Quantum mechanics in phase space
A reformulation of quantum mechanics for a finite system in terms of twisted multiplication of functions on phase space is studied in the framework of Tomita's theory of generalized Hilbert algebras. A classical observable h is quantized when the twisted exponential \(Exp^ 0(-h)\) (which will define \(e^{-\ell (h)}\) with the quantized selfadjoint operator \(\ell (h))\) is defined as a tempered distribution. The latter is the case when h satisfies a certain semi-boundedness condition, linear in h (Corollary 5.2). The condition is the usual boundedness from below if h depends on one canonical variable. This formalism is extended from a function h to what the author calls Hamiltonian weight (Definition 6.3), the correspondence between selfadjoint operators and the corresponding Hamiltonian weight is established (Theorem 7.1) and the functional calculus is developped (Proposition 7.3). 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.
twisted multiplication of functions on phase space
Tomita's theory of generalized Hilbert algebras
inequalities for Wigner functions on phase space
scientific article; zbMATH DE number 3911071
scientific article