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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 1, Pages 137–160 (Mi izv786)  

This article is cited in 1 scientific paper (total in 1 paper)

Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation of quantum averages by Gaussian functional integrals

A. Yu. Khrennikov

Växjö University

Abstract: We study the relation between the mathematical structures of statistical mechanics on an infinite-dimensional phase space (denoted by $\Omega$) and quantum mechanics. It is shown that quantum averages (given by the von Neumann trace formula) can be obtained as the main term of the asymptotic expansion of Gaussian functional integrals with respect to a small parameter $\alpha$. Here $\alpha$ is the dispersion of the Gaussian measure. The symplectic structure on the infinite-dimensional phase space plays a crucial role in our considerations. In particular, the Gaussian measures that induce quantum averages must be consistent with the symplectic structure. The equations of Schrödinger, Heisenberg and von Neumann are images of the Hamiltonian dynamics on $\Omega$.

DOI: https://doi.org/10.4213/im786

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English version:
Izvestiya: Mathematics, 2008, 72:1, 127–148

Bibliographic databases:

UDC: 511.34
MSC: 81P05, 81P20
Received: 06.02.2006

Citation: A. Yu. Khrennikov, “Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation of quantum averages by Gaussian functional integrals”, Izv. RAN. Ser. Mat., 72:1 (2008), 137–160; Izv. Math., 72:1 (2008), 127–148

Citation in format AMSBIB
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\pages 137--160
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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. Khrennikov A., “Quantum correlations from classical Gaussian correlations”, J. Russian Laser Research, 30:5 (2009), 472–479  crossref  mathscinet  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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