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reference request - Path integral methods - MathOverflow
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This is the fifth, expanded edition of the comprehensive textbook published in on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances.
The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations. In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations.
A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions. The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results.
The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena. Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions.
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It only takes a minute to sign up. Are there detailed expositions of the path integral methods in mathematical physics other than Feynman-Hibbs and Glimm-Jaffe?
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Of course the physics literature is full of references on path integrals. One would not know where to start. Kleinert's books suggested by Alexander must be a good choice.
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On the rigorous side you might want to have a look at:. An impressive list of references to complement the answers already given can be found at the bottom of the following Scholarpedia article by Sergio Albeverio and Sonia Mazzucchi. The approach to path integrals based on "cylindrical measures", aka "promeasures", is not discussed in the body of the article but can be read about in the work by DeWitt-Morette and collaborators referenced there.
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