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El. knyga: Large Deviations Applied to Classical and Quantum Field Theory [Taylor & Francis e-book]

  • Formatas: 254 pages
  • Išleidimo metai: 22-Dec-2022
  • Leidėjas: CRC Press
  • ISBN-13: 9781003363248
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  • Taylor & Francis e-book
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Large Deviations Applied to Classical and Quantum Field TheoryDidesnis paveikslėlis
  • Formatas: 254 pages
  • Išleidimo metai: 22-Dec-2022
  • Leidėjas: CRC Press
  • ISBN-13: 9781003363248
Kitos knygos pagal šią temą:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Partnerių interneto svetainė: https://www.taylorfrancis.com/books/9781003363248
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).
1 LDP Problems in Quantum Field Theory
26(8)
1.1 Large Deviations for Supergravity Fields
1(1)
1.2 Rate Function of String Propagator
2(3)
1.3 Large Deviations for p-form Fields
5(2)
1.4 The Dynamics of the Electro-weak Theory
7(1)
1.5 Filtering in Fermionic Noise
7(11)
1.6 Quantum Field Theory is a Low Energy Limit of String Field Theory
18(3)
1.7 The Atiyah-Singer Index Theorem and LDP Problems Associated with It
21(2)
1.8 LDP Problems in General Relativity
23(11)
2 LDP in Biology, Neural Networks, Electromagnetic Measurements, Cosmic Expansion
34(8)
2.1 The Importance of Mathematical Models in Medicine
27(3)
2.2 LDP Related Problems in Neural Networks and Artificial Intelligence
30(1)
2.3 LDP Related Problems to Cosmic Expansion in General Relativity
31(1)
2.4 LDP Problems in Biology
31(3)
2.5 A Sensitive Quantum Mechanical Method for Measuring the Scattered Electromagnetic Fields
34(8)
3 LDP in Signal Processing, Communication and Antenna Design
42(26)
3.1 Review
35(1)
3.2 Large Deviation Problems in SSB Modulation
36(1)
3.3 What is Meant by Estimating a Quantum Field in Space-time
37(1)
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)
38(2)
3.5 The pde's Satisfied by the Quantum Electromagnetic Field Observables in a Cavity Resonator in the Presence of Bath Noise
40(1)
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
41(27)
4 LDP Applied to Quantum Measurement, Classical Markov Chains, Quantum Stochastics and Quantum Transition Probabilities
68(12)
4.1 Some Other Aspects of Measurement of a Quantum Field
43(2)
4.2 Some Parts of the Solution to the Question Paper on Antenna Theory
45(5)
4.3 Some Additional LDP Related Problems in Antenna Theory
50(2)
4.4 LDP for Quantum Markov Chains Using Discrete Time Quantum Stochastic Flows
52(2)
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 the Differential of a Levy Process (ie, a Limit of Compound Poisson Process Plus white Gaussian noise)
54(1)
4.6 LDP Applied to the Electroweak Theory
54(2)
4.7 LDP Problems in Quantum Mechanical Transitions
56(1)
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
57(1)
4.9 Large Deviation Problems in Queueing Theory
58(4)
4.10 Large Deviation Problems Associated with Quantum Filtering Theory
62(18)
5 LDP in Classical Stochastic Process Theory and Quantum Mechanical Transitions
80(26)
5.1 Large Deviations Problems to the Propagation of Noise at the Sigmoidal Computation Nodes Through the Neural Network
69(1)
5.2 Law of the Iterated Logarithm for Sums of iid Random Variables
70(1)
5.3 A version of the LDP for iid Random Variables
71(1)
5.4 The Law of the Iterated Logarithm for Sums of iid Random Variables Having Finite Variance
72(1)
5.5 An Open Problem Relating Applications of LDP to Martingales
72(2)
5.6 Properties of ML Estimators Based on iid Measurements
74(32)
6 LDP in Pattern Recognition and Fermionic Quantum Filtering
106(6)
6.1 Large Deviation Problems in Pattern Recognition
81(2)
6.2 LDP for Estimating the Parameters in Mixture Models
83(1)
6.3 The EM Algorithm
83(2)
6.4 Sanov's Theorem and Gibbs Distributions
85(1)
6.5 Gibbs Distribution in the Interacting Particle Case
86(1)
6.6 Inversion of the Characteristic Function of a Probability Distribution on the Real Line
87(1)
6.7 Infinitely Divisible Distributions, The Levy-Khintchine Theorem
88(1)
6.8 Stationary Distribution for Markov Chains
88(3)
6.9 Lecture on Quantum Filtering in the Presence of Fermionic Noise
91(8)
6.10 Lecture Plan for Pattern Recognition
99(1)
6.11 Review of the Book Stochastics, Control and Robotics
100(12)
Harish Parthasarathy
7 LDP in Spin Field Theory, Anharmonic Perturbations of Quantum Oscillators, Small Perturbations of Quantum Gibbs States
112(12)
7.1 Large Deviation Problems in Spin-field Interaction Theory
107(1)
7.2 Large Deviation Problems Associated with a Quantum Gravitational Field Interacting with a Non-Abelian Gauge Field
108(1)
7.3 Large Deviation Problems in Quantum Harmonic Oscillator Problems with Nonlinear Terms
109(1)
7.4 Formulation of an LDP for Quantum Stochastic Processes
110(14)
8 LDP for Electromagnetic Control of Gravitational Waves, Randomly Perturbed Quantum Fields, Hartree-Fock Approximation, Renewal Processes in Quantum Mechanics
124(18)
8.1 Gravitational Wave Propagating in a Background Curved Space-time, LDP for Reducing the Wave Fluctuations via Electromagnetic Control
113(2)
8.2 The Lehmann Representation of the Propagator
115(1)
8.3 The LDP Problem in this Context
116(1)
8.4 Central Limit Theorem for Renewal Processes
117(1)
8.5 Applications of Renewal Process Theory in Quantum Field Theory
118(2)
8.6 Large Deviation Principle in the Hartree-Fock Method for Approximately Solving Many Electron Problems
120(1)
8.7 LDP in Fuzzy Neural Networks
121(3)
8.8 Large Deviation Analysis of this qnn
124(1)
8.1 LDP Problems in Quantum Field Theory Related to Corrections to ; the Electron, Photon and non-Abelian Gauge Boson Propagators
124(18)
9 LDP in Electromagnetic Scattering and String Theory, Control of Dynamical Systems Using LDP
142(5)
9.1 A Summary of a List of LDP Applications in Physics and Engineering
125(2)
9.2 LDP Problems Related to Scattering of Electromagnetic Waves by a Perfectly Conducting Cylinder
127(2)
9.3 String Theory and Large Deviations
129(2)
9.4 Questions Related to Qualitative Properties of Quantum Noise
131(4)
9.5 Questions on Pattern Recognition
135(5)
9.6 Appendix
140(7)
10 LDP in Markov Chain and Queueing Theory with Quantum Mechanical Applications
147(7)
10.1 Notes on Applications of LDP to Stochastic Processes and Queueing Theory
143(2)
10.2 LDP Problems in Markov Chain Theory
145(1)
10.3 Continuity and Non-differentiability of the Brownian Sample Paths
145(1)
10.4 Renewal Processes in Quantum Mechanics
146(8)
11 LDP in Device Physics, Quantum Scattering Amplitudes, Quantum Filtering and Quantum Antennas
154(8)
11.1 Large Deviations in Vacuum Polarization
149(1)
11.2 An Application of the EKF and LDP to Estimating the Current in a pn Junction
150(1)
11.3 Large Deviation Problems in Quantum Stochastic Filtering Theory
151(1)
11.4 Large Deviation Problems in Quantum Antennas
151(11)
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
162(8)
12.1 Large Deviation Methods in Classical and Quantum Field Theory
155(15)
13 Mathematical Tools for Large Deviations, Neural Networks, LDP in Physical Theories, EM and LDP Algorithms in Quantum Parameter Estimation and Filtering
170(16)
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
163(2)
13.2 Proof of the Prohorov Tightness Theorem
165(1)
13.3 Some Remarks on Neural Networks Related to Large Deviation Theory
166(2)
13.4 LDP Problems in General Relativity and non-Abelian Gauge Field Theory
168(1)
13.5 Large Deviations in String Theoretic Corrections to Field Theories
169(17)
14 Quantum Transmission Lines, Engineering Applications of Stochastic Processes
186(24)
14.1 Introduction
171(1)
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
172(1)
14.3 Dirichlet Series with Image Processing Applications
172(2)
14.4 About the Book
174(1)
14.5 An Application of the EM Algorithm to Quantum Parameter Estimation and Quantum Filtering
175(4)
14.6 The EM Algorithm and Large Deviation Theory
179(1)
14.7 Kolmogorov-Smirnov Statistics
180(4)
14.8 Quantum Transmission Lines, LDP Problems
184(26)
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
210(8)
15.1 On the Amount of Mass that an Electron can Get from the Background Electromagnetic and Gravitational Fields
189(1)
15.2 Electron Propagator Corrections in the Presence of Quantum Noise
190(1)
15.3 Large Deviation Problems for the Schrodinger and Dirac Noisy Channels
190(3)
15.4 Central Limit Theorem for Martingales
193(3)
15.5 More Problems in LDP Applied to Quantum Field Theory
196(1)
15.6 Schrodinger and Klein-Gordon Equations in Quantum Field Theory Based on An Infinite Dimensional Laplacian Operator
197(3)
15.7 Proof of the Prohorov Tightness Theorem
200(2)
15.8 Large Deviation Problems in Field Measurement Analysis
202(1)
15.9 ADM Action for Quantum Gravity and Its Noisy Perturbation with LDP Analysis of the Solution Metric
203(1)
15.10 The Bianchi Identity for non-Abelian Gauge Fields
204(2)
15.11 More Problems in LDP Applied to Quantum Field Theory
206(2)
15.12 Schrodinger and Klein-Gordon Equations in Quantum Field Theory Based on An Infinite Dimensional Laplacian Operator
208(10)
16 Weak Convergence, Sanov's Theorem, LDP in Binary Signal Detection
218(10)
16.1 Prohorov's Tightness Theorem, "Necessity Part"
211(1)
16.2 Sanov's Theorem on the LDP for Empirical Distributions of Discrete iid Random Variables
212(2)
16.3 LDP in Binary Phase Shift Keying
214(1)
16.4 Compactness of the Set of Probability Measures on a Compact Metric Space
215(2)
16.5 Large Deviations for Frequency Modulated Signals
217(11)
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
228(9)
17.1 LDP Theory Applied to Transmission Lines with Line Loading
219(2)
17.2 LDP Formulation of the Quantum Transmission Line
221(1)
17.3 Yang-Mills Gauge Fields, The Euler characteristic, String Theoretic Corrections to the Yang-Mills Anomaly Cancellation Lagrangian Terms
222(4)
17.4 Quantum Averaging Based Derivation of Action Functional for Point Fields from Action Functional of String Fields
226(11)
18 LDP and EM Algorithm, LDP for Parameter Estimates in Linear Dynamical Systems, Philosophical Questions in Quantum General Relativity
237(17)
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
229(1)
18.2 Fundamental Problems in Quantum General Relativity
230(1)
18.3 A Problem in Large Deviations and Lie Algebras
231(1)
18.4 Describing Quantum Gravity Using Holonomy Fields
232(1)
18.5 Square of the Dirac Operator in Curved Spacetime in the Presence of a non-Abelian Yang-Mills Gauge Potential
233(1)
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
234(1)
18.7 Applications
234(1)
18.8 Equivalence of LDP
235(1)
18.9 Lehmann's Representation of the Propagator of the Klein-Gordon Field with Nonlinear Perturbations
236(18)
Chapters Index 254
Harish Parthasarathy received B.tech degree in Electrical Engineering from IIT Kanpur and his PhD degree in Signal Processing from IIT Delhi in 1994. He worked as a visiting fellow in the Indian Institute of Astro Physics, Bangalore, from 1993 to 1994. He has worked as a member of faculty in the Electrical Engineering departments at IIT Bombay and IIT Kanpur. Currently, he is a professor in the Electronics and Communication Division in NSIT, Delhi. He teaches courses on Signal processing, Electromagnetics, transmission lines, wave guides & antennas and electives in quantum field theory, quantum robotics & quantum stochastic processes to undergraduate and postgraduate students. He has published over 70 papers in international journals and conferences and he has also published 12 books on a variety of topics in mathematical physics, signal processing and antenna theory.
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