Symplectic structure of the von Neumann equation

Vladimir I. Man’ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

Andrei Yu. Khrennikov

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 .

D. J. Simms, L. Castel

Kay-Michael Voit

Ya. A. Korennoy, Vladimir I. Man’ko

The von Neumann evolution equation for density matrix and the Moyal equation for the Wigner function are mapped onto evolution equation for optical tomogram of quantum state. The connection with known evolution equation for symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for the optical tomograms. Classical Liouville equation for optical tomogram is obtained. Example of parametric oscillator is considered in detail.

Marmo, G, Vilasi, G

Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula. 1 Introduction. In the past few years there has been a renewed interest in completely integrable Hamiltonian systems, specially in connection with the study of integrable quantum