition property. This remarkable fact was first noted by Belavkin and he used these measurements to obtain a real time filter that generalizes the Kushner-Kallianpur filter.

9.3.2

Derivation of the general Belavkin filter both in observable and in state for quadrature and photon counting measurements using Gough’s reference probability approach

The HP equation has the general form dU (t) = (−(iH + P )dt +

d 

Li dAi + Mi dA∗i + Si dΛi )U (t)

i=1

and using the quantum Ito formulae, the condition for U (t) to be unitary at all times is that  Mi Mi∗ /2, Si∗ + Si + Si∗ Si = 0, Mi∗ + Li + Mi∗ Si = 0, P = i

L∗i + +Mi + Si∗ Mi = 0 the last two of which are equivalent. Here, Li , Mi , Si , H are all system space observables.

9.3.3

Why the Belavkin filter is the non-commutative generalization of the Kushner-Kallianpur filter ?

The Belavkin filter for quadrature measurements in a coherent state has the form dπt (X) = πt (Lt X)dt+(πt (Mt X+XMt∗ )−πt (Mt +Mt∗ )πt (X))(dYo (t) −πt (Mt +Mt∗ ))−−−(1)

147

Quantum Antennas

where Mt , t ≥ 0 is a family of system operators. Taking h = L2 (R), X as the operator of multiplication by a function φ(x) in L2 (R) and jt (φ) = φ(ξ(t)) where ξ(t) is a classical diffusion process dξ(t) = μ(ξ(t))dt + σ(ξ(t))dB(t) we find that by the classical Ito formula, djt (φ) = φ (ξ(t))σ(ξ(t))dB(t) + Lφ(ξ(t))dt = jt (Lφ)dt + jt (θ(φ))dB(t) where

L = μ(x)d/dx + (σ 2 (x)/2)d2 /dx2

is the generator of the diffusion and θ = σ(x)d/dx We now briefly take a look at the Kushner-Kallianpur filter in classical nonlinear filtering theory. The state process x(t) satisfies the above sde and the measurement process has the form dy(t) = h(x(t))dt + σv dv(t) where the measurement noise process v(.) is a Brownian motion independent of the state process noise B(.) which is another Brownian motion. The measurement σ-field upto time t is given by ηt = σ(y(s) : s ≤ t) but now everything in the state and measurement system commute with each other since all processes are defined on a fixed classical probability space (Ω, F, P ). The aim is to obtain a stochastic pde for pt (x|ηt ) defined as the probability density of the state x(t) at time t given measurement upto time t. We first do not make any restrictions about the state process x(t) except that it is a Markov process with generator Kt (In the above special diffusion process case, Kt = μ(x)d/dx+(σ 2 (x)/2)d2 /dx2 but if, for example, x(t) is driven by a Poisson field N (t, dξ) having a rate function λdF (ξ)dt, with the sde given by  dx(t) = μ(t, x(t))dt + g(t, x(t), ξ)N (dt, dξ) ξ∈E

where (E, E) is a measure space on which the Poisson field measure is defined, then x(t) is a Markov process with generator given by Kt φ(x) = limdt→0 dt−1 E(φ(x(t) + dx(t)) − φ(x(t))|x(t) = x)  = μ(t, x)dφ(x)/dx + (φ(x + g(t, x, ξ)) − φ(x))λdF (ξ) ξ∈E

148

Quantum Antennas

The filtering equation is obtained by applying the Bayes rule: p(x(t + dt)|ηt+dt ) = p(x(t + dt), ηt , dy(t))/p(ηt , dy(t))  = p(dy(t)|x(t+dt), x(t))p(x(t+dt)|x(t))p(x(t)|ηt )dx(t)/ numeratordx(t+dt)   = p(dy(t)|x(t))p(x(t + dt)|x(t))p(x(t)|ηt )dx(t)/ numeratordx(t + dt) 

where we’ve made use of the fact that the difference between p(dy(t)|x(t)) and p(dy(t)|x(t + dt), x(t)) is o(dt) and hence can be neglected. Continuing further, we get on multiplying both sides by φ(x(t + dt)) and integrating w.r.t. x(t + dt),  πt+dt (φ) = φ(x(t + dt))p(x(t + dt)|ηt+dt )dx(t + dt) = E(φ(x(t + dt))|ηt+dt ) =



2 2 exp(−(dy(t)−h(x(t))dt) /2σ v dt)(φ(x(t))+Kt φ(x(t))dt)p(x(t)|ηt )dx(t)/num(φ = 1)

 =

exp(h(x(t))dy(t)/σv2 −h(x(t))2 dt/2σv2 )(φ(x(t))+Kt φ(x(t))dt)p(x(t)|ηt )dx(t)/num(φ = 1)

which on application of Ito’s formula for Brownian motion in the form (dy(t))2 = σv2 dt, becomes  (1 + h(x(t))dy(t))(φ(x(t)) + Kt φ(x(t))dt)p(x(t)|ηt )dx(t)/num(φ = 1) = (πt (φ) + πt (hφ)dy(t) + πt (Kt φ)dt)/(1 + πt (h)dy(t)) from which another application of Ito’s formula after using the expansion 1/(1 + πt (h)dy) = 1 − πt (h)dy + πt (h)2 (dy)2 /2 + o(dt) yields the Kushner-Kallianpur filter: dπt (φ) = πt (Kt φ)dt + (πt (hφ) − πt (h)πt (φ))(dy(t) − πt (h)dt) This is easily seen to be a special commutative case of the Belavkin filter (1) once we identify Kt with Lt and h with Mt + Mt∗ assuming now that Mt is a multiplication operator. The conditional expectation operation πt (X) in the case of the Belavkin filter is the same as T r(ρB (t)X) where ρB (t) is the Belavkin filtered state which can be regarded as a random density matrix in the System Hilbert space that is measurable w.r.t the Abelian Von-Neumann algebra ηt = σ(Yo (s) : s ≤ t). In our classical probabilistic scenario, ρB (t) is identified with a random diagonal matrix, ie, a multiplication operator pt (x|ηt ), x ∈ R. The system observable jt (X) in the Belavkin filter is identified with the function φ(x(t)). More precisely, if x(0) = x, we write x(t, x) for x(t). Then X stands for the multiplication operator by the function φ(x) while jt (X) stands for the multiplication operator by the function φ(x(t, x)).

149

Quantum Antennas

9.3.4

Quantum control for Lindblad noise reduction based on Luc-Bouten’s approach

There is another interesting viewpoint that one can take in quantum filtering theory. It is related to quantum communication. Here, we wish to transmit the system state ρs (0) over a noisy channel. The dynamics of the channel is dictated by a noisy Schrodinger equation. So the system state at time t after transmitting it through the noisy channel is ρs (t) = T r2 (U (t)(ρs (0) ⊗ |φ(u) > φ(u)|)U (t)∗ ) where U (t) satisfies the noisy HP-Schrodinger equation. To recover ρs (0) from ρs (t), we can use the recovery operators of the Knill-Laflamme theorem if appropriate conditions are satisfied. This is because, one can express ρs (t) =

p 

Ek (t)ρs (0)Ek (t)∗

k=1

where the system operators Ek (t) are obtained by solving the GKSL equation. However, when the required conditions on the system and noise subspaces for the Knill-Laflamme theorem are not satisfied, then we must adopt a different approach. This involves taking non-demolition measurements and applying the Belavkin filter to obtain an estimate ρB (t) of the dynamically evolving system state. However, this estimate will contain the GKSL noise operators, which along with the Hamiltonian appearing in the HP-Schrodinger equation, dictate the true noisy state evolution. In order to obtain the initial system state, we must therefore apply some sort of control operation that would remove the GKSL noise. Such an algorithm was proposed by Belavkin and developed to perfection by Luc-Bouten in his Ph.D thesis []. We briefly summarize this idea here: At time t = 0, the system state is assume to be ρc (0) which is assumed to be the Belavkin filtered state followed by application of control operations. We note that the Belavkin filter dπt (X) = πt (Lt X)dt+(πt (Mt X+XMt∗ ) −πt (Mt +Mt∗ )πt (X))(dYo (t)−πt (Mt +Mt∗ ))−−−(1) can be expressed as a stochastic Schrodinger equation for ρB (t) by replacing πt (Z) with T r(ρB (t)Z) for any system operator Z and using the arbitrariness of the system observable X as: dρB (t) = L∗t (ρB (t))dt+ (ρB (t)Mt +Mt∗ ρB (t)−T r(ρB (t)(Mt +Mt∗ ))ρB (t))(dYo (t)−T r(ρB (t)(Mt +Mt∗ )) It should be noted that this is a stochastic Schrodinger equation and not a quantum stochastic equation since the driving noise process Yo (.) is commutative. t Further, the innovations process W (t) = Yo (t) − 0 T r(ρB (s)(Ms + Ms∗ ))ds is a scalar multiple of the Wiener process when the measurement is quadrature, ie, ∗

Yo (t) = U (t) (

d  k=1

αk Ak (t) + α ¯ k Ak (t)∗ )U (t)

150

Quantum Antennas

ρc (0) evolves under the Belavkin dynamics to ρB (dt) at time dt given by ρB (dt) = ρc (0)+(L∗t (ρc (0))dt+(ρc (0)M0 +M0∗ ρc (0)−T r(ρc (0)(M0 +M0∗ ))ρc (0)dW (t))

and then we apply an infinitesimal control unitary Uc (dt) = exp(iZdYo (t)) to the Belavkin filtered state where Z is an appropriately chosen system space Hermitian operator. This gives us the filtered and controlled state at time dt as: ρc (dt) = Uc (dt)ρB (dt)Uc (dt)∗ and it is easy to show Bouten) as we shall in chapter [] using the quantum Ito formula, that Z can be chosen so that ρc (dt) is ρc (0) plus terms which have one lesser Lindblad noise component. This was already proved by Bouten but in our thesis, we do something more, ie we also include quantum Poisson noise in our HP dynamics, and define a new control objective, namely to design Z so that the controlled state at time dt, namely ρc (dt) is as close as possible in norm to a given state ρd , in other words, we achieve state tracking.

9.3.5

The Von-Neumann entropy of a state and its significance and properties

If ρ is a state in a separable Hilbert space with spectral resolution ρ=

∞ 

|e(k) > p(k) < e(k)|

k=1

then S(ρ) = −T r(ρ.log(ρ)) = −



p(k)log(p(k))

k

ie, S(ρ) can be regarded as the classical entropy of ρ relative to its eigenbasis. More generally, if M = {Mα } is a measurement system, ie, POVM, then the classical entropy of ρ relative to M is given by  SM (ρ) = − T r(ρMα )log(ρMα ) α

and the maximum of SM (ρ) over all measurement systems M is S(ρ) [Mark Wilde, Hayashi, Quantum information].

9.3.6

Maassen’s Guichardet kernel approach to solving quantum stochastic differential equations

Let (X, F, μ) be a measurable space and assume the measure μ to be nonatomic, ie, μ({x}) = 0, x ∈ X. Denote by Γ the set of all subsets of X. Denote

151

Quantum Antennas

by Γn the set of all subsets of X having n elements. let Gamma0 = φ, the empty set. Thus,  Γ= Γn n≥0

Define a measure μΓ on Γ by the prescription that if f : Γ → C is measurable, then    f dμΓ = 1 + (1/n!) f |Γn (σ)dμn (σ) Γ

Γn

n≥1

where μn is the product measure on Γn , ie, for σ = (x1 , ..., xn ), dμn (σ) = Πnk=1 dμ(xk ) For f : X → C, or more precisely, f ∈ L2 (X, F, μ), define e(f ) ∈ L2 (Γ, μΓ ) by e(f )(σ) = Πx∈σ f (x), σ ∈ Γn , n ≥ 1 and e(f )(φ) = 1. We easily verify that   < e(f ), e(g) >= e¯(f )e(g)dμΓ = exp(< f, g >) = exp( f¯(x)g(x)dμ(x)) X 2

For f ∈ L (X, F, μ) define the operator a(f ) : L2 (Γ, μΓ ) → L2 (Γ, μΓ ) by

 f¯(x)ψ(σ ∪ x)dμ(x), σ ∈ Γn

a(f )ψ(σ) = X

Then, for σ ∈ Γn ,

 f¯(x)e(u)(σ ∪ x)dμ(x)

a(f )e(u)(σ) = X



f¯(x)(Πy∈σ∪σ∪x u(y))dμ(x)

= X



f¯(x)u(x)dμ(x))e(u)(σ) =< f, u > e(u)(σ)

=( X

Equivalently, a(f )e(u) =< f, u > e(u) We calculate the adjoint a(f )∗ of a(f ): a(f )∗ must satisfy < a(f )∗ e(v), e(u) >=< e(v), a(f )e(u) >=< f, u >< e(v), e(u) > or equivalently,   ¯(a(f )∗ e(v))(σ)e(u)(σ)dμn (σ) n!−1 Γ  n n  −1 n! =< f, u > n

e¯(v)(σ)e(u)(σ)dμn (σ) Γn

152

Quantum Antennas

Suppose we try for ψ ∈ L2 (Γ, μΓ ),  f (x)ψ(σ − {x}), σ ∈ Γn (a(f )∗ ψ)(σ) = x∈σ

Then, we get

  

¯(a(f )∗ e(v))(σ)e(u)(σ)dμn (σ) = Γn

 Γn x∈σ

f¯(x)Πy∈σ,y =x v¯(y)Πz∈σ u(z)dμn (σ)



f¯(x)u(x)dx)(Πy∈σ v¯(y)u(y))dμn−1 (σ)

(

=n Γn−1

X

This gives the correct result since n/n! = 1/(n − 1)!. Now we can explain Maasen’s method for solving Hudson-Parthasarathy like qsde’s. Consider first the annihilation process At (f ) = a(f χ[0,t] ) where f ∈ L2 (X, μ) with X = R+ , μ = Lebesgue measure on R+ . Then, we have  At (f )ψ(σ) = f¯(s)χ[0,t] (s)ψ(σ ∪ {s})ds R+



t

= 0

Likewise, At (f )∗ ψ(σ) =

f¯(s)ψ(σ ∪ {s})ds 

f (s)ψ(σ − {s})

s∈σcap[0,t]

It follows that dAt (f )ψ(σ) = f¯(t)ψ(σ ∪ {t})dt and

dAt (f )∗ ψ(σ) = χσ (t)f (t)ψ(σ − {t})

We then find that dAt (f )dAt (f )∗ ψ(σ) = f (t)f¯(t)dtχσ∪{t} (t)ψ(σ) = |f (t)|2 ψ(σ)dt Thus, we deduce the quantum Ito formula dAt (f )dAt (f )∗ = |f (t)|2 dt from Maasen’s kernel theory. Now writing ψt = U (t)ψ0

153

Quantum Antennas ie ψ(σ) = (U (t)ψ0 )(σ), σ ∈ Γn , n ≥ 1 where U (t) satisfies the Hudson-Parthasarathy equation dU (t) = (−(iH + P )dt + L1 dAt (f ) + L2 dAt (f )∗ )U (t) or equivalently, dψt (σ) = (−(iH + P )dt + L1 dAt (f ) + L2 dAt (f )∗ )ψt )(σ) = −(iH + P )ψt (σ)dt + L1 f¯(t)ψ(σ ∪ t)dt + L2 χσ (t)f (t)ψ(σ − {t})

It is not difficult to obtain an explicit solution for ψ in this form. Note that we assume that for any σ ∈ Γn , ψt (σ) ∈ h where h is the system Hilbert space in which the operators H, P, L1 , L2 act.

9.3.7

Unsolved problems in quantum filtering and control

[a] Develop quantum filtering theory when the measurements are obtained as arbitrary independent increment processes. [b] Develop quantum filtering theory in non-coherent states. [c] Develop quantum control theory when the objective cost function to be minimized is the expected value of the integral of any function of the evolving quantum observable in a coherent state. [d] Develop quantum filtering theory for arbitrary Evans-Hudson flows not necessarily obtained via the unitary dynamics of the Hudson-ParthasarathySchrodinger qsde.

9.3.8

Simulating the HP and Belavkin filter using finite matrix algorithms

The HP equation dU (t) = [−(iH + P )dt +

p 

(Lm dAm (t) + Mm dAm (t)∗ + Sm dΛm (t))]U (t)

m=1

is simulated as an ordinary matrix differential equation and hence after discretizing time, as an ordinary matrix difference equation (no random variables/random processes are involved). This equation is given by (after approximating it by a truncation of orthonormal bases) d < ηk ⊗ ξr |U (t)|ηl ⊗ ξs >≈ −

 l ,s

< ηk |iH + P |ηl > δr,s < ηl ⊗ ξs |U (t)|ηl ⊗ ξs > dt

154 +

Quantum Antennas 

< ηk |Lm |ηl > δrs < ηl ⊗ξs |U (t)|ηl ⊗ξs >< ηl ⊗ξs |dAm (t)|ηl ⊗ξs >

m,l ,s ,l ,s

+



< ηk |Lm |ηl > δrs < ηl ⊗ξs |U (t)|ηl ⊗ξs >< ηl ⊗ξs |dAm (t)∗ |ηl ⊗ξs >

m,l ,s ,l ,s

+



< ηk |Lm |ηl > δrs < ηl ⊗ξs |U (t)|ηl ⊗ξs >< ηl ⊗ξs |dΛm (t)|ηl ⊗ξs >

m,l ,s ,l ,s

where we keep in mind the fact that the ηk s form an onb for the system Hilbert space and the ξr s for an approximate onb for the bath space, ie the Boson Fock space. It should be noted that this approximation is in fact very crude since the Boson Fock space is not a separable Hilbert space, however it works well during practical simulation just as the Feynman path integral does not converge on formally replacing real variances in Gaussian distributions by purely imaginary variances and yet it yields the correct physical results like amplitudes for Compton scattering, vacuum polarization and the anomalous magnetic moment. In the above equations, we substitute < ηk ⊗ ξr |dAm (t)|ηl ⊗ ξs >= δkl < ξr |dAm (t)|ξs >= = δkl



c¯(r, r )c(s, s ) < e(ur |dAm (t)|e(us ) >=

r  ,s



= δkl

c¯(r, r )c(s, s )us m (t) < e(ur |e(us > dt

r  ,s

< ηk ⊗ ξr |dAm (t)∗ |ηl ⊗ ξs >= δkl < ξr |dAm (t)∗ |ξs >= 

= δkl

= δkl < dAm (t)ξr |ξs >= c¯(r, r )c(s, s ) < dAm (t)e(ur |e(us ) >=

r  ,s

= δkl



c¯(r, r )c(s, s )¯ ur m (t) < e(ur |e(us > dt,

r  ,s

and finally, < ηk ⊗ ξr |dΛm (t)|ηl ⊗ ξs >= δkl < ξr |dΛm (t)|ξs >= = δkl



c¯(r, r )c(s, s ) < e(ur |dΛm (t)|e(us ) >=

r  ,s

= δkl



c¯(r, r )c(s, s )¯ ur m (t)us m (t) < e(ur |e(us > dt

r  ,s

The details of how this simulation of the HP equation is actually carried out by choosing the functions us (t) as normalized orthogonal sinusoids over a finite time interval [0, T ] are discussed in Chapter 2 of the thesis. It should be noted

155

Quantum Antennas

that as regards the quantum Poisson/conservation processes Λm (t), we can more generally consider processes Λkm (t) defined as Λkm (t) = λ(χ[0,t] |fk >< fm |), 1 ≤ k, m ≤ d defined by its matrix elements < e(u)|Λkm (t)|e(v) >=< χ[0,t] u|fk >< fm |χ[0,t] v = u We then have the generalized Ito rules dAk (t)dΛrs (t) = dAk (t)dAr (t)∗ dAs (t)/dt = δkr dAs (t), dΛrs (t)dAk (t)∗ = dAr (t)∗ dAs (t)dAk (t)∗ /dt = δsk dAr (t)∗ Then the term

d

k=1

Sk dΛk (t) can be replaced by the more general term

d

k,m=1

k Sm dΛm k (t).

However, for simulation purposes, it is easier to consider the above special case of this since the general case is a straightforward extension of this special case. The Belavkin filter may be simulated both the observable and the state domain either as ordinary non-random matrix differential equations or equivalently as classical stochastic differential equations. Specifically, the Belavkin filter for quadrature measurements dπt (X) = πt (Lt X)dt+(πt (Mt X+XMt∗ )−πt (Mt +Mt∗ )πt (X))(dYo (t)−πt (Mt +Mt∗ )dt)

where X ranges over all system observables is equivalent to the following in view of the linearity of the equation w.r.t. X: First choose a basis {X1 , ..., XK } of Hermitian matrices for the vector space of all Hermitian matrices in the system Hilbert space. Then, we can write Lt Xk =

K 

akm (t)Xm , Mt Xk + Xk Mt∗ =

m=1

K 

bkm (t)Xm ,

m=1

Mt + Mt∗ =

K 

ek (t)Xk

k=1

where akm (t), bkm (t), ek (t) are real valued functions. Now define the commutative processes ξk (t) = πt (Xk ), 1 ≤ k ≤ K

156

Quantum Antennas

Then recalling the fact that for quadrature measurements, the process  t πs (Ms + Ms∗ )ds W (t) = Yo (t) − 0

is a multiple of Brownian motion, the Belavkin filter reduces to the following system of K coupled stochastic differential equations driven by the classical Brownian motion process W (t): dξk (t) =

K 

akm (t)ξm (t)dt

m=1

+(

K  m=1

bkm (t)ξm (t) −

K 

em (t)ξm (t)ξk (t))dW (t), k = 1, 2, ..., K

m=1

We may simulate this evolution as a system of classical stochastic differential equations and obtain the correct statistics for the estimate of any observable when the the bath is the coherent state |φ(u) >, however such a simulation would not tell us how the observable estimate depended on the output measurements Yo (.). To obtain this information, we must regard ξk (t) as a pN × pN matrix and also Yo (t) as the pN × pN matrix whose (N (r − 1) + k, N (s − 1) + l)th entry is given by < ηk ⊗ ξr |Yo (t)|ηk ⊗ ξs >

The Belavkin filter can also be simulated in the state domain in the form of a classical sde by regarding the filtered state ρB (t) as a random density matrix in the system Hilbert space that is measurable w.r.t the commutative family of output measurements Yo (s), s ≤ t which in turn, owing to their commutativity, can be regarded as a classical random process when the noisy bath is in a coherent state. This involves writing πt (Z) = T r(ρB (t)Z) where ρB (t) is regarded as a classical random process with values in the space of density matrices in the system Hilbert space and Z is any system operator. From the above observable form of Belavkin’s equation, by applying duality in the Banach space of system Hilbert-Schmidt operators, we deduce the state form of Belavkin’s equation as a stochastic non-linear Schrodinger equation: dρB (t) = L∗t (ρB (t))dt+ (ρB (t)Mt + Mt∗ ρB (t) − T r(ρB (t)(Mt + Mt∗ )))(dYo (t) − T r(ρB (t)(Mt + Mt∗ ))) The simulation of this equation is by direct time discretization. However, this simulation will not, in general, lead to a positive definite matrix of unit trace for ρB (t), t ≥ 0. So, after each iteration, we extract out the part of ρB (t)

157

Quantum Antennas

that satisfies these properties required of a density matrix by performing the transformation  ρB (t)∗ ρB (t)  ρB (t) → T r( ρB (t)∗ ρB (t)) and prove numerical stability using the Dunford-Taylo integral for the square root of an operator (T.Kato[]):  √ √ −1 T = (2πi) z(zI − T )−1 dz Γ

where the contour Γ is chosen appropriately within the resolvent set of T .

9.3.9

Generalizing the Belavkin filter when the measurements are mixture of creation, annihilation and conservation processes, ie superpositions of quantum Brownian motions and quantum Poisson processes or equivalently quadrature plus photon counting

The Belavkin filter has been constructed in the existing literature [J.Gough and Kostler) for the special cases when the measurement noise is either quadrature (ie, classical Brownian motion A(t) + A(t)∗ passed through the HP system and also in the case when the measurement noise is a photon counting process, ie, the quantum Poisson process Λ(t) passed through the HP system. In both the cases, the construction is simple owing to the fact that the output measurement process differential can be expressed in terms of either the creation and annihilation processes differentials whose cubic and higher powers vanish by quantum Ito’s formula, or in terms of quantum Poisson process differentials all of whose powers are proportional to the process differential: (dΛ)n = dΛ, n = 1, 2, .... When however, the measurement is a mixture of quadrature and photon counting processes, ie, Yo (t) = U (t)∗ (

d 

c[k]Ak (t) + c¯[k]Ak (t)∗ + d[k]Λk (t))U (t)

k=1

then application of quantum Ito’s formula shows that dYo (t) can be expressed in the form dYo (t) = jt (N0 )dt +

d 

(jt (N1k )dAk (t) + jt (N2k )dAk (t)∗ + jt (N3k )dΛk (t))

k=1

and it is clear that all the powers of dYo (t) will not be expressible in an elementary way in terms of the fundamental processes. The trick is to assume n

(dYo (t)) = jt (N0 [n])dt+

d  k=1

(jt (N1k [n])dAk (t)+jt (N2k [n])dAk (t)∗ +jt (N3k [n])dΛk (t)), n ≥ 1−−−(2)

158

Quantum Antennas

and calculate the system operators N0 [n], Nmk [n], m = 1, 2, 3 for different n s by applying quantum Ito’s formula to obtain a recursion for these system matrices. Having done so, we assume that the Belavkin filter has the form  dπt (X) = Ft (X)dt + Gkt (X)(dYo (t)k k≥1

where Ft (X), Gkt (X) are functions of the Abelian family ηt = σ(Yo (s), s ≤ t) (because πt (X) = E(jt (X)|ηt ) and then calculate Ft (X), Gkt (X), k ≥ 1 by applying the reference probability method of Gough and Kostler. This method involves choosing arbitrary real valued functions fk (t), k ≥ 1, considering the process C(t) that is ηt measurable because it is assumed to satisfy the sde  C(t)fk (t)(dYo (t))k , t ≥ 0, C(0) = I dC(t) = k≥1

and then applying quantum Ito’s formula to the orthogonality relation (satisfied by the conditional expectation): E[(jt (X) − πt (X))C(t)] = 0 and then using the arbitrariness of the functions fk (t), k ≥ 1 to deduce that E[djt (X) − dπt (X)|ηt ] = 0, E[(jt (X) − πt (X))(dYo (t))k |ηt ] + E[(djt (X) − dπt (X))(dYo (t))k |ηt ] = 0 Further simplifications of these to calculate Ft (X), Gkt (X) are based on the homomorphism property of the map jt , the definition of πt as a conditional expectation, expressions for (dYo (t))k in terms of the fundamental noise processes in the form (2) and further applications of the quantum Ito formula. These aspects as well as the simulation of the Belavkin filter upto second degree terms in the noise differential are discussed in chapter 3.

9.3.10

Quantum control with more generalized objectives: Removing some components of Lindblad noise, state trajectory tracking

We’ve already discussed this aspect above.

9.3.11

Use of the differential of the exponential map and the Baker-Campbell-Hausdorff formula in Lie group theory for evaluating the rate of Von-Neumann entropy increase in the HP, Belavkin and controlled Belavkin state

The HP system state satisfies the generalized GKSL equation obtained after tracing out over the bath state. It is given by ρs (t) = −i[H, ρs (t)] + θt (ρs (t))

159

Quantum Antennas

where θt is a time dependent generalized Linear Lindblad operator depending on the vector u(t) which defines the coherent state vector |φ(u) >= exp(−  u 2 /2)|e(u) > of the bath and also on the system space operators Lk , Mk , Sk , k = 1, 2, ..., d appearing in the HP equation. We can use this equation combined with the standard formula for the differential of the exponential map [V.S.Varadarajan, Lie groups, Lie algebras and their representations] to calculate the rate of change of the Von-Neumann entropy of the system. That would tell us at what rate the environmental bath is pumping in entropy into the system and if we are able to select our system operators so that this rate is positive then we can say that our system dynamics viewed as our complete system is in agreement with the second law of thermodynamics. We can also by regarding the Belavkin equation as a classical stochastic nonlinear Schrodinger equation, compute the average rate of entropy of increase of the Belavkin filtered state when the bath is in a coherent state. This computation would tell us whether conditioning the HP state on the non-demolition measurements increases or decreases the entropy. Ideally, we know that conditioning decreases the entropy: H(X|Y ) ≤ H(X). Our simulations confirm this in the quantum case for a majority of experiments but we are yet to discover a conclusive proof that this is always the case. Finally, after applying quantum control using control unitaries as in the PhD thesis of Luc Bouten, we compute the approximate increase in system entropy after control. We expect that the entropy of the state will decrease after control since control involves Lindblad noise removal.

9.3.12

Performance analysis of the filtering and control algorithms

9.3.13

Simulations of the Hudson-Parthasarathy QSDE and Belavkin filter equations

Study projects [a] Simulating the Hudson-Parthasarathy equation. [b] MATLAB simulations of observable and state evolution in noise. [c] Simulating the Belavkin filtering equation for quadrature noise measurements [d] Plots of the nsr nsr(t) = E(jt (X) − πt (X))2 /E(jt (X)2 ) where expectations are taken in the coherent state.

9.3.14

Simulating the Belavkin filter for mixture of quadrature and photon counting noise

The qsde has all powers of the measurement noise differential. [a] Simulation of the nsr.

160

Quantum Antennas

9.3.15

Study project:Quantum control of the Belavkin filtered state for (a) Lindblad noise removal and (b) state tracking

9.3.16

Study project:Von-Neumann entropy rate for HP,Belavkin and controlled Belavkin filtered states

[a] Theoretical derivations using Lie algebra theory. [b] Simulation of the entropy evolution.

9.4

Filter design for physical applications

Ref:Mridul, Ph.D thesis. An ideal integrator in discrete time has the impulse response u[n]. Thus, when a signal x[n] is passed through an integrator, the output is y[n] = u ∗ x[n] =

n 

x[k]

k=0

The transfer function of an integrator is  n

u[n]z −n =

1 1 − z −1

with the ROC of |z| > 1. An ideal integrator is an unstable system since its impulse response is not summable. We therefore seek a stable approximation to an ideal integrator. This can be obtained either by truncating the impulse response, ie, by approximating 1/(1 − z −1 ) by 1 + z −1 + ... + z −N for some sufficiently large N , or by approximating 1/(1 − z −1 ) by the ratio of two polynomials B(z)/A(z) with the zeros of A(z) constrained to fall inside the unit circle. One way to design such an approximate integrator would be to fix A(z) so that its zeroes are inside the unit circle and to calculate B(z) so that 

π

−π

W (ω)|(1 − exp(−jω))B(ejω ) − A(ejω )|2 dω

is a minimum. Since B is a polynomial in z, minimization of this error energy w.r.t the coefficients of B(z) would amount to solving a linear least squares problem. After designing B(z), we can approximate B(z)/A(z) with a function 1/H(z), where H(z) is obtained by truncating the infinite series for A(z)/B(z). This process is explained below.

161

Quantum Antennas

[1] The aim is to express the ratio B(z)/A(z) of two polynomials in z −1 , namely p p   a[k]z −k , B(z) = 1 + b[k]z −k A(z) = k=0

k=1

as 1/H(z) where H(z) =

∞ 

h[n]z −n

n=0

with z varying over an appropriate region in the complex plane. Suppose we impose the condition p  b[k]z −k | < 1 | k=1

This can be satisfied if for example, p 

|b[k]|z|−k < 1

k=1

which can in turn be guaranteed provided that we choose z so that |z| > 1 and simultaneously p  |b[k]| < 1 |z|−1 k=1

ie |z| > max(1,

p 

|b[k]|)

k=1

Then, we have the convergent geometric series −1

B(z)

= (1 +

p 

b[k]z

−k −1

)

=1+

∞ 

(−1) (

n=0

k=1

n

p 

b[k]z −k )n

k=1

By the multinomial theorem in the form (

p 



xk ) n =

n1 ,...,np ≥0,n1 +...+np =n

k=1

n! xn1 ...xnp p n1 !...np ! 1

we get (

p 



b[k]z −k )n =

k=1

n1 +...+np

it follows that (1 +

p  k=1

n! b[1]n1 ...b[p]np z −(n1 +2n2 +...+pnp ) n !...n ! 1 p =n

b[k]z −k )−1 = 1 +

 l≥0

c[l]z −l

162

Quantum Antennas

where for l ≥ 1. 

c[l] =

(−1)n1 +...+np

n1 ,..,np ≥0,n1 +2n2 +...+pnp =l

(n1 + ... + np )! b[1]n1 ...b[p]np n1 !...np !

Finally, with c[0] = 1, we have H(z) =



h[n]z −n = (

n≥0

p 

a[k]z −k ).(

k=0



c[l]z −l )

l≥0

so that 

min(p,n)

h[n] =

a[k]c[n − k], n ≥ 0

k=0

Example: p = 2. Then 

c[l] =

(−1)n1 +n2

n1 +2n2 =l [l/2]

=



(−1)l−n

n=0

(n1 + n2 )! b[1]n1 b[2]n2 n1 !n2 !

(l − n)! b[1]l−2n b[2]n (l − 2n)!n!

In particular, c[0] = 1, c[1] = −b[1], c[2] =



(−1)2−n

n=0,1

(2 − n)! b[1]2−2n b[2]n = b[1]2 − b[2], (2 − 2n)!n!

Design of filters using transmission line elements. The T matrix of the mth section of a cascade of M finite transmission line elements has the general form Tm (z) = Am + Bm z −1 , m = 1, 2, ..., M where Am , Bm are constant 2 × 2 matrices:   Am (1, 1) Am (1, 2) Am = , Am (2, 1) Am (2, 2)  Bm =

Bm (1, 1) Bm (2, 1)

Bm (1, 2) Bm (2, 2)

 ,

The aim is to express the matrix product T (z) = TM (z)TM −1 (z)...T2 (z)T1 (z) = ΠM m=1 Tm (z)

163

Quantum Antennas as a matrix polynomial in z −1 of degree M , ie, in the form T (z) =

M 

Sm z −m

m=0

where S0 , S1 , ..., SM are constant 2 × 2 matrices. We write Km (z) = Tm (z)Tm−1 (z)...T1 (z), 1 ≤ m ≤ M Then we have the obvious recursion Km+1 (z) = Tm+1 (z)Km (z) Writing Km (z) =

m 

Km [r]z −r

r=0

where Km [r] is a constant 2 × 2 matrix, we have m+1 

Km+1 [r]z −r = (Am+1 + Bm+1 z −1 ).

r=0

m 

Km [r]z −r

r=0

which gives on equating coefficients of the same powers of z −1 , Km+1 [r] = Am+1 Km [r] + Bm+1 Km [r − 1], r = 0, 1, 2, ..., M − 1, Km [−1] = 0 ⎛

Define the matrix Pm

⎜ ⎜ =⎜ ⎜ ⎝

Km [0] Km [1] Km [2] ... Km [M ]

⎞ ⎟ ⎟ ⎟ ∈ C2M +2×2 ⎟ ⎠

where Km [r] = 0, r > m Then, we can write the above recursion as Pm+1 = Cm+1 Pm , 1 ≤ m ≤ M − 1 where Cm+1 is the 2M + 2 × 2M + 2 block structured matrix ⎛ Am+1 0 0 0.. 0 ⎜ Bm+1 Am+1 0 0.. 0 ⎜ 0... 0 0 Bm+1 Am+1 Cm+1 = ⎜ ⎜ ⎝ .. .. .. .. .. 0 0 0 Bm+1 Am+1 We can express this as Cm+1 = IM ⊗ Am+1 + ZM ⊗ Bm+1

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

164

Quantum Antennas

where IM is the M ×M identity matrix and ZM is the M ×M unit delay matrix: ⎛ ⎞ 0 0 0 0.. 0 ⎜ 1 0 0 0.. 0 ⎟ ⎜ ⎟ ⎟ ZM = Cm+1 = ⎜ ⎜ 0 1 0 0... 0 ⎟ ⎝ .. .. .. .. .. ⎠ 0 0 0 1 0 This matrix difference equation has the solution Pm = Cm Cm−1 ...C2 P1 , 2 ≤ m ≤ M ⎛

Note that

⎜ ⎜ ⎜ P1 = ⎜ ⎜ ⎜ ⎝

K1 [0] K1 [1] 0 0 ... 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where K1 [0] = A1 , K1 [1] = B1 Finally, T (z) = KM (z) =

M 

z −r KM [r] = [I2 , z −1 I2 , ..., z −M I2 ]PM

r=0

= ([1, z −1 , z −2 , ..., z −M ] ⊗ I2 )PM To implement this algorithm for computing T (z), first we must write a program for computing PM as a product of matrices.

Chapter 10

Gravity interacting with waveguide quantum fields with filtering and control 10.1

Waveguides placed in the vicinity of a strong gravitational field

In the absence of the gravitational field, let the electromagnetic field tensor (0) within the guide be given by Fμν . The components of this tensor are easily determined using the standard expressions in the frequency domain:  (−γE [n]/hE [n]2 )∇⊥ Ez,n exp(−γE [n]z) E⊥ = n

H⊥ =

−(jωμ/hH [n]2 )∇⊥ Hz,n ׈ z .exp(−γH [n]z))

 (−γH [n]/hH [n]2 )∇⊥ Hz,n exp(−γH [n]z)+(jω/hE [n]2 )∇⊥ Ez,n ׈ z )exp(−γE [n]z)) n

with

(∇2⊥ + hE [n]2 )Ez,n = 0, Ez,n |∂D = 0,

∂Hz,n |∂D = 0 ∂n ˆ In the presence of a gravitational field, let the perturbation to the em field tensor (1) be denote by Fμν . In other words, (∇2⊥ + hE [n]2 )Hz,n = 0,

(1) Aμ = A(0) μ + Aμ , (0) (1) + Fμν , Fμν = Fμν (0) (0) = A(0) Fμν ν,μ − Aμ,ν , (1) (1) = A(1) Fμν ν,μ − Aμ,ν

165

166

Quantum Antennas

Write gμν = ημν + hμν (x) so that g ≈ −(1 + h), h = ημν hμν = hμμ and hence,

√

−g ≈ 1 + h/2

This gives us using the exact Maxwell equations √ (F μν −g),ν = 0 the first order perturbed term √ (0) (1) (Fαβ δ(g μα g νβ −g) + ημα ηνβ Fαβ ),ν = 0 or equivalently, (1)

(ημα ηνβ Fαβ ),ν = √ (0) −(Fαβ δ(g μα g νβ −g),ν Now,

√ δ(g μα g νβ −g) = √ √ (δg μα )(g νβ −g) + δ(g νβ )g μα −g + g μα g νβ h/2

where we substitute δg μα = −g μρ g νσ hρσ δg νβ = −g νρ g βσ hρσ Exercise: Using the above perturbed Maxwell equations along with the gauge condition √ (Aμ −g),μ = 0, or equivalently,

√ (Aν g μν −g),μ = 0

whose first order perturbed term gives μν √ μν √ −g),μ + (A(0) −g)),ν = 0, (A(1) ν g ν δ(g (1)

derive the modified wave equation with source for Aμ .

167

Quantum Antennas

10.2

Some study projects regarding waveguides and cavity resonators in a gravitational field

[1] Rectangular waveguides near a gravitational field of a Schwarzchild blackhole [2] Cylindrical waveguides with cladding near a gravitational field. Inner radius is a and outer radius is b. In the region ρ < a, the permittivity and permeability are (1 , μ1 ) and in the region a < ρ < b, these parameters are (2 , μ2 ). The transverse components of the field are (k)

E⊥ = (−γ/h2k )∇⊥ Ez(k) − (jωμk /h2k )∇⊥ Hz(k) × zˆ (k)

H⊥ = (−γ/h2k )∇⊥ Hz(k) + (jωk /h2k )∇⊥ Ez(k) × zˆ where k = 1, 2. Note that the propagation constant γ for the fields along the z direction must be the same in both the regions in view of the continuity of Ez and Hz at ρ = a. Note that there is no surface current density at the dielectric interface ρ = a and hence the tangential components of H are also continuous. We have h2k = ω 2 k μk + γ 2 , k = 1, 2 Also the z components of the field satisfy the Helmholtz equation (∇2⊥ + h2k )(Ez(k) , Hz(k) ) = 0 Keeping in mind the fact that the second Bessel function Ym (x) is singular at ρ = 0, the solutions are Ez(1) = Jm (h1 ρ)(C1 .cos(mφ) + C2 .sin(mφ)), Hz(1) = Jm (h1 ρ)(C1 .cos(mφ) + C2 .sin(mφ)) Ez(2) = (A1 Jm (h2 ρ) + A2 Ym (h2 ρ)).(C1 .cos(mφ) + C2 .sin(mφ)) Hz(2) = (A1 Jm (h2 ρ) + A2 Ym (h2 ρ)).(C1 .cos(mφ) + C2 .sin(mφ)) These are consistent with the continuity of Ez and Hz at ρ = a provided that Jm (h1 a) = A1 Jm (h2 a) + A2 Ym (h2 a) = (A1 Jm (h2 a) + A2 Ym (h2 a)) − − − (1) Assuming that γ is known, it follows that h1 , h2 are also known and hence these furnish us with two equations for the four constants A1 , A2 , A1 , A2 . The outer (2) surface ρ = b is a perfect conductor. Hence, Ez = 0 at ρ = b and this gives us A1 Jm (h2 b) + A2 Ym (h2 b) = 0 − − − (2a) which is yet another equation for the above four constants. We also have that Hρ vanishes at the perfectly conducting surface ρ = b. This gives (−γ/h22 )∂Hz(2) /∂ρ + (jω2 /bh22 )∂Ez(2) /∂φ = 0 − − − (2b)

168

Quantum Antennas

at ρ = b. This yields us (−γ/h2 )(A1 Jm (h2 b) + A2 Ym (h2 b))(C1 cos(mφ) + C2 sin(mφ)) +(jω2 /bh22 )(A1 Jm (h2 b)+A2 Ym (h2 b))(−mC1 sin(mφ)+mC2 cos(mφ)) = 0−−−(2c) which gives us two equations on equating separately the coefficients of cos(mφ) and sin(mφ). Continuity of Eρ at ρ = a gives us two more equations: (−γ1 /h21 )∂Ez(1) /∂ρ − (jω1 μ1 /ah21 )∂Hz(1) /∂φ = (−γ2 /h22 )∂Ez(2) /∂ρ − (jω2 μ2 /ah22 )∂Hz(2) /∂φ − − − (3a) evaluated at ρ = a, or equivalently,  (−γ1 /h1 )Jm (h1 a)(C1 cos(mφ)+C2 sin(mφ))

−(jω1 μ1 /ah21 )Jm (h1 a)(−mC1 .sin(mφ)+m.C2 .cos(mφ))  = (−γ2 /h2 )(A1 Jm (h2 a) + A2 Ym (h2 a))(C1 cos(mφ) + C2 sin(mφ))

+(jω2 μ2 /ah22 )(A1 Jm (h2 a)+A2 Ym (h2 a))(−mC1 .sin(mφ)+mC2 .cos(mφ))−−−(3b) Equating the coefficients of cos(mφ) and sin(mφ) in these equations gives us two equations for the four constants C1 , C2 , C1 , C2 . Likewise equating μHρ at ρ = a gives us two more equations relating these constants. These are obtained from (−γμ1 /h21 )∂Hz(1) /∂ρ + (jωμ1 1 /ah21 )∂Ez(1) /∂φ = (−γμ2 /h22 )∂Hz(2) /∂ρ + (jωμ2 2 /ah22 )∂Ez(2) /∂φ − − − (4a) evaluated at ρ = a, or equivalently,  (−μ1 γ/h1 )Jm (h1 a)(C1 .cos(mφ) + C2 .sin(mφ))

+(jω1 μ1 /ah21 )Jm (h1 a)(−mC1 .sin(mφ) + mC2 .cos(mφ))  = (−γμ2 /h2 )(A1 Jm (h2 a) + A2 Ym (h2 a))(C1 cos(mφ) + C2 sin(mφ))

−(jω2 μ2 /ah22 )(A1 Jm (h2 a)+A2 Ym (h2 a))(−mC1 .sin(mφ)+mC2 .cos(mφ))−−−(4b) Equating Eφ and Hφ at ρ = a gives us another set of four equations. Specifically, these are (−γ/ah21 )∂Ez(1) /∂φ + jωμ1 ∂Hz(1) /∂ρ = (−γ/ah22 )∂Ez(2) /∂φ + jωμ2 ∂Hz(2) /∂ρ − − − (5) evaluated at ρ = a and (−γ/h21 a)∂Hz(1) /∂φ − (jω1 /h21 )∂Ez(1) /∂ρ = (−γ/ah22 )∂Hz(2) /∂φ − (jω2 /h22 )∂Ez(2) /∂ρ − − − (6) evaluated at ρ = a. Now, the continuity of Ez and Hz at ρ = a implies the continuity of ∂Ez /∂φ and ∂Hz /∂φ at ρ = a. In view of this fact, (6) is equivalent to the equation 2

2

(1 − γ /h1 )a

−1

(1)

2

(1)

∂Hz /∂φ − (jωγ1 /h1 )∂Ez /∂ρ

Quantum Antennas

169

= (1 − γ 2 /h22 )a−1 ∂Hz(2) /∂φ − (jωγ2 /h22 )∂Ez(2) /∂ρ at ρ = a, which in view of the fact that h2k − γ 2 = ω 2 μk k , k = 1, 2 becomes (ω 2 μ1 1 /ah21 )∂Hz(1) /∂φ − (jωγ1 /h21 )∂Ez(1) /∂ρ = (ω 2 μ2 2 /ah22 )∂Hz(2) /∂φ − (jωγ2 /h22 )∂Ez(2) /∂ρ at ρ = a, or after making cancellations, is the same as (3a), namely the equation of continuity of Eρ at ρ = a. Likewise, (5) is equivalent to (4a), namely, the equation of continuity of μHρ at ρ = a. Thus, in all, our only independent equations are (1), (2a), (2c), (3b), (4b) which yield 2+1+2+2+2 = 9 equations for the nine variables γ, A1 , A2 , A1 , A2 , C1 , C2 , C1 , C2 . These yield a characteristic equation for γ which determines discrete set of possible propagation constants.

A neater way to obtain the required characteristic function for γ is to denote the constants A1 C1 , A1 C2 , A2 C1 , A2 C2 by B1 , B2 , B3 , B4 and likewise the constants A1 C1 , A1 C2 , A2 C1 , A2 C2 by B1 , B2 , B3 , B4 so that the continuity of Ez and Hz at ρ = a gives us four equations obtained by equating the coefficients of cos(mφ) and sin(mφ) in place of (1). Then the constants to be determined are C1 , C2 , C1 , C2 , B1 , B2 , B3 , B4 , B1 , B2 , B3 , B4 , namely twelve in number and the number of linear equations for these constants is 4 corresponding to continuity of Ez and Hz at ρ = a plus 4 corresponding to continuity of Ez and Hρ at ρ = b plus four corresponding to continuity of Eρ and μHρ at ρ = a, ie, in all twelve homogeneous linear equations. Setting the determinant of the corresponding 12 × 12 matrix to zero then yield the possible discrete values of γ. [3] cavity resonators in a gravitational field. [4] Waveguides with inhomogeneity and anisotropicity placed close to a strong gravitational field. Lie groups, Lie algebras and differential equations

10.3

A comparison between the EKF and Wavelet based block processing algorithms for estimating transistor parameters in an amplifier drived by the Ornstein-Uhlenbeck process

A problem in group theory related to fluid velocity pattern recognition

170

Quantum Antennas

v(t, r) = (vx (t, r), vy (t, r)), r = (x, y) is a 2-D fluid velocity field. It satisfies the Navier-Stokes equation (v, ∇)v + v,t = −∇p/ρ + ν∇2 v + f where f (t, r) = (fx (t, r), fy (t, r)) is a random driving force field assume to be white Gaussian w.r.t the time variable. We apply a Galilean transformation to this field comprising of a rotation R(φ) in SO(2) followed by a translation a ∈ R2 . The resulting velocity field is w(t, r) and after noise corruption, this rotated and translated velocity field is assumed to satisfy a Navier-Stokes equation with a different driving noise g(t, r) again white Gaussian w.r.t. the time variable: (w, ∇)w + w,t = −∇p1 /ρ + ν∇2 w + g The fluid is incompressible, ie, divv = 0, divw = 0 Thus there exist stream functions ψ1 (t, r) and ψ2 (t, r) such that v = ∇ψ1 × zˆ, w = ∇ψ2 × zˆ

10.4

Computing the Haar measure on a Lie group using left invariant vector fields and left invariant one forms

G is a Lie group and (X1 , ..., Xn ) is a basis of left invariant vector fields on G. Lg denotes left translation on G, ie, Lg h = gh. By left invariance of the vector fields Xk , we have that dLg (Xk ) = Xk ie Xk (f oLg )(x) = Xk (gx), g, x ∈ G or equivalently, Lg∗ Xk (x) = Xk (Lg x) or equivalently, Lg∗ Xk = Xk Lg , g ∈ G where Lg∗ denotes the push-forward map, ie if T : M → N is a differentiable map from a differentiable manifold M into another differentiable manifold N and if X(x) ∈ T Mx , then T∗ X(x) = Y (T (x)) ∈ T NT (x)

171

Quantum Antennas where Y (T (x))(f ) = X(f oT )(x) Equivalently, in terms of local coordinates, Y a (T (x)) =

∂T a (x) b X (x) ∂xb

with the Einstein summation convention adopted. It follows that Xka (gx) =

∂Lag (x) b Xk (x) ∂xb

which gives

∂Lag (e) b Xk (e) ∂xb from which we get on taking determinants, Xka (g) =

det(((Xka (g))) = (detLg (e))det((Xka (e))) so that the unnormalized left invariant Haar measure density (detLg (e))−1 is proportional to 1/det((Xka (g))). Equivalently, by duality of bases, if ωk (g), k = 1, 2, ..., n is a basis of left invariant one forms on G, then the left invariant Haar measure density is proportional to det((ωka (g))).

10.5

How background em radiation affects the expansion of the universe

Terms that are square in the electromagnetic field components determine the first order perturbations to the metric tensor of the expansion universe. The unperturbed metric of the universe is the Robertson-Walker metric in the coordinates t = x0 , r = x1 , θ = x2 , φ = x3 : g00 = 1, g11 = −f (r)S 2 (t), g22 = −r2 S 2 (t), g33 = −r2 sin2 (θ)S 2 (t) The unperturbed Maxwell equations determine the electromagnetic field which in turn drives the metric perturbations via the Einstein field equations. M Rμν = −8πGSμν , M M + Sμν ) δRμν = −8πG(δSμν

where M = (ρ(t) + p(t))v μ v ν − p(t)g μν Tμν

with (v μ ) = (1, 0, 0, 0) being the comoving four-velocity field. M M Sμν = Tμν − T M gμν /2

172

Quantum Antennas M T M = g μν Tμν = ρ(t) − 3p(t) M M δSμν = δTμν − δT M .gμν /2 − T M δgμν /2 M δTμν = (δρ(x) + δp(x))v μ v ν − δp(x)gμν − p(t)δgμν (x)

+(ρ(t) + p(t))(v μ δv ν (x) + v ν δv μ (x)) δT M (x) = δρ(x) − 3δp(x) EM = (−1/4)Fαβ F αβ gμν + Fμα Fνα Sμν

Check of the above formula in special relativity: We use the Maxwell equations F,νμν = −μ0 J μ S EM

μ

ν

= [(−1/4)Fαβ F αβ g μν + F μα Fαν ],ν

EM μν ν S,ν = (−1/2)g μν F αβ Fαβ,ν + F,νμα Fαν + F μα Fα,ν

= (1/2)g μν F αβ (Fβν,α + Fνα,β ) + F,νμα Fαν + μ0 F μα Jα where we have made use of the Maxwell equations Fμν,α + Fνα,μ + Fαμ,ν = 0 which may also be seen as a consequence of the definition Fμν = Aν,μ − Aμ,ν Continuing further, the above equals μ μ + Fα,β ) + F,νμα Fαν + μ0 F μα Jα (1/2)F αβ (Fβ,α μ μ μα = (1/2)F αβ (Fβ,α + Fα,β ) + Fαβ F,β + μ0 F μα Jα μ μ μα = (1/2)F αβ (Fβ,α + Fα,β ) + Fαβ F,β + μ0 F μα Jα μ μ μ = (1/2)F αβ (Fβ,α + Fα,β − Fα,β + μ0 F μα Jα

= (1/2)F αβ g μν (−Fνβ,α + Fνα,β − 2Fνα,β ) + μ0 F μα Jα = (−1/2)F αβ g μν (Fνβ,α + Fνα,β ) + μ0 F μα Jα = μ0 F μα Jα as required by the four-divergence of the energy-momentum tensor of the electromagnetic field.

173

Quantum Antennas

10.6

Stochastic BHJ equations in discrete and continuous time for stochastic optimal control based on instantaneous feedback

. First we consider a general example of a Markov process satisfying a stochastic differential equation driven by Brownian motion and a Poisson field:  ψ(t, X(t), ξ)dN (t, ξ) dX(t) = μ(t, X(t))dt + σ(t, X(t))dB(t) + ξ∈E

where N (., .) is a Poisson field with intensity dF (t, ξ), ie, E[N (dt, dξ)] = dF (t, ξ) Here (E, E) and the space-time Poisson random field N (., .) is defined on the measure space (R × E, B(R) ⊗ E). F is a measure on this space. Ito’s formula for φ(X(t)) gives dφ(X(t)) = Lt φ(X(t))dt + dB(t)T σ(t, X(t))T ∇φ(X(t))  + (φ(X(t) + ψ(t, X(t), ξ)) − φ(X(t)))N (dt, dξ) ξ∈E

from which we deduce that the generator of X(t) defined by Kt φ(x)dt = E(dφ(X(t))|X(t) = x) is given by  (φ(x + ψ(t, x, ξ)) − φ(x))f (t, dξ)

Kt φ(x) = Lt φ(x) + ξ∈E

where

∂F (t, dξ) ∂t and Lt is the generator of the diffusion part of the Markov process X(t), ie, f (t, dξ) = F (dt, dξ)/dt =

Lt φ(x) = μ(t, x)T ∇φ(x) + (1/2)T r(σ(t, x)σ(t, x)T ∇∇T φ(x)) ie, Lt = μ(t, x)T ∇ + (1/2)T r(σ(t, x)σ(t, x)T ∇∇T ) Equivalently, if we interpret the operator exp(xT ∇) by the familiar Taylor expansion formula exp(xT ∇)f (y) = f (y + x) then we can write

 (exp(ψ(t, x, ξ)T ∇) − 1)f (t, dξ)

K t = Lt + E

174

Quantum Antennas

Now consider the optimal control problem in which the generator Kt is a function of the instantaneous input u(t), ie, Kt = Kt (u(t)). We are allowed to take the control input u(t) only as a function of the instantaneous state X(t), ie, u(t) = χt (X(t)), where χt maps the state space to the control input space. Only then the Markovianity of X(t) is not destroyed. In fact the generator of the controlled Markov process X(t) is then given by ˜ t φ(x) = (Kt (χt (x))φ)(x) K We wish to select the feedback control functions χt (.) so that  CT (x) = E[  = E[

0

T

L(X(t), u(t))dt|X(0) = x]

0

T

L(X(t), χt (X(t)))dt|X(0) = x]

is minimized. To this end, we define 

T

C(t, T, x) = minu(s),t≤s≤T E[

L(X(s), u(s))ds|X(t) = x] t

 = minχs ,t≤s≤T E[

T

L(X(s), χs (X(s)))ds|X(t) = x] t

Then, by applying the Markov property, we easily deduce that C(t, T, x) = minχt (.) (L(x, χt (x))dt + E(C(t + dt, T, X(t + dt))|X(t) = x)) = minχt (.) (L(x, χt (x))dt + C(t, T, x) + dt.

∂C(t, T, x) + dt.Kt (χt (x))(C(t, T, x))) ∂t

or equivalently, ∂C(t, T, x) + minχt (.) (L(x, χt (x)) + Kt (χt (x))(C(t, T, x))) = 0 ∂t This is the stochastic BHJ (Bellman-Hamilton-Jacobi) equation. As a by product, it yields the optimal feedback control maps χt (.), 0 ≤ t ≤ T . The final point condition while solving this pde is given by limt→T C(t, T, x) = 0 The discrete time case: Here X(n), n ≥ 0 is a Markov process in discrete time with one step transition probability generator Kn (u(n)) where u(n) = χn (X(n)). Thus, E(φ(X(n + 1))|X(n) = x) = (Kn (χn (x))φ)(x)

175

Quantum Antennas The control maps χn (.), n ≥ 0 are to be chosen so that CN (x0 ) = E[

N 

L(X(n), u(n))|X(0) = x0 ]

n=0

= E[

N 

L(X(n), χn (X(n)))|X(0) = x0 ]

n=0

is a minimum. We define C(n, N, x) = minχk (.),n≤k≤N E[

N 

L(X(k), χk (X(k)))|X(n) = x]

k=n

Using the Markov property, we find that C(n, N, x) = minχn (.) (L(x, χn (x)) + E[C(n + 1, N, X(n + 1))|X(n) = x]) = minχn (.) (L(x, χn (x)), Kn (χn (x))(C(n + 1, N, x))) This is the stochastic BHJ equation in discrete time. This is to be solved with the final point condition χN (.) = argminχ L(x0 , χ(x0 ))

10.7

Quantum stochastic optimal control of the HP-Schrodinger equation

The controlled HP equation is dU (t) = ((−iH+P )dt+LdA(t)+M dA(t)∗ +SdΛ(t)−iK(t)(Xd (t)−πt (X))dt)U (t) where

jt (X) = U (t)∗ XU (t), πt (X) = E(jt (X)|ηt )

Xd (t) is the desired state trajectory to be tracked and K(t) is the controller coefficient. ηt is the Von-Neumann algebra generated by output non-demolition measurements upto time t. Equivalently, we can express the controlled HP equation as dU (t) = [(−i(H + K(t)(Xd (t) − πt (X)) + P )dt + LdA(t) + M dA(t)∗ + SdΛ(t)]dt

Study project: Explain how you can solve the above controlled HP equation coupled with the Belavkin filter equation for πt (X) using perturbation theory.

176

10.8

Quantum Antennas

Bath in a superposition of coherent states interacting with a system

The system dynamics corresponds to that of a fan motor subject to quantum noise: θ (t) + aθ (t) + f (t, θ(t)) = w(t) where w(t) is white noise, a is the damping coefficient and f (t, θ) = −I(t)BLsin(θ(t)). To give a quantum mechanical description of this motor, we introduce a Lagrangian  θ  L(t, θ, θ ) = θ 2 /2 − f (t, θ)dθ + w(t)θ 0

which yields using the Euler-Lagrange equations θ (t) + f (t, θ) − w(t) = 0 namely, the same as that of the fan but with the damping term excluded. To include the damping term using quantum mechanics, we consider the canonical momentum p = ∂L/∂θ and apply the Legendre transformation to obtain the Hamiltonian H(t, θ, p) = pθ − L = p2 /2 + V (t, θ) − w(t)θ where



θ

f (t, θ)dθ

V (t, θ) = 0

and then introduce the Lindblad operators to obtain damping. The resulting master equation is of the form ρ (t) = −i[H(t, θ, p), ρ(t)] − θ(ρ(t)) where

θ(ρ) = (1/2)(L∗ Lρ + ρL∗ L − 2LρL∗ )

where L = αθ + βp

Chapter 11

Basic triangle geometry required for understanding Riemannian geometry in Einstein’s theory of gravity 11.1

Problems in mathematics and physics for school students

[1] A triangle has sides a, b, c and corresponding angles A, B, C. Draw the triangle and mark all the sides and angles. Express b in terms of a, c, C by solving a quadratic equation. Likewise, express c in terms of a, b, B and a in terms of b, c, C. [2] Let ABC be a triangle with side lengths a, b, c and opposite angles A, B, C. Drop the altitude from vertex A onto the side a = BC. Let D denote the intersection point. Draw the diagram marking all the points. Let h = AD denote this altitude. Calculate h in terms of a, B, C. Also calculate h in terms of c and B and finally, h in terms of b and C. [3] Two straight lines having equations y = mx+c and y = m x+c are given. Determine their intersection point coordinates (x, y) by solving the simultaneous equation. For m = tan(60o ) and m = tan(30o ) and c = −5, c = −4, draw these lines using ruler and protractor and verify your result. [4] Factorize: [a] (x + y)2 − x2 − 2zy

177

178 [b]

Quantum Antennas

(x2 − y 2 ) + a(x + y) + c(x − y + a)

[4] Calculate the incircle radius of a triangle ABC with side-lengths a, b, c in terms of B/2, C/2, a. Also calculate the radius of the excircle that touches the side a = BC in terms of B, C, a.

11.2

Geometry on a curved surface, study problems

[1] Define a straight line on a curved surface as the path of shortest Euclidean distance on the surface between two points. This presupposes that the curved surface of dimension p is immersed in an N > p dimensional Euclidean space. [2] Define parallel displacement of a vector on a curved surface by infinitesimally translating the vector parallely in the Euclidean sense and then projecting the resulting vector onto the tangent plane at the neighbouring point. [3] Define a geodesic triangle on a curved two dimensional surface and prove Gauss’m theorem that the sum of the angles of such a triangle equals the integral of the Gauss curvature over the triangle. [4] Show that an alternate equivalent definition of a straight line on a curved surface is given by that path such that when the tangent vector to this curve at any point is parallely displaced along the curve to another point, it continues to be a tangent to the curve at the displaced point.

Chapter 12

Design of gates using Abelian and non-Abelian gauge quantum field theories with performance analysis using the Hudson-Parthasarathy quantum stochastic calculus 12.1

Design of quantum gates using Feynman diagrams

If 1/A denotes the electron propagator, then corrections to it coming from terms like vacuum polarization, external field effects etc. will result in the corrected propagator 1/(A + B). Denoting the parameters of the external fields by θ, we have the corrected propagator (A + B(θ))−1 = A−1 − A−1 BA−1 + A−1 BA−1 BA−1 + ... Let X denote the desired propagator that will yield the desired scattering matrix. Then we must design the parameters θ so that  X − (A + B(θ))−1  is minimized. Writing Z = X − A−1 , the approximate minimization problem is θˆ = argminθ  Z + A−1 B(θ)A−1 − A−1 B(θ)A−1 B(θ)A−1 2

179

180

Quantum Antennas

If the minimization is over all B, then we expand the above propagator error energy upto quadratic terms in B and then the optimal equations will be linear in B which are easily inverted. We leave it as an exercise to show that with this second order approximation, ˆ = argminB (T r(A−1 BA−2 BA−1 )+2Re(T r(ZA−1 BA−1 ) B −2Re(T r(ZA−1 BA−1 BA−1 )) An electron with a four momentum of p1 , a spin of σ1 interacts with a positron with a four momentum of p2 and a spin of σ2 , annihilate each other to produce a γ-ray photon of four momentum k = p1 +p2 which again polarizes into an electron-positron pair in the form of a loop (vacuum polarization) and again a pair annihilation takes place from this loop to result in a γ-ray photon which propagates and again polarizes into an electron positron pair with four momenta p3 , σ3 and p4 , σ4 respectively. To calculate the amplitude for this process, we must assume four momentum conservation, ie, p1 + p2 = p3 + p4 which can be ensured by introducing a momentum conserving δ-function δ 4 (p1 + p2 − p3 − p4 ). Denoting the electron wave function by u(p, σ) and the positron wave function by v¯(p, σ) = v(p, σ)∗ γ 0 , the photon propagator by Dμν (k) and the electron propagator matrix by S(p) = (γ.p − m)−1 = (γ.m + p)/(p2 − m2 ), with p2 = (p0 )2 − (p1 )2 − (p2 )2 − (p3 )2 = p02 − P 2 so that

p2 = m2 = p02 − E(P ), E(P ) = m2 + P 2

we obtain the amplitude for the above process as S(p3 , σ3 , p4 , σ4 |p1 , σ1 , p2 , σ2 ) =  v (p4 , σ4 )γ ν u(p3 , σ3 ) Dμρ (p1 +p2 )Dνα (p3 +p4 ) (¯ v (p2 , σ2 )γ μ u(p1 , σ1 ))(¯ T r(S(p1 +p2 −q)γ α S(q)γ ρ )d4 q = (p1 +p2 )−4¯(v(p2 , σ2 )γρ u(p1 , σ1 ))(¯ v (p4 , σ4 )γα u(p3 , σ3 )T r(S(p1 +p2 −q)γ α S(q)γ ρ )d4 q

Note that this amplitude is to be multiplied by δ 4 (p1 + p2 − p3 − p4 ). Remark 1: To make the integrals converge we insert factors like −λ2 /(q 2 − 2 λ ). In the limit as λ → ∞, this factor becomes unity. The meaning of this factor is that it corresponds to a pseudo-photon propagator with pseudo-photon mass of λ. Remark 2: Consider introducing an external em field in the vacuum polarization loop of a photon. The process is described in the following way. An electron of four momentum p1 and spin σ1 interacts with a positron of four momentum p2 and spin σ2 , annihilating each other to give a photon which propagates and then again polarizes into an electron-positron pair appearing in the form of a loop in the Feynman diagram. An external field Aμ (x) is connected to this loop, ie, it interacts with the electron-positron pair after polarization of the photon.

181

Quantum Antennas

The electron-positron pair again, after this interaction annihilate to produce a photon which propagates and finally polarize into an electron-positron pair of four momenta and spins (p3 , σ3 ) and (p4 , σ4 ) respectively. Using the Feynman rules, the amplitude for this overall correction to the scattering matrix is given by assuming k to be the four momentum of the external photon line S(p3 , σ, p4 , σ4 |p1 , σ1 , p2 , σ2 ) = v (p2 , σ2 )γ μ u(p1 , σ1 ))Dμν (p1 + p2 )Aβ (k) δ 4 (p1 + p2 − p3 − p4 ).(¯  .( T r[S(p1 + p2 − q + k)γ β S(p1 + p2 − q)γ ν S(q)γ ρ ]d4 q) Dρα (p3 + p4 )¯ v (p4 , σ4 )γ α u(p3 , σ3 ) where S(p) = (γ.p − m)−1 is the electron propagator while Dμν (q) = ημν /q 2 is the photon propagator. This scattering amplitude correction can alternately be viewed as coming from a correction to the electron propagator 1/A coming from the external photon line. We can express the contribution of such multiple loops with external field lines by the corrected propagator 1/(A + B) where B depends on the external photon line Aμ (k). Thus, the corrected propagator can be expressed as (A + B)−1 = A−1 − A−1 BA−1 + A−1 BA−1 BA−1 + ... = A−1 +



(−1)n (BA−1 )n

n≥1

For a desired scattering matrix, let X denote the desired propagator, then the external photon field Aμ (k) must be ”controlled” so that  X − (A + B)−1  is a minimum.

12.2 An optimization problem in electromagnetism We consider here the problem of calculating the multipole em radiation fields E, H expressing them in terms of components tangential to the radial direction rˆ and parallel to the radial direction. They satisfy the free space Maxwell equations divE = 0, divH = 0, (∇2 + k 2 )E = 0, (∇2 + k 2 )H = 0 From these equations, it easily follows that (∇2 + k 2 )(r.E) = 0, (∇2 + k 2 )(r.H) = 0

182

Quantum Antennas

This suggests to us the possibility of decomposing em fields into radial an tangential components with each of these components satisfying the Helmholtz equation. To this end we denote by L = r × p = −ir × ∇ the angular momentum vector operator. Let us define Elm = fl (r)LYlm (ˆ r) Then clearly since r.L = 0 or equivalently, rˆ.L = 0, rˆ = r/|r|, we have r.Elm = 0 ie Elm is a purely tangential solution to the Helmholtz equation provided that (∇2 + k 2 )Elm = 0 and (∇2 + k 2 )(r.Elm ) = 0. Note that these two equations guarantee that divElm = 0. Also since r.Elm = 0, the latter equation is already guaranteed to be satisfied. Since L commutes with L2 , we have using ∇2 = r−2

∂ 2 r ∂/∂r − L2 /r2 ∂r

and r) = l(l + 1)Ylm (ˆ r) L2 Ylm (ˆ that (∇2 + k 2 )Elm = fl (r) + (2/r)fl (r) + (−l(l + 1)/r2 + k 2 )fl (r) and for this to vanish, we must have r2 fl (r) + 2rfl (r) + (k 2 r2 − l(l + 1))fl (r) = 0 which has two linearly independent solutions jl (kr), hl (kr) so that fl (r) = c(l, m)jl (kr) + d(l, m)hl (kr) These linearly independent functions are called the modified Bessel functions and they can be expressed in terms of the usual Bessel functions by formulae of the form jl (x) = x−1/2 Jl+1/2 (x) Assuming these, we have that Elm is a valid electric field. The corresponding ˜ lm is given by magnetic field H ˜ lm ∇ × Elm = −jωμH or equivalently, ˜ lm = (j/ωμ)∇ × (fl (r)LYlm (ˆ H r))

183

Quantum Antennas since the operator ∇× commutes with ∇2 , it follows that since (∇2 +k 2 )Elm = 0 that ˜ lm = 0 (∇2 + k 2 )H and it is clear that ˜ lm = 0 r.H In fact, ˜ lm = (j/ωμ)r.(∇ × fl (r)LYlm (ˆ r.H r)) = (j/ωμ)(r × ∇).(fl (r)LYlm (ˆ r)) r)) = (−1/ωμ)fl (r)L2 Ylm (ˆ r) = (−1/ωμ)L.(fl (r)LYlm (ˆ r) = 0 = (−1/ωμ)l(l+1)fl (r)Ylm (ˆ ˜ lm is a non-tangential solution to for the magnetic field. thus proving that H Likewise, we can start with another solution gl (r) to the equation r2 gl (r) + 2rgl (r) + (k 2 r2 − l(l + 1))gl (r) = 0 and construct a tangential solution to the magnetic field as r) Hlm = gl (r)LYlm (ˆ with the corresponding non-tangential solution to the electric field given by ˜lm ∇ × Hlm = jωE or equivalently, ˜lm = (−j/ω)∇ × (gl (r)LYlm (ˆ r)) E Note that since ∇.∇ × F = 0 for any vector field F , we must necessarily have ˜lm = 0, div H ˜ lm = 0 div E which means that the non-tangential components of the electric and magnetic field constructed above satisfy all the requirements for an electric and magnetic field. Superposing both the tangential and non-tangential components, we get the general radiation fields at a given frequency ω as  [flm (r)LYlm (ˆ r) − (j/ω)∇ × (glm (r)LYlm (ˆ r))] E(r) = l,m

H(r) =



[glm (r)LYlm (ˆ r) + (j/ωμ)∇ × (flm (r)Ylm (ˆ r)]

l,m

where

flm (r) = cE (l, m)jl (kr) + dE (l, m)hl (kr), glm (r) = cH (l, m)jl (kr) + dH (l, m)hl (kr)

and the coefficients cE , dE , cH , dH are obtained by measuring the fields on the surface of a sphere and using orthogonality properties of the vector valued complex functions r), ∇ × LYlm (ˆ r) LYlm (ˆ on the unit sphere.

184

12.3

Quantum Antennas

Design of quantum gates using non-Abelian gauge theories

The Yang-Mills equations for the 4N × 1 wave function ψ(x) in the presence of mass terms, external electromagnetic fields Aμ (x) and non-Abelian gauge field terms Bμα (x)τα are given by [γ μ (i∂μ + eAμ (x) + Bμα (x)τα − m]ψ(x) = 0 or more precisely (iγ μ ⊗ IN )∂μ ψ(x) + eAμ (x)(γ μ ⊗ IN )ψ(x) +Bμα (x)(γ μ ⊗ τα )ψ(x) − mψ(x) = 0 Expressing this equation in Hamiltonian form, we get i∂0 ψ(x) = H(x)ψ(x) where H(x) = −iγ 0 γ r ∂r − e(γ 0 γ μ ⊗ IN )Aμ (x) −(γ 0 γ μ ⊗ τα )Bμα (x) + mγ 0 ⊗ IN By regarding the potentials Aμ (x), Bμα (x) as control fields and replacing the wave function ψ(x) by a unitary operator kernel Ut (r, r ) ∈ C4N ×4N so that  Ut (r, r )∗ U (r , r )d3 r = I4N δ 3 (r − r ) our aim is to get as close as possible after time T to a given unitary kernel Ug (r, r ) in the sense that 

 Ug (r, r ) − UT (r, r ) 2 d3 rd3 r

is a minimum. One can also try to introduce quantum noise processes in the sense of Hudson and Parthasarathy into the control fields to model the effects of noise on the designed gate. Specifically, this would involve replacing Aμ (t, r)dt β α by a classical field plus cα μβ (r)dΛα (t) and likewise Bμ (t, r)dt by a classical field μρ σ plus dασ (r)dΛρ (t) where the fundamental noise processes Λα β (t) satisfy the quantum Ito formula μ μ α dΛα β dΛν = ν dΛβ with μν assuming the value zero if either μ or ν is zero and otherwise assuming the value δνμ (μ, ν = 0, 1, 2, ...).

185

Quantum Antennas

12.4

Design of quantum gates using the HudsonParthasarathy quantum stochastic Schrodinger equation

U (t) satisfies the HP equation dU (t) = (−(iH + e2 P )dt + eL1 dA(t) + eL2 dA(t)∗ + e2 SdΛ(t))U (t) e is a perturbation parameter introduced to show that the creation and annihilation process noise are small ie of O(e) while the conservation process term is of order e2 since dΛ = dA∗ dA/dt and finally the coefficient of the quantum Ito correction term P dt is O(e2 ) since P = (eL2 )∗ (eL2 ).

12.5

gravitational waves in a background curved metric (0) gμν (x) = gμν (x) + hμν (x)

hμν = δgμν is the metric of space-time. αβ δ(Γα Γβμν ) = μν ) = δ(g (0)

(δg αβ )Γβμν + g (0)αβ δΓβμν (0)

= −g (0)αρ g (0)βσ hρσ Γβμν +(1/2)g (0)αβ (hβμ,ν + hβν,μ − hμν,β ) = g (0)αβ (1/2(hβμ,ν + hβν,μ − hμν,β ) − hβσ Γ(0)σ μν ) = (1/2)g (0)αβ (hβμ:ν + hβν:μ − hμν:β ) (0)

where the covariant derivative is taken w.r.t the unperturbed metric gμν . Since the covariant derivative of the unperturbed metric is zero, assuming that raising and lowering of metric perturbations and their covariant derivatives are taken w.r.t. the unperturbed metric, we can also write this equation as α α :α δ(Γα μν ) = (1/2)(hμ:ν + hν:μ − hμν )

Now, a straightforward computation shows that the perturbation in the Ricci tensor is α δRμν = δΓα μα:ν − δΓμν:α which implies on substituting the above equation, α α 2δRμν = (hα μ:α:ν + hα:μ:ν − hμ:α:ν )

186

Quantum Antennas α :α −(hα μ:ν:α + hν:μ:α − hμν:α )

which simplifies to α α :α 2δRμν = hα α:μ:ν − (hμ:ν:α + hν:μ:α − hμν:α ) :α or writing h = hα α , and hμν = hμν:α , we get α 2δRμν = hμν + h,μ:ν − hα μ:ν:α − hν:μ:α

Suppose we use harmonic coordinates, ie g μν Γα μν = 0. The perturbed form of this is δ(g μν Γα μν ) = 0 or equivalently, α :α μν (0)α g (0)μν (hα μ:ν + hν:μ − hμν ) + δg Γμν = 0

Instead, we modifiy our coordinate condition by removing the last term to get a modified version of perturbed harmonic coordinates: 2hμα:μ − h,α = 0 Then noting that h,μ:ν = h,ν:μ we get α hα μ:α:ν = hν:α:μ

our perturbed field equations δRμν = 0 in this perturbed coordinate system assume the form α 0 = hμν + h,μ:ν − hα μ:ν:α − hν:μ:α = α = hμν + (hα μ:α:ν − hμ:ν:α ) α +(hα ν:α:μ − hν:μ:α ) = 0

The last two brackets can be expressed in terms of the Riemann curvature tensor of the unperturbed metric and the perturbed metric coefficients hμν not involving their partial derivatives.

Quantum Antennas

12.6

187

Topics for a short course on electromagnetic field propagation at high frequencies

To transmit and detect high frequency electromagnetic waves, our transmitter and receiver antennae must be of very small size, ie of the Angstrom scale where quantum mechanical effects become dominant. This is because, the wavelength of an em wave is inversely proportional to frequency (λ = c/ν or equivalently, k = ω/c, λ = 2π/k). It is therefore impossible to discuss a theory of electromagnetic wave propagation at high frequencies without introducing fundamental quantum mechanical principles such as second quantization, Feynman path integrals for fields, interaction between the electromagnetic field, matter and the gravitational field at the quantum level. Keeping this in mind, we discuss the following topics for a short course on high frequency communication. [1] Quantization of the em field in terms of creation and annihilation operator fields. [2] Canonical commutation relations. [3] Quantization with constraints–The Dirac bracket. [4] Creation, annihilation and conservation processes in the sense of Hudson and Parthasarathy. Quantum Ito’s formula, Derivation of the GKSL equation when the bath is in a coherent state or in a superposition of coherent states. Remark: In the quantum theory of fields, we introduce creation and annihilation operator fields in three momentum space. However these operator fields are time independent. To represent quantum noise, however, we need to make these creation and annihilation fields time dependent in such a way that these time dependent processes behave like classical stochastic processes in certain states of the bath. The Hudson-Parthasarathy quantum stochastic calculus precisely achieves this. Whenever we have an operator field like a(u), a(u)∗ , λ(H) acting in the Boson Fock space of H where u is a vector in the Hilbert space H and H is an operator in the Hilbert space H, we can introduce time dependence by replacing u with χ[0,t] u and H with χ[0,t] H provided that H has the form L2 (R+ ) ⊗ H0 and χ[0,t] denotes multiplication by the indicator function of [0, t] in L2 (R+ ). We must necessarily assume that χ[0,t] commutes with H. This will happen when for example, H acts in H0 . When time dependence of the creation, annihilation and conservation fields is thus introduced, we obtain quantum stochastic processes that satisfy quantum Ito’s formula owing to the non-commutativity of these operator fields. Classical Ito’s formula for Brownian motion and Poisson processes follow as special cases. Since non-commutativity of observables implies Heisenberg uncertainty, ie, impossibility of simultaneously measuring these observables, it follows that Ito’s formula can be traced to the Heisenberg uncertainty principle. [5] Dirac’s equation for the electron in a noisy electromagnetic field. [6] Approximate solution of the GKSL equation using time dependent perturbation theory on system space. [7] Approximate solution of the Hudson-Parthasarathy noisy Schrodinger equation using time dependent perturbation theory on system⊗ bath space.

188

Quantum Antennas

[8] Quantum entropy pumped by a noisy em field into an atomic system— approximate expressions. [9] Filtering in quantum mechanics using V.P.Belavkin’s theory. The notion of non-demolition measurements associated to a given Hudson-Parthasarathy noisy Schrodinger evolution. Examples of non-demolition measurements using quadrature processes, photon counting processes and mixture of quadrature and photon counting processes. [10] Quantum control after filtering with the objective of (a) reducing GKSL noise and (b) state tracking. [11] Comparison of quantum filtering and control with classical filtering and control. [12] Interaction of the gravitational field with the em field–the classical theory based on the Einstein-Maxwell equations. [13] Interaction of the gravitational field with the em field–the quantum theory based on approximate linearization of the Einstein field equations. [14] Gravitational waves in a flat and curved background metric. [15] Proof that gravitons are spin 2 particles. Proof based on choosing a harmonic coordinate system and determining transformation properties of the tensor components of the gravitational wave amplitudes under rotations of the coordinate system around an axis. [16] Energy-momentum tensor of the gravitational field. [17] Noether’s theorem on conserved charges for a classical field theory when the Lagrangian density is invariant under an infinitesimal Lie algebra of field transformations. [18] How to draw Feynman diagrams for scattering, absorption and emission processes involving electrons, positrons, photons, mesons and gravitons. Derivation of the Feynman rules using operator theory, ie using canonical commutation rules for Bosons and canonical anticommutation rules for Fermions. [19] Path integrals for fields with application to Yang-Mills quantization. Derivation of the invariance of the path integral under different gauge fixing conditions whenever the action and path measure are gauge invariant. [20] The Galilean group and its projective unitary representations. Derivation of the energy, position, momentum, angular momentum and velocity operators from the multipliers of the Galilean group. [21] Application of the theory of induced representations for estimating the rotation, translation and uniform velocity of motion of an antenna from noisy em pattern measurements. The initial current density field is J(r) at frequency ω. Let G(|r|) = (μ/4π|r|)exp(−jω|r|/c) Then the initial magnetic vector potential is  A(r) = G(|r − r |)J(r )d3 r After rotation and translation, the current density is ˜ = RJ(R−1 (r − a)), R ∈ SO(3), a ∈ R3 J(r)

189

Quantum Antennas The corresponding magnetic vector potential is  ˜ ˜  )d3 r + w(r) A(r) = G(|r − r |)J(r where w(r) is a noise field. This evaluates to  ˜ A(r) = G(|r − Rr − a|)RJ(r )d3 r + w(r)  =R

G(|R−1 (r − a) − r |)J(r )d3 r + w(r) = RA(R−1 (r − a)) + w(r)

since detR = 1. The initial and final magnetic fields are respectively B(r) = ∇ × A(r), ˜ ˜ B(r) = ∇ × A(r) = ((R−1 ∇) × (RA))(R−1 (r − a)) + ∇ × w(r) Likewise, the electric field can be transformed: The initial electric field is E(r) = −∇V (r) − jωA(r) where so that

V (r) = (jc2 /ω)divA(r) E(r) = (−jc2 /ω)∇(divA(r)) − jωA(r)

This can equivalently be expressed using the Maxwell equation ∇ × B/μ = J + jωE, B = ∇ × A as E(r) = (−j/ω)(∇ × (∇ × A(r))/μ − J(r)) The equivalence of the two expressions follows from the wave equation for A(r): (∇2 + ω 2 /c2 )A(r) = −μJ(r), c2 = 1/μ It is however more convenient to work using special relativistic tensors: Fμν = Aν,μ − Aμ,ν where we use time domain expressions. We get on applying the Lorentz gauge condition Aμ,μ = 0 to the Maxwell equations F,νμν = −μ0 J μ

190

Quantum Antennas

that Aμ (x) = μ0 J μ (x),  = ∂α ∂ α 

with solution Aμ (x) =

J μ (x )G(x − x )d4 x

where G(x − x ) = μ0 δ((x − x )2 ) = (μ0 /2π)δ((t − t )2 − |r − r |2 ) = μ0 δ(t − t − |r − r |)/4π|r − r | assuming t > t . Now suppose, we apply a Poincare transformation, ie, a Lorentz transformation L along with a space-time translation a to the four current density J μ = J. The transformed four current density is then ˜ J(x) = LJ(L−1 (x − a)) where (LJ)μ = Lμν J ν The problem is to estimate the Poincare group element (L, a) where a = (aμ ). We get for the initial andtransformed em four potentials  A(x) = G(x − x )J(x )d4 x  ˜ A(x) =  =L

G(x − x )LJ(L−1 (x − a))d4 x

G(L−1 (x − a) − x )J(x )d4 x = LA(L−1 (x − a))

where we have used the fact that L preserves the space-time Minkowski metric (x − x )2 , ie, (L(x − x ))2 = (x − x )2 .

Chapter 13

Quantum gravity with photon interactions, cavity resonators with inhomogeneities, classical and quantum optimal control of fields 13.1

Quantum control of the HP-Schrodinger equation by state feedback dU (t) = (−(iH + P )dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t) jt (X) = U (t)∗ XU (t), X ∈ L(h)

djt (X) = jt (θ0 (X))dt + jt (θ1 (X))dA(t) + jt (θ2 (X))dA(t)∗ + jt (θ3 (X))dΛ(t) We wish jt (X) to track the noiseless trajectory Xd (t) ∈ L(h). Assume nondemoltion measurements Yo (t) = U (t)∗ Yi (t)U (t) are made and the Belavkin filter for πt (X) = E(jt (X)|ηt ), ηt = σ(Yo (s) : s ≤ t) has been constructed as dπt (X) = Ft (X)dt + Gt (X)dYo (t) trajectory estimation error Xd (t) − πt (X) is given as feedback to the state equations in the form djt (X) = jt (θ0 (X))dt + jt (θ1 (X))dA(t) + jt (θ2 (X))dA(t)∗ + jt (θ3 (X))dΛ(t)

191

192

Quantum Antennas +K(t)(Xd (t) − πt (X))dt

Alternately, if we assume that Xd follows a noiseless trajectory, ie, it evolves according to the equation (0)

Xd (t) = jt (Xd ) = U0 (t)∗ Xd U0 (t), U0 (t) = exp(−itH0 ) then we can give as feedback the estimation error E(t) = Xd (t) − T r2 (πt (X)(I ⊗ |φ(u) >< φ(u)|)) which is a system observable. This error feedback is given to the above state equations, ie, to the Evans-Hudson flow resulting in the dynamics djt (X) = jt (θ0 (X))dt + jt (θ1 (X))dA(t) + jt (θ2 (X))dA(t)∗ + jt (θ3 (X))dΛ(t) +K(t)E(t) Alternately, we can incorporate this error feedback in the original HP equation as dU (t) = (−(iH + P + K(t)E(t))dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t) resulting in the Evans-Hudson flow the same as above but with θ0 (X) replaced by θ0 (X) + iK(t)[E(t), X]. Luc-Bouten’s method of control is slightly different. Here, we choose a system observable Z and give an infinitesimal control unitary Uc (t, t + dt) = U (t + dt)∗ exp(−iZdYi (t))U (t + dt) = exp(iZ(t + dt)dYo (t)) Note that Yo (t) = U (t + dt)∗ Yi (t)U (t + dt), Yo (t + dt) = U (t + dt)∗ Yi (t + dt)U (t + dt) and hence, forming the difference, we get dYo (t) = U (t + dt)∗ dYi (t)U (t + dt) Also, Z commutes with dYi (t) and hence Z(t + dt) = U (t + dt)∗ ZU (t + dt) commutes with U (t + dt)∗ dYo (t)U (t + dt). Let ρc (t) denote the controlled state at time t, ie after applying the Belavkin filter and control upto time t to the HP evolved state. Then, we apply the Belavkin filter from t to t + dt, via ρB (t + dt) = ρc (t) + δρB (t) where δρB (t) = L∗t (ρc (t))dt+(Mt ρc (t)+ρc (t)Mt∗ −T r(Mt +Mt∗ )ρc (t))ρc (t))(dYo (t) −T r(ρc (t)(Mt +Mt∗ ))dt) Finally, the filtered and controlled state at time t + dt is given by ρc (t + dt) = ρc (t) + δρc (t) =

193

Quantum Antennas Uc (t, t + dt)(ρc (t) + δρB (t)).Uc (t, t + dt)∗ = exp(iZ(t + dt)dYo (t))(ρc (t) + δρB (t)).exp(−iZ(t + dt)dYo (t)) Comparison with classical filter/state observer and controller: X  (t) = ψ(t, X(t)) + G(t, X(t))(τc (t) + W (t)) −1 ˆ ˆ ˆ (K(t)(Xd (t) − X(t)) + Xd (t) − ψ(t, X(t))) τc (t) = G(t, X(t))

Xd (t) = ψ(t, Xd (t)) + G(t, Xd (t))τd (t) ˆ  (t) = ψ(t, X(t)) ˆ ˆ X + L(t)(dZ(t) − h(t, X(t))dt) dZ(t) = h(t, X(t)dt + σV dV (t) ˆ e(t) = Xd (t) − X(t), f (t) = X(t) − X(t) Then, e (t) = ψ(t, Xd (t))−ψ(t, X(t))+G(t, Xd (t))τd (t)−G(t, X(t))τc (t)−G(t, X(t))W (t) = ψ(t, Xd (t)) − ψ(t, X(t)) + G(t, Xd (t))τd (t) − G(t, X(t))τc (t) − G(t, X(t))W (t) −1 ˆ = ψ(t, Xd (t))−ψ(t, X(t))+G(t, Xd (t))τd (t)−G(t, X(t))G(t, X(t)) K(t)(e(t)+f (t)) −1 ˆ ˆ −G(t, X(t))G(t, X(t)) (Xd (t) − ψ(t, X(t))) − G(t, X(t))W (t)

ˆ On linearizing this equation around X(t), this equation appears in the form e (t) = A1 (t)e(t) + A2 (t)f (t) + A3 (t)W (t) ˆ only. where A1 (t), A2 (t), A3 (t) are functions of t, X(t)

13.2

Some applications of Poisson processes

Change of measure theorem of Girsanov for Poisson Martingales Consider the process p  c(a)Na (t) X(t) = a=1

where N1 , ..., Np are p independent Poisson processes with rates λ1 , ..., λp respectively. The process Y (t) = X(t) −

p  a=1

c(a)λa t =

p 

c(a)(Na (t) − λa t)

a=1

is a Martingale. Let η(t) be a finite variation adapted process so that exp(Y (t)+ η(t)) is a Martingale. Then, we require to compute η(t). We have with Z(t) = exp(Y (t) + η(t)),

194

Quantum Antennas dZ(t) = d(exp(Y (t) + η(t))) = d(exp(X(t) + η(t) − Z(t)[



(exp(c(a)) − 1)dNa (t) + dη(t) −



a

= Z(t)[



c(a)λa t)) =

a

c(a)λa dt]

a

  (exp(c(a))−1)(dNa (t)−λa dt)+dη(t)+ λa (exp(c(a))−1−c(a))dt) a

a

So for Z(t) to be a Martingale, we require that  λa (exp(c(a)) − 1 − c(a))t η(t) = − a

Thus the exponential Martingale associated with X(t) is given by  λa (exp(c(a)) − 1 − c(a))t) Z(t) = exp(X(t) − a

and, in fact, we have dZ(t) = Z(t).

 (exp(c(a)) − 1)(dNa (t) − λa dt) a

We next define a measure Q so that dQt /dPt = Z(t), t ≥ 0 where Qt , Pt respectively are restrictions of Q and P to Ft . This is a consistent definition since if t > s and B ∈ Fs , then     Qt (B) = Z(t)dPt = Z(t)dP = Z(s)dP = dQs = Qs (B) B

B

B

B

where the martingale property of Z w.r.t P has been used. Now, we wish to determine an adapted process f (t) of finite variation such that U (t) = Y (t)+f (t) is a Q-martingale. Note that Y (t) is a P-martingale. For this, we must have that EP [d(Z(t)U (t))|Ft ] = 0 ie, the process ZU is a P -Martingale, for then, this would imply that for any Ft -measurable; r.v. V , we have EP [d(Z(t)U (t)).V ] = 0 which would in turn imply that EP [(Z(t + dt)U (t + dt) − Z(t)U (t))V ] = 0 or equivalently, EQ [(U (t + dt) − U (t))V ] = 0

195

Quantum Antennas

ie U (t) is a Q-martingale. Now application of Ito’s formula for Poisson processes gives d(ZU )) = ZdU + U dZ + dU.dZ = Z(dY + df ) + U dZ + dY dZ + df.dZ    Since U dZ and ZdY are martingales, we therefore require that (Zdf + dY dZ + df dZ) be a Martingale. But,   Zdf + dY dZ = Zdf + Z. exp(c(a)) − 1)(dNa − λa dt). c(b)(dNb − λb dt) a

b

 (exp(c(a)) − 1)c(a)dNa ] = Z[df + a

Writing f (t) =



d(a)Na (t)

a

we get df.dZ = Z.

 (exp(c(a)) − 1)d(a)dNa (t) a

and hence, we deduce that U is a Q-martingale provided that  (exp(c(a)) − 1)(c(a) + d(a))dNa (t) f (t) = − a

This means that we should have d(a) + (exp(c(a)) − 1)(c(a) + d(a)) = 0 or equivalently, d(a) = −c(a)(1 − exp(−c(a))) In other words, U (t) = Y (t) −



c(a)(1 − exp(−c(a)))Na (t)

a

is a Q-martingale. A Feynman-Kac formula for Poisson processes. Then define  Let as before X(t) = a c(a)Na (t).  t u(t, x) = E[exp( V (X(s))ds)φ(X(t))|X(0) = x] 0

We get using the Markov property of X(.) that u(t + dt, x) = (1 + V (x)dt)E(u(t, X(dt))|X(0) = x)  = (1 + V (x)dt)(u(t, x) + (u(t, x + c(a)) − u(t, x))λa dt) a

196

Quantum Antennas

so that u satisfies the equation  λa (u(t, x + c(a)) − u(t, x)), u(0, x) = φ(x) u,t (t, x) = V (x)u(t, x) + a

Replacing t by −it by defining ψ(t, x) = u(−it, x) we get iψ,t (t, x) = V (x)ψ(t, x) +



λa (ψ(t, x + c(a)) − ψ(t, x)), ψ(0, x) = φ(x)

a

This equation has the following physical interpretation. At time t = 0, φ(x) is the wave function of a quantum particle. The amplitude of the particle to go from x + c(a) to x in time dt is given by −iλa dt, a = 1, 2, ..., p in the absence of an external potential. In the presence of an external potential V (x), the amplitude of the particle to go from x + c(a) to x in time dt is given by −iλa dt and the amplitude to stay at x in time dt is given by 1+i( a λa dt−V (x))dt. In other words, this version of the Feynman path integral assumes that quantum transitions take place by discrete jumps rather than continuous motion. Suppose that an electron moves in a one dimensional crystal. At any time t, there is an amplitude ct (n) for the electron to be at the site nΔ at time t. Transitions can take place only between neighbouring sites. If there is an amplitude −iadt for the electron to make a transition from (n + 1)Δ to nΔ and the same for the transition from (n − 1)Δ to nΔ, then quantum mechanics gives us the equation ct+dt (n) = ct (n)(1 − iλdt) − ct (n − 1)iadt − ct (n + 1)iadt or idct (n)/dt = λct (n) + a(ct (n + 1) + ct (n − 1)) where 1 − iλdt is the amplitude for no transition in time dt. We make the approximation, ct (n + 1) + ct (n − 1) − 2ct (n) ≈ Δ2 ct (x), x = nΔ and then we get Schrodinger’s equation idct (x)/dt = aΔ2 ct (x) + (λ + 2a)ct (x) a should be chosen to be negative. Then a has the interpretation of being −h2 /8π 2 m while λ + 2a has the interpretation of being V (x) the external potential field in which the electron moves. We can obtain the solution to the above equation using the Feynman path integral based on Poisson processes or equivalently, birth-death processes.

197

Quantum Antennas

13.3

A problem in optimal control

The state equations are dX(t) = AX(t)dt + Cu(t)dt + GdW (t) where X(t) ∈ Rn , A ∈ Rn×n , u(t) ∈ Rp , C ∈ Rn×p , G ∈ Rn×d , W (t) ∈ Rd with W (.) being vector valued standard Brownian motion. The control input u(t) is restricted to be of instantaneous feedback type, ie, of the form u(t) = χt (X(t)) where χt : Rn → Rp is a non-random function. The aim is to determine this control input over the time range [0, T ] so that 

T

(1/2)E 0

(X(t)T Q1 X(t) + u(t)T Q2 u(t))dt

is a minimum. We already know from the stochastic Bellman-Hamilton-Jacobi dynamic (SBHJ) programming theory that if we define  V (t, X(t)) = minu(s),s∈[t,T ] E[

T

(X(s)T Q1 X(s) + u(s)T Q2 u(s))ds|X(t)] t

then V (t, x) satisfies the SBHJ equation V,t (t, x) + minu (Kt (u)V (t, x) + (1/2)(xT Q1 x + uT Q2 u)) where Kt (u) is the generator of the Markov process X(t). It is given by Kt (u) = (Ax + Cu)T ∇x + (1/2)T r(GGT ∇x ∇Tx ) Thus, our SBHJ equation is V,t (t, x) + minu ((xT AT ∇x V (t, x) + uT C T ∇x V (t, x) +(1/2)(xT Q1 x + uT Q2 u) + (1/2)T r(GGT ∇x ∇Tx V (t, x))) The minimization is easily carried out and it gives the optimal value of u = χt (x) as T u = −Q−1 2 C ∇x V (t, x) = χ(x) Substituting, we get the SBHJ equation in the form T T V,t (t, x)+xT AT ∇x V (t, x)+(1/2)(−∇x V (t, x))T CQ−1 2 C ∇x V (t, x)+x Q1 x

+T r(GGT ∇x ∇Tx V (t, x))) = 0 We rearrange this equation so that it comprises of three parts. The first part is linear in V and does not involve noise terms, the second part is nonlinear in V and again does not involve noise terms and finally, the third part is linear in V but involves noise terms. The nonlinear part is assumed to be of O(δ) and

198

Quantum Antennas

the noise part is assumed to be of of O(δ 2 ), where δ is a small perturbation parameter: T (V,t + xT AT ∇x V (t, x) + xT Q1 x/2) − (δ/2)(∇x V (t, x))T CQ−1 2 C ∇x V (t, x)

+(δ 2 /2)T r(GGT ∇x ∇Tx V (t, x)) = 0 We solve this equation approximately upto O(δ 2 ) using perturbation theory: V (t, x) = V0 (t, x) + δ.V1 (t, x) + δ 2 .V2 (t, x) + O(δ 3 ) Substituting this and equating coefficients of δ m , m = 0, 1, 2 successively gives us V0,t (t, x) + xT AT ∇x V0 (t, x) + xT Q1 x/2 = 0 T V1,t (t, x) + xT AT ∇x V1 (t, x) = (∇x V0 (t, x))T CQ−1 2 C ∇x V0 (t, x), T V2,t (t, x)+xT AT ∇x V2 (t, x) = ∇x V1 (t, x)T CQ−1 2 C ∇x V0 (t, x)

−T r(GGT ∇x ∇Tx V0 (t, x)) It is clear that the boundary condition V (T, x) = 0 gives us V0 (T, x) = V1 (T, x) = V2 (T, x) = 0 and hence,



T

V0 (t, x) = −

exp((s − t)xT AT ∇x )xT Q1 xds/2 t



T

V1 (t, x) = − t



T

V2 (t, x) = − t

T exp((s − t)xT AT ∇x )(∇x V0 (s, x)T CQ−1 2 C ∇x V0 (s, x)ds

T exp((s−t)xT AT ∇x )(∇x V1 (s, x)T CQ−1 2 C ∇x V0 (s, x)

−T r(GGT ∇x ∇Tx V0 (s, x)))ds

13.4

Interaction between photons and gravitons

The Einstein tensor is Gμν = Rμν − (1/2)Rg μν We write

Gμν = G(1)μν + G(2)μν

where G(1)μν is linear in the metric perturbations hμν (x) and G(2)μν consists of quadratic and higher order terms in the metric perturbations. It is easy to see that =0 G(1)μν ,ν and therefore, the Einstein field equations Gμν = −8πGT μν

199

Quantum Antennas which can also be expressed as G(1)μν ) = −8πG(T μν − G(2)μν /8πG) imply that

(T μν − G(2)μν /8πG),ν = 0

and this is a conservation law. Since T μν is the energy-momentum tensor of the matter and radiation field, we can thus interpret τ μν = −G(2)μν /8πG as the energy-momentum pseudo-tensor of the gravitational field. In what follows we first show that = 0, G(1)μν ,ν then calculate τ μν = −G(2)μν /8πG upto quadratic orders in the hμν and their partial derivatives, then we evaluate the energy of the gravitational field, namely,  HG = τ 00 d3 r in terms of the gravitational field creation and annihilation operators d(K, σ)∗ , d(K, σ) upto second order where in view of the plane wave expansion

 hμν (x) =

[eμν (K, σ)d(K, σ)exp(−ik.x) + e¯μν (K, σ)d(K, σ)∗ exp(ik.x)]d3 K

The form of HG is given by  HG = C(K, σ, σ  )d(K, σ)d∗ (K, σ  )d3 K We then evaluate the interaction Hamiltonian HI (t) between the electromangnetic field and the gravitational field using the energy-momentum tensor of the electromagnetic field S μν = (−1/4)Fαβ F αβ g μν + F μα Fαν Then Interaction energy of the gravitational field with the em field is therefore given by the spatial integral of the (00)th component of S μν that contains terms linear in hμν . It is given by  √ [(−1/4)Fμν Fαβ δ(g muα g νβ −gg 00 ) √ +δ(g 0μ g αβ g 0ρ −g)Fμβ F0α ]d3 r In this expression we note that gμν = ημν + δgμν (x), δgμν (x) = hμν (x) αβ hα = η αμ η βν hμν , μ = ηαβ hβμ , h

200

Quantum Antennas g μν = ημν − ημα ηνβ hαβ + O(h2 ) = ημν − hμν + O(h2 ), √ g = −(1 + h), −g = 1 + h/2, h = hμμ = ημν hμν

where O(h2 ) terms have been neglected. Thus, δg μν = −η μα η νβ hαβ = −hμν The free gravitational field in the absence of gravitational interactions, satisfies the wave equation hμν (x) = 0 provided that we retain only terms linear in the hμν in the Einstein field equations and further assume harmonic coordinates, ie, hμν,μ − h,ν /2 = 0 The solution for the free gravitational field is thus expandable as above as a superposition of plane waves with the coefficient functions eμν (K, σ) satisfying the coordinate condition eμν kμ − eα α kν /2 = 0 These are 4 constraints on the ten coefficients eμν and hence, we have just six degrees of freedom. Actually, these reduce further to five if we ignore an arbitrary scaling factor. That is why a graviton is a spin two particle (l = 2 implies 2l + 1 = 5). Now the energy of the gravitational field is computed as above. Likewise, the interaction energy between the electromagnetic field and the gravitational field upto linear orders in the hμν can be obtained from the plane wave expansion for the free gravitational field discussed above and the plane wave expansion of the free electromagnetic field: Aμ (x) = 0 gives  Aμ (x) =

(eμ (K, s)a(K, s)exp(−ik.x) + e¯μ (K, s)a(K, s)∗ exp(ik.x))d3 K

where s = 1, 2, ie, there are only two degrees of freedom for the polarization of the em field. The first comes from the Lorentz gauge condition Aμ,μ = 0 which results in eμ (K, s)kμ = 0 and the second from the fact that some part of the electromagnetic field is a matter field. This can be seen more clearly from the Coulomb gauge divA = Ar,r = 0 which results in ∇2 A0 = −μJ 0 implying that A0 is a pure matter field.

201

Quantum Antennas

Note that since both the weak gravitational field and the em field satisfy the wave equation, it follows that both photons and gravitons travel at the speed of light, ie k 0 = |K| for both of them. The interaction Hamiltonian between the gravitational field and the electromagnetic field is therefore approximately given by an expression of the form  HGEM (t) = [C1 (K, K  , σ, σ  , s))a(−K − K  , s)d(K, σ)d(K  , σ  )+ C2 (K, K  , σ, σ  , s)a(K − K  , s)d(K, σ)∗ d(K  , σ  ) +C3 (K, K  , σσ  , s)a(K + K  , s)∗ d(K, σ)d(K  , σ  ) +C4 (K, K  , σ, σ  , s)a(K − K  , s)∗ d(K, σ)d(K, σ)∗ d(K  , σ)]d3 K +c.c. where c.c. denotes the complex adjoint of the previous terms. The commutation relations are the usual Bosonic relations: [a(K, s), a(K  , s )∗ ] = δ 3 (K − K  )δs,s , [a(K, s), a(K  , s )] = 0, [d(K, σ), d(K  , σ  )∗ ] = δ 3 (K − K  )δσ,σ [d(K, σ), d(K  , σ  )] = 0, [a(K, s), d(K  , σ  )] = 0, [a(K, s), d(K  , σ  )∗ ] = 0 Computation of G(1)μν and G(2)μν : R(1)μν = (g μα g νβ Rαβ )(1) (1)

= (ημα − hμα )(ηνβ − hνβ )Rαβ (1)

= ημα ηνβ Rαβ

R(1) = (g μν Rμν )(1) = (ημν − hμν )Rμν )(1) (1) = ημν Rμν ,

Thus,

(1)

(1)

G(1)μν = ημα ηνβ Rαβ − (ηαβ Rαβ )ημν G(2)μν = R(2)μν − (1/2)(Rg μν )(2) (1)

(2)

(2) Rμν = [(ημα + hμα )(ηνβ + hνβ )(Rαβ + Rαβ )](2) (2)

(1)

= ημα ηνβ Rαβ + ημα hνβ Rαβ (1)

+ηνβ hμα Rαβ

202

Quantum Antennas (Rg μν )(2) = −R(1) hμν + R(2) ημν , (1) R(1) = ημν Rμν , (2) (1) − hμν Rμν R(2) = (g μν Rμν )(2) = ημν Rμν (1)μν

We now evaluate G(1)μν and show that G,ν

= 0. First observe that

(1) α (1) Rμν = (Γα = μα,ν − Γμν,α ) αβ Γβμν )(1) = (g αβ Γβμα )(1) ,ν − (g ,α

= (1/2)ηαβ (hβμ,αν + hβα,μν − hμα,βν −hβμ,να − hβν,μα + hμν,αβ ) ,α = (1/2)(h,μν + hμν − h,α μα,ν − hνα,μ )

It follows that (1) R(1) = ημν Rμν =

h − h,αβ αβ and hence, (1) (1) G(1) ημν μν = Rμν − (1/2)R ,αβ ,α = (1/2)(h,μν + hμν − h,α μα,ν − hνα,μ ) − ημν h + hαβ ημν )

It follows that G(1)μν = (1/2)(h,μν + hμν − hμα,ν − hνα,μ ,α ,α μν −η μν h + h,αβ αβ η )

from which, we deduce that G(1)μν = ,ν αβ,μ μα να,μ ,μ (1/2)(h,μ + hμν ,ν − h,α − h,να − h + h,αβ ) = 0

203

Quantum Antennas

13.5

A version of quantum optimal control

Abstract: The state equations for the evolution of a quantum system observable in the presence of bath noise is modeled using noisy Heisenberg dynamics defined by a unitary evolution operator in system⊗ bath which satisfies the HudsonParthasarathy noisy Schrodinger equation with the noise processes being families of observables in the bath Boson Fock space. These noise processes are the creation, annihilation and conservation operators that exhibit in some special cases statistical properties like classical Brownian motion and Poisson processes while in the general case, these processes are non-commutative and therefore have no analogue in classical stochastic processes. In fact, these processes do not in general have any joint probability distribution because their values at two different times generally do not commute. Furthermore, the Heisenberg uncertainty principle ensures that general non-commutative measurements are not only impossible but even commutative measurements in general may not commute with the future values of the states. Belavkin thereofore constructed a family of non-demolition measurements which form an Abelian family and which also commute with the future values of the state. Now, Belavkin constructed a quantum filter which provides real time estimates of either noisy Heisenberg states or equivalently of noisy Schrodinger states coming from the HudsonParthasarathy Schrodinger equation. based on non-demolition measurements. These estimates are functions of the output non-demolition measurements and hence are commutative. Owing to the commutativity of all the variables in the Belavkin filter, this filter is also known as a stochastic Schrodinger equation. The Belavkin filter is a non-commutative generalization of the classical Kushner filter in the sense that if the system Hilbert space is L2 (Rn ), the system observable whose evolution is to be studied is multiplication by some function f (x) in L2 (Rn ) and X(t) is a classical Markov process so that we can define the homomorphism jt (f ) = f (X(t)), then based on noisy measurements dz(t) = ht (X(t))dt + σv dV (t) we define the conditional expectation πt (f ) = E(f (X(t))|ηo (t)), ηo (t) = σ(z(s), s ≤ t) then if Kt denotes the generator of X(t), we obtain the classical KushnerKallianpur filter

dπt (f ) = πt (Kt f )dt + σv−2 (πt (ht f ) − πt (ht )πt (f ))(dz(t) − πt (ht )dt) which can also be derived from the Belavkin filter by replacing the non-commutative operators appearing in it with multiplication operators by functions. Now we can formulate a control problem by including in the Hudson-Parthasarathy equation, polynomial functions of the input measurement process Yi (t) (which are superpositions of the fundamental noise processes) with coefficients being operators in the system Hilbert space. These functions may be chosen

204

Quantum Antennas

arbitrarily with the only constraint that the HP evolution operator should be unitary at all times. It should be noted that the system operators commute with the input measurements but not with the output measurements Yo (t) = jt (Yi (t)) = U (t)∗ Yi (t)U (t). Corresponding to this revised version of the HP equation, the unitary evolution gives rise to Heisenberg dynamics for the evolving state jt (X) similar to the Evans-Hudson flow but with functions of the output measurements appearing as coefficients. These functions are the control functions. When one formulates the Belavkin filter for such processes using the reference probability approach of Gough et.al, then one ends up with the Belavkin filter having the form  dπt (X) = Ft (X, u(t))dt + Gkt (X, u(t))(dYo (t))k k≥1

This is a direct consequence of the revised Evans-Hudson flow assuming the form djt (X) = θba (u(t), jt (F ), jt (X))dΛba (t) where F denotes the set of all the system observables appearing in the HudsonParthasarathy equation and θba are the structure maps. Here, u(t) is of the form χt (Yo (t)) ∈ ηo (t) and it is then an easy problem to derive the quantum stochastic Bellman-Hamilton-Jacobi equation for the optimal control χt (.) that would minimize a cost function of the form  E

T 0

L(jt (X), u(t))dt

with the expectation being taken in any initial state of the system⊗ bath. The method for carrying out the minimization is based on the Hamilton-Jacobi function  T

V (t, πt ) = minu(s),t≤s≤T E[

L(js (X), u(s))ds|ηo (t)] t

Problem formulation The process to be controlled is jt which satisfies the qsde djt (X) = jt (θba (X))dΛba (t) where X is in B(h), h being the system Hilbert space and θba : B(h) → B(h) are linear operators which are called structure maps. They satisfy certain relations that guarantee that jt is a ∗-unital homomorphism. Λab (t), a, b ≥ 0 are the fundamental processes of Hudson and Parthasarathy. They satisfy the quantum Ito’s formula: dΛab Λcd = ad dΛcb where ad is zero if either a or d is zero and δda otherwise. The structure maps θba are assumed to depend on a control input u(t) which is restricted to be a

205

Quantum Antennas

function of jt only. More precisely, we choose a basis {Za < a = 1, 2, ...} for B(h) and then any jt (X) is a complex linear combination of jt (Za ), a = 1, 2, ..., so we can regard the homomorphism jt as being equivalent to the family of operators jt (Za ), a = 1, 2, .... choose a basis {ηk } for the system Hilbert space h and an approximate basis {ξr } for the Boson Fock space such that ξr = s c(r, s)|e(us ) > where |e(us ) >, s = 1, 2, ... are exponential vectors in the Boson Fock space. Then we can represent the operator jt (Za ) by the matrix elements < ηk ⊗ ξr |jt (Za )|ηl ⊗ ξs >= Jt (a, k, r, l, s). Now let ρ be a state in the system⊗ bath space, ie, is h ⊗ Γs (H) where H = L( R+ ) ⊗ Cd . Then let Xd (t) be process in this space to be tracked. The θba being dependent on u(t) can be expressed as θba (X) = θba (u(t), X) Then after applying this control input u(t) = F (t, jt ) = F (t, jt (Za ), a = 1, 2, ...) − − − (1) our qsde in the sense of Hudson-Parthasarathy and Evans-Hudson, can be expressed as djt (X) = jt (θba (u(t), X))dΛba (t) − − − (2) The quantum Markovianity is still preserved after applying such a state dependent control. The control input u(t), 0 ≤ t ≤ T is to be selected so that it minimizes the cost function  T L(jt (X), u(t))dt) C(u) 0

where for example we may take L as L(jt (X), u(t)) = T r(ρ.(Xd (t) − jt (X))2 ) where ρ is a state in h ⊗ Γs (H) and Xd (t) is an operator valued process in h ⊗ Γs (H) to be tracked. As in the classical BHJ theory, we introduce the energy function V (t, jt ) = V (t, jt (Za ), a = 1, 2, ...) = V (t, Jt (a, k, r, l, s), a, k, r, l, s = 1, 2, ...)  T L(js (X), u(s))ds = mint≤u(s)≤T s

where jt (X) satisfies the above qsde (2) with u(t) allowed to be of the form (1) only, ie, instantaneous state feedback. Then as in the classical BHJ theory of optimal control, we easily derive the equation V,t (t, jt )) + minu(t) (L(jt (X), u(t)) + (V (t, jt + djt ) − V (t, jt ))/dt) = 0 Now, we can write V (t, jt + djt ) − V (t, jt ) =

∂V (t, jt ) .djt ∂jt

206

Quantum Antennas =

∂V (t, Jt (a, k, r, l, s)) dJt (a, k, r, l, s) ∂Jt (a, k, r, l, s)

where the summation over repeated indices is assumed. We note that dJt (a, k, r, l, s) =< ηk ⊗ ξr |djt (Za )|ηl ⊗ ξs >= =< ηk ⊗ ξr |jt (θqp (u(t), Za ))dΛqp (t)|ηl ⊗ ξs >= umq (t)unp (t)dt =< ηk ⊗ ξr |jt (θqp (u(t), Za ))|ηl ⊗ ξs > c¯(r, m)c(s, n)¯ We can write θqp (u(t), Za ) =



A(u(t), p, q, a, b)Zb

b

where if u(t) is a scalar function, then A(u(t), p, q, b) are complex numbers and if u(t) is a function of jt (Zc ), c = 1, 2, ..., then A(u(t), p, q, b) becomes an operator in h ⊗ Γs (H). We then have by the homomorphism property of jt that jt (θqp (u(t), Za )) =



A(u(t), p, q, a, b)jt (Zb )

b

So we get umq (t)unp (t)A(u(t), p, q, a, b)Jt (b, k, r, l, s) dJt (a, k, r, l, s)/dt = c¯(r, m)c(s, n)¯ and so our quantum BHJ equation assumes the form V,t (t, Jt )+ minu(t) (L(jt (X), u(t))+

∂V (t, Jt ) c¯(r, m)c(s, n)¯ umq (t)unp (t)A(u(t), p, q, a, b)Jt (b, k, r, l, s) ∂Jt (a, k, r, l, s)

=0 In these expressions, Jt corresponds to the set of numbers {Jt (a, k, r, l, s)} and writing  d(a)Za X= a

we can express this quantum stochastic BHJ equation as V,t (t, Jt )+ minu(t) (L(

 a

d(a)Jt (a, .), u(t))+

∂V (t, Jt ) c¯(r, m)c(s, n)¯ umq (t)unp (t)A(u(t), p, q, a, b)Jt (b, k, r, l, s)) ∂Jt (a, k, r, l, s)

=0 In this expression, the solution for u(t) on minimization comes out to be a function of jt or equivalently, of the numbers {Jt (a, k, r, l, s)}

207

Quantum Antennas

A more practically implementable approach to the quantum control problem is first to estimate the Belavkin filter estimate of the state jt (X) at time t based on non-demolition measurements Yo (s) = U (s)∗ Yi (s)U (s), s ≤ t as πt (X) = E(jt (X)|ηo (t)], ηo (t) = σ(Yo (s), s ≤ t) and then choose our control input u(t) in the form u(t) = χt (πt ) where by πt , we mean the family of operators πt (Za ), a = 1, 2, .. where Za , a = 1, 2, ... forms a basis for B(h), We choose the function χt so that  E



T 0

L(t, jt (X), u(t))dt = E

0

T

L(t, jt (X), χt (πt ))dt

is a minimum. The above expectation may for example be taken when the system and bath are in the state |f ⊗ φ(u) >, where |f >∈ h, < f |f >= 1 and |φ(u) >= exp(−  u 2 /2)|e(u) >, u ∈ H is a coherent state of the bath. As in the classical BHJ equation, we therefore seek to minimize  T L(s, js (X), u(s))ds|ηo (t)] E[ t

w.r.t u(s), t ≤ s ≤ T . This minimum will be of the form V (t, πt ) = V (t, πt (Za ), a = 1, 2, ...) and as usual, we have V (t, πt ) = minu(t)=χt (πt ) (E[L(t, jt (X), u(t))|ηo (t)]dt + E(V (t + dt, πt+dt )|ηo (t))) The Belavkin filter when the structure map θba depend on the control input u(t) is given by dπt (X) = Ft (X, u(t))dt + Gt (X, u(t))dYo (t) where the control input u(t) = χt (πt ). Everything is commutative in this Belavkin equation. Thus, V (t + dt, πt+dt ) = V (t, πt ) + V,t (t, πt )dt + T r( +T r(

∂V (t, πt ) dπt (Za )) ∂πt (Za )

∂ 2 V (t, πt ) dπt (Za )dπt (Zb )) ∂πt (Za )∂πt (Zb )

with summation over the repeated index a being implied assuming quadrature noise. This leads to the quantum stochastic BHJ equation V,t (t, πt ) + minu(t)=χt (πt ) (πt (L(t, X, u(t)) + dt−1 .T r(

∂V (t, πt ) E[dπt (Za )|ηo (t)]) ∂πt (Za )

208

Quantum Antennas +dt−1 .T r(

∂ 2 V (t, πt ) E[dπt (Za )dπt (Zb )|ηo (t)]) = 0 ∂πt (Za )∂πt (Zb )

Remark: We have used the following identities. Writing  Lk (t, χt (πt ))jt (X)k L(t, jt (X), u(t)) = L(t, jt (X), χt (πt )) = k≥0

(Note that [jt (X), χt (πt )] = 0since[jt (X), πt (Za )] = 0∀a)   = Lk (t, χt (πt ))πt (X k ) = πt ( Lk (t, χt (πt ))X k ) k≥0

k≥0

= πt (L(t, X, χt (πt ))) = πt (L(t, X, u(t))) Since all the operators πt (Za ), a = 1, 2, ... are commutative, whilst carrying out the above minimization, we may assume that these operators are all real numbers. We also note that in the case of quadrature noise, dYo (t) = dYi (t) + dU (t)∗ dYi (t)U (t) + U (t)∗ dYi (t)dU (t) = dYi (t) + jt (L2 + L∗2 )dt so that ¯(t) + πt (L2 + L∗2 )) dt−1 E[dπt (Z)|ηo (t)] = Ft (X) + Gt (X)(u(t) + u Further,

13.6

dt−1 E[dπt (Za )dπt (Zb )|ηo (t)] = Gt (Za )Gt (Zb )

A neater formulation of the quantum optimal control problem

Let L1 , L2 , S, H, P be functions of the input measurement Yi (t) at time t where we take Yi (t) = c1 A(t) + c¯1 A(t)∗ + c2 Λ(t). We assume that these are polynomial functions of Yi (t) with coefficients being operators in the system Hilbert space h and these functions have been chosen so that U (t) is unitary for all t ≥ 0 where U (t) satisfies the qsde dU (t) = (−(iH + P )dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t), U (0) = I Note that the family of operators Yi (.) commutes with B(h), so we can write H = F1 (Hk , k = 1, 2, ..., p, χt (Yi (t))), P = F2 (Pk , k = 1, 2, ..., p, χt (Yi (t))), L1 = F3 (L1k , k = 1, 2, ..., p, χt (Yi (t))), L2 = F4 (L2k , k = 1, 2, ..., p, χt (Yi (t))),

209

Quantum Antennas S = F5 (S1 , ..., Sp , χt (Yi (t)))

where Hk , Pk , L1k , L2k , Sk are all system space operators, ie, operator in h, we may assume that they are in B(h) and the control input at time t is u(t) = χt (Yo (t)) where χt is an ordinary function of a real variable. Here, Yo (t) = U (t)∗ Yi (t)U (t) is the output measurement process. Clearly, Yo (t) = U (t)∗ Yi (t)U (t) = U (T )∗ Yi (t)U (T ), T ≥ t which follows by taking the differential of the rhs w.r.t. T and using the fact that the unitarity of U (T ) depends only on the operators Hk , Pk , L1k , L2k , Sk , Yi (T ) all of which commute with Yi (t). Defining for any system⊗ bath observable X, jt (X) = −U (t)∗ XU (t) We get that if X is a system observable, djt (X) = dU (t)∗ XU (t) + U (t)∗ XdU (t) + dU (t)∗ XdU (t) = jt (θ0 (w(t), X))dt+jt (θ1 (w(t), X))dA(t)+jt (θ2 (w(t), X))dA(t)∗ +jt (θ3 (w(t), X))dΛ(t)

where w(t) = χt (Yi (t)) and θk (w(t), .) are linear maps that take system observables to system ⊗ bath observables. These maps are functions of w(t) and the system operators L1k , L2k , Sk , Hk , Pk , k = 1, 2, ..., p. We denote the set of system operators Hk , Pk , Sk , L1k , L2k , k = 1, 2, ..., p by F . Then, to be precise, we must write the above qsde as

djt (X) = jt (θ0 (w(t), F, X))dt+ jt (θ1 (w(t), F, X))dA(t) + jt (θ2 (w(t), F, X))dA(t)∗ + jt (θ3 (w(t), F, X))dΛ(t) It is easy to see using the unitarity of U (t) that jt (θk (w(t), X)) = θk (u(t), jt (X)), u(t) = jt (w(t)) = χt (Yo (t)) So, we get the qsde djt (X) = θ0 (u(t), jt (F ), jt (X))dt+θ1 (u(t), jt (F ), jt (X))dA(t)+θ2 (u(t), jt (F ), jt (X))dA(t)∗

+θ3 (u(t), jt (F ), jt (X))dΛ(t) We also note the fact that jt (X) and jt ∗ (F ) both commute with u(t) and that E[θk (u(t), jt (F ), jt (X))|ηo (t)] = πt (θk (u(t), F, X)) which is easily seen by using the above stated commutativity and the fact that jt is a homomorphism. It should be noted that if f (u(t)) is any function of u(t) ∈ ηo (t) and X is a system operator, then by πt (f (u(t))X), we mean

210

Quantum Antennas

f (u(t))πt (X) = f (u(t))E[jt (X)|ηo (t)] and not E[jt (f (u(t))X)|ηo (t)]. Now, we are in a position to formulate and solve the optimal control problem. We define  T V (t, πt ) = minu(s),t≤s≤T E[ L(js (X), u(s))ds|ηo (t)] t

where u(s) = χs (Yo (s)) = js (χs (Yi (s))) = U (s)∗ χs (Yi (s))U (s) = χs (U (s)∗ Yi (s)U (s)) Then, we get V,t (t, πt ) + minu(t) (πt (L(X, u(t)))  ∂ n V(t, πt ) T r( E[(dπt )⊗n |ηo (t)]) +dt−1 ⊗n ∂π t n≥1 =0 πt⊗n ,

where by we mean a lexicographically ordered set of the elements πt (Za1 )...πt (Zan ) with a1 , ..., an = 1, 2, .... The above conditional expectation is easily computed using the Belavkin filtering equations

dπt (X) = Ft (X, u(t))dt +



Gt,k (X, u(t))(dYo (t))k

k≥1

with the functions Ft , Gk,t derived in the usual way using the reference probability method.

13.7

Calculating the approximate shift in the oscillation frequency of a cavity resonator having arbitrary cross section when the medium has a small inhomogeneity

(ω, x, y, z), μ(ω, x, y, z) are the permittivity and permeability. They can be expressed as (ω, r) = 0 (1 + δχe (ω, r)), μ(ω, r) = μ0 (1 + δχm (ω, r)), r = (x, y, z) By virtue of the boundary conditions on a conducting surface and the Maxwell equations, we have the fact that Hz vanishes when z = 0, d, Ez,z vanishes when z = 0, d, E⊥ vanishes when z = 0, d and hence, these fields can be expanded as  Hz (ω, x, y, z) = Hzp (ω, x, y)sin(pπz/d), p

Ez (ω, x, y, z) =

 p

Ezp (ω, x, y)cos(pπz/d),

211

Quantum Antennas E⊥ (ω, x, y, z) =



E⊥p (ω, x, y)sin(pπz/d)

p

Further, the Maxwell curl equations give us Ez,y − Ey,z = −jωμHx , Ex,z − Ez,x = −jωμHy , Ey,x − Ex,y = −jωμHz , Hz,y − Hy,z = jωEx , Hx,z − Hz,x = jωEy , Hy,x − Hx,y = jωEz Combining these with the above boundary conditions implies that H⊥,z vanishes when z = 0, d and hence we have the expansion  H⊥ (ω, x, y, z) = H⊥,p (ω, x, y)cos(pπz/d) p

Substituting these expansions into the Maxwell curl equations expressed in the form ∇⊥ Ez × zˆ + zˆ × E⊥,z = −jωμH⊥ , ∇⊥ × E⊥ = −jωμHz zˆ, ∇⊥ × H⊥ = jωEz zˆ we get   ∇⊥ Ezp (ω, x, y) × zˆ.cos(pπz/d) + (πp/d)ˆ z × E⊥,p (ω, x, y)cos(pπz/d) p

p

−jωμ(ω, r) 



H⊥,p (ω, x, y)cos(pπz/d)

p

∇⊥ Hzp (ω, x, y) × zˆ.sin(pπz/d) −

p

jω(ω, r)





(πp/d)ˆ z × H⊥,p (ω, x, y)sin(pπz/d)

p

E⊥,p (ω, x, y)sin(pπz/d)

p

Multiplying the first equation by (2/d)cos(mπz/d), the second equation by (2/d)sin(mπz/d) and integrating w.r.t. z over [0, d] gives us z × E⊥,m (ω, x, y) ∇⊥ Ezm (ω, x, y) × zˆ + (πm/d)ˆ = −jω

 (

d 0

p

μ(ω, x, y, z)(2/d)cos(pπz/d)cos(mπz/d)dz)H⊥,p (ω, x, y)

and likewise, ∇⊥ Hzm (ω, x, y) × zˆ − (πm/d)ˆ z × H⊥,m (ω, x, y)  ( = jω p

d 0

(ω, x, y, z)(2/d)sin(pπz/d)sin(mπz/d)dz)E⊥,p (ω, x, y)

212

Quantum Antennas

Finally, the z component of the Maxwell curl equations give   ∇⊥ × E⊥,p (ω, x, y)sin(pπz/d) = −jωμ(ω, r) Hz,p (ω, x, y)sin(pπz/d)ˆ z, p

p



∇⊥ × H⊥,p (ω, x, y)cos(pπz/d) = jω(ω, r)



p

Ez,p (ω, x, y)cos(pπz/d)ˆ z,

p

which given in the same manner, ∇⊥ ×E⊥,m (ω, x, y)  ( = −jω p

d 0

∇⊥ ×H⊥,m (ω, x, y)  = jω (

d 0

p

μ(ω, x, y, z)(2/d)sin(pπz/d)sin(mπz/d)dz)Hz,p (ω, x, y)ˆ z,

(ω, x, y, z)(2/d)cos(pπz/d)cos(mπz/d)dz)Ez,p (ω, x, y)]ˆ z,

So far, everything is exact. No approximations have been made. Writing these equations in perturbation theoretic form gives us ∇⊥ Ezm (ω, x, y) × zˆ + (πm/d)ˆ z × E⊥,m (ω, x, y)  = −jωμ0 ( p

+jωμ0 H⊥,m (ω, x, y) d 0

δχm (ω, x, y, z)(2/d)cos(pπz/d)cos(mπz/d)dz)H⊥,p (ω, x, y),

∇⊥ Hzm (ω, x, y) × zˆ − (πm/d)ˆ z × H⊥,m (ω, x, y)  = jω0 ( p

 −jωμ0 ( p

−jω0 E⊥,m (ω, x, y) d

∇⊥ × E⊥,m (ω, x, y) + jωμ0 Hz,m (ω, x, y) = d

δχm (ω, x, y, z)(2/d)sin(pπz/d)sin(mπz/d)dz)Hz,p (ω, x, y)ˆ z,

0

 = jω0 ( p

δχe (ω, x, y, z)(2/d)sin(pπz/d)sin(mπz/d)dz)E⊥,p (ω, x, y),

0

∇⊥ × H⊥,m (ω, x, y) − jω0 Ez,m (ω, x, y) d 0

χe (ω, x, y, z)(2/d)cos(pπz/d)cos(mπz/d)dz)Ez,p (ω, x, y)ˆ z,

These equations can be expressed as z × E⊥,m (ω, x, y) ∇⊥ Ezm (ω, x, y) × zˆ + (πm/d)ˆ

=

 p

+jωμ0 H⊥,m (ω, x, y) δF1 (ω, x, y, m, p)H⊥,p (ω, x, y),

213

Quantum Antennas ∇⊥ Hzm (ω, x, y) × zˆ − (πm/d)ˆ z × H⊥,m (ω, x, y)

=



−jω0 E⊥,m (ω, x, y) δF2 (ω, x, y, m, p)E⊥,p (ω, x, y),

p

∇⊥ × E⊥,m (ω, x, y) + jωμ0 Hz,m (ω, x, y) =  δG1 (ω, x, y, m, p)Hz,p (ω, x, y)ˆ z, p

∇⊥ × H⊥,m (ω, x, y) − jω0 Ez,m (ω, x, y)  = δG2 (ω, x, y, m, p)Ez,p (ω, x, y)ˆ z, p

Let us write the approximate solutions to these equations as (0) (1) (0) (1) + Em , Hm = Hm + Hm , Em = Em

or equivalently, (0) (1) (0) (1) + Ez,m , Hz,m = Hz,m + Hz,m , Ez,m = Ez,m (0)

(1)

(0)

(1)

E⊥,m = E⊥,m + E⊥,m , H⊥,m = H⊥,m + H⊥,m and also assume that the characteristic frequency of oscillation gets perturbed from ω to ω + δω. Then applying perturbation theory, we get the zeroth order perturbation equations as (0)

(0) × zˆ + (πm/d)ˆ z × E⊥,m ∇⊥ Ezm (0)

+jωμ0 H⊥,m = 0, (0)

(0) × zˆ − (πm/d)ˆ z × H⊥,m ∇⊥ Hzm (0)

−jω0 E⊥,m = 0 (0)

(0) zˆ = 0 ∇⊥ × E⊥,m + jωμ0 Hz,m (0)

(0) zˆ ∇⊥ × H⊥,m − jω0 Ez,m

and the first order perturbation equations as (1)

(1) ∇⊥ Ezm × zˆ + (πm/d)ˆ z × E⊥,m (1)

(0)

+jωμ0 H⊥,m + jμ0 δωH⊥,m  (0) = δF1 (ω, x, y, m, p)H⊥,p , p (1)

(1) × zˆ − (πm/d)ˆ z × H⊥,m ∇⊥ Hzm

214

Quantum Antennas (1)

(0)

−jω0 E⊥,m − j0 δωE⊥,m  (0) = δF2 (ω, x, y, m, p)E⊥,p , p (1)

(1) (0) ∇⊥ × E⊥,m + jωμ0 Hz,m zˆ + jμ0 δωHz,m zˆ =  (0) δG1 (ω, x, y, m, p)Hz,p zˆ p (1)

(1) (0) ∇⊥ × H⊥,m − jω0 Ez,m zˆ − j0 δωEz,m zˆ  (0) = δG2 (ω, x, y, m, p)Ez,p zˆ, p

When the zeroth order equations are solved subject to the above mentioned boundary conditions, we get as in standard cavity resonator analysis, we get a discrete set of frequency values for ω, say ω(m, n)(0) , n = 1, 2, ... for each value (0) (0) of m and correspondingly a normalized eigenvector for Em , Hm , we denote (0) (0) these eigenvectors by (Emn , Hmn ), n = 1, 2, .... Specifically, we can split these eigenvector into transverse and longitudinal components: (0)

(0)

(0) (0) (0) (0) Emn = E⊥,mn + Ezmn zˆ, Hmn = H⊥,mn + Hzmn zˆ

where all of these are functions of (x, y) only. There may also be a degeneracy of these eigenvectors. Specifically, for the unperturbed eigen-frequency (0) (0)T (0)T ω(m, n)(0) , we may have K(m, n) eigenvectors, say ψmnk = (Emnk , Hmnk )T , k = 1, 2, ..., K(m, n). We assume that these eigenvectors are all normalized in the standard sense, ie,  (0) (0) (0) (0) ψmnk (x, y)∗ ψmn k (x, y)dxdy = δmm δkk < ψmnk , ψmn k >= D

where D denotes the cross section of the guide. The first order transverse equations give (1) (1) E⊥,m = (−πm/dh(ω, m)2 )∇⊥ Ezm (1) −(jμ0 ω/h(ω, m)2 )∇⊥ Hzm × zˆ (0)

−(jμ0 πmδω/dh(ω, m)2 )ˆ z ×H⊥,m +(πm/dh(ω, m)2



(0)

δF1 (ω, x, y, m, p)ˆ z ×H⊥,p

p (0)

−(ωμ0 0 δω/h(ω, m)2 )E⊥,m  (0) +(jμ0 ω/h(ω, m)2 ) δF2 (ω, x, y, m, p)E⊥,p p

and

(1)

(1) H⊥,m = (πm/dh(ω, m)2 )∇⊥ Hzm (1) +(jμ0 ω/h(ω, m)2 )∇⊥ Ezm × zˆ

215

Quantum Antennas (0)

−(j0 πmδω/dh(ω, m)2 )ˆ z × E⊥,m − (πm/dh(ω, m)2



(0)

δF2 (ω, x, y, m, p)ˆ z × E⊥,p

p (0)

−(ωμ0 0 δω/h(ω, m)2 )H⊥,m  (0) −(j0 ω/h(ω, m)2 ) δF1 (ω, x, y, m, p)H⊥,p p

where

h(ω, m)2 = ω 2 μ0 0 − (πm/d)2

Thus, we get

(1)

(1) zˆ.∇⊥ × E⊥,m = (jμ0 ω/h(ω, m)2 )∇2⊥ Hzm (0)

−(jμ0 πmδω/dh(ω, m)2 )(∇⊥ .H⊥,m )+(πm/dh(ω, m)2 )



(0)

∇⊥ .(δF1 (ω, x, y, m, p)ˆ z ×H⊥,p (x, y))

p

(0)

2

−(μ0 0 ωδω/h(ω, m) )ˆ z .∇⊥ × E⊥,m  (0) +(jμ0 ω/h(ω, m)2 ) zˆ.∇⊥ × (δF2 (ω, x, y, m, p)E⊥,p (x, y) p (1) (0) = −jωμ0 Hz,m − jμ0 δωHz,m +



(0) δG1 (ω, x, y, m, p)Hz,p (x, y)

p

This equation can be rearranged and simplified using the identities (0)

(0) = 0, ∇⊥ .H⊥,m (x, y) + (mπ/d)Hz,m (0)

(0) zˆ.∇⊥ × E⊥,m = −jωμ0 Hz,m

as (0) +2δω.ωμ0 0 Hz,m +

 p

13.8

(1) (x, y) (∇2⊥ + h(ω, m)2 )Hz,m (0)

zˆ.∇⊥ ×(δF2 (ω, x, y, m, p)E⊥,p )+(jh(ω, m)2 /μ0 ω)



(0) δG1 (ω, x, y, m, p)Hz,p =0

p

Optimal control for partial differential equations

. Examples: [a] Controlling the electromagnetic field within a box over a given space-time range with a control current density sources so that the controlled em fields are close in distance to a given em field. [b] Controlling the energy-momentum tensor of matter and radiation in the Einstein field equations of general relativity so that the controlled metric is

216

Quantum Antennas

close in distance to a given metric over a given space-time range. Both using the approximate linearized Einstein field equations and the fully non-linear Einstein field equations. [c] Controlling the stirring force in a fluid so that the fluid velocity pattern matches a given pattern over a given space-time interval. [2] Study of gravitational waves produced by an electromagnetic source. Rμν − (1/2)Rgμν = −8πGSμν Sμν = (−1/4)Fαβ F αβ gμν + Fμα Fνβ This is the energy-momentum tensor of the radiation field. Assume that the em four potential is Aμ = A(0) μ + δAμ , gμν = gμν (0) + δgμν = ημν + hμν (x) (0)

where Aμ satisfies the flat space-time Maxwell equations, ie, if (0) + δFμν , Fμν = Fμν (0) (0) = A(0) Fμν ν,μ − Aμ,ν ,

δFμν = δAν,μ − δAμ,ν then (0)

(0)

(0) Fνβ hαβ δSμν = (−1/4)Fαβ F (0)αβ hμν + Fμα (0)

(0) + 2ηαβ (Fμα δFνβ + Fνβ δFμα )

This is of the general form δSμν (x) = C1 (μναβ, x)hαβ (x) + C2 (μναβ, x)δFαβ (x) where the functions C1 , C2 are completely decided by the unperturbed em wave (0) Fμν (x). We have already seen that the first order perturbation to the Einstein tensor Gμν = Rμν − (1/2)Rg μν has the form

G(1)μν = δGμν = δRμν − (1/2)δR.η μν = C3 (μναβρσ)hαβ,ρσ

whose ordinary four divergence vanishes, ie C3 (μναβρσ)hαβ,ρσν = 0 The first order perturbed Einstein-Maxwell field equations then give C3 (μναβρσ)hαβ,ρσ (x) = C1 (μναβ, x)hαβ (x) +C2 (μναβ, x)δFαβ (x)

217

Quantum Antennas The Maxwell equations are √ (F μν −g),ν = 0 which is to be combined with the gauge condition √ (Aμ −g),μ = 0 or equivalently,

√ (g μν −gAν ),μ = 0

The unperturbed, ie, flat space-time components of these equations are (0)

ημα ηνβ Fαβ,ν = 0, ημν A(0) ν,μ = 0 Substituting

(0) (0) = A(0) Fμν ν,μ − Aμ,ν

these lead to the flat-space-time wave equation α A(0) μ = 0,  = ∂α ∂ = ηαβ ∂α ∂β

The first order perturbed versions of the Maxwell equations and gauge condition are √ (0) (δ(g μα g νβ −g)Fαβ,ν ) +ημα ηνβ δFαβ,ν = 0 and

√ ημν δAν,μ + (δ(g μν −g)A(0) ν ),μ = 0

Chapter 14

Quantization of cavity fields with inhomogeneous media, field dependent media parameters from Boltzmann-Vlasov equations for a plasma, quantum Boltzmann equation for quantum radiation pattern computation, optimal control of classical fields, applications classical nonlinear filtering 14.1

Computing the shift in the characteristic frequencies of oscillation in a cavity resonator due to gravitational effects and the effect of non-uniformity in the medium

Assume first that the unperturbed frequencies are ω0 [p, n], n = 1, 2, ... for a given p ∈ Z+ . p decides the z-dependence of the fields. For example, the variation

219

220

Quantum Antennas (0)

(0)

of Hz with z is sin(πpz/d) while the variation of Ez with z is cos(πpz/d). (0) (0) The variation of E⊥ with z is sin(πpz/d) and the variation of H⊥ with z is (0) (0) cos(πpz/d). This ensures that Hz and E⊥ vanish when z = 0, d. Define h(ω, p)2 = ω 2 μ0 0 − (πp/d)2 Then, the first order perturbed equations for fields and characteristic frequencies are (∇2⊥ + h(ω0 [p, n], p)2 )[Ez(1) , Hz(1) ]T + (0) (0) T , Hz,p,n ] 2μ0 0 ω0 [p, n]δω[Ez,p,n

= [δF1 (p, n, x, y), δF2 (p, n, x, y)]T where [δF1 (p, n, x, y), δF2 (p, n, x, y)]T is of the form (0) (0) δL(Ez,p,n (x, y), Hz,p,n (x, y)]T )

with δL being a linear first order partial differential operator depending on the permittivity and permeability perturbations 0 δχe (ω, x, y), μ0 δχm (ω, x, y) where ω = ω0 [p, n]. We may assume that the unperturbed modes ψp,n (x, y) = (0) (0) [Ez,p,n (x, y), Hz,p,n (x, y)]T , p, n = 1, 2, ... for form a complete onb for L2 (D)2 where D is the cross sectional area of the resonator in the xy plane. This is because these are eigenfunctions of the Laplace operator ∇2⊥ which is Hermitian z when we apply joint Dirichlet and Neumann boundary conditions, ie Ez and ∂H ∂n vanish on the boundary ∂D. Thus, we infer that the shift in the characteristic frequency ω0 [p, n] δω is given by δω = δω[p, n] =  (−1/2μ0 0 ω0 [p, n])

¯ (0) (x, y).δF1 (p, n, x, y)+H ¯ (0) (x, y).δF2 (p, n, x, y))dxdy (E z,p,n z,p,n D

Further since the eigenfunctions ψp,n of ∇2⊥ with the stated boundary conditions are orthogonal, we obtain that its first order perturbation is given by δψp,n (x, y) =

 (p ,n )=(p,n)



 ψp ,n (x, y)

ψp ,n (x, y)∗ [δF1 (p, n, x, y), δF2 (p, n, x, y)]dxdy

D

Here, we are assuming non-degeneracy of the unperturbed modes. If we have a degeneracy of k(p, n) for each p, n, then it means that corresponding to the unperturbed characteristic frequency ω0 [p, n] or equivalently to the unperturbed modal eigenvalue h(ω0 [p, n], p)2 , we have an orthonormal basis of unperturbed eigenfunctions (0) (0) ψn,p,m (x, y) = [Ez,p,n,m (x, y), Hz,p,n,m (x, y)]T , m = 1, 2, ..., k(p, n)

221

Quantum Antennas

and by using the secular determinant theory in time independent perturbation theory developed for solving quantum mechanical problems, the perturbations to the frequency ω0 [p, n] are δω[p, n, m], m = 1, 2, ..., k(p, n), where these are solutions to the determinantal equations det(2μ0 0 ω0 [p, n]δωIk + ((< ψp,n,m , δLψp,n,m >))1≤m,m ≤k(p,n) ) = 0

14.2

Quantization of the field in a cavity resonator having non-uniform permittivity and permeability

We’ve derived the first order eigen-equations which determine the shift in the cavity resonator frequencies when there is a small perturbation in the uniformity of the medium. These equations were found to have the general form (∇2⊥ + h(p, n)2 )δψ(x, y) + a(p, n)δωψp,n (x, y) = δF (p, n, x, y) These equations could be derived from a variational principle using the action functional  Sp,n (δψ) = (1/2) (|∇⊥ δψ(x, y)|2 − h(p, n)2 |δψ(x, y)|2 )+  −a(p, n)

 T

(δψ(x, y) ψp,n (x, y))dxdy +

δψ(x, y)T δF (p, n, x, y)

We assume that the unperturbed system is classical but the perturbed system has quantum fluctuations δψ. Although this variational principle is suitable for applying the finite element method, it is not suitable for quantization since time is not explicitly involved. One approach to quantizing this would be to replace h(p, n)2 by ω 2 μ0 0 where ω = i∂/∂t but that would again land us up in difficulty since we cannot give any physical interpretation for δω in terms of time. So the only way out is start with the basic Maxwell equations with susceptibilities χe and χm defined as functions of omega as χe (iω, x, y), χm (iω, x, y) and replace iω by the operator ∂/∂t at the end of all calculations. Then we would end up with an equation for δψ(ω, x, y) as (∇2⊥ + ω 2 μ0 0 − π 2 p2 /d2 )δψ(ω, x, y) =  δF (ω, p, m, x, y)ψp,m (ω, x, y) m

which can be interpreted in the time domain as (∇2⊥ − μ0 0 ∂t2 − π 2 p2 /d2 )δψ(t, x, y) =

222

Quantum Antennas 

δF (∂t , p, m, x, y)ψp,m (t, x, y)

m

where the operators F (∂t , p, m, x, y) are built out of the operators χe (∂t , x, y, z) and χm (∂t , x, y, z). The above equation can be derived from a variational principle with the action

14.3

Problems in transmission lines and waveguides

[1] Calculate using perturbation theory for partial differential equations, the approximate changes in the line voltage and current when the distributed parameter R of the line gets perturbed by a small non-uniform term δR(z), ie, the line equations are v,z (t, z) + (R + δR(z))i(t, z) + Li,t (t, z) = 0, i,z (t, z) + Gv(t, z) + Cv,t (t, z) = 0 Quantize these line equations by first removing the dissipative terms involving R + δR and G, and deriving the resulting line equations from an appropriate Hamiltonian after expanding the line equations as a Fourier series in the spatial variable z and then introduce Lindblad noise terms that enable us to model the dissipative effects. [2] Given a coaxial cable having inner radius a and outer radius b and with the medium in between the two cylinders having parameters (, μ, σ), calculate the distributed line parameters. [3] A transmission line at a given frequency has characteristic impedance Z0 = R0 + jX0 . The load connected to it has impedance ZL = RL + jXL at that frequency and a propagation constant γ. At a distance of d1 from the load, a stub of characteristic impedance Z01 = R01 + jX01 ,length l1 and propagation constant γ1 is attached. At a distance of d2 > d1 , from the load, another stub of characteristic impedance Z02 = R02 + jX02 , length l2 and propagation constant γ2 is attached. Find the input impedance of the line at a distance d3 > d2 from the load. If this input impedance matches a line of characteristic impedance Z0 = R0 + jX0 attached to it at d3 , then calculate the values of d1 and d2 . Explain how you would solve this problem using the Smith chart. [4] Explain from first principles how you would calculate the location of the nth voltage maximum and voltage minimum from the load end of a transmission line when the load ZL and characteristic impedance R0 of the line are given. Assume that the line is lossless. Also calculate the VSWR of the line. Finally, explain how you would determine the reflection coefficient of the line at the

223

Quantum Antennas

load end (both magnitude and phase) and the wavelength/propagation constant given the location of the nth voltage maximum, the distance between successive voltage maxima and the VSWR. [5] Draw the constant r and constant x circles in the Re(Γ) − Im(Γ) plane from the defining relation for the reflection coefficient Γ=

14.4

r + jx − 1 r + jx + 1

Problems in optimization theory

[1] State and prove Appolonius’ theorem in an infinite dimensional Hilbert space H and use it to establish the orthogonal projection theorem for closed convex subsets of H: If W is a closed and convex subset of H, then for each x ∈ H, there exists a unique vector P x ∈ W such that  x − P x = infw∈W  x − w 

[2] Let X = C 2 [0, 1] denote the normed linear space of all twice continuously differentiable functions on [0, 1]. For f : X → R and x ∈ X, we say that Df (x) : X → X ∗ exists (X ∗ denotes the Banach space of all bounded linear functions on X), if there exists a y ∈ X ∗ such that for all z ∈ X, one has lim→0 |(f (x + z) − f (x))/ − y(z)| = 0 In this case, we set Df (x) = y. Suppose f is given by  1 f (x) = L(x(t), x (t))dt, x ∈ X 0

where L : R × R → R is a twice continuously differentiable function of its arguments. Then prove that Df (x) exists and is given by  t d ∂L(x(t), x (t)) ∂L(x(t), x (t)) − z(t)( )dt Df (x)(z) = ∂x dt ∂x 0 Justify all the conditions imposed on the spaces X and the function L. [3] Write down the optimal Bellman-Hamilton-Jacobi equation for minimizing the expected cost  T C(u) = E L(x(t), x (t), u(t))dt 0

224

Quantum Antennas

where x(t) satisfies the equations dx(t) = x (t)dt, dx (t) = −γx (t)dt − Kx(t) + u(t) + σdB(t) where B(.) is standard Brownian motion and 

L(x(t), x (t), u(t) = x 2 (t)/2 + x2 (t)/2 + u2 (t)/2 The control function is u(t) and it is constrained to be of the instantaneous feedback type, ie u(t) = χt (x(t), x (t)). Before doing this problem, you must first prove that the bivariate process (x(t), x (t))T is a Markov process and calculate its infinitesimal generator. [4] Consider a 2-D image field g(x, y) obtained by rotating and translating a given image f (x, y) and further adding an additive white Gaussian noise w(x, y) to it with autocorrelation 2 δ(x − x )δ(y − y  ) E(w(x, y)w(x , y  )) = σw

The transformed image is therefore given by g(x, y) = f ((x−a)cos(θ)+(y −b)sin(θ), −(x−a)sin(θ)+(y −b)cos(θ))+w(x, y) Using two dimensional Fourier transforms and one dimensional Fourier series, derive an algorithm for estimating the rotation angle θ and the translation vector (a, b) from measurements of g(x, y) and f (x, y) over the entire plane using the maximum likelihood method. Finally, calculate an approximate formula for the covariance matrix of the estimation error: ˆM L − a, ˆbM L − b)) Cov((θˆM L − θ, a in terms of f, a, b, θ, σw . The approximation involves retaining only linear terms in the noise field. [5] State the Peter-Weyl theorem. Let χ1 , χ2 be the characters of two inequivalent irreducible representations of a compact group G. Prove using the Peter-Weyl theorem that   |χ1 (g)|2 dg = 1, χ ¯1 (g)χ2 (g)dg = 0 G

14.5

G

Another approach to quantization of wavemodes in a cavity resonator having nonuniform medium based on the scalar wave equation

The wave field ψ(r) satisfies (∇2 + h(ω, r)2 )ψ(r) = 0, r ∈ B, ψ(r) = 0, r ∈ ∂B

225

Quantum Antennas We write h(ω, r)= ω 2 /c2 + δλ(ω, r) and ψ(r) = ψ0 (r) + δψ(r), ω = ω0 + δω Then, application of standard first order perturbation theory gives (∇2 + ω02 /c2 )ψ0 (r) = 0, r ∈ B, ψ(r) = 0, r ∈ ∂B (∇2 + ω02 /c2 )δψ(r) + (2ω0 δω/c2 )ψ0 (r) + δλ(ω0 , r)ψ0 (r) = 0, r ∈ B, δψ(r) = 0, r ∈ ∂B The solutions to the unperturbed equation are ω0 = ω0 [n], n = 1, 2, ..., ψ0 (r) = ψ0,n,k (r), k = 1, 2, ..., d(n), n = 1, 2, ... ie (∇2 + ω0 [n]2 /c2 )ψ0,n,k (r) = 0, k = 1, 2, ..., d(n)

where we may assume that {ψ0,n,k , 1 ≤ k ≤ d(n), n ≥ 1} forms an onb for L2 (B) since ∇2 is a self-adjoint operator. In other words, the unperturbed mode ω0 [n] has a degeneracy of d(n). Then, it follows by writing 

d(n)

ψ0 (r) =

c(n, k)ψ0,n,k (r), ω0 = ω0 [n]

k=1

that ((ω0 [n]2 −ω0 [m]2 )/c2 ) < ψ0,m,s , δψ >  c(n, k) < ψ0,m,s , ψ0,n,k > +(2ω0 [n]/c2 )δω +



k

c(n, k) < ψ0,m,s , δλ(ω0 [n], r)ψ0,n,k >= 0

k

or equivalently, ((ω0 [n]2 − ω0 [m]2 )/c2 ) < ψ0,m,s , δψ > +(2ω0 [n]/c2 )δωc(n, s)δm,n +



c(n, k) < ψ0,m,s , δλ(ω0 [n], r)ψ0,n,k >= 0

k

In particular, considering the cases m = n and m = n separately gives us det(2ω0 [n]/c2 )δωId(n) + ((< ψ0,n,s , δλ(ω) [n], r))ψ0,n,k >))1≤s,k≤d(n) ) = 0 which has solutions δω = δω[n, k], k = 1, 2, ..., d(n)

226

Quantum Antennas d(n)

and with the corresponding secular eigenvectors ((ck (n, s)))s=1 , k = 1, 2, ..., d(n), and for m = n,  c(n, k) < ψ0,m,s , δλ(ω) [n], r)ψ0,n,k > < ψ0,m,s , δψ >= (ω0 [m]2 − ω0 [n]2 )−1 k

where we substitute c(n, k) = cl (n, k), l = 1, 2, ..., d(n) to get d(n) solutions for δψ(r). Thus, corresponding to the oscillation frequency ω0 [n] + δω[n, k], the wave field is d(n)  ck (n, s)ψ0,n,s + δψ(r) s=1

where δψ(r) =





ψ0,m,s (r)(ω0 [m]2 −ω0 [n]2 )−1

m=n,1≤s≤d(m)

ck (n, l) < ψ0,m,s , δλ(ω0 [n], r)ψ0,n,l >

l

To carry out the quantization, we derive the above generalized wave equation from the action principle δψ S[ψ] = 0 

where S[ψ] =

ψ(t, r)(∇2 + h(−i∂t , r)2 )ψ(t, r)dtd3 r = 0

Quantization can be performed by considering the Feynman path integral corresponding to this action between two states |i > and |f >:  < f |S|i >= exp(iS[ψ])Dψ To quantize this using Heisinberg operators, we expand h(ω, r)2 = ω 2 /c2 + δλ(ω, r) as a power series in jω: 

h(i∂t , r)2 = (−1/c2 )∂t2 +

δλm (r)∂tm

m≥0

If we truncate this series at m = N , then we have the approximation (c = 1) h(∂t , r)2 = −∂t2 +

N 

δλm (r)∂tm

m=0

and our modified wave equation in the time domain becomes (∇2 − ∂t2 +

N  m=0

δλm (r)∂tm )ψ(t, r) = 0

227

Quantum Antennas

The Lagrangian density for this equation is conveniently written by introducing a dual field φ(t, r): L(φ, ∂tm ψ, 0 ≤ m ≤ N, ∇ψ, ∇2 ψ) = (1/2)φ(∇2 − ∂t2 +

N 

δλm (r)∂tm )ψ

m=0

which is equivalent to (ie differs by a partial derivative) (1/2)[∂t φ.∂t ψ − (∇φ, ∇ψ) + φ

N 

λm (r)∂tm ψ]

m=0

The variational equations



Ld4 x = 0, δψ

δφ



Ld4 x = 0

give us the generalized wave equation along with its dual: N 

[∂t2 − ∇2 −

δλ(r)∂tm ]ψ(tr, r) = 0,

m=0

[∂t2 − ∇2 −

N 

λm (r)(−1)m ∂tm ]φ(t, r) = 0

m=0

To cast this Lagrangian in a form from which the Hamiltonian can be derived, we must allow only first order partial derivatives w.r.t. time to appear in the picture. To this end, we define the auxiliary fields ψk = ∂tk ψ, k = 1, 2, ..., N − 1 and then the field equation can be expressed as a sequence of first order in time equations: ∂t ψk = ψk+1 , k = 0, 1, ..., N − 2, ψ0 = ψ, λN (r)∂t ψN −1 +

N −1 

λm (r)ψm (t, r) + ∇2 ψ(t, r) − ψ1 = 0

m=0

These equations can be derived from a Lagrangian density after introducing dual fields φk , k = 0, 1, ..., N : L(φk , ψk , ∂t ψk , k = 0, 1, ..., N − 1) = N −2 

φk (∂t ψk − ψk+1 )

k=0

+φN −1 (λN (r)∂t ψN −1 +

N −1 

λm (r)ψm (t, r) + ∇2 ψ(t, r) − ψ1 )

m=0

Problem: By applying the Legendre transformation to this Lagrangian density, write down the corresponding Hamiltonian density and discuss the constraints involved.

228

14.6

Quantum Antennas

Derivation of the general structure of the field dependent permittivity and permeability of a plasma

. We start with the Boltzmann equation for the particle distribution function f (t, r, v) with the relaxation time approximation used in place of the collision term: f,t (t, r, v)+(v, ∇r )f (t, r, v)+(q/m)(E(t, r)+v×B(t, r), ∇v )f (t, r, v) = (f0 (t, r, v)−f (t, r, v))/τ (v) We write the frequency domain solution as fˆ(ω, r, v) = τ (v)−1 [jω+1/τ (v)+(v, ∇r )+(q/m)(E(j∂/∂ω, r)+v×B(j∂/∂ω, r), ∇v )]−1 fˆ0 (ω, r, v)

and the current density in the frequency domain is given by  ˆ r) = q v fˆ(ω, r, v)d3 v J(ω, while the charge density is ρˆ(ω, r) = q



fˆ(ω, r, v)d3 v

The Maxwell equations are ˆ ˆ ∇.E(ω, r) = ρˆ(ω, r)/0 , ∇.B(ω, r) = 0, ˆ ˆ ˆ ˆ r) + jωμ0 0 E(ω, ˆ ∇ × E(ω, r) = −jω B(ω, r), ∇ × B(ω, r) = μ0 J(ω, r) ˆ r) as a functional of E ˆ and B. ˆ Equating and solving these equations gives us J(ω, ˆ r) to (σ + jω)E(ω, ˆ ˆ and J(ω, r) gives us σ and  as nonlinear functionals of E ˆ B.

14.7

Other approaches to calculating the permittivity and permeability of a plasma via the use of Boltzmann’s kinetic transport equation

Let the unperturbed electrostatic potential be Φ(r). The corresponding equilibrium particle density in phase space is given by f0 (r, v) = N.exp(−β(mv 2 /2 + qΦ(r)))/Z(β) where N is the total number of particles and Z(β) is the classical partition function  Z(β) = exp(−β(mv 2 /2 + qΦ(r)))d3 rd3 v

229

Quantum Antennas It satisfies the equilibrium kinetic equation [(v, ∇r ) + (q/m)(−∇Φ(r), ∇v )]f0 (r, v) = 0

On application of an external electromagnetic field E(t, r), B(t, r) assumed to be small, the particle distribution function gets perturbed to f (t, r, v) = f0 (r, v) + δf (t, r, v) which satisfies upto first order terms, the perturbed kinetic equation iδf,t (t, r, v)+(v, ∇r )δf (t, r, v)−(q/m)(∇Φ(r), ∇v )δf +(q/m)(E(t, r)+v×B(t, r), ∇v )f0 (r, v) = −δf (t, r, v)/τ (v) The term involving the magnetic field cancels out and and in the special case when Φ = 0, this simplifies to iδf,t + (v, ∇r )δf + (q/m)(E(t, r), ∇v )f0 + δf /τ (v) = 0 which gives on Fourier transforming, ˆ (ω, r, v) = (qβ)(E(ω, ˆ [(iω + 1/τ (v)) + (v, ∇r )]δf r), v)f0 (r, v) The solution to this equation is given by  ˆ r ), v)f0 (r , v)d3 r δf (ω, r, v) = K(ω, v, r − r )(E(ω, where qβ(2π)−3



K(ω, v, r) = exp(ik.r)[iω + 1/τ (v) + i(v, k)]−1 d3 k

The current density in the plasma medium is therefore given by  ˆ (ω, r, v)d3 v = J(ω, ˆ r) = μ−1 ∇ × B(ω, ˆ ˆ q v δf r) − iω0 E(ω, r) 0 We have in fact, ˆ r) = q J(ω,



ˆ r ))f0 (r , v)d3 r d3 v vK(ω, v, r − r )(v, E(ω,

from which the complex anisotropic permittivity matrix (ω, r) defined by  ˆ J(ω, r) = jω (ω, r − r )E(ω, r )d3 r can be read of immediately. This is the linear theory of permittivity. It should be noted that the complex permittivity contains the plasma conductivity as a

230

Quantum Antennas

component. When Φ = 0, things are a little more complicated Formally, we can write the solution as ˆ (ω, r, v) = qβ[iω+1/τ (v)+(v, ∇r )−(q/m)(∇Φ(r), ∇v )]−1 ((E(ω, ˆ δf r), v)f0 (r, v)) Formally, denoting the kernel of the operator (iω + 1/τ (v) + (v, ∇r ))−1 by K = K(ω, v, r − r ), we can write the kernel of the operator (q/m)[iω + 1/τ (v) + (v, ∇r ) − (q/m)(∇Φ(r), ∇v )]−1 as  (qβ)[K + (KT )n K] n≥1

where T = (q/m)(∇Φ(r), ∇v ) In this way the current density in the frequency domain is   ˆ ˆ r) = (qβ) v[K + (KT )n K](E(ω, r), ∇v )f0 (r, v)d3 v J(ω, n≥1

from which we easily read out the inhomogeneous frequency dependent permittivity. Taking non-linearities into account. Let J(ω, r) be the current density. We write the Boltzmann equation in perturbation theoretic form: δf,t + (v, ∇r )(f0 + δf ) + (q/m)(−∇Φ(r) + δE(t, r), ∇v )(f0 + δf ) +(q/m)(δE(t, r) + v × δB(t, r), ∇v )δf + δf /τ (v) = 0 Expanding δf =



δn f

n≥1

we have on equating order zero terms, (v, ∇r )f0 − (q/m)(∇Φ(r), ∇v )f0 = 0 which is satisfied by the above Gibbsian f0 . Equating nth order terms gives us δn f,t + (v, ∇r )δn f − (q/m)(∇Φ(r), ∇v )δn f +(q/m)(δE + v × δB, ∇v )δn−1 f + δn f /τ (v) = 0, n ≥ 1 where δ0 f = f0 whose solution can be expressed as δn f (t, r, v) = (q/m)[i∂/∂t+1/τ (v)+(v, ∇r )−(q/m)(∇Φ(r), ∇v )]−1 (δE(t, r)+v×δB(t, r), ∇v )δn−1 f (t, r, v)], n ≥ 1  By iterating this equation, we can express δf (t, r, v) = δn f (t, r, v) as a n≥1 Volterra series in the electric and magnetic fields δE(t, r), δB(t, r) and hence determine the current density as a similar Volterra series in the electric and magnetic fields:

231

Quantum Antennas

14.8

Derivation of the permittivity and permeability functions using quantum statistics

In the presence of an external electric and magnetic field and taking into account quantum noise in the sense of Hudson and Parthasarathy, the Schrodinger evolution equation is dU (t) = (−(iH(t) + P )dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t) where H(t) = (α, P + eA(t, r)) + βm − eV (t, r) Solving this equation gives us the state of the system at time t as ρs (t) = T r2 (U (t)(ρs (0) ⊗ ρenv (0))U (t)∗ ) and hence, the average electric dipole moment and magnetic dipole moment can be computed using respectively the equations p(t) = −eT r(ρs (t)r), m(t) = (e/2m)T r(ρs (t)(L + gσ/2)) where L = r × P = −ir × ∇.

14.9

Approximate discrete time nonlinear filtering for non-Gaussian process and measurement noise

(Summary of Rohit Singh’s Ph.D work). The process to be estimated on a real time basis is a Markov process X(n), n ≥ 0 with one step transition probability distribution Pn (x, dy) = pn (x, y)dy, ie, P r(X(n + 1) ∈ dy|X(n) = x) = pn (x, y)dy The measurement process is y(n) = hn (X(n) + v(n) where v(n) is iid noise with pdf pv . The measurement data upto time n is given by Yn = σ(y(m), m ≤ n) and the MAP estimate of the process X at time n is ˆ X(n) = argmaxX(n) p(X(n)|Yn ) ˆ We wish to calculate X(n) recursively, ie, on a real time basis. Using Bayes’ rule, and the Markov property, we get p(X(n + 1)|Yn+1 ) = p(y(n + 1), Yn , X(n + 1))/P (y(n + 1), Yn ) = A/B

232

Quantum Antennas  A=

p(y(n + 1)|X(n + 1))p(X(n + 1)|X(n))p(X(n)|Yn )dX(n)  B=

AdX(n + 1)

so the MAP estimate of X(n + 1) given Yn+1 is given by ˆ + 1) = −argmaxX(n+1) A X(n Write X(n + 1) = fn (X(n)) + W (n + 1) where W is an iid process. Then,  A = pv (y(n + 1) − hn+1 (X(n + 1)))pw (X(n + 1) − fn (X(n))p(X(n)|Yn )dX(n) Write ˆ + δX X(n + 1) = fn (X(n)) Then, we have approximately, ˆ + δX) = hn+1 (X(n + 1)) = hn+1 (fn (X(n)) ˆ ˆ ˆ + hn+1 (fn (X(n))δX + (1/2)hn+1 (fn (X(n))(δX ⊗ δX) hn+1 (fn (X(n))) and writing ˆ e(n + 1) = y(n + 1) − hn+1 (fn (X(n))) we get approximately. pv (y(n + 1) − hn+1 (X(n + 1)) = T  ˆ ˆ [pv (e(n+1))hn+1 (fn (X(n))) pv (e(n+1))+pv (e(n+1))hn+1 (fn (X(n))))δX+(1/2)δX T  ˆ ˆ +hn+1 (fn (X(n)) pv (e(n + 1))hn+1 (fn (X(n)))]δX

Likewise, ˆ + δX − fn (X(n)) pw (X(n + 1) − fn (X(n)) = pw (fn (X(n))   ˆ ˆ ˆ = pw (fn (X(n))−f n (X(n))+pw (fn (X(n))−fn (X(n)))δX+(1/2)pw (fn (X(n))

−fn (X(n)))δX⊗δX and hence upto O(|δX|2 ), we have  A = A(X(n))p(X(n)|Yn )dX(n) where ˆ A(X(n)) = pv (e(n + 1))pw (fn (X(n)) − fn (X(n)))+ ˆ ˆ + pw (fn (X(n)) − fn (X(n)))]δX [pv (e(n + 1))hn+1 (fn (X(n)))) ˆ ˆ +(1/2)δX T [[pv (e(n + 1))hn+1 (fn (X(n))) + pw (fn (X(n)) − fn (X(n)))

233

Quantum Antennas ˆ ˆ +2pw (fn (X(n)) − fn (X(n)))T pv (e(n + 1))hn+1 (fn (X(n))))]δX ˆ ˆ = P1 (n, e(n + 1), X(n), X(n))T δX + (1/2)δX T P2 (n, e(n + 1), X(n), X(n))δX say, plus terms that are independent of δX. Maximizing this over δX gives us −1 ˆ ˆ ˆ ˆ + 1) = fn (X(n)) − P2 (n, e(n + 1), X(n)) P1 (n, e(n + 1), X(n)) X(n

where  ˆ = P1 (n, e(n + 1), X(n))

ˆ P1 (n, e(n + 1), X(n), X(n))p(X(n)|Yn )dX(n), 

ˆ = P2 (n, e(n + 1), X(n))

ˆ P2 (n, e(n + 1), X(n), X(n))p(X(n)|Yn )dX(n)

We can now also compute the error covariance matrix update equation: ˆ + 1) = fn (X(n)) + w(n + 1) − fn (X(n)) ˆ ˆ X(n + 1) − X(n + ψ(n, e(n + 1), X(n)) where −1 ˆ ˆ ˆ P1 (n, e(n + 1), X(n)) ψ(n, e(n + 1), X(n)) = −P2 (n, e(n + 1), X(n))

and then we get approximately with ˆ X(n) − X(n) = E(n), ˆ ˆ ˆ + w(n + 1) + ψ(n, y(n + 1) − hn+1 (fn (X(n))), X(n)) E(n + 1) = fn (X(n))E(n) ˆ ˆ ˆ = fn (X(n))E(n)+w(n+1)+ψ(n, hn+1 (fn (X(n))+w(n+1))−hn+1 (fn (X(n)))+v(n+1), X(n))  ˆ ˆ ˆ (hn+1 ofn ) (X(n)))E(n)+h = fn (X(n))E(n)+w(n+1)+ψ(n, n+1 (fn (X(n))w(n+1)+v(n+1), X(n))  ˆ ˆ ˆ ˆ = fn (X(n))E(n)+w(n+1)+ψ(n, 0, X(n))+ψ ,2 (n, 0, X(n))((hn+1 ofn ) (X(n))E(n)+

hn+1 (fn (X(n))w(n + 1) + v(n + 1)) If we assume that E(n) is orthogonal to the σ algebra generated by Yn , then we get approximately for the covariance P (n + 1) of E(n + 1), P (n + 1) = Q1 [n]P [n]Q1 [n]T + Q2 [n]Pv [n]Q2 [n]T + Q3 [n]Pw [n]Q3 [n]T where  ˆ ˆ ˆ + ψ,2 (n, 0, X(n))(h Q1 [n] = fn (X(n)) n+1 ofn ) (X(n))

234

Quantum Antennas

14.10

Quantum theory of many body systems with application to current computation in a Fermi liquid

The Fermi operator field is ψa (r). They satisfy the anticommutation relations [ψa (r), ψb (r )∗ ]+ = δab δ 3 (r − r ), [ψa (r), ψb (r )]+ = 0, [ψa (r)∗ , ψb (r )∗ ]+ = 0 The Hartree-Fock Hamiltonian of the fluid taking into account interactions with external fields as well as internal interactions is   ∗ 2 3 ψa (r) (−∇ /2m)ψa (r)d r + Vab (r, r )ψa (r)∗ ψb (r )d3 rd3 r H= a

+



ab

Vabcd (r1 , r2 , r3 , r4 )ψa (r1 )∗ ψb (r2 )ψc (r3 )∗ ψd (r4 )d3 r1 d3 r2 d3 r3 d3 r4

abcd

A special case of this Hamiltonian is   H= ψa (r)∗ (−∇2 /2m)ψa (r)d3 r + Va (r)ψa (r)∗ ψa (r)d3 r a

a

+



Vab (r, r )ψa (r)∗ ψa (r)ψb (r )∗ ψb (r )d3 rd3 r

a,b

The potential terms in this latter Hamiltonian correspond to the number ψ (r)∗ ψ a(r)d3 r

a

of Fermions of type a interacting with an external potential Va (r) and numbers ψa (r)∗ ψa (r)d3 r of particles of type a at r interacting with numbers ψb (r )∗ ψb (r )d3 r of particles of type b at r via an interaction potential Vab (r, r ). We shall work with this latter Hamiltonian as it has a nice physical interpretation. The above Fermion anticommutation rules give us on writing ψa (t, r) = exp(itH)ψa (r).exp(−itH), the same anticommutation rules for the time dependent field operators ψa (t, r) for a fixed time t and the fact that H is a constant of the motion, ie, exp(itH).H.exp(−itH) = H so that H is unchanged when ψa (r) is replaced by ψa (t, r) etc. We may also allow the potentials Va , Vab to depend explicitly on time in situations describing the application of a time varying voltage to a Fermi liquid or to a superconductor. In such a case, however, the Hamiltonian would not be a constant, it would depend explicitly on time and we write H(t) for it. However the anticommutation relations between the field operators would still remain the same since ψa (t, r) = U (t)∗ ψa (r)U (t)∀a, r where U (t) is a unitary operator defined by  t U (t) = T {exp(−i 0 H(s)ds)}

235

Quantum Antennas

The aim is to compute the average current in the liquid. For this, we assume that at the start, the liquid is in a Gibbs state ρ(0) = exp(−βH0 )/Z(β), Z(β) = T r(exp(− βH0 )) and H0 is obtained by replacing Va (t, r) by Va0 (r) and Vab (t, r, r ) by Vab0 (r, r ) with Va (t, r) = Va0 (r)+δVa (t, r), Vab (t, r, r ) = Vab0 (r, r )+δVab (t, r, r ). We regard δVa (t, r) and δVab (t, r, r ) as being the applied external potentials to the system. In the linear response theory, we compute the average current as a linear functional of these external potentials. The Hamiltonian can thus be expressed as H(t) = H0 + δH(t) and the state of the system as ρ0 (t) + δρ(t) where upto linear orders, ρ0 (t) = −i[H0 , ρ0 (t)] which is satisfied by ρ0 (t) = ρ(0) and δρ (t) = −i[H0 , δρ(t)] − i[δH(t), ρ(0)] whose solution is  δρ(t) = −i

t 0

exp(−i(t − s)H0 )[δH(t), ρ(0)].exp(i(t − s)H0 )ds

The Heisenberg field operators ψa (t, r) satisfy ψa,t (t, r) = i[H0 + δH(t), ψa (t, r)] and we can also apply first order perturbation theory to this to get ψa (t, r) = ψa0 (t, r) + δψa (t, r), ψa0,t (t, r) = i[H0 , ψa0 (t, r)] so that ψa0 (t, r) = exp(itH0 )ψa (r).exp(−itH0 ) δψa (t, r) = i[H0 , δψa (t, r)] + i[δH(t), ψa0 (t, r)] with solution  δψa (t, r) = i

t 0

exp(i(t − s)H0 ).[δH(s), ψa0 (s, r)].exp(−i(t − s)H0 )ds

The current density operator at time t is J(t, r) = (−ie/2m)(ψa (t, r)∗ ∇ψa (t, r) − ψa (t, r)∇ψa (t, r)∗ ) and in the presence of an external magnetic field described by the magnetic vector potential A(r), this current gets modified to J(t, r) = (e/2m)(ψa (t, r)∗ (−i∇+eA(r))ψa (t, r)−ψa (t, r)(−i∇−eA(r))ψa (t, r)∗ )

236

Quantum Antennas

summation over the repeated index a being implied. The Hamiltonian used in this case should be modified to H(t) = T + V1 (t) + V2 (t) 

where T = 

ψa (r)∗ ((−i∇ + eA(r))2 /2m)ψa (r)d3 r+

Va (t, r)ψa (r)∗ ψa (r)d3 r +



Vab (r, r )ψa (r)∗ ψa (r)ψb (r )∗ ψb (r )d3 rd3 r

and the average current is < J > (t, r) = T r(ρ(0)J(t, r)) which can be calculated using the linear response theory described above. The Heisenberg equations for ψa (t, r) = U (t)∗ ψa (r)U (t) with

 U (t) = T {exp(−i

t

H(s)ds)} 0

are ψ,t (t, r) = U (t)∗ i[H(t), ψa (r)]U (t) = i[U (t)∗ H(t)U (t), U (t)∗ ψa (r)U (t)] ˜ = i[H(t), ψa (t, r)] ˜ where H(t) is obtained from H(t) by replacing ψa (r) with ψa (t, r), and the commutation and anticommutation relations between the operators ψa (t, r), ψa (t, r)∗ are the same as those between the ψa (r), ψa (r) after premultiplying the result with U (t)∗ and postmultiplying it with U (t). We therfore compute [H(t), ψa (r)] = [T, ψa (r)] + [V1 (t), ψa (r)] + [V2 (t), ψa (r)]   T ψa (r)] = (−1/2) ψb (r )∗ ∇ 2 ψb (r )d3 r ψa (r)  = (−1/2)  = (1/2)  = (1/2)



ψb (r )∗ ∇ 2 ψb (r )ψa (r)d3 r 

ψb (r )∗ ψa (r)∇ 2 ψb (r )d3 r 

(δab δ 3 (r − r ) − ψa (r)ψb (r )∗ )∇ 2 ψb (r )d3 r = (1/2)∇2 ψa (r) + ψa (r)T

Equivalently,

[T, ψa (r)] = (1/2)∇2 ψa (r)

237

Quantum Antennas Next,

 V1 (t)ψa (r) = (  =− 

Vb (t, r )ψb (r )∗ ψb (r )d3 r )ψa (r) =

Vb (t, r )ψb (r )∗ ψa (r)ψb (r )d3 r =

Vb (t, r )(δab δ 3 (r − r ) − ψa (r)ψb (r )∗ )ψb (r )d3 r

−

= −Va (t, r)ψa (r) + ψa (r)V1 (t) or equivalently, [V1 (t), ψa (r)] = −Va (t, r)ψa (r) Thus,

[T + V1 (t), ψa (r)] = ((1/2)∇2 − Va (t, r))ψa (r)

Finally,  V2 (t)ψa (r) =

14.11

Vbc (t, r , r )ψb (r )∗ ψb (r )ψc (r )∗ ψc (r )ψa (r)d3 r d3 r

Optimal control of gravitational, matter and em fields

The perturbed Einstein-Maxwell equations are δRμν − (1/2)δR.η μν = −8πG(δT μν + δS μν ) where gμν = ημν + δgμν is a small perturbation of the flat space-time metric, δT μν = (1+p (ρ0 ))δρ.V μ V ν +(ρ0 +p(ρ0 ))(V μ δv ν +V ν δv μ )−p (ρ0 )δρη μν −p(ρ0 )η μα η νβ δgαβ is a small perturbation of the energy-momentum tensor of the matter field and finally (0)

δS μν = (1/4)F (0)αβ Fαβ η μρ η νσ δgρσ − (1/2)η μν F (0)αβ δFαβ is a small perturbation of the energy-momentum tensor of the em field. We can write Gμν = Rμν − (1/2)Rg μν , δGμν = C1 (μναβρσ)δgαβ,ρσ δS μν = C2 (μναβ, x)δgαβ (x) + C3 (μναβ, x)δFαβ (x)

238

Quantum Antennas (0)

where C2 and C3 are functions of the unperturbed em field tensor Fαβ (x). Finally, we can write δT μν (x) = C4 (μνα, x)δv α (x) + C5 (μν, x)δρ(x) + C6 (μναβ, x)δgαβ (x) As a result of the Einstein field equations, we have the perturbed fluid dynamical equations (MHD) δ((T μν + S μν ):ν ) = 0 Also the perturbed Maxwell equations are √ √ δ(F μν −g)),ν = δ(J μ −g) which can be expressed in the form √ (0) (δ(g μα g νβ −g),ν Fαβ + √ (0) +δ(g μα g νβ −g)Fαβ,ν + √ η μα η νβ δFαβ,ν = J (0)μ δ −g + δJ μ Since μν S:ν = F μν Jν

we can write the perturbed MHD equations as δ(T:νμν ) = δ(F μν Jν ) Now, μν μν = T,α + Γμαρ T νρ + Γναρ T μρ T:α

so that since the unperturbed Christoffel symbols are zero, we get δ(T:νμν ) = (δT μν ),ν +T (0)νρ δΓμαρ + T (0)μρ δΓναρ We note that these equations have to be modified if the background space-time, ie, unperturbed space-time is curved. We also note that δΓμαβ = δ(g μν Γναβ ) (0)

= η μν δΓναβ − η μρ η νσ Γναβ δgρσ μ μ ,μ = (1/2)δgα,β + δgβ,α − δgαβ )

since

(0)

Γναβ = 0 because the unperturbed space-time is flat. We note that δFαβ = δAβ,α − δAα,β

239

Quantum Antennas

Now, denoting the set of fields δgμν (x), δAμ (x), δv μ (x), δρ(x) by the symbols φk (x) and the control current sources δJ μ (x) by sm (x), the above linearized field equations have the general structure C1 (r, k, μν, x)φk,μν (x)+C2 (r, k, μ, x)φk,μ (x)+C3 (r, k, x)φk (x) = sr (x), r = 1, 2, ..., p with the summation being over the repeated indices. In arriving at this general form, we have chosen a specific coordinate system so that only six out of the ten metric coefficients δgμν (x) are independent and further that only three out of the four δv μ (x) are independent, these being related by 0 = δ(gμν v μ v ν ) = 2ημν V μ δv ν + V μ V ν δgμν Subject to these equations of motion, we wish to select the control input fields sr (x) so that the response field functions φk are as close as possible in distance to given fields. Using Lagrange multiplier fields λr (x), we may therefore consider the problem of minimizing  L(φk , sk , λk ) = wk (x)(φk (x) − φdk (x))2 d4 x −



k

(C1 (r, k, μν, x)φk,μν (x)+C2 (r, k, μ, x)φk,μ (x)+C3 (r, k, x)φk (x)

k

−sr (x))λr (x)d4 x

with summation over the repeated indices being implied.

14.12

Calculating the modes in a cylindrical cavity resonator with a partition in the middle

. The cylinder is over the length 0 ≤ z ≤ d = d1 + d2 and the partition is at z = d1 . The radius of the cylinder is R and the parameters of the medium for 0 ≤ z ≤ d1 are (1 , μ1 ) and those of the medium for d1 < z < d1 + d2 are (2 , μ2 ). The side walls at ρ = R and the top and bottom walls at z = 0, d are perfect conductors. The standard formulae relating the transverse components of the electric and magnetic fields to the longitudinal components are Ek⊥ = (1/h2k )∂/∂z(∇⊥ Ekz ) − (jωμk /h2 )∇⊥ Hkz × zˆ, Hk⊥ = (1/h2k )∂/∂z(∇⊥ Hkz ) + (jωk /h2k )∇⊥ Ekz × zˆ k = 1, 2. k = 1 stands for the bottom medium and k = 2 for the top medium. The boundary conditions are that Hz and E⊥ vanish at z = 0, d, μHz , Ez , H⊥ , E⊥ are continuous at z = d1 and Ez , Hρ vanish at ρ = R. These give us (∇2⊥ + h2k )(Ekz , Hkz ) = 0, k = 1, 2

240

Quantum Antennas

with different hk s for the electric and magnetic field. The solutions that match these boundary conditions are E1z = A1 Jm (h1 ρ)(P1 cos(mφ) + P2 sin(mφ))cos(α1 z) E2z = A2 Jm (h2 ρ)(P1 cos(mφ) + P2 sin(mφ))cos(α2 (d − z)) H1z = B1 Jm (h1 ρ)(Q1 cos(mφ) + Q2 sin(mφ))sin(β1 z), H2z = B2 Jm (h2 ρ)(Q1 cos(mφ) + Q2 sin(mφ))sin(β2 (d − z)) with 1 A1 cos(α1 d1 ) = 2 A2 cos(α2 d2 ), hk = αm [n], hk = βm [n], k = 1, 2  (x). Matching where αm [n] are the roots of Jm (x) while βm [n] are the roots of Jm of E⊥ at z = d1 gives us

α1 A1 sin(α1 d1 ) = −α2 A2 sin(α2 d2 ) so we get α2 1 cos(α1 d1 )sin(α2 d2 ) + α1 2 sin(α1 d1 )cos(α2 d2 ) = 0 Further, μ1 B1 sin(β1 d1 ) = μ2 B2 sin(β2 d2 ), β1 μ1 B1 cos(β1 d1 ) = −β2 μ2 B2 cos(β2 d2 ) which gives β2 μ1 sin(β1 d1 )cos(β2 d2 ) + β1 μ2 cos(β1 d1 )sin(β2 d2 ) = 0 Thus, if we know α1 , β1 , we can determine α2 , β2 . But what are the possible values of α1 , β1 that guarantee oscillations ? For the TM mode, we have using the wave equation for Ez , ω 2 μ1 1 − α12 = αm [n]2 , ω 2 μ2 2 − α22 = αm [n]2 Note that h1 and h2 are the same for a given mode because of the radial part of fields Ez at the interface z = d1 , ie, Jm (h1 ρ) = Jm (h2 ρ), 0 ≤ ρ ≤ R implies h1 = h2 = αm [n]. Likewise, considering the TE mode we have ω 2 μ1 1 − β12 = βm [n]2 , ω 2 μ2 2 − β22 = βm [n]2 These equations imply that we have two additional equations relating αk , βk , k = 1, 2 which can be solved to determine the characeteristic frequencies of oscillations of the TE and TM modes: (α12 + αm [n]2 )/μ1 1 = (α22 + αm [n]2 )/μ2 2 ,

241

Quantum Antennas (β12 + βm [n]2 )/μ1 1 = (β22 + βm [n]2 )/μ2 2 ,

Exercise: Using the above formulas, write down the complete expansions for the electric and magnetic fields in the time domain for the TE and TM modes of oscillation in a cylindrical DRA. Quantization of the fields inside the cylindrical DRA with the two partitions. We can write E1z (t, r) = Re



Jm (αm [n]ρ/R)(C1 (n, m, p)cos(mφ)+D1 (n, m, p)sin(mφ))sin(α1 [n, m, p]z)exp(jω[n, m, p]t),

nmp



=

ψ1nmp (t, r), 0 < z < d1

n,m,p

say and E2z (t, r) = Re



Jm (αm [n]ρ/R)(C2 (n, m, p)cos(mφ)+D2 (n, m, p)sin(mφ))sin(α2 [n, m, p](d−z))exp(jω[n, m, p]t)

nmp

=



ψ2nmp (t, r), d1 < z < d

nmp

say. It should be noted that C1 (n, m, p) : D1 (n : m, p) = C2 (n, m, p) : D2 (n, m, p) Assume that the TM modes only are sustained by the resonator. Then, the transverse components of the electric and magnetic fields are given by  αm [n]−2 ∇⊥ ∂ψk,nmp (t, r)/∂z, k = 1, 2 Ek⊥ (t, r) = nmp

Hk,⊥ (t, r) =



αm [n]−2 k ∇⊥ ∂ψk,nmp (t, r)/∂t × zˆ, k = 1, 2

nmp

The average energy stored within the resonator is  T  H(C1 , D1 , C2 ) = (1 /2T ) dt (E1z (t, r)2 + |E1⊥ (t, r)|2 )d3 r 0 /dt = Hk |ψkt > + m =k



−

< ψmt ⊗ I|V |ψkt ⊗ ψmt >

m =k

which in coordinate form can be expressed as   idψkt (r)/dt = Hk ψkt (r) + ( V (r, r )|ψmt (r )|2 d3 r )ψkt (r) m =k

  ( V (r, r )ψ¯mt (r )ψkt (r )d3 r )ψmt (r) = 0 − m =k

These equations are special cases of nonlinear Schrodinger equations of the form idψkt (r)/dt = −∇2 ψkt (r)/2m+V0 (r)ψkt (r)+



Vkm (r, ψ1t , ..., ψN t )ψmt (r), k = 1, 2, ..., N

m

The charge density is ρ(t, r) = −e



|ψkt (r)|2

k

and its rate of increase is given by  ρ,t (r) = −e (ψ¯kt,t (r).ψkt (r) + ψ¯kt (r).ψkt,t (r) k

= (ie/2m)

 k

= div((ie/2m)

[ψkt (r)∇2 ψ¯kt (r) − ψ¯kt (r)∇2 ψkt (r)]



[ψkt (r)∇ψ¯kt (r) − ψ¯kt (r)∇ψkt (r)])

k

= −divJ(t, r)

257

Quantum Antennas where the current density J(t, r) is given by  J(t, r) = (−ie/2m) [ψkt (r)∇ψ¯kt (r) − ψ¯kt (r)∇ψkt (r)] k

It should be noted that this expression for the current density is the same as that derived using the linear Schrodinger equations but the wave functions ψkt are calculated by solving the nonlinear Schrodinger equation. It should be noted that in deriving the above formula for the current density, we have made used of the relation V¯km (r, ψ1 , ..., ψN ) = Vmk (r, ψ1 , ..., ψN ) which is readily verified.

16.2

Controlling the current produced by a single quantum charged particle quantum antenna

The wave function is ψ(t, r). The external magnetic vector potential is A(t, r) and the external electric potential is V (t, r). The Hamiltonian of the electron after it interacts with the external em field is given by H(t) = (1/2m)(∇ + ieA(t, r))2 ψ(t, r) + (V0 (r) − eV (t, r)) Writing down the Schrodinger equation iψ,t (t, r) = H(t)ψ(t, r) and calculating the rate of change of the smeared charge density ρ(t, r) = −e|ψ(t, r)|2 ¯  ρ,t = −eψ¯ ψ − eψψ ¯ ¯ = −divJ ¯ ψ)) = ie(ψ¯ψHψ − ψ(H gives us ¯ + ieA)ψ − ψ(∇ − ieA)ψ) ¯ J(t, r) = (−ie/2m)(ψ(∇ ¯ ¯ + (e2 /2m)A|ψ|2 = (−ie/2m)(ψ∇ψ − ψ∇ψ) The external em field potentials A(t, r), V (t, r) are controlled by an external current source Jc (t, r). Apart from this external current source, there is the current J produced by the charged particle after quantum smearing. Thus, Maxwell’s equations for the em four potential are ∇2 A − (1/c2 )A,tt = −μ0 (J + Jc ),

258

Quantum Antennas ∇2 V − (1/c2 )V,tt = −(ρ + ρc )/0

where

 ρc = −

t 0

divJc dt

The solutions to these equations are obtained by the retarded potential formula  A(t, r) = (μ0 /4π) (J(t − |r − r |/c, r ) + Jc (t − |r − r |/c, r ))d3 r /|r − r |  V (t, r) = (1/4π0 )

(ρ(t − |r − r |/c, r ) + ρc (t − |r − r |/c, r ))d3 r /|r − r |

The aim is to design the control current density Jc (t, r) over a given time range t ∈ [0, T ] within a box [0, L]3 , so that the generated quantum current density J( t, r) tracks a desired current density Jd (t, r) in the sense that  |Jd (t, r) − J(t, r)|2 d3 rdt [0,T ]×[0,L]3

is a minimum. The second quantized picture: Assume that there are two species of Fermions described by the Fermionic operator wave fields ψa (t, r), a = 1, 2 satisfying the canonical anticommutation relations [ψa (t, r), ψb (t, r )∗ ]+ = δab δ 3 (r − r ) The unperturbed Hamiltonian operator of this Fermionic field is   H0 = (−1/2m) ψa (r)∗ ∇2 ψa (r)d3 r + Vab (r, r )ψa (r)∗ ψb (r )d3 rd3 r The quantum statistical Gibbsian density operator is then ρG = exp(−βH0 )/Z(β), Z(β) = T r(exp(−βH0 )) In the presence of an external electromagnetic field described by a magnetic vector potential A(t, r) and scalar electric potential V (t, r), the second quantized Hamiltonian of the Fermi liquid assumes the form   ∗ 2 3 H(t) = (−1/2m) ψa (r) (∇ + ieA(t, r)) ψa (r)d r − e V (t, r)ψa (r)∗ ψa (r)d3 r  +

Vab (r, r )ψa (r)∗ ψb (r )d3 rd3 r

and we may also add to this a ”Cooper pair” term  Uab (r, r )ψa (r)∗ ψa (r)ψb (r )∗ ψb (r )d3 rd3 r

259

Quantum Antennas

coming from the interaction between ψa (r)∗ ψa (r)d3 r Fermions (number operator) at d3 r with ψb (r )∗ ψb (r )d3 r Fermions at d3 r . In this, case, the unperturbed Hamiltonian for the Fermionic field would be taken as H0 =  ψa (r)∗ ∇2 ψa (r)d3 r + Vab (r, r )ψa (r)∗ ψb (r )d3 rd3 r

 (−1/2m) 

Uab (r, r )ψa (r)∗ ψa (r)ψb (r )∗ ψb (r )d3 rd3 r

while the interaction Hamiltonian between the Fermionic field and the external electromagnetic field is  HI (t) = (−ie/2m) ψa (r)∗ (2(A(t, r), ∇) + divA(t, r))ψa (r)d3 r  −e

V (t, r)ψa (r)∗ ψa (r)d3 r

where the canonical anticommutation relations are satisfied by the Fermionic field operators: [ψa (r), ψb (r )∗ ]+ = δab δ 3 (r − r ), [ψa (r), ψb (r )]+ = 0, [ψa (r)∗ , ψb (r )∗ ]+ = 0 The Fermionic fields satisfy Heisenberg’s equation of motion: ψa,t (t, r) = i[H(t), ψa (t, r)] where H(t) is obtained from the previous H(t) = H0 + HI (t) by replacing ψa (r), ψa (r)∗ with ψa (t, r), ψa (t, r)∗ respectively and using the same anticommutation relations for the time dependent field operators. The Fermionic current density operator is  J(t, r) = (−ie/2m) [ψa (t, r)∗ (∇+ieA(t, r))ψa (t, r) −ψa (t, r)∗ (∇−ieA(t, r))ψa (t, r)d3 r] The average current density is then Jav (t, r) = T r(ρG J(t, r)) and the objective of our Fermi-liquid quantum antenna is to make this current density track a desired current density Jd (t, r) by controlling the external electromagnetic fields A, V .

Chapter 17

photons in a gravitational field with gate design applications and image processing in electromagnetics 17.1

Some remarks on quantum blackhole physics

[1] Time always flows in the forward direction, the entropy of a blackhole always increases and the mass of a blackhole always increases since any particle is always attracted towards itself and absorbed by the massive gravitational field of the blackhole. This is why computing the entropy of a blackhole becomes important. At time t = 0, assume that a system of particles in the vicinity of a blackhole is in the pure state |ψm >. We assume that |ψm > in position space is concentrated within the critical radius of the blackhole. Let Hm denote the Hamiltonian of the system of particles, HG , the Hamiltonian of the gravitational field of the blackhole and HI the interaction Hamiltonian between the particles and the gravitational field of the blackhole. Hm acts in the system Hilbert space Hm , HG acts in the gravitational field Hilbert space HG while HI acts in the tensor product space Hm ⊗ HG . The initial state of the blackhole gravitational field and the system particles is a pure state |ψ(0) >= |ψm (0) ⊗ ψG (0) >. After time t, it evolves to the state |ψ(t) >= exp(−it(Hm + HG + HI ))|ψ(0) >

261

262

Quantum Antennas

The states after time t of the system of particles and the gravitational field of the blackhole are both mixed and are respectively given by ρm (t) = T r2 (|ψ(t) >< ψ(t)|), ρG (t) = T r1 (|ψ(t) >< ψ(t)|) The initial entropy of the system is zero and so also of the initial gravitational field of the blackhole since both of these states are pure. The final entropies of these are non-zero in general and are respectively given by Sm (t) = −T r(ρm (t)log(ρm (t)), SG (t) = −T r(ρG (t)log(ρG (t))) In particular, this shows that by interacting with material particles, the entropy of the blackhole increases. We can also understand Hawking radiation via the tunneling phenomena in quantum mechanics. Consider the case of a Schwarzchild blackhole. Classically, a particle within the critical radius m = 2GM/c2 of this blackhole cannot escape outside in finite coordinate time. Quantum mechanically there is a small probability of such an escape taking place. This can be computed as follows. The KG equation for a particle in the metric gμν is given by (g μν ψ,ν )):μ + (2πmc2 /h)2 ψ = 0 or writing

β = 2πmc2 /h

this equation becomes √ √ (g μν −gψ,ν ),μ + β 2 −gψ = 0 In the case of a Schwarzchild metric and when the wave function depends only on the radial coordinate r, this equation reduces to √ √ √ g 00 −gψ,00 + (g 11 −gψ,1 ),1 + β 2 −gψ = 0 or equivalently, α(r)−1 r2 ψ,00 (t, r) − (α(r)r2 ψ,1 (t, r)),1 + β 2 r2 ψ(t, r) = 0 This equation is solved using separation of variables. If we take the initial KG wave function as ψ(0, r) = Kχr rc . This result means that although classically the particle cannot tunnel through the critical radius, yet quantum mechanically, there is a small probability of this happening. Instead of KG particles, we could also work with photons which are governed by the Maxwell equations in curved space-time. Let Aμ (x) denote the four potential of the photon. Assume that at time t = 0, all the photons are contained within the Schwarzchild radius. For example we may take Aμ (0, r) = ψμ (r), Aμ,0 (0, r) = φμ (r)

263

Quantum Antennas Then the Aμ (t, r) satisfy the Maxwell equations in the Schwarzchild metric √ (g μα g νβ −gFαβ ),ν = 0 with the gauge condition

√ (Aμ −g),μ = 0

We find therefore that Aμ satisfies a second order pde in the space-time variables and the two specified initial conditions suffice to solve for Aμ for all values of space-time. After rearrangement, the Maxwell equations give us √ √ (g μα g νβ −g),ν Fαβ + g μα g νβ −gFαβ,ν = 0 The gauge condition gives us √ √ (g μα −g),μ Aα + g μα −gAα,μ = 0 A better approach to this problem is to work directly with covariant derivatives, ie, F:νμν = 0, Aμ:μ = 0 and then get μ:ν Aν:μ :ν − A:ν = 0

Now, μα ν Aν:μ A:α:ν = :ν = g

g μα (Aν:α:ν − Aν:ν:α ) (in view of the gauge condition Aν:ν = 0) ρ Aρ = g μα g νβ Rβαν

Thus, the Maxwell equations reduce to ρ Aρ Aμ = g μα g νβ Rβαν

where  denotes the Laplace-Beltrami wave operator of curved space-time acting on four vector fields. We can write

17.2

EM field pattern produced by a rotated and translated antenna with noise deblurring

A transmitter antenna is completely specified by its current density J(ω, r) at the frequency ω. If the antenna is rotated and translated by (R, a) ∈ SO(3)×R3 , the resulting current density is J1 (ω, r) = J(ω, R−1 (r − a)). The electric field

264

Quantum Antennas

patterns in space due to the original antenna and the transformed antenna are respectively given by  F (ω, r) = K(ω, r − r )J(ω, r )d3 r , 

and F1 (ω, r) =

K(ω, r − r )J1 (ω, r )d3 r + w(r)

where K is the vector valued Green’s function defined by F (ω, r) = ∇ × ∇ × A(ω, r)/jωμ where



A(ω, r) = (mu/4π)

J(ω, r )exp(−jω|r − r |/c)d3 r /|r − r |, c = (μ)−1/2

Here, we are assuming the medium to be linear, homogeneous and isotropic. The aim is to estimate the rotation-translation pair (R, a) from measurements on F and F1 . More generally, if the medium is nonlinear, inhomogeneous and anisotropic, then we can write with J, J1 , F, F1 being 3 × 1 vector fields,  Kn (ω, ω1 , ..., ωn , r, r1 , ..., rn )(⊗nk=1 J(ωk , rk )d3 r1 ...d3 rn , F (ω, r) = n≥1

F1 (ω, r) = =





Kn (ω, ω1 , ..., ωN , r, r1 , ..., rn )(⊗nk=1 J1 (ωk , rk ))d3 r1 ...d3 rn ,

n≥1

Kn (ω, ω1 , ..., ωn , r, r1 , ..., rn )(⊗nk=1 J(ωk , R−1 (rk − a))d3 r1 ...d3 rn ,

n≥1

=



Kn (ω, ω1 , ..., ωn , r, Rr1 + a, ..., Rrn + a) ⊗nk=1 J(ωk , rk ))d3 r1 ...d3 rn

n≥1

In these expressions, Kn is a matrix valued complex function of size 3×3n , n ≥ 1. By measuring the untransformed em field F at different frequencies and at different spatial locations, we can get to estimate the kernels Kn (ω, ω1 , ..., ωn , r, r1 , ..., rn ) and by measuring the transformed em field F at different frequencies and at different spatial locations, we can get to estimate the kernels Kn (ω, ω1 , ..., ωn , r, r1 , ..., rn ) = = Kn (ω, ω1 , ..., ωn , r, Rr1 + a, ..., Rrn + a) From the knowledge of these two kernels, the rotation translation pair (R, a) may be determined by applying a combination of 3-D spatial Fourier transforms and the Peter-Weyl theory for SO(3) based on spherical harmonic expansions.

265

Quantum Antennas

17.3

Estimation the 3-D rotation and translation vector of an antenna from electromagnetic field measurements

Original antenna current density: J(ω, r) Rotated and translated antenna current density: J1 (ω, r) = J(ω, R−1 (r−a)). Original electromagnetic field:  F (ω, r) = G1 (ω, r−r1 )J(ω, r1 )d3 r1  + G2 (ω1 , ω−ω1 , r−r1 , r−r2 )J(ω1 , r1 )J(ω−ω1 , r2 )d3 r1 d3 r2 Electromagnetic field after rotating and translating the antenna:  F1 (ω, r) = G1 (ω, r−r1 )J1 (ω, r1 )d3 r1  + G2 (ω1 , ω−ω1 , r−r1 , r−r2 )J1 (ω1 , r1 )J1 (ω−ω1 , r2 )d3 r1 d3 r2  = G1 (ω, r − a − Rr1 )J1 (ω, r1 )d3 r1 +  G2 (ω1 , ω−ω1 , r−a−Rr1 )G2 (ω1 , ω−ω1 , r−a−Rr2 )J1 (ω1 , r1 )J1 (ω−ω1 , r2 )d3 r1 d3 r2 Assuming that G1 , G2 depend on r only via |r|, it follows that after taking noise into account, F1 (ω, r) = F (ω, R−1 (r − a)) + w(ω, r) where w is the noise field. (R, a) can thus be identified from measurements of the original and final em field patterns using Fourier transforms in R3 and the Peter-Weyl theory for SO(3). More generally, consider a set of N antennae having current densities Jk (ω, r), k = 1, 2, ..., N . These antennae are permuted, rotated and translated so that the resulting sequence of current densities becomes Jσk (ω, R−1 (r − a)), k = 1, 2, ..., N . The resulting em field pattern before applying this set of transformations (R, a, σ) is F (ω, r) = F (ω, r|r1 , ..., rN ) and the em field pattern after applying the rotation, translation and permutation transformation is given by F1 (ω, r|r1 , ..., rN ) = F (ω, R−1 (r − a)|rσ1 , ..., rσN ) + w(ω, r) We wish to estimate the group element (R, a, σ) from measurements of F1 and F2 at different r s. Let π1 denote a unitary representation of Sn -the permutation group of n objects and π2 a unitary representation of SO(3). We have assuming zero noise,  F1 (ω, r|r1 , ..., rN ).exp(−jk.r)d3 r = 

F (ω, r|rσ1 , ..., rσN )exp(−jk.(Rr + a))d3 r

266

Quantum Antennas  = exp(−jk.a)

F (ω, r|rσ1 , ..., rσN )exp(−j(RT k).r)d3 r

We express this identity as Fˆ1 (ω, k|r1 , ..., rN ) = exp(−jk.a)Fˆ (ω, RT k|rσ1 , ..., rσN ) from which, we deduce that |Fˆ1 (ω, k|r1 , ..., rN ) = |Fˆ (ω, RT k|rσ1 , ..., rσN )| Calculating the spherical harmonic coefficients on both sides on the sphere k ∈ k0 S 2 gives us  |Fˆ1 (ω, k0 n ˆ |r1 , ..., rN )Y¯lm (ˆ n)dΩ(ˆ n)  =

|Fˆ (ω, k0 n ˆ |rσ1 , ..., rσN )|Y¯lm (Rˆ n)dΩ(ˆ n)

 =

|Fˆ (ω, k0 n ˆ |rσ1 , ..., rσN )|



(¯ πl )m ,m (R−1 )Y¯lm (ˆ n)dΩ(ˆ n)

m

This equation can be expressed in matrix form after noting that πl (R−1 ) = (πl (R))∗ , |Fˆ1 (ω, k0 |r1 , ..., rN )|l = πl (R)|Fˆ (ω, k0 |rσ1 , ..., rσN )|l

17.4

Mackey’s theory of induced representations applied to estimating the Poincare group element from image pairs

The Poincare group is the semidirect product of R4 (space-time translations) with the proper orthochronous Lorentz group (spatial rotations and boosts). Essentially, this group can be expressed as P = R4 ⊗s SL(2, C) where we identify SL(2, C) as the double cover of the proper orthochronous Lorentz group via the map X → gXg ∗ with X being a Hermitian matrix that represents the space-time coordinates and g ∈ SL(2, C). For a given character χ0 of R4 and the corresponding stability subgroup H0 of SL(2, C) with a given irreducible representation L of H0 , we wish to determine the irreducible representation of P obtained by inducing χ0 ×L from R4 ⊗s H0 to P = R4 ×s SL(2, C). It is easy to see that if n ∈ R4 and h ∈ SL(2, C), then the representation U of P induced by χ0 and L is defined by U (nh)f (χ) = χ(n)L(γ(χ)−1 hγ(h−1 χ))f (h−1 χ), χ ∈ Oχ0

267

Quantum Antennas

where the representation space of U consists of all functions f defined on the orbit Oχ0 of χ0 under SL(2, C) with values in the representation space V of L. Here, γ is a cross-section map in the sense that for each χ ∈ Oχ0 , γ(χ) ∈ SL(2, C) is such that γ(χ)χ0 = χ and that if χ, χ are two distinct elements of Oχ0 , then γ(χ) = γ(χ ). The representation property is easily checked: For h1 , h2 ∈ SL(2, C), (U (h1 )(U (h2 )f ))(χ) = −1 L(γ(χ)−1 h1 γ(h−1 1 χ))(U (h2 )f )(h1 χ) −1 −1 −1 −1 −1 h2 γ(h−1 = L(γ(χ)−1 h1 γ(h−1 1 χ))L(γ(h1 χ) 2 h1 χ))f (h2 h1 χ)

= L(γ(χ)−1 h1 h2 γ((h1 h2 )−1 χ)f ((h1 h2 )−1 χ) = U (h1 h2 )f (χ) We now take an image field f1 (x) defined on a manifold x ∈ M on which the group P acts. Here, M = R4 . The transformed image field is f2 (x) = f1 (g −1 x) + w(x) where g = nh ∈ P. The left invariant Haar measure on P is dndh where dn is the standard Lebesgue measure on R4 while dh is the left invariant Haar measure on SL(2, C). Our aim is to estimate g = nh from measurements of both f1 and f2 . We have in the absence of noise,       −1    f2 (g x0 )U (g )dg = f1 (g g x0 )U (g )dg = f1 (g  x0 )U (gg  )dg   = U (g)

f1 (g  x0 )U (g  )dg 

which can be written in the language of group theoretic Fourier transforms as fˆ2 (U ) = U (g)fˆ1 (U ) From this equation, U (g) can be estimated accurately using a linear least squares algorithm provided that we have a sufficient large number of image field pairs (f1 , f2 ), all related through U . We now observe that for ψ defined on the orbit Oχ0 of χ0 under SL(2, C), we have  ˆ f1 (U )ψ(χ) = f1 (gx0 )U (g)ψ(χ)dg =  R4 ×SL(2,C)

f1 (nhx0 )χ(n)L(γ(χ)−1 hγ(h−1 χ))ψ(h−1 χ)dndh

where χ ∈ Oχ0 is arbitrary. This expression by left invariance of the measure dh on SL(2, C), can equivalently be expressed by the change of variables h → γ(χ)h as  f1 (nγ(χ)hx0 )χ(n)L(hγ(h−1 χ0 ))ψ(h−1 χ0 )dh We therefore have the following relations for determining the operator U (g). We write ψ1k (χ) = fˆ1 (U )ψk (χ), ψ2k (χ) = fˆ2 (U )ψk (χ), k = 1, 2, ..., n

268

Quantum Antennas

Then, U (g)ψ1k (χ) = ψ2k (χ), k = 1, 2, ..., n or equivalently, with g = nh, ψ2k (χ) = χ(n)L(γ(χ)−1 hγ(h−1 χ))ψ1k (h−1 χ), k = 1, 2, ..., n − − − (1) This gives us on taking the norm in the representation space V in which the unitary irreducible representation L of the stability group H0 of χ0 acts,  ψ2k (χ) = ψ1k (h−1 χ) , k = 1, 2, ..., n from which h may be determined by taking the ordinary group theoretic Fourier transform on SL(2, C). Assuming that h has thus be determined, n is easily determined using the equations (1) for different χ ∈ Oχ0 . A remark on the different orbits in R4 of SL(2, C) action. (a) Corresponding to (1, 0, 0, 0)T (mass m = 1), the corresponding representing matrix in H, the space of 2 × 2 Hermitian matrices on which SL(2, C) acts by adjoint action, is X 0 = I2 The stability subgroup H0 for X0 is the set of all g ∈ SL(2, C) for which gX0 g ∗ = X0 , ie, gg ∗ = I2 . This means that g ∈ SU (2), ie, H0 = SU (2). (b) Corresponding to (0, 0, 0, 1)T (imaginary mass m = i), the element of H is   1 0 X1 = 0 −1 The stability group of X1 consists of all g ∈ SL(2, C) for which gX1 g ∗ = X1 or equivalently,

gX1 = X1 g ∗−1 , g ∈ SL(2, C)

Writing

 g=

a c

b d

we get g g

−1

∗−1

 =  =

Thus,

 gX1 =

 , ad − bc = 1 d −c



−b a

d¯ −¯ c −¯b a ¯ a c

−b −d



 ,

269

Quantum Antennas X1 g ∗−1 =



d¯ −¯ c ¯b −¯ a



So the stability group H0 of X1 is the set of all g ∈ SL(2, C) for which d=a ¯, c = ¯b Thus, H0 , consists of all matrices of the form   a b , |a|2 − |b|2 = 1 ¯b a ¯ We now consider the stability group H0 of (1, 0, 0, 1) (zero mass). The element X2 of H corresponding this is   1 0 X2 = 2 0 0 Thus, the condition for  g=

a c

b d

 ∈ SL(2, C)

to leave this element fixed, ie, that gX2 g ∗ = X2 is that g have the form   exp(iθ) z g= , θ ∈ [0, 2π), z ∈ C 0 exp(−iθ) Denoting this element by g(z, θ), we see that g(z, θ).g(z  , θ ) = g(z  .exp(iθ) + z.exp(−iθ ), θ + θ ) The group H0 is isomorphic to the semidirect product C ⊗s T where the isomoprhism takes g(z, θ) to the element (z.exp(iθ), θ) with the action of T on C being defined by θ[z] = exp(2iθ).z Indeed, we then have for this semidirect product (z.exp(iθ), θ).(z  .exp(iθ ), θ ) = (z  .exp(i(θ + 2θ)) + z.exp(iθ), θ + θ ) = (z  .exp(iθ) + z.exp(−iθ )).exp(i(θ + θ )), θ + θ ) which corresponds to the matrix g(z  .exp(iθ) + z.exp(−iθ ), θ + θ ), thereby demonstrating the required group isomorphism. Remark: We wish to demonstrate that the two definitions of the induced representation of the semidirect product of an Abelian group N with another group H such that H normalizes N are equivalent. The first definition is based

270

Quantum Antennas

ˆ and let H0 be its stability on the following construction: Choose a χ0 ∈ N subgroup in H. Let Oχ0 be the orbit of χ0 under H. Let L be an irreducible ˜ : nh → representation of H0 in a vector space V . It is easy to see that L χ0 (n)L(h) is an irreducible representation of G0 = N ⊗s H0 in V where n ∈ ˜ N, h ∈ H0 . The conventional method of constructing U = IndG G0 L is to define the representation space Y of U to be the set of all maps f : G → V for which ˜ 0 )−1 f (g) for all g ∈ G, g0 ∈ G0 and then define f (gg0 ) = L(g U (g)f (x) = f (g −1 x), g, x ∈ G, g ∈ Y We now give another construction and prove that the two methods of constructing the induced representation are isomorphic. For f ∈ Y , define ψf : Oχ0 → V ˆ. by ψf (χ) = f (γ(χ)) where γ(χ) ∈ H is such that γ(χ)χ0 = χ for each χ ∈ N Now, define W (nh)ψf (χ) = χ(n)L(γ(χ)−1 hγ(h−1 χ))ψf (h−1 χ), n ∈ N, h ∈ H We observe that ψf1 = ψf2 for f1 , f2 ∈ Y implies f (γ(χ)) = 0 for all χ ∈ Oχ0 where f = f1 − f2 ∈ Y . This implies in turn that −1 ˜ 0 = L(nh) f (γ(χ)) = f (γ(χ)nh), n ∈ N, h ∈ H0 , χ ∈ Oχ0

This in turn implies that f (g) = 0 for all g ∈ G since γ(χ) runs over one element from each coset of G/G0 = H/H0 as χ runs over Oχ0 Thus, the map f → ψf is a bijection from Y onto the set of all functions on Oχ0 . Note that if ψ : Oχ0 → V is a map, we define f (nh) = ψ(h.χ0 ), n ∈ N, h ∈ H. We then have ψf (χ) = f (γ(χ)) = ψ(γ(χ)χ0 ) = ψ(χ) which demonstrates the bijection once we observe that for n ∈ N, h ∈ H, h0 ∈ H0 , we have f (nh(n0 h0 )−1 ) = ψ(hh−1 0 χ0 ) = ψ(hχ0 ) = f (nh) implying that f ∈ Y . If ψ : Oχ0 → C, define T ψ : G → C by (T ψ)(nh) = A(n, h).ψ(h.χ0 ), n ∈ N, h ∈ H It is easily seen that T ψ = 0 iff ψ = 0. Further, for n0 , n ∈ N, h0 ∈ H0 , h ∈ H, we have (T ψ)(nhn0 h0 ) = (T ψ)(nhn0 h0 ) = (T ψ)(nhn0 h−1 hh0 ) = A(nhn0 h−1 , hh0 )ψ(h.χ0 ) In order that T ψ ∈ Y , we require that A(nhn0 h−1 , hh0 ) = χ ¯0 (n0 ).L(h−1 0 )A(n, h)

271

Quantum Antennas This happens provided that ¯ 0 ](n)L(h−1 γ(hχ0 )) A(n, h) = h[χ for then, ¯ 0 [χ0 ](nh[n0 ])L(h−1 h−1 γ(h[χ0 ])) A(nhn0 h−1 , hh0 ) = hh 0 ¯ 0 ](nh[n0 ])L(h−1 )L(h−1 (γ(h[χ0 ]) = h[χ 0 =χ ¯0 (n0 )L(h−1 0 )A(n, h)

We further note that any f ∈ Y is of the form T ψ for some function ψ on Oχ0 , the reason being that if f ∈ Y , then we define ψ(χ) = f (γ(χ)). Then, T ψ(nh) = A(n, h)f (γ(nhχ0 )) = A(n, h)f (γ(hh−1 nhχ0 )) = A(n, h)f (γ(hχ0 )) = A(n, h)f (h.γ(χ0 )γ(χ0 )−1 .h−1 γ(hχ0 )) = A(n, h)L(γ(hχ0 )−1 h)f (h.γ(χ0 )) ¯ 0 ](n)L(h−1 γ(hχ0 ))L(γ(hχ0 )−1 h)f (h) = h[χ ¯ 0 ](n)f (h) = χ0 (h−1 n−1 h)f (h) = f (nh) = h[χ Finally, we show that the two definitions of the induced representation are equivalent. To se this, we note that −1 −1 −1 −1 U (n1 h1 )T ψ(nh) = T ψ(h−1 1 n1 nh) = T ψ(h1 n1 nh1 h1 h) −1 −1 −1 = A(h−1 1 n1 nh1 , h1 h)ψ(h1 hχ0 )

−1 −1 −1 n1 h1 )L(h−1 h1 γ(h−1 = h−1 1 h[χ0 ](h1 n 1 hχ0 ))ψ(h1 hχ0 ) −1 −1 −1 = h−1 n1 ])L(h−1 h1 γ(h−1 1 h[χ0 ](h1 [n 1 hχ0 ))ψ(h1 hχ0 ) −1 = h[χ0 ](n−1 n1 )L(h−1 h1 γ(h−1 1 hχ0 ))ψ(h1 hχ0 )

while on the other hand, −1 W (n1 h1 )ψ(χ) = χ(n1 )L(γ(χ)−1 h1 γ(h−1 1 χ))ψ(h1 χ)

and therefore, T W (n1 h1 )ψ(nh) = A(n, h)(W (n1 h1 )ψ)(h.χ0 ) −1 = A(n, h)h[χ0 ](n1 )L(γ(h[χ0 ])−1 h1 γ(h−1 1 h[χ0 ]))ψ(h1 h[χ0 ])

¯ 0 ](n)L(h−1 γ(hχ0 ))h[χ0 ](n1 )L(γ(h[χ0 ])−1 h1 γ(h−1 h[χ0 ]))ψ(h−1 h[χ0 ]) = h[χ 1 1 −1 = h[χ0 ](n−1 n1 )L(h−1 h1 γ(h−1 1 h[χ0 ]))ψ(h1 h.χ0 )

thus proving that U (n1 h1 )T = T W (n1 h1 ), n1 ∈ N, h1 ∈ H ie,

U (g) = T W (g)T −1

thereby establishing the equivalence of the two definitions of the induced representation for the semidirect product.

272

17.5

Quantum Antennas

Effect of electromagnetic radiation on the expanding universe

First the unperturbed em field is calculated by solving Maxwell’s equations in the Robertson-Walker metric: g00 = 1, g11 = −S 2 (t)f (r), f (r) = 1/(1−kr2 ), g22 = −S 2 (t)r2 , g33 = −S 2 (t)r2 sin2 (θ) The solution to Maxwell’s equations in this metric will result in a generalized wave equation for the unperturbed em four potential Aμ (x). The coefficients of this wave equation will depend upon t, r, θ, with the dependence on t coming via the term S(t). This wave equation will be a second order partial differential equation in the space and time variables. Hence, if we know the em four potential Aμ as well as its time derivative Aμ,0 at time t = 0, then we can in principle get a unique solution for all t ≥ 0. We now assume that at time t = 0, Aμ and Aμ,0 are random functions of the spatial variable and we calculate the ensemble averaged energy-momentum tensor Sμν (t, r, θ, φ) = (−1/4) < Fαβ F αβ > gμν + g αβ < Fμα Fνβ > The probability distribution of Aμ (0, r, θ, φ), Aμ,0 (0, r, θ, φ), μ = 0, 1, 2, 3 must be chosen in such a way that Sμν is a homogeneous and isotropic (0, 2) tensor. Once this has been done, we solve the perturbed Einstein field equations by considering the background to be Robertson-Walker and a small perturbation to this metric begin given by δgμν (x) which satisfies the electromagnetically perturbed Einstein field equations: δRμν − (1/2)Rδgμν − (1/2)gμν δR = −8πGSμν The question therefore arises is that what should be the general form, ie, spacetime dependence of Sμν for it to be regarded as a homogeneous and isotropic tensor? It is easy to see that for this to be the case, Sμν must have the form F (t, r)gμν where F (t, r) is a scalar dependent only on time and the comoving radial coordinate. The condition that g να Sμν:α = 0 then implies F,0 (t, r)gμ0 + F (t, r)gμ0:0 + F (t, r)g km gμk:m + g 11 F,1 (t, r)gμ1 which gives since gμν:α = 0 the equation F (t, r) = F0 = constt. Such a solution is not of much value since it is equivalent to adding a cosmological constant term to the Einstein field equations. So we drop the condition that the unperturbed em field have an averaged energy-momentum tensor that

273

Quantum Antennas

is homogeneous and isotropic. More generally, solving the Maxwell equations in the Robertson-Walker space-time gives with the above stated initial conditions,  Aμ (t, r) =

Kμν (t, r, r )Aν (0, r )d3 r +



Lνμ (t, r, r )Aν,0 (0, r )d3 r

where K and L are uniquely determined kernels expressible in terms of S(t) and k. Now assuming < Aμ (0, r)Aν (0, r ) >= Pμν (r, r ), < Aμ (0, r)Aν,0 (0, r ) >= Qμν (r, r ), < Aμ,0 (0, r)Aν,0 (0, r ) >= Mμν (r, r ) We have < Aμ (t, r)Aν (t , r ) >= intKμα (t, r, r1 )Kνβ (t , r , r1 )Pαβ (r1 , r1 )d3 r1 2d3 r1 (1) = Nμν (t, r, t , r )

say. Next, < Aμ,ν (t, r)Aα,β (t , r ) >= ∂2 ∂xν ∂xβ  =

(< Aμ (x).Aα (x ) >

∂2 N (1) (x, x ) ∂xν ∂xβ  μα (2)

= Nμανβ (x, x ) say, where x = (t, r), x = (t , r ) Now, the ensemble averaged energy-momentum tensor of the unperturbed em field is given by Sμν (x) = (−1/4)g αρ (x)g βσ (x)gμν (x) < Fαβ (x)Fρσ (x) > + < Fμα(x) Fνβ (x) > g αβ (x)

274

Quantum Antennas

17.6

Photons inside a cavity

. Assume that there are N photons, so that, for example, a state of the em field within the box may be |k1 , s1 , ..., kN , sN > corresponding to the fact that the lth photon has a four momentum kl and spin/helicity sl . This photon field interacts with the electron positron field coming from a probe inserted within the cavity. The state of the electron-positron field within the box is then described  > corresponding to the fact that there by |p1 , σ1 , ..., pM , σM , p1 , σ1 , ..., pK , σK are M electrons with four momenta pk and spin σk , k = 1, 2, ..., M and K positrons with four momenta pk and spin σk s, k = 1, 2, ..., K. The interaction Hamiltonian between the photon field and the electron-positron field is then   HI (t) = J μ Aμ d3 r = −e ψ(x)∗ αμ ψ(x)Aμ (x)d3 r where x = (t, r). After interaction, we wish to describe the final state of the electrons, positrons and photons after time T . This state is given in the interaction picture by  T HI (t)dt)}|φ(0) > |φ(T ) >= T {exp(−i 0

where |φ(0) >= |kj , sj , j = 1, 2, ..., N, pj , σj , j = 1, 2, ..., M, pj , σj , j = 1, 2, ..., K > is the initial state of the photons, electrons and positrons. We note the expansions  ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v¯(P, σ)b(P, σ)∗ exp(ip.x))d3 P  Aμ (x) =

(2|K|)−1/2 [eμ (K, s)c(K, s).exp(−ik.x)+¯ eμ (K, s)c(K, s)∗ exp(ik.x)]d3 K

After time T , the current operator in the interaction picture within the box is J μ (T, r) = −eψ(T, r)∗ αμ ψ(T, r) where ψ(T, r) = exp(−iHD T )ψ(0, r)exp(iHD T ) 

with HD =  ψ(0, r) =

ψ(0, r)∗ ((α, −i∇) + βm)ψ(0, r)d3 r,

(u(P, σ)a(P, σ)exp(iP.r) + v¯(P, σ)b(P, σ)∗ exp(−iP.r))d3 P

The radiation field produced by this current density outside the box in the far-field zone is given by  Aμ1 (t, r) = (μ0 /4πr) J μ (t − r/c + rˆ.r /c, r )d3 r

275

Quantum Antennas Apart from this, there is a surface current density operator on the walls of the n × H where cavity resonator. This surface current density operator is JS = −ˆ H is the magnetic field on the boundary. This is determined from the em four potential within the box in the interaction picture: Aμ (T, r) = exp(iT Hem )Aμ (0, r).exp(−iT Hem ) where Aμ (0, r) =



(2|K|)−1/2 [eμ (K, s)c(K, s).exp(iK.r)+¯ eμ (K, s)c(K, s)∗ exp(−iK.r)]d3 K 

and

|K|c(K, s)∗ c(K, s)d3 K

Hem =

The far field em four potential operator produced by the surface electric current density on the resonator boundaries is given by  A2 μ(T, r) = (μ/4πr) JS (t − r/c + rˆ.r /c, r )d3 r It follows that the far-field em four potential operator (Aμ1 + Aμ2 )T, r) is expressible in terms of the creation and annihilation operators of the electrons, positrons and photons. Denoting this operator by Aμ3 (t, r), we can compute the far field Poynting vector operator S(t, r) in terms of the electron-positronphoton creation and annihilation operators in the interaction picture. Thus, the mean, mean square and more generally, any higher order moment of the far field Poynting vector or more generally of the far field electromagnetic field which we denote by F (Aμ3 , r) can be evaluated as < φ(T )|F (Aμ3 , r)|φ(T ) > We can also consider other initial states of the photon field like the coherent states and calculate the higher moments of the far field em potentials in such a state. Remark: When we solve the Maxwell equations with the boundary conditions required by the walls of the cavity resonator, the solution for the four potential will not be expressed as linear combinations of exp(±ik.x) but rather in terms of certain eigenfunctions ηk (t, r) = ηk (x), k = 1, 2, ... that satisfy the boundary conditions. Likewise the solution of the free Dirac equation with wave function vanishing on the boundary will cause the solution to this confined Dirac equation to be expandable in terms of certain vector valued eigenfunctions χ1k (x), χ2k (x), k = 1, 2, .... So, ideally speaking, we should express the em four potential operator field Aμ (x) and the Dirac electron-positron vector operator field as  c(k)ηk (x) + c(k)∗ η¯k (x) Aμ (x) = k

ψ(x) =

 k

a(k)χ1k (x) + b(k)∗ χ2k (x)

276 where

Quantum Antennas

[c(k), c(m)∗ ] = δ[k − m], [c(k), c(m)] = 0, [a(k), a(m)∗ ]+ = δ[k − m], [a(k), a(m)]+ = 0, [b(k), b(m)∗ ]+ = δ[k − m], [a(k), b(m)]+ = 0, [a(k), b(m)∗ ]+ = 0, [c(k), a(m)] = 0, [c(k), a(m)∗ ] = 0, [c(k), b(m)] = 0, [c(k), b(m)∗ ] = 0

It follows that the Dirac current density field operator within the cavity at time t given by J μ (x) = −eψ(x)∗ αμ ψ(x) is expressible in terms of a(k), a(k)∗ , b(k), b(k)∗ , k = 1, 2, .... We can calculate the far field em four potential operator in terms of these operators using the standard retarded potential formula. Further, the magnetic field operator on the walls and hence the surface current density field operator Js on the walls can be computed as n ˆ × H where the magnetic field is calculated as components of Ar,s − As,r . Thus, Js on the walls is expressible in terms of c(k), c(k)∗ and hence the far field four potential operator generated by the surface current density can be computed using the standard retarded potential method. The state of the electron-positron-photon field within the cavity after time T with the evolution in the interaction picture taking place in accordance with the  interaction Hamiltonian HI (t) = −e ψ(x)∗ αμ ψ(x)Aμ (x)d3 r is given by  t  t ρ(t) = T {exp(−i HI (s)ds)}.ρ(0).T {exp(−i HI (s)ds)})∗ 0

0

and hence we can calculate the moment T r(ρ(T ).F (A3μ )) where A3μ is the far field em four potential generated by the current density within the cavity and the surface current density on the walls of the cavity and F is some functional of this far field em four potential.

17.7

Justification of the Hartree-Fock Hamiltonian using second order quantum mechanical perturbation theory

First consider a first quantized system with unperturbed Hamiltonian H0 and (0) perturbation V . Let En , n = 1, 2, ... denote the unperturbed energy levels (0) and |ψn > the corresponding unperturbed eigenfunctions. The perturbation of these levels and eigenfunctions upto second order in V is given by En = En(0) + En(1) + En(2) , |ψn >= |ψn(0) > +|ψn(1) > +|ψn(2) >

277

Quantum Antennas Substituting these into the stationary state Schrodinger equation (H + V )(|ψn >= En |ψn > and equating terms of the same order of magnitude gives us (H0 − En(0) )|ψn(0) >= 0, (H0 − En(0) )|ψn(1) > +V |ψn(0) > −En(1) |ψn(0) >= 0, (H0 − En(0) )|ψn(2) > +V |ψn(1) > −En(1) |ψn(1) > −En(2) |ψn(0) >= 0 (0)

Using the orthonormality of the ψn , n = 1, 2, ..., we get En(1) =< ψn(0) |V |ψn(0) >, |ψn(1) >=



(0) |ψm >

m =n

and

(0)

(0)

(0)

(0)

< ψm |V |ψn > En − Em

En(2) = En(1) < ψn(0) |ψn(1) > − < ψn(0) |V |ψn(1) > =

(0)  | < ψn(0) |V |ψm > |2 m =n

(0)

(0)

Em − En

(1)

(0)

Thus, En is a quadratic function of the wave functions ψn and their complex (2) conjugates while En is a fourth degree function of the same wave functions and their complex conjugates. This means that in the second quantization process, (0) En will be replaced by  < ψn(0) |H0 |ψn(0) >= ψn(0)∗ (r)H0 ψn(0) (r)d3 r (0)

(1)

where ψn (r) s are now quantum fields and En will be replaced by the operator  ψn(0)∗ (r)V (r)ψn(0) (r)d3 r (2)

while finally, En will be replaced by a fourth degree polynomial function in the operator wave fields and their conjugates

17.8

Tetrad formulation of the Einstein-Maxwell field equations

Tetrad: eμa (x). Metric is g = ηab ω a ⊗ ω b , ω a = eaμ dxμ

278

Quantum Antennas

where (ηab )) = diag[1, −1, −1, −1] Thus, dxμ = eμa ω a Let ∇ denote the metrical connection. Then its torsion is zero. We write ∇X ea = ωab (X)eb for any vector field X. ωba is a one form. We have ∇ ea g = 0 Since the torsion is zero, we have ∇ea eb − ∇eb ea − [ea , eb ] = 0 From this, we can easily deduce Cartan’s first equation of structure: dω a + ωba ∧ ω b = 0 a Likewise, the curvature tensor R has tetrad Rbcd given by a Rbcd ea = [∇eb , ∇ec ]ed − ∇[eb ,ec ] ed

and we easily deduce using this Cartan’s second equation of structure: a Rbcd ec ∧ ed = dωba + ωca ∧ ωbc a So in order to determine Rbcd , we must first express the one forms {ωba } in terms a of {ω }. The equation ∇X g = 0 gives

X(g(eb , ec )) + g(∇X eb , ec ) − g(eb , ∇X ec ) = 0 for any vector field X and since ηbc = g(eb , ec ) are constants (by the definition of tetrad), X(g(eb , ec ) = 0. Thus, we get g(ωbd (X)ed , ec ) + g(eb , ωcd (X)ed ) = 0 or equivalently, ωbd (X)ηdc + ωcd (X)ηbd = 0 which is experssed as ωcb + ωbc = 0 where ωab is the one form defined by ωab = ηac ωbc Now writing ω a = ea μdxμ

279

Quantum Antennas we get dω a = eaμ,ν dxν ∧ dxμ = eaμ,ν eνb eμc ω b ∧ ω c and hence, the first equation of structure gives eaμ,ν eνb eμc ω b ∧ ω c + ωca ∧ ω c = 0 Comparing the coefficient of ω c on both sides gives us ωca = −eaμ,ν eνb eμc ω b + λac ω c

where in the last term, there is no summation over c. Lowering the index a using the η-metric then gives us ωac = −eaμ,ν eνb eμc ω b + λac ω c It follows that the rhs must be skew-symmetric in the indices (a, c). In other words λac ω c + λca ω a = ecμ,ν eνb eμa ω b +eaμ,ν eνb eμc ω b = (ecμ,ν eμa + eaμ,ν eμc )eνb ω b Now, eaμ,ν eμc = −eaμ eμc,ν = −eρa gμρ eμc,ν = −eρa (ecρ,ν − eμc gμρ,ν ) Thus, we get λac ω c + λca ω a = gμρ,ν eρa eμc eνb ω b for all a, c. Equivalently, λac ω c + λca ω a = −gμρ (eρa eμc ),ν eνb ω b = −gμρ (eρa,ν eμc + eρa eμc,ν )eνb ω b = −(eρa,ν ecρ + eaμ eμc,ν )eνb ω b −2eaμ eμc,ν eνb ω b These equations may be used to calculate the one forms ωba in terms of the one forms ω a . The same logic may be applied to the determine the tetrad components of the curvature tensor using Cartan’s second equation of structure: a ω c ∧ ω d = dωba + ωca ∧ ωdc Rbcd

We’ve seen that ωba = f (a, b, c, x)ω c , dω a = g(a, b, c, x)ω b ∧ ω c

280

Quantum Antennas

for some appropriate functions f, g determined by the tetrad. Thus, dωba = f,mu (a, b, c, x)dxμ ∧ ω c +f (a, b, c, x)dω c = f,μ (a, b, c, x)eμd ω d ∧ ω c +f (a, b, c, x)g(c, k, m, x)ω k ∧ ω m = h(a, b, c, d, x)ω c ∧ ω d where the function h is easily identified in terms of f, g. This gives a ω c ∧ ω d = h(a, b, c, x)ω c ∧ ω d + ωca ∧ ωbc Rbcd

= h(a, b, c, x)ω c ∧ ω d + f (a, c, k, x)f (c, b, m, x)ω k ∧ ω m = P (a, b, c, d, x)ω c ∧ ω d where the function P is easily identified in terms of h, f . From this identity, it easily follows that a = (P (a, b, c, d, x) − P (a, b, d, c, x))/2 Rbcd

Exercise: Compute the tetrad components of the curvature tensor for the generalized Kerr metric defined by g = ηab ω a ∧ ω b , where ω 0 = f0 (x)dx0 , ω 1 = f1 (x)(dx1 − a0 (x)dx0 − a2 (x)dx2 − a3 (x)dx3 ), ω 2 = f2 (x)dx2 , ω 3 = f3 (x)dx3

17.9

Optimal quantum gate design in the presence of an electromagnetic field propagating in the Kerr metric

. The Kerr metric has the form dτ 2 = a0 (r, θ)dt2 − a1 (r, θ)dr2 − a2 (r, θ)dr2 − a3 (r, θ)(dφ − ω(r, θ)dt)2

281

Quantum Antennas so that g00 = a0 − a3 ω 2 , g11 = −a1 , g22 = −a2 , g33 = −a3 , g03 = g30 = a3 ω

We first write down the Maxwell equations in this metric in the tetrad formalism: Aμ eμa = Aa , Aμ = Aa eaμ eaμ eνb Aμ:ν = eaμ eνb (eμc Ac ):ν = eaμ eνb (eμc Ac,ν + eμc:ν Ac ) = eνb Aa,ν + eaμ eμc:ν eνb Ac a c = Aa,b + γcb A a where γcb are the spin coefficients. Note that that Aa are scalars. We now write down the Maxwell equations in the tetrad basis. Here X,b means eμb X,μ .

Fab = eμa eνb Fμν are the Maxwell scalars. We have F:νμν = 0 for the Maxwell equations which give (F ab eμa eνb ):ν = 0 or equivalently, F,νab eμa eνb + F ab eμa:ν eνb + F ab eμa eνb:ν = 0 or, F,bab eμa + F ab eμa:ν eνb + F ab eμa eνb:ν = 0 Thus, F,bab + F cb eaμ eμc:ν eνb + F ab eνb:ν = 0 or in terms of the spin coefficients, a + F ab edν eνc:ρ eρd = 0 F,bab + F cb γcb

or a d + F ab γcd F,bab + F cb γcb

This is the first part of the Maxwell equations in tetrad formalism. The second part is to express the Maxwell equation Fμν,ρ + Fνρ,μ + Fρμ,ν = 0 in tetrad formalism. The Maxwell equation Fμν,ρ + Fνρ,μ + Fρμ,ν = 0

282

Quantum Antennas

is equivalent to Fμν:ρ + Fνρ:μ + Fρμ:ν = 0 We have Fμν:ρ = (Fab eaμ ebν ):ρ = Fab,ρ eaμ ebν + Fab (eaμ:ρ ebν + eaμ ebν:ρ ) Thus, eμa eνb eρc Fμν:ρ = n m Fab,c + Fnm eμa eνb eρc (enμ:ρ em ν + eμ eν:ρ ) ρ = Fab,c + Fnb eμa enμ:ρ eρc + Fam eνb em ν:ρ ec n m = Fab,c + Fnb γac + Fam γbc

or more precisely, this is written as Fab,c + Fnb η nm γamc + Fam η mn γbnc = Fab,c + η nm (γamc Fnb + γbnc Fam ) where the spin coefficients are defined by γabc = eμa ebμ:ν eνc Thus, the homogeneous components of the Maxwell equations can be expressed in tetrad notation as Fab,c + Fbc,a + Fca,b + η nm (γamc Fnb + γbnc Fam + γbma Fnc + γcna Fbm +γcmb Fna + γanb Fcm ) = 0 Calculating the spinor connection of the gravitational field for the Schwarzchild and Kerr metrics. Let Vaμ be the tetrad and J ab = (1/4)[γ a , γ b ] The spinor connection of the gravitational field is b Γμ = (1/2)J ab Vνa Vν:μ

An easy calculation shows that by taking the local Lorentz matrix Λ(x) to be infinitesimal, ie, I + ω(x) where ωab = −ωba , we have D(Λ) = I + dD(ω) = I + ωab J ab Then, we require that if Γμ and Γμ are respectively the spinor connections in the original and in locally Lorentz transformed frames, we must have D(Λ)γ a Vaμ (∂μ + Γμ )D(Λ)−1 = Vaμ D(Λ)γ a D(Λ)−1 D(Λ)(∂μ + Γμ )D(Λ)−1

283

Quantum Antennas = Vaμ Λab γ b (D(Λ)(∂μ D(Λ)−1 ) + D(Λ)Γμ D(Λ)−1 + ∂μ ) = Λab Vaμ γ b (∂μ + Γμ ) where

Γμ = D(Λ)(∂μ D(Λ)−1 ) + D(Λ).Γμ .D(Λ)−1

or equivalently when Λ = I + ω is inifintesimal, we must have Γμ − Γμ = −ωab,μ J ab + ωab [J ab , Γμ ] This is satisfied by choosing Γμ as above and using the fact that the tetrad under local Lorentz transformations transforms as Vaμ → Vaμ + ωab V μb We leave this as an exercise for the reader to prove. Now we compute the connection for the Kerr metric in the form dτ 2 = a0 (x1 , x2 )2 dx02 −a1 (x1 , x2 )2 dx12 −a2 (x1 , x2 )2 dx22 −a3 (x1 , x2 )2 (dx3 −ω(x1 , x2 )dx0 )2

where

x0 = t, x1 = r, x2 = θ, x3 = φ

The tetrad basis for this problem with the Minkowski metric ((ηab )) = diag[1, −1, −1, −1] is given by

ω 0 = Vμ0 dxμ = a0 dx0 , ω 1 = Vμ1 dxμ = a1 dx1 , ω 2 = Vμ2 dxμ = a2 dx2 , ω 3 = Vμ3 dxμ = a3 (dx3 − ωdx0 ) Thus, in terms of components, V00 = a0 , Vr0 = 0, r = 1, 2, 3, V11 = a1 , Vμ1 = 0, μ = 0, 2, 3, V22 = a2 , Vμ2 = 0, μ = 0, 1, 3, V03 = −ωa3 , V33 = a3 , Vμ3 = 0, μ = 1, 2 Vaν:μ = Vaν,μ − Γρνμ Vaρ Treating the gravitational field and the electromagnetic field as a perturbation to the Dirac equation and noting that the Dirac equation can be expressed as [γ a Vaμ (i∂μ + iΓμ + eAμ ) − m]ψ = 0 we have the following perturbation equations: ψ = ψ0 + ψ1 + ... + ψn + ...

284

Quantum Antennas

where γ μ (i∂μ − m)ψ0 = 0, [iγ μ ∂μ − m]ψ1 + iγ a (Vaμ − δaμ )∂μ + iγ μ Γμ + eγ μ Aμ ]ψ0 = 0 upto first order. In general, taking higher order terms into consideration, we have [iγ μ ∂μ − m]ψn+1 + iγ a (Vaμ − δaμ )∂μ + iγ μ Γμ + eγ μ Aμ ]ψn = 0, n ≥ 0 or equivalently, if S(x) = (2π)−4



[iγ μ pμ − m]−1 exp(ip.x)d4 p

is the electron propagator, we can write ψn+1 (x) =  −

S(x − x )[iγ a (Vaμ (x ) − δaμ )∂μ + iγ μ Γμ (x ) + eγ μ Aμ (x )]ψn (x )d4 x

We assume that the em four potential Aμ satisfies the Maxwell equation in curved space-time with metric gμν (x) which is weak perturbation of flat spacetime. So we can write taking into account the gauge condition, √ √ (g μν Aν −g),ν = 0, (g μα g νβ −gFαβ ),ν = 0 We write gμν = ημν + hμν (x) where hμν is of the first order of smallness and likewise expand Aμ as (1) Aμ = A(0) μ + Aμ

Then, since upto first order, √ −g = 1 + h/2, h = hμμ = ημν hμν , g μν = ημν − hμν , hμν = ημα ηνβ hαβ we get upto the first order, the following Maxwell equations: (0)

(ημα ηνβ Fαβ ),ν = 0, (0)

[(−hμα ηνβ − hνβ ημα + hη μα η νβ /2)Fαβ ],ν + (1)

(ημα ηνβ Fαβ ),ν = 0 Considerable simplification of this equation is achieved by using the gauge condition upto first order: (1) ((ημν − hμν )(1 + h/2)(A(0) ν + Aν ),μ = 0

285

Quantum Antennas

On equating zeroth and first order of smallness terms, this gauge condition gives (ημν A(0) ν ),μ = 0, μν (0) (ημν A(1) ν ),μ + ((hημν /2 − h )Aν ),μ = 0

Now, where

(0) (1) Fμν = Fμν + Fμν , (0) (0) Fμν = A(0) ν,μ − Aμ,ν , (1) (1) Fμν = A(1) ν,μ − Aμ,ν ,

Now using the first order gauge condition, (1)

(ημα ηνβ Fαβ ),ν = (1)

(1)

[ημα ηνβ (Aβ,α − Aα,β )],ν = (1)

−ημα ηνβ Aα,βν −ημα ((hηνβ /2 − hνβ )A(0) ν ),αβ

For the Schwarzchild space-time, the tetrad can be chosen as  Vμ0 dxμ = α(r)dt, α(r) = 1 − 2m/r Vμ1 dxμ = α(r)−1/2 dr, Vμ2 dxμ = rdθ, Vμ3 dxμ = r.sin(θ)dφ To compute the spinor gravitational connection, we need to evaluate the covariant derivatives of the tetrad. These are as follows. First we evaluate the connection components Γμαβ : Γ000 = 0, Γ010 = Γ001 = (1/2)g 00 g00,1 = α (r)/2α(r) = (1/2)(1/(r − 2m) − 1/r) Γ011 = 0, Γ022 = 0, Γ033 = 0, Γ0rs = 0, r, s = 1, 2, 3 Γ002 = Γ020 = (1/2α(r))g00,2 = 0, Γ003 = Γ030 = 0 Γ110 = Γ101 = (1/2)g 11 g11,0 = 0 Γ220 = Γ202 = 0, Γ100 = (−α(r))g00,1 = −α (r)α(r)

286

17.10

Quantum Antennas

Maxwell’s equations in the Kerr metric in the tetrad formalism

The tetrad of the Kerr metric is ω 0 = a0 (r, θ)dt = e0μ dxμ , ω 1 = a1 (r, θ)dr = e1μ dxμ , ω 2 = a2 (r, θ)dθ = e2μ dxμ ω 3 = a3 (r, θ)(dφ − ω(r, θ)dt) = e3μ dxμ so that the metric is (ω 0 )2 − (ω 1 )2 − (ω 2 )2 − (ω 3 )2 = ηab ω a ⊗ ω b We have eaμ eμb = δba or equivalently, if the top index denotes the row index and the bottom index the column index, then ((eaμ )) = ((eμa ))−1 (a)

We shall use the notations eμ(a) and eμ respectively in place of eμa and eaμ in order to avoid confusion. Then, we have a0 e0(b) = δb0 , a1 e1(b) = δb1 , a2 e2(b) = δb2 , a3 e3(b) − a3 ωe0(b) = δb3 from which, we deduce that (e0(0) , e0(1) , e0(2) , e0(3) ) = (1/a0 , 0, 0, 0), (e1(0) , e1(1) , e1(2) , e1(3) ) = (0, 1/a1 , 0, 0), (e2(0) , e2(1) , e2(2) , e2(3) ) = (0, 0, 1/a2 , 0), (e3(0) , e3(1) , e3(2) , e3(3) ) = (ω/a0 , 0, 0, 1/a3 ) The Maxwell scalars F(a)(b) are related to the Maxwell tensor Fμν by the relations F(a)(b) = eμa eμb Fμν We thus evaluate: F(0)(1) = eμ(0) eν(1) Fμν = (1/a0 a1 )F01 , F(0)(2) =

eμ(0) eν(2) Fμν

= (1/a0 a2 )F02 ,

F(0)(3) = eμ(0) eν(3) Fμν = (1/a0 a3 )F03

287

Quantum Antennas F(1)(2) = eμ(1) eν(2) Fμν = (1/a1 a2 )F12 F(1)(3) = eμ(1) eν(3) Fμν = (1/a1 a3 )F13 + (ω1 /a1 a0 )F10 F(2)(3) = eμ(2) eν(3) Fμν = (1/a2 a3 )F23 + (ω1 /a2 a0 )F20 The Maxwell complex scalars in the Newman-Penrose formalism. The Newman-Penrose tetrad of null geodesics l, n, m, m ¯ satisfy l.n = 1, m.m ¯ = −1, l.l = n.n = m.m = m. ¯ m ¯ = l.m = l.m ¯ =0 where by a.b we mean gμν aμ bν . We define the complex Maxwell scalars by φ1 = Fμν lμ mμ , φ2 = Fμν lμ m ¯ μ, ¯ ν) φ3 = Fμν (lμ nν + mμ m

The six components of Fμν can be recovered from these three complex scalars and their complex conjugates. The em field when passed through the Kerr-metric:Design of quantum gates based on this idea The em four potential is (1) Aμ = A(0) μ (x) + Aμ (x)

The metric is gμν (x) = ημν + hμν (x) The zeroth order Maxwell equations are after taking into account the Lorentz gauge condition for the unperturbed potential, α A(0) μ = 0,  = ∂α ∂

The first order Maxwell equations are after taking into account the first order perturbed gauge condition derived from √ (g μν −gAμ ),ν = 0 of the form

(0) A(1) μ = C1 (μνρσαβ)Aν hρσ,αβ (0)

+C2 (μνρσαβ)Aν,αβ hρσ +C3 (μνρσαβ)A(0) ν,α hρσ,β If the Dirac current field J μ = −eψ(x)∗ αμ ψ(x) is taken into account as a first order perturbation, then the above equation gets modified to (0) A(1) μ = C1 (μνρσαβ)Aν hρσ,αβ

288

Quantum Antennas (0)

+C2 (μνρσαβ)Aν,αβ hρσ +C3 (μνρσαβ)A(0) ν,α hρσ,β +μ0 eψ(x)∗ αμ ψ(x) The Dirac field taking into account gravity and electromagnetic interactions satisfies the first order perturbed equation: Vaμ (x) = δaμ + faμ (x) where (ημν + hμν ) = ηab (δμa + fμa )(δνb + fνb ) so that hμν = ημb fνb + ηνa fμa = fμν + fνμ So upto first order, we can take fμν = hμν /2 or equivalently, faμ = hμa /2 Likewise, the spinor gravitational connection upto first order terms is Γμ = (1/2)Vaν Vbν:μ J ab = (1/2)(δaν + faν )(ηbν + fbν ):μ J ab ab = (1/2)(δaν + faν )(−Γα νμ ηbα + fbν,μ )J

= (1/2)(−Γbaμ + fba,μ )J ab = (1/2)((−1/2)(hba,μ + hbμ,a − haμ,b ) + fba,μ )J ab = (1/2)(−fbμ,a + faμ,b )J ab The Dirac equation taking em field interactions and gravitational field interactions into consideration after neglecting nonlinear terms in the metric perturbations hμν is therefore [γ a (δaμ + faμ )(i∂μ + eAμ + (i/2)J ab (faμ,b − fbμ,a )) − m]ψ = 0  s to which further simplifies on neglecting quadratic terms in the faμ

[γ μ (i∂μ + eAμ + (i/2)J ab (faμ,b − fbμ,a )) + iγ a faμ ∂μ − m]ψ = 0 or equivalently, written in the form of first order perturbation theory, (γ μ (i∂μ − m)ψ = −[γ μ (eAμ + (i/2)J ab (faμ,b − fbμ,a )) + iγ a faμ ∂μ ]ψ

Chapter 18

Quantum fluid antennas interacting with media 18.1

Quantum MHD antenna in a quantum gravitational field

The MHD equations in the curved space-time of general relativity are derived using (T μν ):ν = F μν Jν − ΔT:νμν = σF μν Fνα v α − ΔT:νμν where ΔT μν is the contribution to the energy-momentum tensor of the fluid matter coming from viscous and thermal effects. Here, T μν = (ρ + p)v μ v ν − pg μν is the energy-momentum tensor of the fluid matter without taking into account viscous and thermal effects. These equations may also be derived from the Einstein-Maxwell field equations: Rμν − (1/2)Rg μν = −8πG(T μν + S μν ), S μν = (−1/4)F αβ Fαβ g μν + gαβ F μα F νβ , F:νμν = −μ0 J μ , J μ = σF μν vν √ Exercise: The coefficient of −gδgμν in the variation of the action of a field in background curved space-time gives the energy-momentum tensor of the field. For the Dirac field in a background curved space-time with an electromgnetic field, the action is given by  √ S[ψ] = Re[ψ(x)∗ [Vaμ (x)γ 0 γ a (i∂μ + iΓμ + eAμ ) − mγ 0 ]ψ(x)] −gd4 x

289

290

Quantum Antennas

where Vaμ (x) is a tetrad for the metric gμν and Γμ is the spinor connection of the gravitational field: Γμ = (1/2)J ab Vνa V:μbν , J ab = (1/4)[γ a , γ b ] Using the above discussion, evaluate the energy-momentum tensor of the Dirac field and denote this by Tμν . Hence set up the Einstein-Maxwell-Dirac field equations: Rμν − (1/2)Rgμν = −8πG(Tμν + Sμν ) where Sμν = (−1/4)Fαβ F αβ gμν + g αβ Fμα Fνβ is the energy-momentum tensor of the electromagnetic field. The vanishing four divergence of the Einstein tensor implies the field equations (T μν + S μν ):ν = 0 which can be derived from the Maxwell equations with Dirac current: F:νμν = −μ0 J μ where J μ is the coefficient of δAμ in the variation of the above Dirac action functional w.r.t Aμ , ie, J μ (x) = Vaμ (x)ψ(x)∗ γ 0 γ a ψ(x) We now explain how a perturbation theoretic analysis of these equations can be performed to obtain dispersion relations for the perturbed quantities, ie, we linearize the Einstein-Maxwell-Dirac equations around fixed values of these fields and study oscillations in these perturbed quantities.

18.2

Applications of scattering theory to quantum antennas

The basic idea here is the following. We have a projectile coming from time t = −∞ at an infinite distance from the scatterer with an energy of H0 = P 2 /2m = −∇2 /2m in the non-relativistic case or H0 = (α, P ) + βm in the relativistic case. This projectile interacts with the scatterer with an interaction potential of V so that during the interaction period, the total energy of the projectile becomes H = H0 + V . After interaction, the projectile goes at time t = ∞ to infinity. According to first order Born scattering theory, the incident state of the projectile of a steady projectile current is given by ψi (r) = C.exp(−ik.r) so that

H0 ψi = E(k)ψi , E(k) = k 2 /2m

291

Quantum Antennas

in the non-relativistic case and in the relativistic case, C = C(k) becomes a 4 × 1 complex vector satisfying ((α, k) + βm)C(k) = E(k)C(k) √ where E(k) = k 2 + m2 . Let ψf (r) denote the final state of the projectile after it gets scattered. By first order perturbation theory, since the energy of the projectile is conserved, this final state will satisfy the exact equation (H0 + V )ψf = E(k)ψf (r) which approximates to H0 (ψf − ψi ) + V ψi = E(k)(ψf − ψi ) or equivalently, (H0 − E(k))(ψf − ψi ) = −V ψi In the non-relativistic case, this equation is the same as (∇2 + k 2 )(ψf − ψi ) = 2mV (r)ψi (r) which has the solution  ψf (r) = ψi (r) − (m.exp(ikr)/4πr)

V (r )ψi ((r )exp(−ikˆ r.r )d3 r

Equivalently, defining  F (ˆ r) = F (θ, φ) = (−m/4π)

r.r )d3 r V (r )exp(−ik.r)exp(−ikˆ

we can write approximately, ψf (r) = C(exp(−ik.r) + F (ˆ r)exp(ikr)/r) in the non-relativistic case. We may assume that the incident flux of particles is parallel to the z axis and directed from z = −∞ to z = 0. In this case k.r = −kz = −krcos(θ) and we have  F (ˆ r) = (−m/4π) V (r , θ , φ )exp(ik(rcos(θ) − rˆ.r )r2 sin(θ)drdθdφ The number of particles getting scattered per unit solid angle at ∞ is therefore proportional to |F (ˆ r)|2 = |F (θ, φ)|2 . The smeared out scattered charge density r)|2 and the smeared out current is therefore proportional to ρ(r, rˆ) = (q/r2 )|F (ˆ density corresponding to the scattered particles is proportional to J(r, rˆ) = q|F (ˆ r)|2 kˆ r/mr2

292

Quantum Antennas

Note that the smeared out charge and current densities corresponding to a wave function ψ(r) are respectively given by ρ(r) = q|ψ(r)|2 , J(r) = (−i/2m)(ψ(r)∗ (∇ + ieA(r))ψ(r) − ψ(r)(∇ − ieA(r))ψ(r)∗ ) The magnetic vector potential the electric scalar potentials produced by the scattered charges are respectively given by  A(t, r) = (μ/4π) cos(ω(t − |r − r |/c))J(r )d3 r /|r − r | 

and Φ(t, r) = (1/4π)

cos(ω(t − |r − r |/c))ρ(r )d3 r /|r − r |

The question is how to control the scattering potential V subject to constraints such that the resulting EM field pattern generated by the scattered charged particles is as close as possible to a desired pattern. More, generally, we can pose the following problem. Control the electromagnetic field Aμ (x) falling on a charged quantum particle with Hamiltonian H(Q, P ) so that if ψ(t, Q) denotes the Schrodinger wave function satisfying iψ,t (t, Q) = [H(Q, −i∇ − qA(t, Q)) + qΦ(t, Q)]ψ(t, Q) then we require the smeared out charge and current densities corresponding to this wave function to generate an EM field pattern as close as possible to a desired one.

18.3

Wave function of a quantum field with applications to writing down the Schrodinger equation for the expanding universe

Main idea: Assume that we have a set of fields φn (x), n = 1, 2, ..., N whose classical dynamics is described by a Lagrangian density L(x, φn , φn,μ , n = 1, 2, ..., N, μ = 0, 1, 2, 3) We write φn (x) = φn0 (x) + δφn (x) where φn0 (x) are the unperturbed classical fields and δφn (x) are the perturbed quantum field fluctuations. We expand L around φn0 and obtain thereby an infinite series for the Lagrangian density:   L = L0 (x) + a1 (n, x)δφn (x) + b1 (n, μ, x)δφn,μ (x)+ n

n,μ

293

Quantum Antennas 

...+

akm (n1 , n2 , ..., nk , r1 , μ1 , ..., rm , μm , x)δφn1 (x)

n1 ,...,nk ,r1 ,...,rm ,μ1 ,...,μm

...δφnk (x)δφn1 ,μ1 (x)...δφnk ,μk (x)+... If we expand around the stationary value of the Lagrangian then the first order terms do note appear, ie, a1 = 0, b1 = 0 and then we have for example upto third order terms,  a20 (n1 , n2 , x)δφn1 (x)δφn2 (x)+a11 (n1 , r1 , μ1 , x)δφn1 (x)δφr1 ,μ1 (x) L = L0 (x)+ n1 ,n2

+a02 (r1 , μ1 , r2 mu2 )δφr1 ,μ1 (x)δφr2 ,μ2 (x)  a30 (n1 , n2 , n3 , x)δφn1 (x)δφn2 (x)δφn3 (x)

+

n1 ,n2 ,n3



+

a21 (n1 , n2 , r1 , μ1 , x)δφn1 (x)δφn2 (x)δφr1 ,μ1 (x)

n1 ,n2 ,r1 ,μ1



+

a12 (n1 , r1 , μ1 , r2 , μ2 , x)δφn1 (x)δφr1 ,μ1 (x)δφr2 ,μ2 (x)

n1 ,r1 ,μ1 ,r2 ,μ2

+



a03 (r1 , mu1 , r2 , μ2 , r3 , μ3 , x)δφr1 ,μ1 (x)δφr2 ,μ2 (x)δφr3 ,μ3 (x)

r1 ,μ1 ,r2 ,μ2 ,r3 ,μ3

The Hamiltonian density corresponding to this Lagrangian density can be written down by performing the Legendre transformation. Retaining only upto cubic terms in the joint position and momentum fields, we have the following approximation for the Hamiltonian density:  b20 (n1 , n2 )δφn1 (x)δφn2 (x)+ H= n1 ,n2



+

b11 (n1 , r1 )δφn1 (x)δπr1 (x)+

n1 ,r1 ,μ1



c11 (n1 , k1 , n2 )δφn1 ,k1 (x)δφn2 (x)

n1 ,k1 ,n2

+



d11 (n1 , k1 , n2 , k2 )δφn1 ,k1 (x)δφn2 ,k2 (x)

n1 ,k1 ,n2 ,k2

+



f11 (r1 , r2 )δπr1 (x)δπr2 (x)

r1 ,r2

+



g11 (n1 , k1 , r1 )δφn1 ,k1 (x)δπr1 (x)

n1 ,k1 ,r1

+cubic terms involving the products δφn1 δφn2 δφn3 , δφn1 δφn2 δφn3 ,k3 , δφn1 δφn2 ,k2 δφn3 ,k3 , δφn1 δφn2 δπr3 ,

294

Quantum Antennas δφn1 δφn2 ,k2 δπr3 , δφn1 ,k1 δφn2 ,k2 δπr3 , δφn1 δπr2 δπr3 , δφn1 ,k1 δπr2 δπr3 , δπr1 δπr2 δπr3

After this, we replace δπr(x) by the variational differential −iδ/δ(δφn (x)) in the Hamiltonian density Hd3 x and write the Schrodinger equation for the quantum field fluctuations as  [ H(t, rδφn (r), δφn,k (r), −iδ/δ(δφn (r))d3 r]ψ(t, δφn ()) = i∂ψ(t, δφn (.))/∂t By applying this idea to the Einstein-Hilbert-Maxwell-Dirac Lagrangian density described in terms of the quantum fields δgμν (x), δAμ (x), δψl (x), we can arrive at the Schrodinger equation for the universe. A slight variant of this idea has been proposed by Hawking (See Hawking and Penrose ”The nature of space and time”, Oxford,1996.) The idea used by Hawking and Penrose is to expand all the metric tensor coefficients as linear combinations of tensor harmonics, expand the coordinates (which may be chosen arbitrarily) or more specifically, the gauge functions specifying the coordinate system and the velocity field of the matter and the electromagnetic potentials as linear combinations of vector harmonics and finally expand the density field as linear combinations of scalar harmonics. The coefficients in these expansions will be functions of time only while the harmonic functions will be functions of the spatial coordinates only. After substituting these expansions into the overall spatial integral of the Lagrangian density for the gravitational field interacting with the matter field and the EM field, we obtain the Lagrangian of this total field as a nonlinear function of the linear combination coefficients as well as their time derivatives. From this expression, by performing a Legendre transformation, we can obtain the Hamiltonian of the total field as a nonnlinear function of the linear combination coefficients and the corresponding canonical momenta which are simply the partial derivatives of the Lagrangian w.r.t the time derivatives of the linear combination coefficients. Once this discrete form of the Hamiltonian has been set up, the Schrodinger equation for the universe can be immediately set up.

18.4

Simple exclusion process and antenna theory

The generator of a simple exclusion process ηt : Z3N → {0, 1} is given by  η(x)(1 − η(y))(f (η (x,y) ) − f (η)) Lf (η) = x =y

The velocity of a particle at site x when it jumps to y in time dt may be defined as (y − x)/dt ∈ R3 . This is provided that ηt (x) = 1, ηt (y) = 0 and in time

295

Quantum Antennas

[t, t + dt] the jump takes place with p(x, y)dt being the jump probability. Thus, the average velocity associated with a jump from the site x is  E(ηt (x)(1 − ηt (y)))p(x, y)(y − x) v(t, x) = y:y =x

If q denotes the charge placed on each particle in this process, then the average current density is proportional to qv(t, x) and one can in principle calculate the EM field produced by the exclusion process field. More generally, conditioned on the state ηt at time t, the velocity process of a particle at the site x is V (t, x) = ηt (x)(1 − ηt (y))p(x, y)(y − x) and we can calculate the EM field produced by the associated random current qV (t, x) and then compute the moments of this EM field, like the mean EM field and the space-time correlations of the resulting EM field.

18.5

MHD and quantum antenna theory

The basic MHD equations for a conducting fluid are the Maxwell equations along with the Navier-Stokes equation with EM force terms. Assuming incompressibility, these equations are: divJ + ρq,t = 0, divv = 0 J = σ(E + v × B), v,t + (v, ∇)v = −∇p/ρ + ν∇2 v + J × B/ρ divB = 0, curlE = −B,t , curlB = μJ + μE,t , divE = ρq / These equations cannot be derived from a field theoretic Lagrangian or a Hamiltonian because of the terms involving the conductivity σ which cause damping. They may however, be quantized using an appropriate Hamiltonian combined with Lindblad damping terms to take into account the dissipative effects of the conductivity. Consider the special case when the EM field is given and the velocity and pressure field are to be determined using the incompressibility equation and the MHD-Navier-Stokes equation. For quantization purposes, we first analyze how a first order state variable system X  (t) = F (t, X(t)), X(t) ∈ Rn may be derived from a Lagrangian and then from a Hamiltonian. The Lagrangian is elementary, simply introduce a Lagrange multiplier λ(t) ∈ Rn and the Lagrangian L(X, X  , λ, λ ) = λ(t)T (X  (t) − F (t, X(t)))

296

Quantum Antennas

Then the Euler-Lagrange equations are λ (t) + F (t, X(t)) = 0, X  (t) − F (t, X(t)) = 0 These are respectively the co-state and state equations. However, if we try to introduce a Hamiltonian by defining the canonical momenta pλ =

∂L = 0, ∂λ

∂L =λ ∂X  then we cannot construct the Legendre transormation pX =

H = pTλ λ − L since there is no way to solve for λ in terms of X, pX , pλ . However, we note that by introducing a small perturbation parameter δ and modifying the Lagrangian to  L(X, X  , λ, λ ) = λT (X  − F (t, X)) + δ.λ T X  we get so that

pX = λ + δX  , pλ = δX  X  = δ −1 pλ , λ = (pX − pλ )

we can write the Hamiltonian as H(X, λ, pX , pλ ) = pTX X  + pTλ λ − L = 

pTX X  + pTλ λ − λT (X − F ) − δλ T X  = δ −1 pTX pλ − λT X + λT F (t, X) and this Hamiltonian can be quantized immediately and δ set to zero at the end of all the calculations. By adopting the same approach, we shall write down the Lagrangian density of the velocity and pressure field as Lv = λT (v,t + (v, ∇)v + ∇p/ρ + ν∇2 v − α(E + v × B) × B) +δ(λT,t v,t + μ,t p,t ) + μ.divv where v = v(t, r), p = p(t, r), λ = λ(t, r), μ = μ(t, r) are fields. To this we add the standard Lagrangian density of the electromagnetic field interacting with the current density: LEM = (/2)|E|2 − (1/2μ)|B|2 − ρq Φ + J.A

297

Quantum Antennas where we substitute  J = σ(E + v × B), ρq = −

t

divJdt 0

and E = −∇Φ − A,t , B = ∇ × A The total Lagrangian density must be taken as the sum of both the Lagrangian densities and the position fields are v, p, λ, μ, A, Φ. It is instructive to carry out the variation in the resulting action and see how the resulting field equations differ from the basic MHD equations coupled to the Maxwell equations. Once we have quantized the fluid and electromagnetic fields, we can regard the electric, magnetic and fluid velocity fields as field operators and calculate the far field antenna Poynting vector pattern and its expectation, correlations and more generally higher order moments in any given initial state of the fluid and electromagnetic field. This would be another way of looking at a quantum MHD antenna.

18.6

Approximate Hamiltonian formulation of the diffusion equation with applications to quantum antenna theory

The Lagrangian density of our theory is L(u, u,t , u,x , λ, λ,t ) = λ(u,t − Du,xx ) + δλ,t u,t This is equivalent to L = λu,t + Dλ,x u,x + δλ,t u,t The canonical momenta are pu =

∂L = λ + δλ,t ∂u,t

pλ =

∂L = δu,t ∂λ,t

Thus, the Hamiltonian density is given by H(u, λ, pu , pλ ) = pu u,t + pλ λ,t − L = δ −1 pu pλ + δu,t λ,t − λu,t − Dλ,x u,x − δλ,t u,t = δ −1 pλ (pu − λ) − Dλ,x u,x Let us write down the canonical Hamiltonian equations: u,t = δH/δpu = δ −1 pλ

298

Quantum Antennas λ,t = δH/δpλ = δ −1 (pu − λ) pu,t = −δH/δu = ∂x (∂H/∂u,x ) = −Dλ,xx , pλ,t = −δH/δλ = −∂H/∂λ + ∂x (∂H/∂λ,x ) = δ −1 pλ − Du,xx

From these equations, we can arrive at the modified heat equation: u,t = δ −1 pλ = pλ,t + Du,xx = δu,tt + Du,xx which in the limit of δ → 0 becomes the heat equation u,t = Du,xx

18.7

Derivation of the damped wave equation for the electromagnetic field in a conducting media in quantum mechanics using the Lindblad formalism

Consider first the damped wave equation in one dimension u,tt + γu,t − c2 u,xx = 0 This equation for u(t, x) is valid for x ∈ [0, L] and to quantize it, we expand the solution as a Fourier series in x:  cn (t)exp(2πinx/L) u(t, x) = n∈Z

Substituting this into the damped wave equation gives us a sequence of ode’s cn (t) + (2πinγ/L)cn (t) + (2nπc/L)2 cn (t) = 0, n ∈ Z We wish to derive these equations from a quantum mechanical formalism. To this end, we express cn (t) = an (t) + ibn (t) and express the above as two real equations: an (t) + ωn2 an (t) − γωn bn (t) = 0, bn (t) + ωn2 bn (t) + γωn an (t) = 0 We take as our unperturbed Hamiltonian H = (p2n + ωn2 qn2 )/2

299

Quantum Antennas and Choose the Lindblad operators as Ln = α n q n + β + β n p n where [qn , pm ] = iδnm

Then apply the Heisenberg form the the Lindblad equations for any observable X:  (L∗n Ln X + XL∗n Ln − 2L∗n XLn ) dX/dt = i[H, X] − (1/2) n

= i[H, X] − (1/2)



(L∗n [Ln , X] + [X, L∗n ]Ln )

n

Taking X = qn gives dqn /dt = pn − (1/2)(−iβn L∗n + iβ¯n Ln ) αn qn + β¯n pn )) = pn − (i/2)(β¯n (αn qn + βn pn ) − βn (¯ = pn + Im(αn β¯n )qn Taking X = pn gives α n Ln ) dpn /dt = −ωn2 qn − (1/2)(iαn L∗n − i¯ αn qn + β¯n pn ) − α ¯ n (αn qn + βn pn )) = −ωn2 qn − (i/2)(αn (¯ ¯ n )pn = −ω 2 qn − (i/2)(αn β¯n − βn α n

= −ωn2 qn + Im(αn β¯n )pn Writing γn = Im(¯ αn βn ), we can express the above equations in the form pn = −ωn2 qn − γn pn , qn = pn − γn qn so that

qn = pn − γn qn = −ωn2 qn − γn pn − γn qn = −ωn2 qn − γn (qn + γn qn ) − γn qn = −(ωn2 + γn2 )qn − 2γn qn

which is precisely the damped harmonic oscillator equation. We note that in the limit of large γn , this differential equation approximates to a first order differential equation in time like the heat equation: qn ≈ −(γn /2)qn More general kinds of damped oscillator equations involving vector operator valued functions of time as considered above by Fourier series expansion of the wave equation with damping can be obtained by considering q(t) = (q1 (t), ..., qN (t))T , p(t) = (p1 (t), ..., pn (t))T

300

Quantum Antennas with [qi , pj ] = δij and using the Hamiltonian H = (pT p + q T Kq)/2 = (1/2)(

n 

p2k +

n 

K(r, s)qr qs )

r,s=1

k=1

with Lindblad operators Ln = αnT q + βnT p We find that dqn /dt = i[H, qn ] − (1/2)



(L∗m Lm qn + qn L∗m Lm − 2L∗m qn Lm )

m

dpn /dt = i[H, pn ] − (1/2)



(L∗m Lm pn + pn L∗m Lm − 2L∗m pn Lm )

m

We find that T [Lm , qn ] = [βm p, qn ] = −iβm [n], T p] = iβ¯m [n] [qn , L∗m ] = [qn , β¯m

[Lm , pn ] = iαm [n], ¯ m [n] [pn , L∗m ] = −iα Further, [H, qn ] = −ipn , [H, pn ] = i



K(n, r)qr

r

So we get for the noisy Heisenberg dynamics,  dqn /dt = pn − (1/2) (−iβm [n]L∗m + iβ¯m [n]Lm ) m

dpn /dt = −(Kp)n − (1/2)

 (iαm [n]L∗m − i¯ αm [n]Lm ) m

Equivalently, dqn /dt = pn − (1/2)

 ∗ ∗ T T [(−iβm [n](αm q + βm p) + iβ¯m [n](αm q + βm p)] m

= pn −



∗ T (Im(βm [n]αm )q − Im(β¯m [n]βm )p)

m

or equivalently in vector operator notation,  T ∗ dq/dt = p + (Im(β¯m βm )p − Im(βm αm )q) m

301

Quantum Antennas and likewise, dp/dt = −Kp +

 ∗ T (Im(αm αm )q − Im(¯ αm βm )p) m

Define the matrices A11 = −



∗ Im(βm αm ), A12 =



m

A21 =



T Im(β¯m βm )=−

m

∗ Im(αm αm ), A22 = −

m





∗ Im(βm βm )

m

T Im(¯ αm β m )=

m



∗ Im(αm βm )

m

Then, the above noisy Heisenberg dynamics can be expressed as dq/dt = A11 q + (A12 + I)p, dp/dt = A21 q + (A22 − K)p

Now consider a system of n charged quantum particles with (q1 , ..., qn ) describing their positions and (p1 , ..., pn ) their momenta. We assume that each qk and each pk is a 3-vector operator valued observable and that ek is the charge placed on the k th particle. The equations of motion of these particles are described by the above noisy Heisenberg model of a system of damped harmonic oscillators. We wish to calculate the statistics of the radiation field produced by these particles when the initial wave function of the particles is ψ0 (q1 , ..., qn ) or more generally, when the initial state of this system of particles is the mixed state with kernel ρ0 (q|q  ) with q = (q1 , ..., qn ) and q  = (q1 , ..., qn ). The electromagnetic field produced by these particles is described by the four-potential A(t, r) =



 (μek /4π)

Jk (t − |r − r |/c, r )d3 r /|r − r |

k

Φ(t, r) =



 (ek /4π)

ρk (t − |r − r |/c, r )d3 r /|r − r |

k

where Jk (t, r) and ρk (t, r) are Heisenberg operator fields defined by Jk (t, r) = (ek /2)(qk (t)δ 3 (r − qk (t)) + δ 3 (r − qk (t))qk (t)) ρk (t, r) = ek δ 3 (r − qk (t))

302

Quantum Antennas

18.8

Boson-Fermion unification in quantum stochastic calculus

Let At , A∗t , Λt denote the canonical noise processes in the Hudson-Parthasrathy quantum stochastic calculus. Let W (u, U ), u ∈ L2 (R+ ), U ∈ U (L2 (R+ )) denote the Weyl operator in Γs (L2 (R+ )). Define dBt = (−1)Λt dAt , dBt∗ = (−1)Λt dA∗t We have

(−1)Λt = exp(iπΛt ) = exp(iπλ(χ[0,t] I) = W (0, exp(iπχ[0,t] I))

so that dBt |e(f ) >= f (t)dt(−1)λt |e(f ) >= f (t)dt|e(exp(iπχ[0,t] )f > = f (t)dt|e(−χ[0,t] f + χ(t,∞) f ) > dBs dBt |e(f ) >= f (t)dt.(−χ[0,t] (s)f (s) + χ(t,∞) (s)f (s))ds. ×|e((−χ[0,s] + χ(s,∞) )(−χ[0,t] + χ(t,∞) )f ) > Thus, for any s, t, we have (dBs dBt + dBt dBs )|e(f ) >= (f (t)dt.(−χ[0,t] (s)f (s)+χ(t,∞) (s)f (s))ds +f (s)ds(−χ[0,s] (t)f (t)+χ(s,∞) (t)f (t))dt ×|e((−χ[0,s] + χ(s,∞) )(−χ[0,t] + χ(t,∞) )f ) > =0

since

χ[0,t] (s) + χ[0,s] (t) = 1a.s., χ(t,∞) (s) + χ(s,∞) (t) = 1, a.s. This proves that for all s, t, B s Bt + Bt Bs = 0 Taking the adjoint gives

Bs∗ Bt∗ + Bt∗ Bs∗ = 0

Now, for s ≤ t, < e(g)|dBs∗ dBt |e(f ) >=< dBs e(g), dBt |e(f ) > = g¯(s)f (t)dsdt < e(−χ[0,s] g + χ(s,∞) g), e(−χ[0,t] f + χ(t,∞) f ) > and

< e(g)|dBt dBs∗ |e(f ) >=< e(g)|(−1)Λt (−1)Λs dAt dA∗s |e(f ) >

303

Quantum Antennas When s = t, by quantum Ito’s formula, this equals < e(g), e(f ) > dt while if s < t, it equals = f (t)dt < e(g)|(−1)Λt (−1)Λs dA∗s |e(f ) > = f (t)dt < (−1)Λt e(g), (−1)Λs dA∗s |e(f ) > = f (t)dt < dAs (e(−χ[0,t] g + χ(t,∞) g)), (−1)Λs |e(f ) > = f (t)dt(−¯ g (s)ds) < e(−χ[0,t] g + χ(t,∞) g)), |e(−χ[0,s] f + χ(s,∞) f ) > It follows that for s < t, < e(g)|dBs∗ dBt + dBt dBs∗ |e(f ) >= 0 and

< e(g)|dBt∗ dBt + dBt dBt∗ |e(f ) >= dt < e(g), e(f ) >

From these relations, by quantum stochastic integration, we easily deduce the canonical anticommutation relations: For all t, s ∈ R+ , we have [Bt , Bs ]+ = 0, [Bt∗ , Bs∗ ]+ = 0, [Bt , Bs∗ ]+ = min(t, s)

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Quantum Antennas

References [1] Steven Weinberg, ”The quantum theory of fields, vols. I,II,III. Cambridge University Press. [2] Steven Weinberg, ”Gravitation and Cosmology:Principles and applications of the general theory of relativity, Wiley. [3] H.Parthasarathy, ”General relativity and its engineering applications”, Manakin press. [4] K.R.Parthasarathy, ”An introduction to quantum stochastic calculus, Birkhauser, 1992. [5] Constantine Balanis, ”Antenna theory”, Wiley. [6] P.A.M.Dirac, ”Principles of quantum mechanics”, Oxford University Press. [7] V.S.Varadarajan ”Harmonic analysis on semisimple Lie groups, Cambridge University Press. [8] Naman Garg and H.Parthasarathy, ”Belavkin filter applied to estimating the atomic observables from non-demolition quantum electromagnetic field measurements”, Technical report, NSIT, 2017. [9] Mark Wilde, ”Quantum Information Theory”. [10] K.R.Parthasarathy, ”Coding theorems of classical and quantum information theory”, Hindustan Book Agency. [11] M.Hayashi, ”Quantum Information”. [12] D.Revuz and M.Yor, ”Continuous Martingales and Brownian Motion”, Springer. [13] A.V.Skorohod, ”Controlled Stochastic Processes”, Springer. [14] D.Stroock and S.R.S.Varadhan, ”Multidimensional Diffusion Processes”, Springer. [15] Pushkar Kumar, Kumar Gautam, Navneet Sharma, Naman Garg and Harish Parthasarathy, ”Design of quantum gates using the quantum stochastic calculus of Hudson and Parthasarathy”, Technical report, NSIT, 2018. [16] Vijay Mohan and Harish Parthasarathy, ”Some versions of quantum stochastic optimal control”, Technical report, NSIT, 2018. [17] K.R.Parthasarathy, ”An introduction to quantum stochastic calculus”, Birkhauser, 1992.