Hilbert space representations of generalized canonical commutation relations
type: Item
instance of: scholarly article
title: Hilbert space representations of generalized canonical commutation relations
zbMATH Open document ID: 1280.46030
DOI: 10.1155/2013/308392
author: Asao Arai
published in: Journal of Mathematics
publication date: 2013-07-25T00:00:00Z
review text: Summary: We consider Hilbert space representations of a generalization of canonical commutation relations (CCRs): \([X_j, X_k] := X_jX_k - X_kX_j = i\Theta_{jk}I\) \((j, k = 1, 2, \dots, 2n)\), where the \(X_j\)'s are elements of an algebra with identity \(I\), \(i\) is the imaginary unit, and \(\Theta_{jk}\) is a real number with antisymmetry \(\Theta_{jk} = -\Theta_{kj}\) \((k, j = 1, 2, \dots, 2n)\). Some basic aspects of Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with \(n\) degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.
Mathematics Subject Classification ID: 46H15
Mathematics Subject Classification ID: 47A67
zbMATH DE Number: 6191138
zbMATH Keywords: Hilbert space representations
zbMATH Keywords: canonical commutation relations
Wikidata QID: Q59015325
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/308392
OpenAlex ID: W1969929475
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description: scientific article; zbMATH DE number 6191138