Quantum mechanics in phase space
type: Item
instance of: scholarly article
title: Quantum mechanics in phase space
zbMATH Open document ID: 0571.46046
DOI: 10.1016/0034-4877(84)90008-9
published in: Reports on Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 3911071
zbMATH Keywords: semiboundedness
zbMATH Keywords: essential selfadjointness
zbMATH Keywords: twisted multiplication of functions on phase space
zbMATH Keywords: Tomita's theory of generalized Hilbert algebras
zbMATH Keywords: twisted exponential
zbMATH Keywords: quantized selfadjoint operator
zbMATH Keywords: Hamiltonian weight
zbMATH Keywords: functional calculus
zbMATH Keywords: inequalities for Wigner functions on phase space
zbMATH Keywords: dilations of the twisted structure
author: Friedhelm Hansen
reviewed by: Huzihiro Araki
MaRDI profile type: Publication
full work available at URL: 0034-4877(84)90008-9
OpenAlex ID: W1977144108
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description: scientific article; zbMATH DE number 3911071