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Modern Physics Letters A

ALTERNATIVE HAMILTONIAN DESCRIPTION FOR QUANTUM SYSTEMS

Boris Dubrovin, Giuseppe Marmo, Alberto Simoni

Published Jun 20, 1990• 27 citations
Publisher

Abstract

The existence of time-invariant Kähler structures is analyzed in both Classical and Quantum Mechanics. In Quantum Mechanics, a family of such Kähler structures is found, in the finite-dimensional case it is proven that this family is complete.
DOI
10.1142/s0217732390001384

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My PhD thesis is focused on geometric Hamiltonian formulation of Quanum Mechanics and its interplay with standard formulation. The main result is the construction of a general prescription to set up a quantum theory as a classical-like theory where quantum dynamics is given by a Hamiltonian vector field on a complex projective space with Kähler structure. In such geometric framework quantum states are represented by classical-like Liouville densities. After a complete characterization of classical-like observables in a finite-dimensional quantum theory, the observable C*-algebra is described in geometric Hamiltonian terms. In the final part of the work, the classical-like Hamiltonian formulation is applied to the study of composite quantum systems providing a notion of entanglement measure.
Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.