Advanced Classical and Quantum Probability Theory with Quantum Field Theory Applications

Cover
Half Title
Title Page
Copyright Page
Preface
Table of Contents
Detailed Contents
1. Matrix Theory
1.1 Perequisites of Linear Algebra
1.2 Quotient of a Vector Space
1.3 Triangularity of Comuting Operators
1.4 Simultaneous Diagonability of a Family of Comuting Normal Operators w.r.t an onb in a Finite Dimensional Complex Inner Product Space
1.5 Tensor Products of Vectors and Matrices
1.6 The Minimax Variational Principle for Calculating all the Eigenvalues of a Hermitian Matrix
1.7 The Basic Decompostition Theorems of Matrix Theory
1.8 A Computational Problems in Lie Group Theory
1.9 Primary Decompostition Theorem
1.10 Existence of Cartan Subalgebra
1.11 Exercises in Matrix Theory
1.12 Conjugancy Classes of Cartan Subalgebras
1.13 Exercises
1.14 Appendix: Some Applications of Matrix Theory to Control Theory Problems
1.15 Controllability of Supersymmetric Field the Oretic Problems
1.16 Controllability of Yang-Mills Gauge Fields in the Quantum Context Using Feynman’s Path Integral Approach to Quantum Field Theory
1.17 Large Deviations and Control Theory
1.18 Approximate Contollability of the Maxwell Equations
1.19 Controllability Problems in Quantum Scatering Theory
1.20 Kalman’s Notion of Controllability and Its Extension to pde’s
1.21 Controllability in the Context of Representations of Lie Groups
1.22 Irreducible Representations and Maximal Ideals
1.23 Controllability of the Maxwell-Dirac Equations Using External Classical Current and Field Sources
1.24 Controllability of the EEG Signals on the Brain Surface Modeled as a Spherical Surface by Influencing the Infinitesimal Dipoles in the Cells of the Brain Cortex to Vary in Accord to Sensory Perturbations
1.25 Control and Relativity
2. Probability Theory
2.1 The Basic Axioms of Kolmogorov
2.2 Exercises
2.3 Exercises on Stationary Stochastic Processes, Spectra and Polyspectra
2.4 A Research Problem Based on Problem
2.5 Exercises on the Construction of the Integral w.r.t a Probability Measure
2.6 Exercises on Stationarity, Dynamical Systems and Ergodic Thery
3. Antenna Theory
3.1 Course Outline
3.2 The Far Field Poynting Vector
3.3 Exercises
3.4 Order of Magnitudes in quantum Antenna Theory
3.5 The Notion of a Fermionic Coherent State and its Application to the Computation of the Quantum Statistical Moments of the Quantum Electromagnetic Field Generated by Electrons and Positors Within a Quantum Antenna
3.6 Calculating the Moments of the Radiation Field Produced by Electrons and Positrons in the Far Field when the Fermions are in a Coherent State
3.7 Controlling the Classical em Fields Interacting with the Dirac Field so that the Mean Value of the em Field Radiated by the Resulting Dirac Second Quantized Current in a Fermionic Coherent State is as Close as Possible to a Given Deterministic Pattern in Space and Simultaneously the Mean Square Fluctuations of this Field in a Fermionic Coherent State are Minimized
3.8 Approximate Analysis of a Rectangular Quantum Antenna
3.9 Remark on the Perturbation in the Quantum Dirac Field and the Quantum Electromangetic Field Interacting with Each Other Caused by Further Interaction of the Dirac Field with a Classical Control em Field and Interaction of the Quantum Electromagnetic Field with a Control Classical Current
3.10 Quantum Antennas Constructed Using Supersymmetric Field Theories
3.11 Quantization of the Maxwell and Dirac Field in a Background Curved Metric of Spacetime.
3.12 Relationship Between the Electron Self Energy and the Electron Propagator
3.13 Electron Self Energy Corrections Induced by Quantum Gravitational Effects
4. Miscellaneous Problems
4.1 A Problem in Robotics
4.2 More on Root Space Decompostion of a Semisim-ple Lie Algebra
4.3 A Project Proposal for Developing an Ex-perimental Setup for Transmitting Quantum States Over a Channel in the Presence of An Eavesdropper
4.4 A Problem in Lie Group Theory
5. More Problems in Linear Algebra and Functional Analysis
5.1 Riesz Representation Theorem
5.2 Lie’s Theorem on Solvable Lie Algebras
5.3 Engel’s Theorem on nil-representation of a Lie Algebra
5.4 Aperture Antenna Pattern Fluctuations
5.5 Spectral Theorem Using Gelfand-Naimark Theorem
5.6 The Atiyah-Singer Index Theorem: A supersymmetric Proof
5.7 Replicas, Regular Elements, Jordan Decomposition and Cartan Subalgebras
5.8 Lecture Plan, Matrix Theory
5.9 More Assignment Problems in Probability Theory
5.10 Multiple Choice Questions on Probability Theory
5.11 Design of a Quantum Unitary Gate Using Superstring Theory with Noise Analysis Based on the Hudson-Parthasarathy Quantum Stochastic Calculus
5.12 Study Projects in Probability Theory: Construction of Brownian Motion, Law of the Iterarted Logarithm
5.13 Quantum Boltzmann Equation for a Systerm of Particles Interacting with a Quantum Electromagnectic Field
5.14 Device Physics in a Semiconductor Using the Classical Boltzmann Transport Equation
5.15 Describing the Value of a Point Charge and Its Location in Space in Terms of the Electrostatic Potential Generated by It
5.16 Calculating the Masses of N Gravitating Particles and Their Postitons and Their Trajectories from Measurement of the Gravitational Potential Distribution in Space-time Using the Newtonian Theory
5.17 The Quantum Boltzmann Equation for a Plasma
5.18 Some Other Remarks on Lie Algebras
5.19 Question Paper on Matrix Theory
5.20 Study Project on Quantum Antennas
5.21 Heat and Mass Transfer Equations in a Fluid
5.22 Quantum Electodynamics in a Background Medium Described by a Permittivity and Permeability Function
5.23 Temperature and Field Dependence of Re-fractive Index
5.24 Quantum Statistical Field Theory
5.25 Root Space Decompositions of the ComplexClassical Lie Algebras
6. Models for the Refractive Index of Materials and Liquids
6.1 Quantum Electrodynamics with the Electronic Charge Expressed in Terms of the Quantum Fields
6.2 Calculating the Masses of N Gravitating Paticles and Their Positions and Their Trajectories from Measurement of the Gravitational Potential Distribution in Space-time Using the Newtonian Theory
6.3 The Quantum Boltzmann Equation for a Plasma
6.4 Quantum Electrodynamics in a Background Medium Described by a Permittivity and Per-meability Function
6.5 Models for the Refractive Index of a Material Based on Classical and Quantum Physics
6.6 Quantum Statistical Field Theory
6.7 Relating the Refractive Index of a Material to the Metric Tensor of Space-time
6.8 Cosmologiccal Effects on the Refractive Index
7. More Problems in Probability Theory, Antennas and Refractive Index of Materials
7.1 Levy’s Modulus of Continuity for Brownian Motion
7.2 Test 2: Antennas and Wave Propagation
7.3 Article Submitted to the Quantum Information Processing Journals for Publication