Quantum mechanics in phase space
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A reformulation of quantum mechanics for a finite system is given using twisted multiplication of functions on phase space and Tomita's theory of generalized Hilbert algebras. Quantization of a classical observable h is achieved when the twisted exponential Exp0(-h) is defined as a tempered distribution. We show that h is in the domain of a generalized Weyl map and define Exp0(-h) as a tempered distribution provided h satisfies a certain semi-boundedness condition. The condition given is linear in h; it coincides with usual boundedness from below if h depends only on one canonical variable. Generalized Weyl-Wigner maps related to the notion of Hamiltonian weight are studied and used in the formulation of a twisted spectral theory for functions on phase space. Some inequalities for Wigner functions on phase space are proven. A brief discussion of the classical limit obtained through dilations of the twisted structure is added
Original language | English |
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Journal | Reports on Mathematical Physics |
Volume | 19 |
Issue number | 3 |
Pages (from-to) | 361-381 |
ISSN | 0034-4877 |
DOIs | |
Publication status | Published - 1984 |
ID: 158020