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Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions. (English) Zbl 1335.81103
There exist many different approaches to the subject of Feynman path integrals, often called functional integrals by mathematicians. Very often the sequential approach, which is used by the authors, defines a functional integral as the limit of finite dimensional integrals when the dimension tends to infinity. However, it has been noticed in the past that the path integrals are very sensitive to our approximation choice, and so the question arises whether the quantization of a classical system might be non-unique. Berezin has posed this problem in an article. The present paper tries to provide an answer Berezin’s problem using what the authors call the method of Feynman formulae. By this method one obtains representations of semigroups solving evolution equations by limits of \(n\)-fold iterated integrals over the phase space of a physical system. Some representations are based on integral operators. Feynman formulae appear to be suitable for numerical computations. Different quantizations now correspond to path integrals with different (pseudo)measures, though they have the same integrand. This answers Berezin’s problem. Finally, so-called symbols of Lévy-Khintchine type are considered with respect to models of one particle interacting with some scalar potential and a magnetic field.

MSC:
81S40 Path integrals in quantum mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47D08 Schrödinger and Feynman-Kac semigroups
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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