Hilbert space representations of generalized canonical commutation relations
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.
scientific article; zbMATH DE number 6191138
scientific article