Large Deviations Applied to Classical and Quantum Field Theory

Cover
Half Title
Title Page
Copyright Page
Preface
Brief Contents
Table of Contents
1. LDP Problems in Quantum Field Theory
1.1 Large Deviations for Supergravity Fields
1.2 Rate Function of String Propagator
1.3 Large Deviations for p-form Fields
1.4 The Dynamics of the Electro-weak Theory
1.5 Filtering in Fermionic Noise
1.6 Quantum Field Theory is a Low Energy Limit of String Field Theory
1.7 The Atiyah-Singer Index Theorem and LDP Problems Associated with It
1.8 LDP Problems in General Relativity
2. LDP in Biology, Neural Networks, Electromagnetic Measurements, Cosmic Expansion
2.1 The Importance of Mathematical Models in Medicine
2.2 LDP Related Problems in Neural Networks and Artificial Intelligence
2.3 LDP Related Problems to Cosmic Expansion in General Relativity
2.4 LDP Problems in Biology
2.5 A Sensitive Quantum Mechanical Method for Measuring the Scattered Electromagnetic Fields
3. LDP in Signal Processing, Communication and Antenna Design
3.1 Review
3.2 Large Deviation Problems in SSB Modulation
3.3 What is Meant by Estimating a Quantum Field in Space-time
3.4 Write Down the Noisy Schrodinger Equation for an N Particle System in the Formalism of Hudson and Parthasarathy and Derive
by Partial Tracing, the Approximate Nonlinear Stochastic Boltzmann
Equation for the State Evolution of a Single Particle (Evolution of
the Marginal Density with Noise)
3.5 The pde’s Satisfied by the Quantum Electromagnetic Field Observables in a Cavity Resonator in the Presence of Bath Noise
3.6 Estimating the Quantum State of a Single Particle in a System of N Indistinguishable Particles Interacting with Each Other and with
an External Bath Field
4. LDP Applied to Quantum Measurement, Classical Markov Chains, Quantum Stochastics and Quantum Transition Probabilities
4.1 Some Other Aspects of Measurement of a Quantum Field
4.2 Some Parts of the Solution to the Question Paper on Antenna Theory
4.3 Some Additional LDP Related Problems in Antenna Theory
4.4 LDP for Quantum Markov Chains Using Discrete Time Quantum Stochastic Flows
4.5 LDP Applied to the Analysis of the Error Process in Stochastic Filtering Theory of a Continuous Time Markov Process when the Measurement Noise is White Gaussian and More Generally when the Measurement Noise is White Gaussian and More Generally when the Measurement Noise is the Differential of a Levy Process (ie, a Limit of Compound Poisson Process Plus white Gaussian noise)
4.6 LDP Applied to the Electroweak Theory
4.7 LDP Problems in Quantum Mechanical Transitions
4.8 Large Deviation Principle for Quantum Gaussian States in Infinite Dimensions Using Quantum Moment Generating Functions,
Quantum Gaussian States Obtained by Perturbing a Harmonic
Oscillator Hamiltonian by Small Anharmonic Terms
4.9 Large Deviation Problems in Queueing Theory
4.10 Large Deviation Problems Associated with Quantum Filtering Theory
5. LDP in Classical Stochastic Process Theory and Quantum Mechanical Transitions
5.1 Large Deviations Problems to the Propagation of Noise at the Sigmoidal Computation Nodes Through the Neural Network
5.2 Law of the Iterated Logarithm for Sums of iid Random Variables
5.3 A version of the LDP for iid Random Variables
5.4 The Law of the Iterated Logarithm for Sums of iid Random Variables Having Finite Variance
5.5 An Open Problem Relating Applications of LDP to Martingales
5.6 Properties of ML Estimators Based on iid Measurements
6. LDP in Pattern Recognition and Fermionic Quantum Filtering
6.1 Large Deviation Problems in Pattern Recognition
6.2 LDP for Estimating the Parameters in Mixture Models
6.3 The EM Algorithm
6.4 Sanov’s Theorem and Gibbs Distributions
6.5 Gibbs Distribution in the Interacting Particle Case
6.6 Inversion of the Characteristic Function of a Probability Distribution on the Real Line
6.7 Infinitely Divisible Distributions, The Levy- Khintchine Theorem
6.8 Stationary Distribution for Markov Chains
6.9 Lecture on Quantum Filtering in the Presence of Fermionic Noise
6.10 Lecture Plan for Pattern Recognition
6.11 Review of the Book Stochastics, Control and Robotics, by Harish Parthasarathy
7. LDP in Spin Field Theory, Anharmonic Perturbations of Quantum Oscillators, Small Perturbations of Quantum
Gibbs States
7.1 Large Deviation Problems in Spin-field Interaction Theory
7.2 Large Deviation Problems Associated with a Quantum Gravitational Field Interacting with a Non-Abelian Gauge Field
7.3 Large Deviation Problems in Quantum Harmonic Oscillator Problems with Nonlinear Terms
7.4 Formulation of an LDP for Quantum Stochastic Processes
8. LDP for Electromagnetic Control of Gravitational Waves, Randomly Perturbed Quantum Fields, Hartree-Fock
Approximation, Renewal Processes in Quantum Mechanics
8.1 Gravitational Wave Propagating in a Background Curved Space-time, LDP for Reducing the Wave Fluctuations via
Electromagnetic Control
8.2 The Lehmann Representation of the Propagator
8.3 The LDP Problem in this Context
8.4 Central Limit Theorem for Renewal Processes
8.5 Applications of Renewal Process Theory in Quantum Field Theory
8.6 Large Deviation Principle in the Hartree-Fock Method for Approximately Solving Many Electron Problems
8.7 LDP in Fuzzy Neural Networks
8.8 Large Deviation Analysis of this qnn
8.9 LDP Problems in Quantum Field Theory Related to Corrections to the Electron, Photon and non-Abelian Gauge Boson Propagators
9. LDP in Electromagnetic Scattering and String Theory, Control of Dynamical Systems Using LDP
9.1 A Summary of a List of LDP Applications in Physics and Engineering
9.2 LDP Problems Related to Scattering of Electromagnetic Waves by a Perfectly Conducting Cylinder
9.3 String Theory and Large Deviations
9.4 Questions Related to Qualitative Properties of Quantum Noise
9.5 Questions on Pattern Recognition
9.6 Appendix
10. LDP in Markov Chain and Queueing Theory with Quantum Mechanical Applications
10.1 Notes on Applications of LDP to Stochastic Processes and Queueing Theory
10.2 LDP Problems in Markov Chain Theory
10.3 Continuity and Non-differentiability of the Brownian Sample Paths
10.4 Renewal Processes in Quantum Mechanics
11. LDP in Device Physics, Quantum Scattering Amplitudes, Quantum Filtering and Quantum Antennas
11.1 Large Deviations in Vacuum Polarization
11.2 An Application of the EKF and LDP to Estimating the Current in a pn Junction
11.3 Large Deviation Problems in Quantum Stochastic Filtering Theory
11.4 Large Deviation Problems in Quantum Antennas
12. How the Electron Acquires Its Mass, Estimating the Electron Spin and the Quantum Electromagnetic Field Within a Cavity
in the Presence of Quantum Noise
12.1 Large Deviation Methods in Classical and Quantum Field Theory
13. Mathematical Tools for Large Deviations, Neural Networks, LDP in Physical Theories, EM and LDP Algorithms
in Quantum Parameter Estimation and Filtering
13.1 Lecture on the Ascoli Arzela Theorem and Prohorov’s Tightness Theorem with Applications to Proving Weak Convergence of
Probability Distributions on the Space of Continuous Functions
on a Compact Interval
13.2 Proof of the Prohorov Tightness Theorem
13.3 Some Remarks on Neural Networks Related to Large Deviation Theory
13.4 LDP Problems in General Relativity and non- Abelian Gauge Field Theory
13.5 Large Deviations in String Theoretic Corrections to Field Theories
14. Quantum Transmission Lines, Engineering Applications of Stochastic Processes
14.1 Introduction
14.2 Kolmogorov’s Existence for Stochastic Processes Applied to the Problem of Describing Infinite Image Fields, ie, Image Fields with a Countably Infinite Number of Pixels
14.3 Dirichlet Series with Image Processing Applications
14.4 About the Book
14.5 An Application of the EM Algorithm to Quantum Parameter Estimation and Quantum Filtering
14.6 The EM Algorithm and Large Deviation Theory
14.7 Kolmogorov-Smirnov Statistics
14.8 Quantum Transmission Lines, LDP Problems
15. More Tools in Probability, Electron Mass in the Presence of Gravity and Electromagnetic Radiation, More on LDP in Quantum Field
Theory, Non-Abelian Gauge Field Theory and Gravitation
15.1 On the Amount of Mass that an Electron can Get from the Background Electromagnetic and Gravitational Fields
15.2 Electron Propagator Corrections in the Presence of Quantum Noise
15.3 Large Deviation Problems for the Schrodinger and Dirac Noisy Channels
15.4 Central Limit Theorem for Martingales
15.5 More Problems in LDP Applied to Quantum Field Theory
15.6 Schrodinger and Klein-Gordon Equations in Quantum Field Theory Based on An Infinite Dimensional Laplacian Operator
15.7 Proof of the Prohorov Tightness Theorem
15.8 Large Deviation Problems in Field Measurement Analysis
15.9 ADM Action for Quantum Gravity and Its Noisy Perturbation with LDP Analysis of the Solution Metric
15.10 The Bianchi Identity for non-Abelian Gauge Fields
15.11 More Problems in LDP Applied to Quantum Field Theory
15.12 Schrodinger and Klein-Gordon Equations in Quantum Field Theory Based on An Infinite Dimensional Laplacian Operator
16. Weak Convergence, Sanov’s Theorem, LDP in Binary Signal Detection
16.1 Prohorov’s Tightness Theorem, “Necessity Part”
16.2 Sanov’s Theorem on the LDP for Empirical Distributions of Discrete iid Random Variables
16.3 LDP in Binary Phase Shift Keying
16.4 Compactness of the Set of Probability Measures on a Compact Metric Space
16.5 Large Deviations for Frequency Modulated Signals
17. LDP for Classical and Quantum Transmission Lines, String Theoretic Corrections to Classical Field Lagrangians,
Non-Abelian Gauge Theory in the Language of
Differential Forms
17.1 LDP Theory Applied to Transmission Lines with Line Loading
17.2 LDP Formulation of the Quantum Transmission Line
17.3 Yang-Mills Gauge Fields, The Euler characteristic, String Theoretic Corrections to the Yang-Mills Anomaly
Cancellation Lagrangian Terms
17.4 Quantum Averaging Based Derivation of Action Functional for Point Fields from Action Functional of String Fields
18. LDP and EM Algorithm, LDP for Parameter Estimates in Linear Dynamical Systems, Philosophical Questions in
Quantum General Relativity
18.1 Large Deviations and the EM Algorithm: Large Deviation Properties of Parameter Estimates Derived Using the EM
Algorithm in the Presence of Noise Relative to ML Parameter
Estimates Obtained in the Absence of Noise when There are
Latent Random Parameter Vectors in the Measurement Model
18.2 Fundamental Problems in Quantum General Relativity
18.3 A Problem in Large Deviations and Lie Algebras
18.4 Describing Quantum Gravity Using Holonomy Fields
18.5 Square of the Dirac Operator in Curved Spacetime in the Presence of a non-Abelian Yang-Mills Gauge Potential
18.6 Exponential Equivalence, The Dawson-Gartner Theorem on LDP on Projective Limits with Applications to Estabilishing LDP
for Processes or More Generally, LDP on the Projective Limit of
Topological Spaces
18.7 Applications
18.8 Equivalence of LDP
18.9 Lehmann’s Representation of the Propagator of the Klein-Gordon Field with Nonlinear Perturbations
Chapters Index