Although more than 60 years have passed since their first appearance, Feynman path integrals have yet to lose their fascination and luster. They are not only a formidable instrument of theoretical physics, but also a mathematical challenge; in fact, several mathematicians in the last 40 years have devoted their efforts to the rigorous mathematical definition of Feynman's ideas. This volume provides a detailed, self-contained description of the mathematical difficulties as well as the possible techniques used to solve these difficulties. In particular, it gives a complete overview of the mathematical realization of Feynman path integrals in terms of well-defined functional integrals, that is, the infinite dimensional oscillatory integrals. It contains the traditional results on the topic as well as the more recent developments obtained by the author. Mathematical Feynman Path Integrals and Their Applications is devoted to both mathematicians and physicists, graduate students and researchers who are interested in the problem of mathematical foundations of Feynman path integrals.
Author(s): Sonia Mazzucchi
Publisher: World Scientific Publishing Company
Year: 2009
Language: English
Pages: 225
Contents......Page 8
Preface......Page 6
1. Introduction......Page 10
1.1 Wiener's and Feynman's integration......Page 13
1.2 The Feynman functional......Page 19
1.3 Infinite dimensional oscillatory integrals......Page 21
2.1 Finite dimensional oscillatory integrals......Page 24
2.2 The Parseval type equality......Page 28
2.3 Generalized Fresnel integrals......Page 32
2.4 Infinite dimensional oscillatory integrals......Page 41
2.5 Polynomial phase functions......Page 51
3.1 The anharmonic oscillator with a bounded anharmonic potential......Page 66
3.2 Time dependent potentials......Page 77
3.3 Phase space Feynman path integrals......Page 85
3.4 Magnetic field......Page 95
3.5 Quartic potential......Page 97
4.1 Asymptotic expansions......Page 110
4.2 The stationary phase method. Finite dimensional case......Page 115
4.3 The stationary phase method. In nite dimensional case......Page 124
4.4 The semiclassical limit of quantum mechanics......Page 137
4.5 The trace formula......Page 147
5.1 Feynman path integrals and open quantum systems......Page 156
5.2 The Feynman-Vernon inuence functional......Page 164
5.3 The stochastic Schr odinger equation......Page 174
6.1 Analytic continuation of Wiener integrals......Page 186
6.2 The sequential approach......Page 190
6.3 White noise calculus......Page 193
6.4 Poisson processes......Page 197
6.5 Further approaches and results......Page 198
A.1 General theory......Page 200
A.2 The classical Wiener space......Page 204
Bibliography......Page 206
Index......Page 224
