Format:
Online-Ressource (XVI, 282p. 9 illus, digital)
Edition:
1
ISBN:
9780817647919
Series Statement:
Applied and Numerical Harmonic Analysis
Content:
Preface -- Introduction -- Part I: Stochastic Theory -- Probability -- Infinite-Dimensional Integrals -- Stochastic Variable Theory -- Part II: Quantum Theory -- Background to an Analysis of Quantum Mechanics -- Quantum Mechanical Path Integrals -- Coherent State Path Integrals -- Continuous-Time Regularized Path Integrals -- Classical and Quantum Constraints -- Part III: Quantum Field Theory -- Application to Quantum Field Theory -- A Modern Approach to Nonrenormalizable Models -- References -- Index.
Content:
This text takes advantage of recent developments in the theory of path integration to provide an improved treatment of quantization of systems that either have no constraints or instead involve constraints with demonstratively improved procedures. Strong emphasis is placed on the coherent state form of the path integral, which typically is only briefly mentioned in the textbook literature. Also of prime importance, a key focus of the book will be on the projection operator method of enforcing quantum constraints. Additionally, several novel proposals are introduced to deal with highly singular quantum field theories. The modern treatment used by the author is an attempt to make a major paradigm shift in how the art of functional integration is practiced. The techniques developed in the work will prove valuable to graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics who find it necessary to deal with solutions to wave equations, both quantum and beyond. Key topics and features: * A thorough grounding in the Gold Standard of path integrals: the Wiener measure * Formulation of all path integral construction from abstract principles * A review of coherent state fundamentals * A critical comparison of several path integral versions with emphasis on the virtues of the coherent state version * A construction of the Wiener-measure regularized phase space path integral, its emergence as a coherent state path integral, and its superior definition and connection to the classical theory underlying the quantization * A review of classical and quantum constraints and some of their traditional treatments * Introduction of the projection operator method to deal with quantum constraints, its many virtues as compared to traditional methods, and how it can be incorporated into a conventional or coherent state phase space path integral * An extension of the book’s principal discussion into the realm of quantum field theory with a special emphasis on highly singular examples A Modern Approach to Functional Integration offers insight into these contemporary research topics, which may lead to improved methods and results that cannot be found elsewhere in the textbook literature. Exercises are included in most chapters, making the book suitable for a one-semester graduate course on functional integration; prerequisites consist mostly of some basic knowledge of quantum mechanics.
Note:
Description based upon print version of record
,
""A Modern Approach to Functional Integration""; ""Preface""; ""Contents""; ""1 Introduction""; ""1.1 Overview and Latest Developments""; ""1.2 Topics Receiving Particular Emphasis""; ""1.3 Basic Background""; ""1.4 Elementary Integral Facts""; ""Part I Stochastic Theory""; ""2 Probability""; ""2.1 Random Variables""; ""2.1.1 Probability distributions""; ""2.2 Characteristic Functions""; ""2.2.1 Convergence properties - 1""; "" 2.2.2 Convergence properties - 2""; ""2.2.3 Characteristic function for a Cantor-like measure""; ""2.2.4 An application of the characteristic function""
,
""2.3 Infinitely Divisible Distributions""""2.3.1 Divisibility""; ""2.3.2 Infinite divisibility""; ""2.4 Central Limit Theorem-and Its Avoidance""; ""3 Infinite -Dimensional Integrals""; ""3.1 Basics""; ""3.2 Support Properties""; ""3.3 Characteristic Functional""; ""3.4 Tightest Support Conditions""; ""3.5 From Sequences to Functions""; ""3.6 Bochner-Minlos Theorem""; ""3.7 Functional Derivatives""; ""3.8 Functional Fourier Transformations""; ""3.9 Change of Variables""; ""3.9.1 Change of infinitely many variables""; ""Exercises""; ""4 Stochastic Variable Theory""; ""4.1 General Remarks""
,
""4.1.1 Stationary processes""""4.1.2 Ergodic processes""; ""4.1.3 Gaussian examples""; ""4.2 Wiener Process, a.k.a. Brownian Motion""; ""4.2.1 Definition of a standard Wiener process""; ""4.2.2 Continuity of Brownian paths""; ""4.2.3 Stochastic equivalence""; ""4.2.4 Independent increments""; ""4.2.5 Some joint and conditional probability densities""; ""4.2.6 ItÃ? calculus""; ""4.2.7 Stochastic integrals""; ""4.3 Wiener Measure""; ""4.3.1 General Wiener process""; ""4.3.2 Pinned Brownian motion""; ""4.3.3 Generalized Brownian bridges""; ""4.3.4 Alternative Brownian bridges""
,
""4.4 The Feynman� Kac Formula""""4.5 Ornstein� Uhlenbeck Process""; ""4.5.1 Addition of a potential to an O-U process""; ""4.6 Realization of a General Gaussian Process""; ""4.7 Generalized Stochastic Process""; ""4.7.1 Gaussian white noise""; ""4.8 Stochastic Differential Equations a.k.a. Langevin Equations""; ""4.9 Poisson Process""; ""Exercises""; ""Part II Quantum Theory""; ""5 Background to an Analysis of Quantum Mechanics""; ""5.1 Hilbert Space and Operators: Basic Properties""; ""5.1.1 Hilbert space""; ""5.1.2 Fourier representation""; ""5.1.3 L² representatives""
,
""5.1.4 Segal-Bargmann representation""""5.1.5 Reproducing kernel Hilbert spaces""; ""5.1.6 Operators for Hilbert space""; ""5.2 Hilbert Space and Operators: Advanced Properties""; ""5.3 Basic Lie Group Theory""; ""5.3.1 Lie algebras""; ""5.3.2 Invariant group measures""; ""5.3.3 Group representations""; ""5.4 Outline of Abstract Quantum Mechanics""; ""5.4.1 Schrödinger picture""; ""5.4.2 Heisenberg picture""; ""Exercises""; ""6 Quantum Mechanical Path Integrals""; ""6.1 Conufiguration Space Path Integrals""; ""6.1.1 SchrÅ?odinger equation (special case)""; ""6.1.2 The free particle""
,
""6.1.3 Quadratic path integrals""
Additional Edition:
ISBN 9780817647902
Additional Edition:
Buchausg. u.d.T. Klauder, John R., 1932 - A modern approach to functional integration New York, NY : Birkhäuser, 2011 ISBN 9780817647902
Language:
English
Subjects:
Mathematics
Keywords:
Integration
;
Quantentheorie
;
Wiener-Maß
;
Integration
;
Quantentheorie
;
Wiener-Maß
;
Lehrmittel
;
Lehrbuch
DOI:
10.1007/978-0-8176-4791-9
URL:
Volltext
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URL:
Volltext
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