Developments in Mathematical and Conceptual Physics

Harish Parthasarathy

Concepts and Applications for Engineers

Developments in Mathematical and ConceptualPhysics

Harish Parthasarathy

Developmentsin Mathematicaland Conceptual PhysicsConcepts and Applications for Engineers

123

Harish ParthasarathyDepartment of Electronics andCommunication EngineeringNetaji Subhas University of TechnologyDelhi, India

ISBN 978-981-15-5057-7 ISBN 978-981-15-5058-4 (eBook)https://doi.org/10.1007/978-981-15-5058-4

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Preface

This book contains some special topics of recent research interest in most of theimportant topics in theoretical physics. It will be of value to the practising theo-retical physicist as well as to applied mathematicians and engineers who wish toapply the techniques developed by physicists to engineering problems especially innonlinear systems, signal processing and probabilistic analysis of systems. Thechapter on classical mechanics talks about the Lagrangian formalism, then intro-duces the theory of the motion of particles in a potential subject to damping andrandom forces and sets up the Fokker–Planck equation that governs the evolutionof the probability density of the particle or systems of particles in phase space usingwhich the classical fluctuation–dissipation theorem of Kubo is derived as a con-dition for balance between the random fluctuating forces and dissipative forces sothat at equilibrium the density is Gibbsian. Possible extensions to a quantumfluctuation–dissipation theorem based on the noisy Schrodinger equation for den-sity operators have been indicated although this is a chapter on classical mechanics.The next two chapters on classical electrodynamics and general classical generalrelativity (referred to by Landau and Lifshitz as the “Classical theory of fields”) dealwith a variety of problems involving antennas, waveguides, transmission lines andelectromagnetic wave propagation in inhomogeneous, anisotropic andfield-dependent media, some of which include the standard material covered inundergraduate courses, some of which depart from the standard syllabi to includemethods for computing numerically the surface current density induced on anantenna sheet when an electromagnetic wave is incident upon it by the method ofintegral equations and some include the analysis of waveguide and cavity resonatormodes when the medium filling these structures is inhomogeneous and frequencydependent and methods involving the application of eigenvalue perturbation theoryfor linear partial differential operators with boundary conditions have been usedhere. This chapter also contains some material on the behaviour of electromagneticwaves in a gravitational field described by the metric tensor of general relativity.This involves setting up the covariant Maxwell equations in a metric that departsslightly from flat space–time Minkowskian and then using perturbation theory forpde’s to construct approximate solutions for the electromagnetic waves perturbed

v

by a gravitational field. This problem has recently become very important indetecting gravitational waves produced by blackhole collision using their effect onthe electromagnetic wave generated by a laser. This chapter also talks about howconfined electromagnetic waves in structures like waveguides and resonator cavitiesbehave near a strong gravitational field. Our method of analysis involves assumingsome sort of symmetry of the metric and then setting up the covariant Maxwellequations for the em field inside the structure directly in terms of the electromag-netic antisymmetric field tensor and imposing boundary conditions on the com-ponents of this tensor to directly arrive at curved space–time generalizations of thestandard flat space–time waveguide/resonator formulas which express the trans-verse components of the electromagnetic field in terms of their longitudinal com-ponents. This chapter also contains some new material on how to analysewaveguides and resonators whose cross sections have aribitrary shape by using theexpressions for the operators of vector calculus in orthogonal curvilinear coordinatesystems. Derivation of the expressions for the transverse field components in termsof the longitudinal components in orthogonal curvilinear systems as well as the 2-DHelmholtz equation with boundary conditions for the longitudinal em field com-ponents in orthogonal curvilinear systems has been presented. This chapter alsoexplains how, when the boundary is a small perturbation of a circular boundary, wecan apply perturbation theory to solve the associated waveguide or cavity resonatorproblem by the application of a conformal transformation to the boundary. Thecurved boundary value problem has been reduced to a non-curved boundary valueproblem by absorbing the curvilinear boundary effect into the Helmholtz operator.Propagation of electromagnetic waves in the standard Robertson–Walker cosmo-logical metric has also been studied here. The metric is again assumed to be a smallperturbation of the flat space–time metric and hence the expansion factor of theuniverse with time appears as a small time-dependent perturbation to the waveequation satisfied by electromagnetic fields. Formulation of the Einstein–Maxwellfield equations in the synchronous reference system (to which any metric can betransformed by passing over to an appropriate coordinate system) has been men-tioned. The methodology of taking into account rotational effects of a distribution ofmatter and how to setup the Einstein–Maxwell equations in such a rotating metrichas been indicated. It is well known that there is an exact solution for the metricproduced by a rotating sphere of matter, namely the Kerr metric. Thus our dis-cussion would enable us to determine how waveguides and cavity resonatorsbehave near a rotating blackhole. The chapter on general relativity and cosmologyalso discusses two very important problems in considerable detail. First, theproblem of galactic evolution based on solving the linearized Einstein field equa-tions in the presence of viscous matter, taking into account contributions to theenergy-momentum tensor due to viscosity and thermal effects. The idea is to firstsolve the Einstein field equations in the absence of inhomogeneities leading to theRobertson–Walker metric which describes a homogeneous and isotropic universe.

vi Preface

We then select an appropriate coordinate system so that metric perturbations aroundthis RW metric have only six independent components, and taken along withvelocity and density perturbations, the linearized field equations involve only tenindependent components. We obtain linear partial differential equations for thesecomponents using which we describe galactic evolution as an evolution of smallinhomogeneities in the metric, velocity and density perturbations. The expandinguniverse as described by the RW metric causes these inhomogeneities to expandwith time, and it is possible to derive dispersion relations that describe theseevolutions in spatial wave number and time space. It should be borne in mind thatthe derivation of the energy-momentum tensor due to viscous and thermal effectsinvolves entropy considerations and the first two laws of thermodynamics. Thesecond important problem discussed in the chapter on general relativity ispost-Newtonian hydrodynamics which presents a perturbative algorithm for solvingthe Einstein field equations in the presence of a matter fluid. The metric tensor,density, pressure and velocity are expanded in powers of the square root of the mass(which is proportional to the characteristic velocity scale from Newton’ law ofgravitation) and then successively equate terms of equals orders in the Einstein fieldequations and in the hydrodynamic equation which is nothing but the equation ofenergy-momentum conservation (which is obtained by forming the covariantdivergence of the Einstein field equations and making use of the Bianchi identity).Apart from these two salient features of this chapter, we’ve mentioned the problemof calculating the effect of the cosmic microwave background radiation on theexpanding universe and conversely the effect of the expanding universe on thecosmic microwave background radiation. To do this, we must perturbatively solvethe Einstein–Maxwell equations with the energy-momentum tensor of the electro-magnetic field being replaced by its ensemble average value. The next two chaptersare on quantum mechanics and quantum field theory wherein we discuss someinteresting new topics of recent research interest like the Hudson–Parthasarathytheory of quantum noise, quantum Gaussian states, the behaviour of atoms andelectrons in a gravitational field, i.e. formulation of the Schrodinger and Dirac waveequations in curved space–time and how to calculate the effect of gravitation onatomic phenomena. We also discuss quantum electrodynamics in which compu-tation of the electron and photon propagators are introduced with application toquantum scattering theory involving photons, electrons, positrons and gravitons.The Feynman diagrammatic rules for calculating the amplitudes for these scatteringprocesses are discussed. Everywhere, we start with the total action for the gravi-tational field, the Dirac field and the Maxwell photon field and derive a Hamiltonianfrom it via the Legendre transformation followed by the Dyson series interactionpicture for the scattering matrix leading finally to the Feynman diagrammatic rules.Quantization of the classical Hamiltonian in order to obtain the Dyson seriesinvolves the introduction of the creation and annihilation operators for photons andgravitons which are bosonic fields and for the electrons and positrons which are

Preface vii

Fermionic fields, noting that bosonic operator fields satisfy the canonical commu-tation relations (CCR) while the Fermionic fields satisfy the canonical anticom-mutation relations (CAR) in view of the Pauli exclusion principle. Quantum noiseand quantum Ito’s formula for creation, annihilation and conservation processes asfirst conceived by R. L. Hudson and K. R. Parthasarathy in their path-breaking 1984paper are introduced along with techniques for solving the Schrodinger and Diracequations perturbatively in the presence of these quantum noise processes. Theeffect of quantum noise as described by Hudson and Parthasarathy on waveguideand resonator electromagnetic fields is also described. Also the effect these quantumnoise fields on Einstein’s theory of gravitation is discussed. The effect of quantumnoise on a general quantum field theory has been emphasized. We have proposedtechniques to design quantum unitary gates of large sizes like the quantum Fouriertransform gate by perturbing the Hamiltonian density of the field with quantum fieldtheoretic potentials and matching the unitary dynamics of such an evolution with agiven unitary gate. The design of unitary quantum gates of large size usingFeynman diagrams has also been mentioned. To do this, we write down the scat-tering matrix for the scattering of two electrons taking a control external field intoconsideration. The evaluation of this diagram uses the electron and photon prop-agator, and we match this scattering matrix to a desired matrix by adjusting theexternal control field. This problem is of importance for the computer scientist whois interested in designing gates for quantum computers that perform classicalcomputational jobs like order finding, phase estimation, searching and Fouriertransforming with much lesser computational complexity than classical algorithms.The chapter on quantum field theory also presents some material on non-Abeliangauge theories for matter and gauge fields as generalizations of the Uð1Þ electro-magnetic gauge theory interacting with a Schrodinger or Dirac particle. Thesenon-Abelian gauge theories also known as the Yang–Mills theory are important indescribing weak and strong nuclear forces and electro-weak-strong unification.Symmetry breaking caused by the interaction of this non-Abelian field with thescalar Higgs field gives masses to the gauge bosons that communicate the weak andstrong forces but does not give any mass to the gauge boson for the electromagneticfield which communicates electric and magnetic forces between charges, i.e. thephoton. The idea of spontaneous symmetric breaking which produces masslessGoldstone bosons when the field goes to the ground state and symmetry breakingwhich arises by corrections to the Lagrangian which gives masses to masslessparticles as in the electro-weak theory has been explained with care. Quantization ofnon-Abelian gauge theories using path integrals has also been dealt with in thisbook. The idea is to start with a gauge-invariant action and path, measure andincorporate a gauge fixing functional of the fields into this path integral and thenprove that such a path integral is essentially independent of our choice of the gaugefixing functional. We then introduce auxiliary and ghost fields to represent the effectof the gauge fixing functional and draw Feynman diagrams for the matter, gauge,

viii Preface

auxiliary and Ghost fields. The action functional for the gauge field is of fourthdegree owing to the gauge covariant derivative, so we must during the evaluationof the path integrals approximate such actions by second degree terms and this canbe achieved by assuming that the quantum gauge field is a small perturbation of abackground classical gauge field. Finally, the more recent supersymmetry theoryhas been discussed in the chapter on quantum field theory. Supersymmetry treatsBosons and Fermions on the same footing and thus along with the Bosonic gen-erators, namely four momenta and angular momenta, i.e. generators of the Poincaregroup, it also introduces Fermionic generators into the Lie algebra whose com-mutators are Bosonic generators along with operators that describe the internaldegrees of freedom. Construction of representations of the Fermionic generators ofthis algebra was first accomplished by Salam and Strathdhee who introduced supervector fields in terms of Bosonic space–time derivatives and Fermionic derivativesw.r.t Majorana spinors. There are four Bosonic variables, namely the commutingspace–time coordinates and four anticommuting Fermionic variables, namelyMajorana spinors. The commutators of the super vector fields of Salam andStrathdhee are linear combinations of the Bosonic space–time derivatives and hencewe get a complete representation of this super-algebra. These super vector fields acton superfields that are smooth functions of the four Bosonic space–time variablesand the four Fermionic variables. Since the Fermionic variables all anticommute, asuperfield can be expressed as a fourth-degree polynomial in the Fermionic vari-ables whose coefficients are all ordinary space–time functions, and these functionsare called component fields. The super vector field of Salam and Strathdhee whenacting on a superfield induces infinitesimal changes in the component fields knownas infinitesimal supersymmetry transformations. The objective of supersymmetrytheory is to construct Lagrangian densities from the component superfields thatchange only by a total of four divergences under an infinitesimal supersymmetrytransformation. Such Lagrangian densities constructed for matter fields whenanalysed carefully, correspond to combinations of Dirac Lagrangians, Scalar fieldLagrangians and non-Abelian matter field Lagrangians. We then define thesuper-gauge fields defined in terms of the component superfields and define asuper-gauge transformation acting on the matter and gauge superfields. Theobjective is that gauge Lagrangian should be supersymmetric and also super-gaugesymmetric, i.e. under supersymmetry transformations, it should change at most by atotal space–time differential, and under gauge transformations, it should beinvariant. The matter component of the total Lagrangian should also, of course, beinvariant under this gauge transformation apart from its integral being invariantunder supersymmetry. After all these conditions are accounted for, one finds thatthe gauge part of the Lagrangian contains the electromagnetic field, the non-Abeliangauge field and some other Fermionic gauge field Lagrangians apart from anauxiliary gauge Lagrangian. In this way, many new features are observed like themass of the Dirac field particle depends on the scalar field, etc. One can also include

Preface ix

a gravitational component into this Lagrangian in the form of a metric superfield.While formulating a quantum theory of superfields, we may make use of super pathintegrals of superfield, i.e. express the supersymetric and gauge-invariant actionfunctional as the integral of a superfield w.r.t the Bosonic and Fermionic variables.Quantum unitary gates may also be designed using quantization of superfields. Theidea is that if a system of Bosonic and Fermionic particles actually follows equa-tions of motion that are invariant under supersymmetry, then we can naturally usesuch a physical system to design a quantum gate by writing down the Hamiltoniancorresponding to the supersymmetric Lagrangian and so on. Finally, the chapter onnonlinear systems focuses on application of statistical methods like large deviationtheory and signal processing algorithms to problems of theoretical physics likecomputation of small fluctuating random forces acting on matter and influencing,thereby the gravitational field in accord with the Einstein field equations, classicaland quantum filtering theory, etc.

Delhi, India Harish Parthasarathy

x Preface

Contents

1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 General Relativity and Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Quantum Mechanics and Quantum Stochastic Processes . . . . . . . . . 117

6 Quantum Field Theory and Quantum Gravity . . . . . . . . . . . . . . . . . 173

7 A Lecture on Quantum Field Theory with EngineeringApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8 The General Theory of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . 355

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

xi

Detailed Table of Contents

Chapter 1: Classical mechanicsSome problems in classical mechanics

[1] Lagrangian of a free particle.[2] Fluctuation–dissipation theorem for a particle moving in one dimension in a

potential field subject to damping and fluctuating forces.[3] Fluctuation–dissipation theorem for n particles moving in a potential field

subject to linear velocity damping and fluctuating forces.[4] Quantum Fluctuation–dissipation theorem for open quantum systems

described by Hamiltonian and Lindblad operators with equilibrium Gibbsstate for the density matrix.

[5] Classical and quantum fluctuation–dissipation theorems for particles movingin an electromagnetic field subject to dissipation and fluctuating forces.

Chapter 2: Fluid dynamicsA. Study Projects

[1] Derivation of the Navier–Stokes equation from Boltzmann’s kinetic transportequation.

[2] Derivation of the Navier–Stokes equation from an action principle byintroducing costates via Lagrange multipliers.

[3] Navier–Stokes equation with generalized stress tensor in different coordinatesystems.

[4] Energy-momentum tensor of a fluid in the special and general theories ofrelativity taking viscous and thermal effects into account with application tothe study of galactic evolution from linearized Einstein’s field equations.

[5] Energy equation for an adiabatic fluid taking internal energy and enthalpyinto consideration via the first and second laws of thermodynamics.

[6] Sphere moving in a viscous fluid; Stokes formula for the viscous force on thesphere and the velocity field of the fluid relative to the sphere.

xiii

[7] Vorticity equation and stream function equation with application to using theextended Kalman filter for estimating the fluid velocity field from discretespatial measurements.

[8] Magnetohydrodynamic equations for a conducting fluid with J � B forcingterm included. Applications to plasma physics.

B. Problems and remarks

[1] Basic equations of non-relativistic fluid dynamics using the energy-momentum tensor. How mass generation terms are introduced.

[2] Basic equations of relativistic fluid dynamics from velocity moments of theBoltzmann kinetic transport equation.

[3] Velocity correlations and higher moments.[4] Kolmogorov’s statistical theory of turbulence from partial differential

equations satisfied by the velocity moments.

Chapter 3: Electrodynamics

[1] Perturbation theory applied to the derivation of integral equations for thesurface current density on an antenna surface placed in a nonlinear (fielddependent) inhomogeneous and anisotropic medium taking general rel-ativistic (gravity/curvature of space–time) corrections into account.Frequency domain analysis in a static synchronous reference system.

[2] Perturbative analysis of Maxwell’s equations in an inhomogeneous,anisotropic and nonlinear medium taking gravitational effects intoaccount.

[3] Special topics in transmission lines and waveguides[1] Transmission lines with distributed parameters being random functions

of space and time coordinates, analysis using perturbation theory forpartial differential equations taking into account random line loading.

[2] Transmission lines with non-uniform distributed parameters which donot vary with time. Analysis using spatial Fourier series.

[3] Transmission lines modelled using an infinite-dimensional linearstochastic differential equation taking random voltage and current lineloading into account.

[4] Some general classroom questions in Electromagnetics[1] Retarded potentials of a point charge moving along an arbitrary

trajectory.[2] Snell’s laws for plane waves incident on a boundary separating two

media with different permittivity and permeability; expressions for thereflected and transmitted power flux.

[3] Gauss’ divergence theorem for elliptic curves in elliptic coordinates.[4] General analysis of magnetic circuits.[5] Maxwell’s equation inside a cylindrical capacitor with parallel circular

plates at arbitrary frequencies.

xiv Detailed Table of Contents

[6] Laplace’s equation within a shell bounded by concentric spherical sur-faces general solution for the potential using spherical harmonics.

[7] Necessity for introducing the displacement current correction term inAmpere’s law.

[8] [a] Uniqueness theorem for Poisson’s equation under general mixedDirichlet and Neumann boundary conditions.

[b] Electromagnetic brakes and eddy currents for heating as applicationexamples of Faraday’s law of induction.

[c] General solution to the wave equation in three dimensions as a super-position of plane waves.

[d] Approximate solution to Poisson’s equation using the finite-elementmethod and the finite-difference’ method.

[e] Derivation of the Biot–Savart law for magnetostatics and its applicationto calculating the magnetic field produced by a DC current-carrying wireof arbitrary shape.

[f] Boundary conditions on the electric and magnetic field at a planarinterface with equation px + qy + rz = d when the plane carries aconstant surface current with density Js and a constant surface chargewith density r. The dielectrics on the two sides have parametersð�k; lkÞ; k ¼ 1; 2, respectively.

[5] Nonlinear effects in transmission lines and wave guides with their impacton quantum phenomena: A study project.

[6] Gradient, divergence and curl in an arbitrary orthogonal curvilinearcoordinate system with applications to waveguides and cavity resonatorshaving arbitrary boundaries.

[7] A problem in cladded cylindrical waveguides: Calculating the modes ofpropagation within two concentric cylinders of radii a < b. Region q <a has different permittivity and permeability from region a < q < b.

[8] Cavity resonators of arbitrary cross section: Analysis using formulas forthe vector operations in arbitary orthogonal curvilinear coordinatesystems.

[9] Waveguides having cross section of arbitrary shape with inhomogeneousmedium, analysis using analytic functions of a complex variable.

[10] Quantum mechanical particle in a 2-D box with curved boundary q1 = c:Analysis using Laplacian in orthogonal curvilinear coordinates.

[11] Waveguides and cavity resonators with arbitrary boundary in the pres-ence of a strong gravitational field: Analysis using Maxwell’s equationsin orthogonal curvilinear coordinates in a curved space–time metric.

[12] Some aspects of plasma physics via the Boltzmann kinetic transportequation: Computing the conductivity of the plasma when the electro-magnetic is a plane wave based on first-order perurbation theory appliedto the Boltzmann equation.

[13] Hamiltonian formulation of electromagnetic field theory in a curvedbackground space–time

Detailed Table of Contents xv

[14] On a problem suggested by Dr. Shailesh Mishra regarding computationof the transfer function between the incident electric field on the kthtransmitted antenna surface and the nth receiver antenna surface whenthere are N transmitter antennas and M antennas with a mediumdescribed by a matrix Green function.

[15] Problems in Transmission line and waveguide theory[1] Computing the distributed parameters of a transmission line comprising

two cylindrical conductors placed parallely to each other.[2] Two stub matching of transmission lines.[3] Computing the propagation constant and reflection coefficient from

locations of voltage maxima, current minima and VSWR along the line.[4] Resonant frequencies within a cylindrial cavity resonator in terms of the

zeros of the Bessel functions and its derivatives.[5] Computing the resonant modes within a cylindrical cavity of height

d with water filled till height d1 < d.[6] 2-D Helmholtz equation in a waveguide of arbitrary cross section

derived from expressions of the transverse field components in termsof the longitudinal components using an orthogonal curvilinear coordi-nate system.

[7]

[a] Boundary conditions for calculating resonant frequencies in a cavityresonator.

[b] Q-factor of a rectangular cavity resonator.[c] Calculating the line voltage and line current in the time domain for a lossless

transmission line terminated with a load given by a series combination of acapacitor and resistor.

[d] Distortion-less transmission lines.[e] Quarter-wave transformer.

Chapter 4: General Relativity and Cosmology

[1] The synchronous reference system, geodesic equations and the Einstein–Maxwell field equations in such a system

[2] Perturbations of the flat and curved isotropic model: The Einstein fieldequations for the general curved isotropic model leading to ordinary dif-ferential equations for the scale factor, the density and pressure in theexpanding universe. Small perturbations of this model lead to the linearizedEinstein–Maxwell equations that describe the evolution of inhomogeneitieslike the velocity and density field of galaxies and the perturbations of thecosmic microwave background radiation field.

[3] A particular solution of the Maxwell equations in the Robertson–Walkermetric involves setting up the Covariant Maxwell equations in theRobertson–Walker metric and examining some symmetries of the resultingpartial differential equations.

xvi Detailed Table of Contents

[4] Gravitational collapse of a dust sphere in general relativity involves writingdown the Einstein field equations in a radially symmetric metric with metriccoefficients dependent only on time and the radial coordinate. The matterdensity within the collapsing dust sphere depends on (t; r) while outside thedust sphere, it is zero. Boundary conditions are applied to the coordinates atthe spherical surface of the dust sphere.

[5] Derivation of the Einstein field equations for the Kerr metric. Involvessetting up the general axially symmetric metric, i.e. a rotating metric havingazimuthal symmetry and calculating the connection and curvature compo-nents by applying Cartan’s equations of structure.

[6] Tetrad formulation of the Einstein–Maxwell equations involves writing thecovariant derivative in tetrad notation by noting that a tetrad is a locallyinertial frame that projects a four-vector into four scalar fields.

[7] Galactic evolution. Describes firstly the viscous and thermal contribution tothe energy-momentum tensor of the matter field based on the second lawof thermodynamics and secondly, small inhomogeneous perturbations ofthis tensor in terms of density, velocity and pressure perturbations and thedynamics of these perturbations using the linearized Einstein fieldequations.

[8] Perturbation of the Einstein field equations for k 6¼ 0 in the absence ofviscous and thermal effects.

[9] Maxwell’s equations in the Robertson–Walker space–time with the metricexpressed in synchronous form in terms of cartesian coordinates. The metrichas the form dt2 − S2(t)cab(r)dxa dxb.

[10] Problems in communication systems and signal analysis for gravitationalwave detection. Discusses delta modulation, large deviation propertiesof the signal estimation error process, application of delta modulation togravitational wave detection using interaction with the photon field from alaser by storing the photon field in compressed form as the error sequence ofa delta modulator. Also describes the form of the far field metric pertur-bations in the frequency domain generated by a fluctuating matter sourceusing the retarded potential obtained by solving the linearized Einstein fieldequations with matter energy-momentum tensor forcing. From this metricperturbation, we compute the energy-momentum pseudo-tensor of thegravitational field and hence obtain formulas for the power flux of gravi-tational radiation in the far field zone caused by a matter source in thefrequency domain.

[11] Post-Newtonian hydrodynamics: Perturbative analysis of the Einstein fieldequations with matter dynamics. Involves expanding the metric tensor, thematter density, pressure and velocity field in powers of the characteristicvelocity and equating terms of each perturbative order to derive linearsolvable equations for terms of each order in terms of the lower orderperturbation terms.

Detailed Table of Contents xvii

[12] Post-Newtonian celestial mechanics and hydrodynamics. Perturbativesolutions to the geodesic equation and to the fluid dynamic equation in abackground metric.

[13] Some problems in general relativity1. Spinorial description of field theories, the work of Roger Penrose and

Wolfgang Rindler.2. The Schwarzchild and Robertson–Walker solutions:

Explain how the RW metric can be brought into the standard form by achange of coordinates and use this to explain gravitational collapse of adustlike sphere.

3. Axial and polar perturbations to the Schwarzchild solution and a studyof the Maxwell and MHD equations in such a background metric.Simultaneous perturbation of the metric, the Maxwell field and the MHDvelocity field in the Einstein field equations for a plasma.

[14] Yang–Mills non-Abelian gauge field contribution to the Einstein fieldequations

Chapter 5: Quantum mechanics and Quantum stochastic processes

[1] On a problem posed by Prof. K. R. Parthasarathy regarding quantumGaussian states. Involves evaluating the quantum Fourier transform(QFT) of a Gaussian state specified in terms of position and momentumoperators using the Stone-Von Neumann theorem and Williamson’s theo-rem on the diagonalization of a positive definite matrix using symplecticmatrices. To evaluate the QFT, we make use of the action of annihilationoperators on coherent vectors and the Glauber–Sudarshan non-orthogonalresolution of the identity operator in Boson Fock space using coherentstates. We also discuss how to invert the QFT using the same idea.

[2] Some problems in Brownian motion and Poisson processes[1] Proof of the infinite variation of the Brownian motion process over a finite

interval.[2] Construction of the stochastic integral of an adapted process w.r.t Brownian

motion.[3] Proof of the almost sure existence and uniqueness of the solution to Ito’s

stochastic differential equation using Picard’s iteration method and Doob’smartingale inequality

[4] Submartingale upcrossings inequality and its application to proving thesubmartingale convergence theorem.

[3] Some remarks about the Dirac equation in curved space–time Focuses onthe construction of the spinor connection of the gravitational field using atetrad basis for the metric.

[4] Some remarks about quantum Gaussian states and the quantum Boltzmannequation

xviii Detailed Table of Contents

[1] Construction of the Wigner particle distribution function and its dynamicsunder fourth-degree Hamiltonians in the creation and annihilation operatorswith the ansatz that the state is quantum Gaussian and hence fourthmoments of the creation and annihilation operators factor into products ofsecond moments.

[2] Diagonalizing a Gaussian state and proving that the QFT of a Gaussian stateis the exponential of a quadratic function using the commutation rules forcreation and annihilation operators.

[5] Estimating the parameters of a superposition of quantum Gaussian states.Focuses on (a) computing the non-Gaussian perturbation of a Gaussian statecaused by a small anharmonic perturbation of the Hamiltonian of an openquantum system described by a harmonic Hamiltonian and Lindbladoperators that are linear in the creation and annihilation operators and(b) estimating the parameters of a state by taking repeated measurements onits evolution under the open quantum system GKSL dynamics and using thecollapse postulate followed by the the maximum likelihood method.

[6] Evolution of quantum Gaussian states under anharmonic perturbationstaking quantum noise into account. We look at how a system Gaussian stateevolves under the unitarily dilated version of the GKSL equation, namely,the Hudson–Parthasarathy noisy Schrodinger equation when theHamiltonian is harmonic plus a small anharmonic perturbation and theLindblad operators are linear functions of the system creation and annihi-lation operators.

[7] Evolution of quantum Gaussian states under anharmonic perturbations.Discusses semigroups of transformations on the space of operators in sys-tem Hilbert space whose dual preserves Gaussianity of a state. The semi-group is described in terms of its action on the Weyl operators. Alsodiscusses how to realize this semigroup by tracing out the Hudson–Parthasarathy noisy Schrodinger equation with Harmonic Hamiltonian andLindblad operators that are linear in the creation and annihilation operatorsover the bath Hilbert space.

[8] Glauber–Sudarshan P representation for solving the GKSL equation withharmonic oscillator Hamiltonian with anharmonic perturbations. Focuses onexpanding the density operator in terms of non-orthogonal coherent stateand deriving from the Schrodinger-Liouville equation a linear pde satisfiedby the coefficient function in this non-orthogonal expansion. The pde isderived using the action of creation and annihilation operators in the systemHamiltonian and in the Lindblad operators on coherent states.

[9] Some remarks on application of quantum Gaussian states to image pro-cessing. (a) Transform the classical image field into a pure quantum state bythe standard C ! Q process.

(b) Approximate the pure quantum state by a mixed Gaussian state.

Detailed Table of Contents xix

(c) Do this for both the noisy and noiseless image fields.(d) Purify both of these Gaussian states by taking as reference space the bath

Boson Fock space.(e) Design an optimal unitary operator that transforms the noisy purification to

the noiseless purification as best as possible for each pair of noisy–noiselessstates. This is the training process.

(f) Apply the trained unitary operator to the purification of the Gaussian stateobtained from a given noisy image field to recover the noiseless versionof the purified state. Then by partial tracing of this processed purified stateover the bath, obtain a mixed state in the system Hilbert space and thenapproximate this mixed state by a pure state and finally transform this purestate to a classical image field by standard Q ! C process.

[10] Schrodinger and Dirac Quantum mechanics in the Robertson–Walkermetric

[a] Schrodinger equation in the RW metric using the expression for the spatialLaplace–Beltrami operator. The spatial metric is derived by synchronizinglight signals between neighbouring points.

[b] Dirac equation in the RW metric: Based on constructing a tetrad for the RWmetric followed by using the standard formulae for the spinor connectionof the gravitational field.

[11] Klein–Gordon Quantum mechanics in the RW background metric[a] Transforming between radially symmetric metrics in the standard form and

in the RW form.[b] Setting up the KG equation in the RW metric taking electromagnetic

interactions into account by replacing partial derivatives by covariantderivatives to get scalar diffeomorphism invariant equations.

[12] Quantum mechanics in a metric that deviates slightly from Minkowskian:Setting up the general relativistic KG equation with the metric being aperturbed version of the flat space–time version and applyingtime-dependent and time-independent quantum mechanical perturbationtheory to obtain the deviation in the wave function and the energy levelscaused by the gravitational field.

[13] Quantum mechanics in a metric that deviates slightly from a given curvedbackground metric. The KG equation is set up for small perturbations of themetric from a given background metric and quantum perturbation theory isapplied to study the deviation of the energy levels and the wave functionfrom that corresponding to the background metric.

[14] Quantum white noise calculus. This discussion is based on theGuichardet-Maasen kernel approach to quantum stochastic calculus. Theidea is to represent exponential vectors in Fock space using functions offinite subsets of a set and to define the creation and annihilation operatorprocesses by actions on such functions.

[15] Some identities in quantum white noise calculus

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[16] Quantum Gaussian and non-Gaussian processes from GKSL equations.How a Gaussian state and its quantum Fourier transform evolve under theGKSL equation for an open quantum system when the Hamiltonian andLindblad operators are functions of creation and annihilation operators.

[17] A new proof of Quantum Ito’s formula of Hudson and Parthasarathy basedon the harmonic oscillator algebra is based on representing anexponential/coherent vector as a superposition of joint eigenstates of asequence of independent quantum harmonic oscillators. Also we express thecreation, annihilation and conservation processes in terms of the sequenceof creation and annihilation operators. Creation processes are linearsuperpositions of creation operators, annihilation processes are linearsuperpositions of annihilation operators and conservation processes arerepresented as quadratic functions of the creation and annihilation operators.

[18] Simulating a quantum stochastic differential equation using MATLAB. Theidea is based on taking matrix elements on both sides of the qsde w.r.t thetensor product of a system orthonormal basis and an approximateorthonormal basis for the Boson Fock space constructed from the expo-nential vectors and using the easy formulas for the matrix elements of thecreation, annihilation and conservation processes w.r.t exponential vectors.

[19] Different versions of the quantum Boltzmann equation. First version isbased on representing the quantum distribution function as the mean valueof the product of a creation operator and an annihilation operator field inmomentum space w.r.t. a density matrix evolving under a Hamiltoniancomprising of quadratic and fourth-degree terms in the creation and anni-hilation fields. The fourth-degree terms contribute to the “collision term” inBoltzmann’s equation. The second version is based on constructing partialtraces of the Schrodinger-Liouville equation for an N-particle density matrixevolving under the sum of identical one particle Hamiltonians andtwo-particle interaction Hamiltonians and assuming that all partial traces ofa given order are identical copies in the tensor product of single-particleHilbert spaces.

[20] Some aspects of classical and quantum Brownian motion and Poissonprocesses. Discuss representing classical stochastic differential equationsdriven by Brownian and Poisson processes in the language of quantumstochasic flows of homomorphisms, namely, as Evans–Hudson flows on acommutative Banach algebra.

[21] Proof of the existence of the positron. Discusses invariance of the Diracequation under charge conjugation, i.e. under conjugation of the wavefunction followed by conjugation followed by replacing the electroniccharge −e by +e. The proof of the invariance is based on standard identitiessatisfied by the Dirac Gamma matrices.

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[22] Invariance of the Dirac equation under Lorentz transformations[23] Some properties of quantum Gaussian states. Discusses the GKSL equation

satisfied by the state of a quantum system obtained by bath tracing of theHudson–Parthasarathy equation when the bath is in a non-vacuum coherentstate and the Hamiltonian is harmonic while the Lindblad operators arelinear in the creation and annihilation operators. For such GKSL equations,we prove invariance of Gaussianity of a state under the dynamics.

Chapter 6: Quantum field theory and quantum gravity; Standard problems inquantum field theory

[1] Computing electron and photon propagators in quantum electrodynamics[2] Restricted quantum gravity. Evaluating the Lagrangian density up to cubic

terms[3] Other versions of the quantum Boltzmann equation based on Fermionic

operator fields.[4] Calculating the S-matrix for the interaction between gravitons and photons[5] Calculating the S-matrix for scattering between gravitons, photons, elec-

trons and positrons based on the Dirac Lagrangian in curved space–time.[6] Use of Feynman diagrams for realizing large size quantum gates[7] The effect of gravity on classical and quantum phenomena[1] Gravity as a curvature of the space–time manifold.[2] Why the laws of nature should be expressed as tensor equations?[3] The equations of motion of a particle in a gravitational field as a geodesic on

the curved space–time manifold.[4] The covariant derivative, its importance and meaning in the formulation of

tensor laws.[5] The equations of motion of a particle in the language of covariant

derivatives.[6] The Maxwell equations in covariant form.[7] The physical interpretation of the covariant derivative of a tensor field: It

causes interaction terms between gravity and the concerned physicalphenomenon.

[8] Maxwell equations in covariant language lead to interaction terms betweenthe gravitational field and the electromagnetic field. Thus, gravity causeslight to bend and also has an effect on the wave field of photons.

[9] Fluid motion in a gravitational field taking into account viscous and thermalterms. Expression of the fluid equations as the vanishing of the covariantdivergence of the energy-momentum tensor of the fluid.

[10] The motion of a conducting fluid in a gravitational field, i.e. the generalrelativistic MHD equations derived from the vanishing of the sum of theenergy-momentum tensors of the matter and electromagnetic fields.

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[11] The effect of gravity on quantum phenomena: The Klein–Gordon equations,the Dirac relativistic wave equations and the Yang–Mills non-Abeliantheory for matter and gauge fields in a background gravitational field basedon the Dirac spinor connection of the gravitational field.

[12] The path integral approach to quantum electrodynamics. Calculation of thephoton and electron propagators. Corrections to the propagators producedby a curved space–time metric, i.e. background gravitational corrections onthe photon and electron propagators.

[13] Maxwell’s equations in an inhomogeneous and anisotropic medium takinginto account gravitational effects via the covariant derivative. Modelling theanisotropy and inhomogeneity of the medium using a (2, 2) tensor called thepermittivity–permeability–conductivity tensor. Approximate solution usingperturbation theory.

[14] The effect of other physical phenomena on gravity. Calculation of theenergy-momentum tensors of the matter field, the electromagnetic field, theDirac field and the Yang–Mills non-Abelian matter and gauge fields anddefining the sum of these tensors as the driving force for the Einstein fieldequations.

[15] Approximate linearized solution to the Einstein field equations coupled tothe Maxwell, Dirac and Yang–Mills fields. Higher order approximationsbased on perturbation theory.

[16] The propagation of gravitational waves in background at space–time met-rics and in background curved space–time metrics.

[17] Gravitational waves produced by a given matter and radiation fieldenergy-momentum tensors.

[18] The energy-momentum pseudo-tensor of the gravitational field, approxi-mate calculation of this tensor up to quadratic terms in the metric pertur-bations for a given matter field energy-momentum tensor. The far fieldgravitational radiation approximation total power radiated by a matter fieldin the form of gravitational waves.

[19] The propagators of non-Abelian matter and gauge fields in the presence of abackground gravitational field—an approximate calculation.

[20] Perturbative study of the evolution of galaxies by linearizing the Einsteinfield equations around the Robertson–Walker metric taking into accountdensity, pressure and velocity perturbations along with the metric pertur-bations in the energy-momentum tensor of the matter field corrected withviscous and thermal terms.

[21] Quantization of the gravitational field. The ADM action obtained byembedding a spatial surface in ℝ4 at different times.

[22] An introduction to supersymmetry. The Salam-Strathdee super vector fieldsthat generate supersymmetry transformations. The general superfield forfour anticommuting Fermionic variables. The notion of Chiral superfields.Left and right Chiral superfields. Invariance of Chirality under a super-symmetry transformation.

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[23] Supergravity: Superpartners of Gravitons, the Gravitino. Derivation of thesuper-Einstein field equations.

[8] Project proposal for the design of quantum unitary gates of large size usingsupersymmetric field theories.

[1] The Dirac equation for the four-component wave function appears with amass term depending upon the scalar field /.

[a] All component matter fields appear in a single left Chiral superfield.[b] All component gauge fields appear in another gauge superfield.

The totality of matter and gauge field Lagrangians are supersymmetric as well assuper-gauge invariant.

[c] From the supersymmetric and gauge-invariant total Lagrangian density formatter and gauge fields, we can derive a Hamiltonian density and formulatethe Schrodinger equation.

[2] The gravitational field comprising the graviton and its superpartner thegravitino, can also be brought into the supersymmetric theory by definingthe metric superfield along with superpotentials required for supersymmetrybreaking.

[3] If a quantum unitary gate based on the natural laws of physics of elementaryparticles is to be designed, then it should be derived either from the totalLagrangian or Hamiltonian of all the elementary particles and their super-partners and these laws should either be supersymmetric or they shouldinvolve supersymmetry breaking potential terms which gives masses tomassless particles.

[4] If supersymmetry is broken, then the minimum energy coming from thesuperpotential terms must be positive, while if supersymmetry is unbroken,then the minimum energy coming from the superpotential terms must bezero.

[5] Examples of how supersymmetric theories can be used to design quantumgates. Based on supersymmetric, Lorentz and gauge-invariant actions formatter and gauge fields.

[6] As a prelude to designing quantum gates using supersymetric field theories,we shall start with designing large size quantum gates using quantumYang–Mills field theories based on the Feynman path integral for Yang–Mills field quantization.

[9] Superconductivity via Feynman path integrals[10] Lecture given at the AMITY University, Noida during a conference orga-

nized by Springer:1. Scattering theory in QFT2. Electrons, Positrons and Photons interacting with each other3. Free field solutions

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Second quantizationHamiltonian for the second quantized-free Dirac field.

[a] The total charge operator[b] The total momentum operator[c] Remarks on the first quantized Dirac field[d] Lorentz invariance of the Dirac equation[e] The Hamiltonian of the free quantum electromagnetic field[f] Propagator via the Feynman path integral for photons:[g] Differential equations for the exact photon and electron propagators[h] Computation of the photon propagator using the Green’s function[i] The Dyson–Schwinger differential equations for the exact electron and

photon propagators:[j] Feynman diagram for vacuum polarization[k] Computation of the electron self-energy[l] Scattering of two electrons

[m] Dimensional regularization for the self-energy[n] Electron self-energy[o] Schwinger’s calculation of the anomalous magnetic moment of the electron[p] Another example of renormalization techniques[q] Solution to Dirac’s equation in a radial potential[r] Dirac equation in an em four potential comprising the sum of a purely

classical part, a purely quantum field theoretic part and a purely quantumnoisy part

[s] Quantization of Yang–Mills fields[t] Some general problems in quantum field theory and superconductivity[u] The Gorkov equations[v] The Ward–Takahashi identities[w] Feynman diagrams for the Klein–Gordon field with Higgs perturbation and

an external line[x] How Feynman diagrams can be used to design a quantum gate

Chapter 7: A lecture on quantum field theory with engineering applicationsLecture on Classical and quantum field theories and nonlinear filtering withapplications to the design of quantum gates, quantum communication andantenna and medium parameter estimation

[1] The Lagrangian and Hamiltonian approaches to classical mechanics.[2] The Lagrangian and Hamiltonian approaches to classical field theories.[3] Einstein–Maxwell–Dirac equations of general relativity, electromagnetics

and relativistic quantum mechanics[4] The energy-momentum tensor of a field in curved space–time.[5] Quantum general relativity based on the Hamiltonian method.[6] The phenomenon of symmetry breaking in field theory.[7] The World of Supersymmetry

Detailed Table of Contents xxv

(1) Dirac equation with scalar field-dependent mass term.(a,b,c) Supersymmetric Lagrangians for matter and gauge fields applied tothe design of quantum gates.

(2) Supergravity with graviton and gravitino, supersymmetry breaking termsused to design quantum gates.

(3) Supersymmetric Lagrangians give more degrees of freedom to designlarger sized quantum gates. If nature obeys supersymmetry, then design ofquantum gates using physical systems on the atomic scale must naturallybe based on supersymmetric Lagrangians, i.e. Lagrangians in whichBosonic fields, Fermionic fields and their corresponding superpartnersappear.

[8] As a prelude to designing quantum gates using supersymmetric field the-ories, we shall start with designing large size quantum gates using quantumYang–Mills field theories based on the Feynman path integral for Yang–Mills field quantization.

[9] Superconductivity via Feynman path integrals: Derivation of the Cooperpair field and the quantum effective action as a function of the pair fieldand the electromagnetic potentials after performing the path integral overthe unpaired electron field.

[10] Propagation of em waves in a random inhomogeneous and anisotropicnonlinear medium, estimating parameters of the medium from em fieldmeasurements.

[11] Quantization of the electromagnetic field and the electron–positron Diracfield.

[12] Interactions between photons and electrons–positron described by theS-matrix. Construction of the S-matrix using the Dyson series in theinteraction picture, interpretation of various terms in the Dyson seriesusing Feynman diagrams.

[13] Use of the Feynman diagrammatic rules for calculating probabilities ofCompton scattering, explanation of the self-energy of the electron viaradiative corrections to the electron propagator, explanation of theanomalous magnetic moment of the electron, corrections to the photonpropagator via vacuum polarization.

[14] Quantum communication.[15] Quantum Error Correcting Codes:[16a] Quantum teleportation an example[16b] Quantum stochastic calculus and quantum filtering theory.

[a] Some typical applications of real-time nonlinear filtering theory to circuittheory and electromagnetism.

[b] Feynman diagrams for the interaction between electrons, positrons, pho-tons and gravitons

[c] Feynman–Kac formula and large deviations[17] Curved waveguides in a gravitational field.

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[18] Quantization of the electromagnetic field inside a cavity resonator withoutand with a background gravitational field. Focuses on quantization of TMmodes in a cavity resonator.

[19] Quantum noise in an electromagnetic field[20] Differential equations for the exact photon and electron propagators[21] Waveguides with noisy current source[22] Belavkin filter formulation for a noisy electromagnetic field[23] Gravitons interacting with a quantum noisy photon field using quantum

stochastic calculus[24] Dimensional regularization for the self-energy[25] Solution to Dirac’s equation in a radial potential[26] Dirac equation in an em four potential comprising the sum of a purely

classical part, a purely quantum field theoretic part and a purely quantumnoisy part

[27] Quantization of Yang–Mills fields; proof of the invariance of the pathintegral under the choice of the gauge fixing functional.

[28] Some general problems in quantum field theory and superconductivity[29] An introduction to Feynman diagrams for computing the S-matrix for

various processes in quantum electrodynamics[30] Quantum Image Processing via the Hudson–Parthasarathy noisy

Schrodinger equation[31] The interaction terms for electrons, positrons and gravitons[32] The effect of a background gravitational field on the photon propagator[33] Approximate expression for the Lagrangian density of the free gravita-

tional field[34] The effect of quantum stochastic noise in the current density on the elec-

tron and photon propagators[35] Feynman diagrams for interaction between gravitational field, electron–

positron field and the electromagnetic field

Remarks on the little group method for constructing representations of theLorentz group in quantum field theory.

[36] Feynman diagrams for describing scattering, absorption and emissionprocesses in non-Abelian gauge theories

(a) Computing the approximate propagator for the gauge field.[37a] Miscellaneous problems in quantum field theory[1] Transmit a pure state over a noisy quantum channel, the channel also adds

noise to this state and at the receiver end, we use the same preprocessor asthe post-processor to optimally decode the message. The channel is a bathin the coherent state

[2] Superconductivity with Fermions of arbitrary half-integral spin.[3] Proof of current conservation in Yang–Mills non-Abelian gauge theories[4] Quantum filtering theory applied to the Yang–Mills field equations

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[37b] Problems in quantum gravity[1] Einstein field equations in the presence of an external quantum noisy

electromagnetic field:

Why is the propagator so important in quantum field theory?

[38] The quantum gravitational field interacting with the quantum electromag-netic field. Discussion focuses on setting up the total Lagrangian of thegravitational field, and the Maxwell and Dirac fields in curved space–timetaking covariant derivatives and spinor connections for the gravitationalfield into account (to preserve diffeomorphism covariance and local Lorentzcovariance of the equations) and applying the path integral method to cal-culate scattering amplitudes or the operator method based on the canonicalHamiltonian formalism after appropriate approximation of the Lagrangians.

Chapter 8: The general theory of nonlinear systems

[1] Summary in brief of the research work carried out by some of the author’sresearch students during the past 15 years.

[1] Applications of nonlinear filtering theory to certain problems in classicalmechanics like the stochastic two-body gravitational problem.

[2] Parameter estimation algorithms in nonlinear systems using nonlinear LMSalgorithm with a study of the behaviour of the Lyapunov exponents ofautonomous nonlinear systems for small initial perturbations around a fixedpoint. Convergence analysis of the LMS algorithm is also performed.

[3] Applications of stochastic nonlinear filtering theory to trajectory maneou-vering of spacecrafts and convergence analysis of least mean phase algo-rithms using stochastic differential equations driven by Brownian motion.

[4] Modelling and parameter estimation in nonlinear transistor circuits usingVolterra approximations combined with wavelet-based compression fordata storage for the purpose of estimation.

[5] Finite-element method for determining the modes in waveguides havingvarious kinds of cross section and with inhomogeneous and anisotropicmedia filling the guide taking into account background gravitational per-turbations in the form of a curved space–time metric.

[6] Study of higher harmonic generation in nonlinear transistor circuits usingFourier series and perturbation theory.

[7] Image modelling, smoothing and enhancement using partial differentialequations with emphasis on diffusion equations with intensity-dependentdiffusion matrix coefficient.

[8] Magnetohydrodynamic antenna construction analysis using Navier–Stokesand Boltzmann kinetic transport equation.

[9] Studies in transmission line and waveguide analysis taking hysteresis andcapacitive nonlinearities and quantum mechanical effects of transmissionline and waveguide fields on atoms and quantum harmonic oscillators.

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[10] Quantum gate design by perturbing real quantum systems with electro-magnetic fields.

[11] Studies in robot trajectory tracking and dynamic parameter estimation in thepresence of noise and in master–slave teleoperation-based tracking usingadaptive control algorithms.

[12] Quantum parameter estimation using search algorithms with applications toquantum communication and quantum gate design.

[13] Antenna design using numerical solution of integral equations arising fromthe boundary conditions on the antenna surface.

[14] Studies in electromagnetic wave propagation in inhomogeneous, aniso-tropic and field-dependent (nonlinear) media with applications to estimatingthe medium parameters from discrete measurements of the electromagneticfield at different space–time points. Applications to antenna design for wavepropagation in nonlinear media are also considered. For this, perturbativeexpansions of the medium permittivity and permeability as Taylor series inthe electric and magnetic fields as well as expansion of the field independentcoefficients of the permittivity and permeability in terms of basis functionsis performed. These expansions are substituted into the Maxwell equationsto obtain a sequence of linear equations for each perturbative order.Boundary conditions of the electromagnetic fields on the antenna surfaceare applied to derive integral equations for the induced surface currentdensity.

[15] Some new results in Belavkin quantum filtering and Lec-Bouten controlusing the Hudson–Parthasarathy quantum stochastic calculus with appli-cations to estimating the spin of the electron and other quantumobservables.

[16] Classical and quantum image field reconstruction using the EKF applied tonoisy quantum measurements and Hudson–Parthasarathy optimal unitaryprocessor applied to quantum states obtained from a classical noisy imagefield.

[2] Linear algebra in signal processing, Questions.These problems deal primarily with the linearization of nonlinear ordinaryand partial differential and difference equations, especially applied to thoseproblems which are important in the general theory of dynamical systemslike population growth models, predator–prey dynamics of fish in a pondand mathematical physics. The basics discussed include the Jordancanonical form of a matrix proved using the primary decomposition theo-rem, the spectral theorem for finite- and infinite-dimensional self-adjointoperators in a Hilbert space, the polar and singular value decompositionsand linear prediction theory.

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[3] School problems in algebra and geometry. This section is aimed at gettingfamiliar with non-Euclidean geometry required for the study of generalrelativity and basic algebra and differential equations required for the studyof nonlinear dynamical systems and quantum mechanics.

[4] Some problems in signal processing.[1] Design an adaptive line enhancer for separating out the signal from the

noise process when the two processes are uncorrelated and the signal pro-cess has long-range correlations while the noise process has short-rangecorrelations.

[2] Design an adaptive echo canceller at one end of a telephone line based onexploiting the correlations between the speaker speech at one end and hisown echo.

[3] Approximate statistical performance analysis of the LMS algorithm basedon linearizing a multivariate stochastic difference equation for the weightprocess and computing the evolution of the first two moments of theresulting weight error process.

[4] Deriving the optimal Kushner–Kallianpur nonlinear filtering equations for aMarkov process with measurement noise using the reference probabilityapproach of Gough and Kostler used in their derivation of the Belavkinfilter in the theory of quantum stochastic processes.

[5] Syllabus for the M.Tech, course “Advanced Signal Processing”[1] Classical and quantum probability spaces—a comparison.[2] Calculating expectations of observables and probabilities of events in

classical and quantum probability.[3] Linear prediction of stationary time series in classical and quantum prob-

ability. Discussion of innovations process and the Levinson–Durbin algo-rithm for order recursive computation of the prediction filter.

[4] Nonlinear filters for signal estimation and prediction. Nonlinear RLS andRLS lattice algorithms.

[5] Definition of the conditional expectation in classical probability as aRadon–Nikodym derivative of absolutely continuous measures and also asan orthogonal projection in Hilbert space.

[5] Quantum information theory: Classical and quantum entropy, Shannon’scoding theorems in the classical and quantum contexts, state collapse fol-lowing measurement in the quantum theory, estimating the parameters of aquantum mixed state by repeated measurements with joint outcome prob-abilities computed based on collapse postulate, quantum Cramer–Rao lowerbound on state parameter estimate variance.

[6] Nonlinear differential equations, linearization, perturbation solution, con-vergence of perturbation series. Nonlinear partial differential equations,solution using perturbation series. Examples of linearization of pde’s takenfrom general relativity.

[7] Linearization applied to plasma physics via the MHD-Navier–Stokes andcoupled Boltzmann–Maxwell equations.

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[8] Quantum stochastic calculus and quantum filtering theory.(1) The Boson and Fermion Fock spaces associated with a Hilbert space.(2) The Harmonic oscillator algebra for a countable number of independent

Harmonic oscillators.(3) The creation and annihilation and conservation/number operator fields

associated with the harmonic oscillator algebra.(4) Coherent states constructed using the eigenstates of a countably infinite

number of harmonic oscillators.(5) The matrix elements of creation, annihilation and conservation operator

fields with respect to coherent states.(6) The creation, annihilation and conservation processes constructed from the

corresponding fields.(7) The exponential vector and Weyl operator approach to the construction of

creation, annihilation and conservation processes. Hilbert space isomorphicequivalence of the two approaches.

[6] Algorithms for system identification[1] Prony’s and Shank’s method for identifying parameters of an ARMA time

series model based on noisy impulse response measurements with statisticalperformance analysis.

[7] Galactic simulation using a super-computer by discretizing the Newtonianinverse square-law-based differential equations, galactic simulation in thepresence of random noise, i.e. when the dynamical equations for thevelocities contain an additional random component, we simulate theFokker–Planck equation for the joint probability density of the positionsand velocities of the stars in a galaxy.

[8] The fundamental equations of Kalman and nonlinear filtering theory[a] Kalman filter in discrete time: Derivation based on the formula for the

conditional expectation and conditional covariance of one random vectorgiven another when the two are jointly multivariate Gaussian.

[9] Some statistical problems in deep neural networks.[a] Single-layered dilated convolutional neural network. Calculating the opti-

mal weights that match a given input–output signal vector pair by mini-mizing the mean square output error.

[10] Some aspects of the classical and quantum Boltzmann kinetic transportequation for a plasma

[a] Binary collision, scattering cross section for the classical case.[b] The classical Boltzmann equation.[c] Boltzmann’s H-theorem.[d] The classical Boltzmann equation in an electromagnetic field.[e] The Vlasov equations: Coupling of the Boltzmann equation for p ion spe-

cies with the Maxwell equations for the electromagnetic field.[f] The quantum Boltzmann equation.

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