Large Deviations Applied to Classical and Quantum Field Theory

This book deals with a variety of problems in Physics and Engineering where the large deviation principle of probability finds application. Large deviations is a branch of probability theory dealing with approximate computation of the probabilities of rare events. It contains applications of the LDP to pattern recognition problems like analysis of the performance of the EM algorithm for optimal parameter estimation in the presence of weak noise, analysis and control of non-Abelian gauge fields in the presence of noise, and quantum gravity wherein we are concerned with perturbation to the quadratic component of the Einstein-Hilbert Hamiltonian caused by higher order nonlinear terms in the position fields and their effect on the Gibbs statistics and consequently quantum probabilities of events computed using the quantum Gibbs state. The reader will also find in this book applications of LDP to quantum filtering theory as developed by Belavkin based on the celebrated Hudson-Parthasarathy quantum stochastic calculus. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan and Bhutan). This book deals with a variety of problems in Physics and Engineering where the large deviation principle of probability finds application. Large deviations is a branch of probability theory dealing with approximate computation of the probabilities of rare events. anharmonic,perturbation;,binary,signal,detection;,classical,field,Lagrangian;,classical,stochastic,process,theory;,cosmic,expansion;,electromagnetic,measurements;,Hartree-Fock,approximation;,large,deviation,principle;,Markov,chains;,neural,networks;,non-abelian,gauge,theory;,probability,theory;,quantum,field,theory;,quantum,general,relativity;,quantum,measurement;,quantum,scattering,amplitudes;,quantum,stochastics;,queueing,theory;,Sanov’s,theorem;,spin,field,theory anharmonic perturbation,binary signal detection,classical field Lagrangian,classical stochastic process theory,cosmic expansion,electromagnetic measurements,Hartree-Fock approximation,large deviation principle,Markov chains,neural networks,non-abelian gauge theory,probability theory,quantum field theory,quantum general relativity,quantum measurement,quantum scattering amplitudes,quantum stochastics,queueing theory,Sanov’s theorem,spin 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