Mathematical Feynman path integrals and their applications
WORLDSHARE SUB RECORDS
Mazzucchi, Sonia.
creator
World Scientific (Firm)
text
bibliography
si
Singapore
Hackensack, N.J
World Scientific Pub. Co.
©2009
2009
monographic
eng
1 online resource (viii, 216 pages)
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.
1. Introduction. 1.1. Wiener's and Feynman's integration. 1.2. The Feynman functional. 1.3. Infinite dimensional oscillatory integrals -- 2. Infinite dimensional oscillatory integrals. 2.1. Finite dimensional oscillatory integrals. 2.2. The Parseval type equality. 2.3. Generalized Fresnel integrals. 2.4. Infinite dimensional oscillatory integrals. 2.5. Polynomial phase functions -- 3. Feynman Path Integrals and the Schrödinger equation. 3.1. The anharmonic oscillator with a bounded anharmonic potential. 3.2. Time dependent potentials. 3.3. Phase space Feynman path integrals. 3.4. Magnetic field. 3.5. Quartic potential -- 4. The stationary phase method and the semiclassical limit of quantum mechanics. 4.1. Asymptotic expansions. 4.2. The stationary phase method. Finite dimensional case. 4.3. The stationary phase method. Infinite dimensional case. 4.4. The semiclassical limit of quantum mechanics. 4.5. The trace formula -- 5. Open quantum systems. 5.1. Feynman path integrals and open quantum systems. 5.2. The Feynman-Vernon influence functional. 5.3. The stochastic Schrödinger equation -- 6. Alternative approaches to Feynman path integration. 6.1. Analytic continuation of Wiener integrals. 6.2. The sequential approach. 6.3. White noise calculus. 6.4. Poisson processes. 6.5. Further approaches and results.
Sonia Mazzucchi.
Includes bibliographical references (pages 197-213) and index.
Feynman integrals
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