A modern approach to functional integration
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New York [u.a.]
Birkhäuser
2011
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| Schriftenreihe: | Applied and numerical harmonic analysis
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| 100 | 1 | |a Klauder, John R. |d 1932- |e Verfasser |0 (DE-588)136019986 |4 aut | |
| 245 | 1 | 0 | |a A modern approach to functional integration |c John R. Klauder |
| 264 | 1 | |a New York [u.a.] |b Birkhäuser |c 2011 | |
| 300 | |a XV, 279 S. |b graph. Darst. | ||
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Datensatz im Suchindex
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| adam_text |
Titel: A modern approach to functional integration
Autor: Klauder, John R.
Jahr: 2011
Contents
Preface.VII
1 Introduction. 1
1.1 Overview and Latest Developments. 1
1.2 Topics Receiving Particular Emphasis. 2
1.3 Basic Background. 3
1.4 Elementary Integral Facts. 5
Part I Stochastic Theory
2 Probability. 9
2.1 Random Variables. 9
2.1.1 Probability distributions . 10
2.2 Characteristic Functions. 13
2.2.1 Convergence properties - 1. 14
2.2.2 Convergence properties - 2. 15
2.2.3 Characteristic function for a Cantor-like measure. 16
2.2.4 An application of the characteristic function. 17
2.3 Infinitely Divisible Distributions. 18
2.3.1 Divisibility. 19
2.3.2 Infinite divisibility. 20
2.4 Central Limit Theorem—and Its Avoidance. 21
3 Infinite-Dimensional Integrals . 27
3.1 Basics. 27
3.2 Support Properties. 29
3.3 Characteristic Functional. 30
3.4 Tightest Support Conditions. 32
3.5 From Sequences to Functions. 33
3.6 Bochner-Minlos Theorem. 36
XII Contents
3.7 Functional Derivatives. 38
3.8 Functional Fourier Transformations. 39
3.9 Change of Variables . 41
3.9.1 Change of infinitely many variables. 43
4 Stochastic Variable Theory. 47
4.1 General Remarks. 47
4.1.1 Stationary processes. 48
4.1.2 Ergodic processes . 48
4.1.3 Gaussian examples. 51
4.2 Wiener Process, a.k.a. Brownian Motion. 52
4.2.1 Definition of a standard Wiener process. 52
4.2.2 Continuity of Brownian paths. 53
4.2.3 Stochastic equivalence . 55
4.2.4 Independent increments. 56
4.2.5 Some joint and conditional probability densities. 57
4.2.6 Itô calculus. 58
4.2.7 Stochastic integrals. 59
4.3 Wiener Measure. 61
4.3.1 General Wiener process. 62
4.3.2 Pinned Brownian motion. 63
4.3.3 Generalized Brownian bridges. 63
4.3.4 Alternative Brownian bridges. 64
4.4 The Feynman-Kac Formula.*. 64
4.5 Ornstein-Uhlenbeck Process. 66
4.5.1 Addition of a potential to an O-U process. 67
4.6 Realization of a General Gaussian Process. 67
4.7 Generalized Stochastic Process. 69
4.7.1 Gaussian white noise. 69
4.8 Stochastic Differential Equations a.k.a. Langevin Equations . 71
4.9 Poisson Process. 73
Part II Quantum Theory
5 Background to an Analysis of Quantum Mechanics. 79
5.1 Hubert Space and Operators: Basic Properties. 79
5.1.1 Hilbert space. 79
5.1.2 Fourier representation. 81
5.1.3 L2 representatives. 81
5.1.4 Segal-Bargmann representation. 83
5.1.5 Reproducing kernel Hilbert spaces. 83
5.1.6 Operators for Hilbert space. 86
5.2 Hilbert Space and Operators: Advanced Properties . 90
5.3 Basic Lie Group Theory. 95
Contents XIII
5.3.1 Lie algebras . 96
5.3.2 Invariant group measures. 97
5.3.3 Group representations . 98
5.4 Outline of Abstract Quantum Mechanics.102
5.4.1 Schrödinger picture.103
5.4.2 Heisenberg picture.104
Quantum Mechanical Path Integrals.107
6.1 Configuration Space Path Integrals.107
6.1.1 Schrödinger equation (special case).107
6.1.2 The free particle .109
6.1.3 Quadratic path integrals .110
6.1.4 Harmonic oscillator.112
6.1.5 Eigenfunctions and eigenvalues.113
6.1.6 Connection with operators.114
6.1.7 Validity of the lattice space regularization.116
6.1.8 Classical symptoms of quantum illnesses .116
6.1.9 The question of a measure for the Feynman
path integral.118
6.1.10 Proposal of Gel'fand and Yaglom to introduce
a measure.118
6.1.11 Proposal of Itô to introduce a measure.120
6.2 Phase Space Path Integrals.121
6.2.1 Momentum space propagator.121
6.2.2 Physical interpretation of path integrals.123
6.2.3 Selected applications.124
6.2.4 Choice of canonical coordinates.128
6.3 Action Principle—and Equations of Motion.130
Coherent State Path Integrals.133
7.1 Canonical Coherent States and Their Properties.133
7.1.1 Coherent states—what are they?.133
7.1.2 Diagonal coherent state matrix elements.137
7.2 Coherent State Propagator .138
7.2.1 Change of coordinates .140
7.2.2 Metrics from coherent states.146
7.2.3 Coherent state path integrals—a one form and
a metric.147
7.2.4 Alternative coherent state path integral construction. 148
7.3 Many Degrees of Freedom.149
7.3.1 Configuration space path integrals.149
7.3.2 Phase space path integrals.150
7.3.3 Coherent state path integrals.151
7.4 Spin Coherent State Path Integrals.152
7.4.1 Spin coherent states.152
XIV Contents
7.4.2 Spin dynamics and the spin coherent state
path integral.154
7.5 Affine Coherent State Path Integrals.156
7.5.1 Affine coherent states.156
7.5.2 Affine dynamics and the affine coherent state path
integral.158
7.6 Coherent State Path Integrals without a Resolution of Unity . 159
8 Continuous-Time Regularized Path Integrals.161
8.1 Wiener Measure Regularization of Phase Space Path Integrals. 161
8.1.1 Covariance under canonical coordinate transformations. 163
8.1.2 Proof of Wiener measure path integral regularization . 164
8.1.3 Multivariable Wiener measure regularization of path
integrals .167
8.2 Continuous-Time Regularization of Spin Variable Path
Integrals .168
8.3 Continuous-Time Regularization of Affine Variable Path
Integrals .170
8.4 Quantization as Geometry.172
9 Classical and Quantum Constraints.175
9.1 Classical Systems with Constraints.175
9.1.1 General classical construction.177
9.1.2 Anomalous constraint situations.180
9.2 Quantum Theory of Constrained Systems.184
9.2.1 Dirac's procedure for quantization of systems with
constraints.185
9.3 The Projection Operator Method.187
9.3.1 Observables and the classical limit.189
9.3.2 Basic examples of the projection operator method.190
9.3.3 Additional examples of the projection operator method 195
9.3.4 Representation of the projection operator.199
9.3.5 A universal representation for the projection operator . 201
9.4 Constrained Dynamics in Operator Form.203
9.5 Coherent State Path Integrals for Systems with Constraints. 205
Part III Quantum Field Theory
10 Application to Quantum Field Theory.211
10.1 Introduction and Overview.211
10.1.1 Classical preliminaries .211
10.2 Relativistic Free Fields.212
10.2.1 A brief survey of classical and quantum properties.212
10.3 Functional Integral Formulation.215
Contents XV
10.4 Euclidean-Space Functional Integral Formulation.219
10.4.1 Local products.221
10.5 Interacting Scalar Fields.223
10.5.1 Perturbation theory .224
10.5.2 Spacetime dimension ç = 3.226
10.5.3 Spacetime dimension n = 4.228
10.5.4 Spacetime dimension ç 5.231
10.6 Euclidean-Space Lattice Regularization.233
11 A Modern Approach to Nonrenormalizable Models.235
11.1 Introduction.235
11.2 Nonrenormalizable Classical Models .236
11.2.1 Relativistic models.236
11.2.2 Ultralocal models .236
11.2.3 Independent value models.237
11.2.4 Classical pseudofree models.237
11.3 Euclidean Space Functional Integrals—Preliminary Remarks . 239
11.3.1 Independent value models.239
11.3.2 Ultralocal models .240
11.3.3 Solution of independent value models.241
11.3.4 Solution of ultralocal models.244
11.3.5 An alternative approach to both the IV and UL
models—an overview with details to follow.248
11.4 An Alternative Method to Solve the IV and UL Models.249
11.4.1 A reexamination of IV models .249
11.4.2 A reexamination of UL models.253
11.5 Relativistic Nonrenormalizable Scalar Models.254
11.5.1 Motivation for alternative studies of relativistic
nonrenormalizable models.255
11.5.2 The free ground state distribution.256
11.5.3 Properties of the matrix Ak-i.257
11.5.4 Mass-like moments in the ground state distribution-----258
11.5.5 Mashing the relativistic measure.258
11.5.6 Lattice Hamiltonian for the free and pseudofree models 260
11.5.7 The importance of sharp time moments.262
11.5.8 Statement of the fundamental problem.263
11.6 The Continuum Limit, and Term-by-Term Finiteness of a
Perturbation Analysis.264
11.6.1 Field strength renormalization.264
11.6.2 Mass and coupling constant renormalization.265
11.7 Conclusion .266
References.269
Index. .275 |
| any_adam_object | 1 |
| author | Klauder, John R. 1932- |
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| discipline | Physik Mathematik |
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| id | DE-604.BV025603790 |
| illustrated | Illustrated |
| indexdate | 2024-11-25T17:37:10Z |
| institution | BVB |
| isbn | 9780817647902 9780817647919 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020198960 |
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| physical | XV, 279 S. graph. Darst. |
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| publishDateSearch | 2011 |
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| publisher | Birkhäuser |
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| series2 | Applied and numerical harmonic analysis |
| spellingShingle | Klauder, John R. 1932- A modern approach to functional integration Quantenphysik (DE-588)4266670-3 gnd Integration Mathematik (DE-588)4072852-3 gnd Mathematische Physik (DE-588)4037952-8 gnd Funktionalintegration (DE-588)4155674-4 gnd Quantentheorie (DE-588)4047992-4 gnd |
| subject_GND | (DE-588)4266670-3 (DE-588)4072852-3 (DE-588)4037952-8 (DE-588)4155674-4 (DE-588)4047992-4 |
| title | A modern approach to functional integration |
| title_auth | A modern approach to functional integration |
| title_exact_search | A modern approach to functional integration |
| title_full | A modern approach to functional integration John R. Klauder |
| title_fullStr | A modern approach to functional integration John R. Klauder |
| title_full_unstemmed | A modern approach to functional integration John R. Klauder |
| title_short | A modern approach to functional integration |
| title_sort | a modern approach to functional integration |
| topic | Quantenphysik (DE-588)4266670-3 gnd Integration Mathematik (DE-588)4072852-3 gnd Mathematische Physik (DE-588)4037952-8 gnd Funktionalintegration (DE-588)4155674-4 gnd Quantentheorie (DE-588)4047992-4 gnd |
| topic_facet | Quantenphysik Integration Mathematik Mathematische Physik Funktionalintegration Quantentheorie |
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