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 TMF, 1994, Volume 100, Number 1, Pages 3–13 (Mi tmf1623)

Transformations of Feynman integral under some nonlinear transformations of the phase space

O. G. Smolyanova, M. O. Smolyanova

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The Feynman measure is defined as a linear continuous functional on a space of test functions (which is introduced in paper), having a given Fourier transformation. Both the case of a positevely defined and the case of an arbitrary correlation operators are considered (the last one is related to the so-called symplectic, or Hamiltonian, Feynman measure). The Feynman integral is the value of the Feynman measure on a function (from the space of test functions). The transformations of the Feynman measure under the nonlinear transformations of the phase space which are included by both the translations along vector fields and along the integral curves of vector fields are described. The formulae that are similar to Cameron–Martin, Girsanov–Maruyama and Ramer formulae (from the theory of Gaussian measures) are obtained. The results can be considered as formulae of change of variables in the Feynman integral.

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English version:
Theoretical and Mathematical Physics, 1994, 100:1, 803–810

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Citation: O. G. Smolyanov, M. O. Smolyanova, “Transformations of Feynman integral under some nonlinear transformations of the phase space”, TMF, 100:1 (1994), 3–13; Theoret. and Math. Phys., 100:1 (1994), 803–810

Citation in format AMSBIB
\Bibitem{SmoSmo94} \by O.~G.~Smolyanov, M.~O.~Smolyanova \paper Transformations of Feynman integral under some nonlinear transformations of the phase space \jour TMF \yr 1994 \vol 100 \issue 1 \pages 3--13 \mathnet{http://mi.mathnet.ru/tmf1623} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1305786} \zmath{https://zbmath.org/?q=an:1041.81549} \transl \jour Theoret. and Math. Phys. \yr 1994 \vol 100 \issue 1 \pages 803--810 \crossref{https://doi.org/10.1007/BF01017317} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994QC09900001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. G. Smolyanov, A. Trumen, “Change of variable formulas for Feynman pseudomeasures”, Theoret. and Math. Phys., 119:3 (1999), 677–686
2. Smolyanov, OG, “Hamiltonian Feynman path integrals via the Chernoff formula”, Journal of Mathematical Physics, 43:10 (2002), 5161
3. “Introduction”, Mathematical Theory of Feynman Path Integrals: An Introduction, 523 (2008), 1
4. Fedorov A.K., Yurchenko S.O., “Simplekticheskie tomogrammy v predstavlenii feinmanovskimi integralami po traektoriyam”, Vestnik moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. seriya: estestvennye nauki, 2012, no. 2, 29–37
5. Orlov Yu.N. Sakbaev V.Zh. Smolyanov O.G., “Feynman Formulas For Nonlinear Evolution Equations”, Dokl. Math., 96:3 (2017), 574–577
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