Symplectic Structures and Quantum Mechanics

Details Symplectic Structures and Quantum Mechanics Symplectic Structures and Quantum Mechanics

1996-05-28

arxiv.org/abs/hep-th/9605191v1

Mod.Phys.Lett. B10 (1996) 545-553

Physics

High Energy Physics

High Energy Physics - Theory

11 pages, LaTex file

Scientific paper

10.1142/S0217984996000602

Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a {\sl 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.

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