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Physics Letters A

An infinite-dimensional classical integrable system and the Heisenberg and Schrödinger representations

Anthony M. Bloch

Published Jul 1, 1986• 23 citations
Publisher
DOI
10.1016/0375-9601(86)90054-x

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We study the relation between the mathematical structures of statistical mechanics on an infinite-dimensional phase space (denoted by ) 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 . Here 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 Schrodinger, Heisenberg and von Neumann are images of the Hamiltonian dynamics on .