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. 1. 1. Introduction. 2. 2. Twisted multiplication. 3. 3. The dynamical representation. 4. 4. Wigner functions. 5. 5. Quantization. 6. 6. Hamiltonian weights. 7. 7. Twisted functional calculus. 8. 8. The classical limit.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics