Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation
Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 127-148
Voir la notice de l'article provenant de la source Math-Net.Ru
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 $\Omega$.
@article{IM2_2008_72_1_a6,
author = {A. Yu. Khrennikov},
title = {Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation},
journal = {Izvestiya. Mathematics },
pages = {127--148},
publisher = {mathdoc},
volume = {72},
number = {1},
year = {2008},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_1_a6/}
}
TY - JOUR AU - A. Yu. Khrennikov TI - Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation JO - Izvestiya. Mathematics PY - 2008 SP - 127 EP - 148 VL - 72 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/IM2_2008_72_1_a6/ LA - en ID - IM2_2008_72_1_a6 ER -
A. Yu. Khrennikov. Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation. Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 127-148. http://geodesic.mathdoc.fr/item/IM2_2008_72_1_a6/