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Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

### Abstract

Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.

© 2016 AIP Publishing LLC

Received 29 October 2014
Accepted 11 January 2016
Published online 02 February 2016

Acknowledgments:
The authors thank the referee for helpful remarks. Financial support by the DFG through the Project No. GR-1809/10-1, by the Ministry of Education and Science of the Russian Federation through the Project No. 14.B37.21.0370, by the President of the Russian Federation through the Project No. MK-4255.2012.1, and by DAAD is gratefully acknowledged.

Article outline:

I. INTRODUCTION
II. NOTATION AND PRELIMINARIES
A. Pseudo-differential operators, their symbols, and tau-quantization
B. The Chernoff theorem and Feynman formulae
III. FEYNMAN FORMULAE FOR TAU-QUANTIZATION OF SOME LÉVY-KHINTCHINE TYPE HAMILTON FUNCTIONS
IV. PROOFS OF FEYNMAN FORMULAE
V. THE CONSTRUCTION OF PHASE SPACE FEYNMAN PATH INTEGRALS VIA A FAMILY OF HAMILTONIAN FEYNMAN PSEUDOMEASURES FOR *τ* ∈ [0, 1]
VI. PHASE SPACE FEYNMAN PATH INTEGRALS FOR EVOLUTION SEMIGROUPS GENERATED BY *τ*-QUANTIZATION OF SOME LÉVY-KHINTCHINE TYPE HAMILTON FUNCTIONS FOR *τ* ∈ [0, 1]
VII. CONCLUSION

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/content/aip/journal/jmp/57/2/10.1063/1.4940697

2016-02-02

2016-09-29

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