Advanced Classical and Quantum Probability Theory with Quantum Field Theory Applications

263 Pages·2023·5.813 MB·English

by  Harish Parthasarathy

Advanced Classical and Quantum

Probability Theory with Quantum

Field Theory Applications

Advanced Classical and Quantum

Probability Theory with Quantum

Field Theory Applications

Harish Parthasarathy
Professor

Electronics & Communication Engineering

Netaji Subhas Institute of Technology (NSIT)

New Delhi, Delhi-110078
First edition published 2023
by CRC Press
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
ISBN: 9781032405124 (hbk)
ISBN: 9781032405148 (pbk)
ISBN: 9781003353430 (ebk)
DOI: 10.4324/9781003353430
Typeset in Arial, BookManOldStyle, CourierNewPS, MS-Mincho, Symbol, and
TimesNewRoman
by Manakin Press, Delhi
1
Preface

Preface
The contents of this book are based on three undergraduate and postgradu­
ate courses taught by the author on Matrix theory, probability theory and an­
tenna theory over the past several years. The portion on matrix theory covers
basic linear algebra including quotient vector spaces, variational principles for
computingeigenvaluesofa matrix,primary andjordandecompositiontheorems
for non­diagonable matrices, simultaneous triangulability, and basic matrix de­
composition theorems useful in statistics, signal processing and control. It also
covers Lie algebra theory culminating the celebrated root space decomposition
introduced by E.Cartan of a semisimple Lie algebra in terms of Cartan subal­
gebras and root vectors and also some interesting topics in control theory like
controllability ofpartialdifferential equations of mathematicalphysics including
Maxwell’s equations, Dirac equation and their quantum versions. By control of
aquantumsystemcomprisingelectrons,positronsandphotonsdescribedbythe
second quantized Maxwell and Dirac equations, we mean the design of classical
control fields like current and electromagnetic fields so that the resulting ra­
diation pattern genererated by the electron­positron field will have space­time
momentsthatprovideagood match toagiven setof moments. Somediscussion
on large deviation theory in control has also been included involving designing
control fields for pde’s driven by weak stochastic noise so that the probability
of deviation of the controlled field from a prescribed set of fields by an amount
greater than a given threshold is a minimum. For constructing the irreducible
finite dimensional representation of a semisimple idea, we discuss the notion of
maximal ideals of the universal enveloping algebra of a semisimple Lie algebra
the one­one correspondence between the irreducible representations and maxi­
mal ideals. The second part of the book deals with probability theory and we
introduce Brownian motion, Poisson process and some of the features assoo­
ciated with such processes. The third part of the book covers basic antenna
theory including far field radiation pattern by a current source at a given fre­
quency, quantum electrodynamics within a cavity described by the coupling of
the second quantized Maxwell and Dirac fields and how to control these cavity
fields so as to get a far field radiation pattern having prescribed statistics in a
coherent state of cavity photons and Fermions. This portion of the book also
introduces the quantum Boltzmann equation in the presence of electromagnetic
fields and how from the resulting nonlinear evolution of the density matrix, we
can compute the refractive index of materials from quantum averages of electric
and magnetic dipole moments. We demonstrate how the refractive index com­
putedin thisquantummechanical way willgenerallybefielddependent. Wealso
show that if gravitational effects are taken into account, for example the metric
tensor of space­time in an expanding universe, then Dirac’s equation will have
to be modified by the presence of background curvature and hence the resulting
quantum Boltzmann equation will contain gravitational terms thereby causing
the refractive index to depend on the space­time metric of gravitation. We also
discuss an interesting notion in electrodynamics namely that of specified the
charges in space in terms of the singularities of the electric and magnetic fields.
Thisfactleadstotheimportant conclusionthatall chargesareinfactgenerated
v
2
by the singularities of the electromagnetic field and hence one can postulate that
the electron’s mass and charge is of electromagnetic origin. In this context, we
explain how to compute the corrected electron propagator due to external elec­
tromagnetic and gravitational fields by formulating an appropriate differential
equation for the electron propagator and hence from this corrected propagator,
how to determine the shift in the electron’s mass due to electromagnetic and
gravitational fields.
Author
vi
Table of Contents

1. Matrix Theory 1–50
2. Probability Theory 51–64
3. Antenna Theory 65–98
4. Miscellaneous Problems 99–108
5. More Problems in Linear Algebra and Functional Analysis 109–184
6. Models for the Refractive Index of Materials and Liquids 185–216
7. More Problems in Probability Theory, Antennas and
Refractive Index of Materials 217–248
vii

Detailed Contents

1. Matrix Theory 1–50
1.1 Perequisites of Linear Algebra 3

1.2 Quotient of a Vector Space 4

1.3 Triangularity of Comuting Operators 5

1.4 Simultaneous Diagonability of a Family of Comuting

Normal Operators w.r.t an onb in a Finite Dimensional

Complex Inner Product Space 6

1.5 Tensor Products of Vectors and Matrices 7

1.6 The Minimax Variational Principle for Calculating

all the Eigenvalues of a Hermitian Matrix 8

1.7 The Basic Decompostition Theorems of Matrix Theory 8

1.8 A Computational Problems in Lie Group Theory 12

1.9 Primary Decompostition Theorem 13

1.10 Existence of Cartan Subalgebra 15

1.11 Exercises in Matrix Theory 24

1.12 Conjugancy Classes of Cartan Subalgebras 28

1.13 Exercises 29

1.14 Appendix: Some Applications of Matrix Theory to

Control Theory Problems 30

1.15 Controllability of Supersymmetric Field the Oretic Problems 38

1.16 Controllability of Yang-Mills Gauge Fields in the Quantum

Context Using Feynman’s Path Integral Approach to

Quantum Field Theory 39

1.17 Large Deviations and Control Theory 40

1.18 Approximate Contollability of the Maxwell Equations 42

1.19 Controllability Problems in Quantum Scatering Theory 43

1.20 Kalman’s Notion of Controllability and Its Extension to pde’s 43

1.21 Controllability in the Context of

Representations of Lie Groups 44

1.22 Irreducible Representations and Maximal Ideals 45

1.23 Controllability of the Maxwell-Dirac Equations

Using External Classical Current and Field Sources 46

ix