Record:   Prev Next
作者 Lörinczi, József
書名 Feynman-Kac-Type Theorems and Gibbs Measures on Path Space : With Applications to Rigorous Quantum Field Theory
出版項 Berlin/Boston : De Gruyter, Inc., 2011
©2011
國際標準書號 9783110203738 (electronic bk.)
9783110201482
book jacket
說明 1 online resource (520 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
系列 De Gruyter Studies in Mathematics Ser. ; v.34
De Gruyter Studies in Mathematics Ser
附註 Intro -- Preface -- I Feynman-Kac-type theorems and Gibbs measures -- 1 Heuristics and history -- 1.1 Feynman path integrals and Feynman-Kac formulae -- 1.2 Plan and scope -- 2 Probabilistic preliminaries -- 2.1 An invitation to Brownian motion -- 2.2 Martingale and Markov properties -- 2.2.1 Martingale property -- 2.2.2 Markov property -- 2.2.3 Feller transition kernels and generators -- 2.2.4 Conditional Wiener measure -- 2.3 Basics of stochastic calculus -- 2.3.1 The classical integral and its extensions -- 2.3.2 Stochastic integrals -- 2.3.3 Itô formula -- 2.3.4 Stochastic differential equations and diffusions -- 2.3.5 Girsanov theorem and Cameron-Martin formula -- 2.4 Lévy processes -- 2.4.1 Lévy process and Lévy-Khintchine formula -- 2.4.2 Markov property of Lévy processes -- 2.4.3 Random measures and Lévy-Itô decomposition -- 2.4.4 Itô formula for semimartingales -- 2.4.5 Subordinators -- 2.4.6 Bernstein functions -- 3 Feynman-Kac formulae -- 3.1 Schrödinger semigroups -- 3.1.1 Schrödinger equation and path integral solutions -- 3.1.2 Linear operators and their spectra -- 3.1.3 Spectral resolution -- 3.1.4 Compact operators -- 3.1.5 Schrödinger operators -- 3.1.6 Schrödinger operators by quadratic forms -- 3.1.7 Confining potential and decaying potential -- 3.1.8 Strongly continuous operator semigroups -- 3.2 Feynman-Kac formula for external potentials -- 3.2.1 Bounded smooth external potentials -- 3.2.2 Derivation through the Trotter product formula -- 3.3 Feynman-Kac formula for Kato-class potentials -- 3.3.1 Kato-class potentials -- 3.3.2 Feynman-Kac formula for Kato-decomposable potentials -- 3.4 Properties of Schrödinger operators and semigroups -- 3.4.1 Kernel of the Schrödinger semigroup -- 3.4.2 Number of eigenfunctions with negative eigenvalues -- 3.4.3 Positivity improving and uniqueness of ground state
3.4.4 Degenerate ground state and Klauder phenomenon -- 3.4.5 Exponential decay of the eigenfunctions -- 3.5 Feynman-Kac-Itô formula for magnetic field -- 3.5.1 Feynman-Kac-Itô formula -- 3.5.2 Alternate proof of the Feynman-Kac-Itô formula -- 3.5.3 Extension to singular external potentials and vector potentials -- 3.5.4 Kato-class potentials and Lp-Lq boundedness -- 3.6 Feynman-Kac formula for relativistic Schrödinger operators -- 3.6.1 Relativistic Schrödinger operator -- 3.6.2 Relativistic Kato-class potentials and Lp-Lq boundedness -- 3.7 Feynman-Kac formula for Schrödinger operator with spin -- 3.7.1 Schrödinger operator with spin -- 3.7.2 A jump process -- 3.7.3 Feynman-Kac formula for the jump process -- 3.7.4 Extension to singular potentials and vector potentials -- 3.8 Feynman-Kac formula for relativistic Schrödinger operator with spin -- 3.9 Feynman-Kac formula for unbounded semigroups and Stark effect -- 3.10 Ground state transform and related diffusions -- 3.10.1 Ground state transform and the intrinsic semigroup -- 3.10.2 Feynman-Kac formula for P(f)1-processes -- 3.10.3 Dirichlet principle -- 3.10.4 Mehler's formula -- 4 Gibbs measures associated with Feynman-Kac semigroups -- 4.1 Gibbs measures on path space -- 4.1.1 From Feynman-Kac formulae to Gibbs measures -- 4.1.2 Definitions and basic facts -- 4.2 Existence and uniqueness by direct methods -- 4.2.1 External potentials: existence -- 4.2.2 Uniqueness -- 4.2.3 Gibbs measure for pair interaction potentials -- 4.3 Existence and properties by cluster expansion -- 4.3.1 Cluster representation -- 4.3.2 Basic estimates and convergence of cluster expansion -- 4.3.3 Further properties of the Gibbs measure -- 4.4 Gibbs measures with no external potential -- 4.4.1 Gibbs measure -- 4.4.2 Diffusive behaviour -- II Rigorous quantumfield theory
5 Free Euclidean quantum field and Ornstein-Uhlenbeck processes -- 5.1 Background -- 5.2 Boson Fock space -- 5.2.1 Second quantization -- 5.2.2 Segal fields -- 5.2.3 Wick product -- 5.3 ℒ -spaces -- 5.3.1 Gaussian random processes -- 5.3.2 Wiener-Itô-Segal isomorphism -- 5.3.3 Lorentz covariant quantum fields -- 5.4 Existence of ℒ -spaces -- 5.4.1 Countable product spaces -- 5.4.2 Bochner theorem and Minlos theorem -- 5.5 Functional integration representation of Euclidean quantum fields -- 5.5.1 Basic results in Euclidean quantum field theory -- 5.5.2 Markov property of projections -- 5.5.3 Feynman-Kac-Nelson formula -- 5.6 Infinite dimensional Ornstein-Uhlenbeck process -- 5.6.1 Abstract theory of measures on Hilbert spaces -- 5.6.2 Fock space as a function space -- 5.6.3 Infinite dimensional Ornstein-Uhlenbeck-process -- 5.6.4 Markov property -- 5.6.5 Regular conditional Gaussian probability measures -- 5.6.6 Feynman-Kac-Nelson formula by path measures -- 6 The Nelson model by path measures 293 6.1 Preliminaries -- 6.2 The Nelson model in Fock space -- 6.2.1 Definition -- 6.2.2 Infrared and ultraviolet divergences -- 6.2.3 Embedded eigenvalues -- 6.3 The Nelson model in function space -- 6.4 Existence and uniqueness of the ground state -- 6.5 Ground state expectations -- 6.5.1 General theorems -- 6.5.2 Spatial decay of the ground state -- 6.5.3 Ground state expectation for second quantized operators -- 6.5.4 Ground state expectation for field operators -- 6.6 The translation invariant Nelson model -- 6.7 Infrared divergence -- 6.8 Ultraviolet divergence -- 6.8.1 Energy renormalization -- 6.8.2 Regularized interaction -- 6.8.3 Removal of the ultraviolet cutoff -- 6.8.4 Weak coupling limit and removal of ultraviolet cutoff -- 7 The Pauli-Fierz model by path measures -- 7.1 Preliminaries -- 7.1.1 Introduction -- 7.1.2 Lagrangian QED
7.1.3 Classical variant of non-relativistic QED -- 7.2 The Pauli-Fierz model in non-relativistic QED -- 7.2.1 The Pauli-Fierz model in Fock space -- 7.2.2 The Pauli-Fierz model in function space -- 7.2.3 Markov property -- 7.3 Functional integral representation for the Pauli-Fierz Hamiltonian -- 7.3.1 Hilbert space-valued stochastic integrals -- 7.3.2 Functional integral representation -- 7.3.3 Extension to general external potential -- 7.4 Applications of functional integral representations -- 7.4.1 Self-adjointness of the Pauli-Fierz Hamiltonian -- 7.4.2 Positivity improving and uniqueness of the ground state -- 7.4.3 Spatial decay of the ground state -- 7.5 The Pauli-Fierz model with Kato class potential -- 7.6 Translation invariant Pauli-Fierz model -- 7.7 Path measure associated with the ground state -- 7.7.1 Path measures with double stochastic integrals -- 7.7.2 Expression in terms of iterated stochastic integrals -- 7.7.3 Weak convergence of path measures -- 7.8 Relativistic Pauli-Fierz model -- 7.8.1 Definition -- 7.8.2 Functional integral representation -- 7.8.3 Translation invariant case -- 7.9 The Pauli-Fierz model with spin -- 7.9.1 Definition -- 7.9.2 Symmetry and polarization -- 7.9.3 Functional integral representation -- 7.9.4 Spin-boson model -- 7.9.5 Translation invariant case -- 8 Notes and References -- Bibliography -- Index
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
鏈接 Print version: Lörinczi, József Feynman-Kac-Type Theorems and Gibbs Measures on Path Space : With Applications to Rigorous Quantum Field Theory Berlin/Boston : De Gruyter, Inc.,c2011 9783110201482
主題 Integration, Functional.;Stochastic analysis.;Quantum field theory -- Mathematics
Electronic books
Alt Author Hiroshima, Fumio
Betz, Volker
Lörinczi, József
Record:   Prev Next