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, briefly mentioned in the textbook literature. Potentially even more important, a key focus of the book will be on the projection operator method of enforcing quantum constraints.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 quantum principles* A review of coherent state fundamentals* A critical comparison of the 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 as well as 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* A brief extension of the book’s principal discussion into the realm of field theoryFunctional Integration: A Modern Treatment 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 each chapter, making the book suitable for a one-semester graduate course on functional integration; prerequisites consist mostly of some basic knowledge of quantum mechanics.

Preface	p. vii
Introduction	p. 1
Overview and Latest Developments	p. 1
Topics Receiving Particular Emphasis	p. 2
Basic Background	p. 3
Elementary Integral Facts	p. 5
Stochastic Theory
Probability	p. 9
Random Variables	p. 9
Probability distributions	p. 10
Characteristic Functions	p. 13
Convergence properties - 1	p. 14
Convergence properties - 2	p. 15
Characteristic function for a Cantor-like measure	p. 16
An application of the characteristic function	p. 17
Infinitely Divisible Distributions	p. 18
Divisibility	p. 19
Infinite divisibility	p. 20
Central Limit Theorem-and Its Avoidance	p. 21
Infinite-Dimensional Integrals	p. 27
Basics	p. 27
Support Properties	p. 29
Characteristic Functional	p. 30
Tightest Support Conditions	p. 32
From Sequences to Functions	p. 33
Bochner-Minlos Theorem	p. 36
Functional Derivatives	p. 38
Functional Fourier Transformations	p. 39
Change of Variables	p. 41
Change of infinitely many variables	p. 43
Stochastic Variable Theory	p. 47
General Remarks	p. 47
Stationary processes	p. 48
Ergodic processes	p. 48
Gaussian examples	p. 51
Wiener Process, a.k.a. Brownian Motion	p. 52
Definition of a standard Wiener process	p. 52
Continuity of Brownian paths	p. 53
Stochastic equivalence	p. 55
Independent increments	p. 56
Some joint and conditional probability densities	p. 57
Itô calculus	p. 58
Stochastic integrals	p. 59
Wiener Measure	p. 61
General Wiener process	p. 62
Pinned Brownian motion	p. 63
Generalized Brownian bridges	p. 63
Alternative Brownian bridges	p. 64
The Feynman-Kac Formula	p. 64
Ornstein-Uhlenbeck Process	p. 66
Addition of a potential to an O-U process	p. 67
Realization of a General Gaussian Process	p. 67
Generalized Stochastic Process	p. 69
Gaussian white noise	p. 69
Stochastic Differential Equations a.k.a. Langevin Equations	p. 71
Poisson Process	p. 73
Quantum Theory
Background to an Analysis of Quantum Mechanics	p. 79
Hilbert Space and Operators: Basic Properties	p. 79
Hilbert space	p. 79
Fourier representation	p. 81
L² representatives	p. 81
Segal-Bargmann representation	p. 83
Reproducing kernel Hilbert spaces	p. 83
Operators for Hilbert space	p. 86
Hilbert Space and Operators: Advanced Properties	p. 90
Basic Lie Group Theory	p. 95
Lie algebras	p. 96
Invariant group measures	p. 97
Group representations	p. 98
Outline of Abstract Quantum Mechanics	p. 102
Schrödinger picture	p. 103
Heisenberg picture	p. 104
Quantum Mechanical Path Integrals	p. 107
Configuration Space Path Integrals	p. 107
Schrödinger equation (special case)	p. 107
The Free particle	p. 109
Quadratic path integrals	p. 110
Harmonic oscillator	p. 112
Eigenfunctions and eigenvalues	p. 113
Connection with operators	p. 114
Validity of the lattice space regularization	p. 116
Classical symptoms of quantum illnesses	p. 116
The question of a measure for the Feynman path integral	p. 118
Proposal of Gel'fand and Yaglom to introduce a measure	p. 118
Proposal of Itô to introduce a measure	p. 120
Phase Space Path Integrals	p. 121
Momentum space propagator	p. 121
Physical interpretation of path integrals	p. 123
Selected applications	p. 124
Choice of canonical coordinates	p. 128
Action Principle-and Equations of Motion	p. 130
Coherent State Path Integrals	p. 133
Canonical Coherent States and Their Properties	p. 133
Coherent states-what are they?	p. 133
Diagonal coherent state matrix elements	p. 137
Coherent State Propagator	p. 138
Change of coordinates	p. 140
Metrics from coherent states	p. 146
Coherent state path integrals-a one form and a metric	p. 147
Alternative coherent state path integral construction	p. 148
Many Degrees of Freedom	p. 149
Configuration space path integrals	p. 149
Phase space path integrals	p. 150
Coherent state path integrals	p. 151
Spin Coherent State Path Integrals	p. 152
Spin coherent states	p. 152
Spin dynamics and the spin coherent state path integral	p. 154
Affine Coherent State Path Integrals	p. 156
Affine coherent states	p. 156
Affine dynamics and the affine coherent state path integral	p. 158
Coherent State Path Integrals without a Resolution of Unity	p. 159
Continuous-Time Regularized Path Integrals	p. 161
Wiener Measure Regularization of Phase Space Path Integrals	p. 161
Covariance under canonical coordinate transformations	p. 163
Proof of Wiener measure path integral regularization	p. 164
Multivariable Wiener measure regularization of path integrals	p. 167
Continuous-Time Regularization of Spin Variable Path Integrals	p. 168
Continuous-Time Regularization of Affine Variable Path Integrals	p. 170
Quantization as Geometry	p. 172
Classical and Quantum Constraints	p. 175
Classical Systems with Constraints	p. 175
General classical construction	p. 177
Anomalous constraint situations	p. 180
Quantum Theory of Constrained Systems	p. 184
Dirac's procedure for quantization of systems with constraints	p. 185
The Projection Operator Method	p. 187
Observables and the classical limit	p. 189
Basic examples of the projection operator method	p. 190
Additional examples of the projection operator method	p. 195
Representation of the projection operator	p. 199
A universal representation for the projection operator	p. 201
Constrained Dynamics in Operator Form	p. 203
Coherent State Path Integrals for Systems with Constraints	p. 205
Quantum Field Theory
Application to Quantum Field Theory	p. 211
Introduction and Overview	p. 211
Classical preliminaries	p. 211
Relativistic Free Fields	p. 212
A brief survey of classical and quantum properties	p. 212
Functional Integral Formulation	p. 215
Euclidean-Space Functional Integral Formulation	p. 219
Local products	p. 221
Interacting Scalar Fields	p. 223
Perturbation theory	p. 224
Spacetime dimension n = 3	p. 226
Spacetime dimension n = 4	p. 228
Spacetime dimension n ≥ 5	p. 231
Euclidean-Space Lattice Regularization	p. 233
A Modern Approach to Nonrenormalizable Models	p. 235
Introduction	p. 235
Nonrenormalizable Classical Models	p. 236
Relativistic models	p. 236
Ultralocal models	p. 236
Independent value models	p. 237
Classical pseudofree models	p. 237
Euclidean Space Functional Integrals-Preliminary Remarks	p. 239
Independent value models	p. 239
Ultralocal models	p. 240
Solution of independent value models	p. 241
Solution of ultralocal models	p. 244
An alternative approach to both the IV and UL models-an overview with details to follow	p. 248
An Alternative Method to Solve the IV and UL Models	p. 249
A reexamination of IV models	p. 249
A reexamination of UL models	p. 253
Relativistic Nonrenormalizable Scalar Models	p. 254
Motivation for alternative studies of relativistic nonrenormalizable models	p. 255
The free ground state distribution	p. 256
Properties of the matrix A_k-l	p. 257
Mass-like moments in the ground state distribution	p. 258
Mashing the relativistic measure	p. 258
Lattice Hamiltonian for the free and pseudofree models	p. 260
The importance of sharp time moments	p. 262
Statement of the fundamental problem	p. 263
The Continuum Limit, and Term-by-Term Finiteness of a Perturbation Analysis	p. 264
Field strength renormalization	p. 264
Mass and coupling constant renormalization	p. 265
Conclusion	p. 266
References	p. 269
Index	p. 275
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A Modern Approach to Functional Integration

9780817647902

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