Geometric Hamiltonian formulation of quantum mechanics in complex projective spaces
Abstract
In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory.
References
- AIP Conf. Proc. 342 , 471 ( 1995 ) . Crossref, Google Scholar
- Ann. Math. 37 , 415 ( 1936 ) . Crossref, Google Scholar
- J. Geom. Phys. 38 , 19 ( 2001 ) . Crossref, Web of Science, Google Scholar
- J. Math. Mech. 6(6), 885 (1957). Web of Science, Google Scholar
- Commun. Math. Phys. 65 , 189 ( 1979 ) . Crossref, Web of Science, Google Scholar
-
V. Moretti , Spectral Theory and Quantum Mechanics ( Springer , 2013 ) . Crossref, Google Scholar - Ann. Henri Poincaré 14 , 1435 ( 2013 ) . Crossref, Web of Science, Google Scholar
- V. Moretti and D. Pastorello, Frame functions in finite-dimensional quantum mechanics and its Hamiltonian formulation on complex projective spaces, preprint (2013) , arXiv: 1311.1720 . Google Scholar
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