Accardi, L.; Skeide, M. Hilbert module realization of the square of white noise and finite difference algebras. (English. Russian original) Zbl 1029.46120 Math. Notes 68, No. 6, 683-694 (2000); translation from Mat. Zametki 68, No. 6, 803-818 (2000). The white noise is a system of operator-valued distributions indexed by real numbers which fulfill the canonical commutation relations. In the paper under review, the algebra of the square of white noise is represented on a Hilbert module over the algebra of number operators. For this reason, the theory of pre-Hilbert modules over \(\ast\)-algebras and symmetric Fock spaces associated with them is developed. The unique Fock representation is founded and it is shown that the representation space is the usual symmetric Fock space. A surprising relation of the representation obtained to finite difference algebras is discussed. Reviewer: Jan Hamhalter (Praha) Cited in 2 ReviewsCited in 2 Documents MSC: 46N50 Applications of functional analysis in quantum physics 46L60 Applications of selfadjoint operator algebras to physics 81S25 Quantum stochastic calculus 39A99 Difference equations Keywords:white noise; Hilbert module; finite difference algebra; Fock space PDFBibTeX XMLCite \textit{L. Accardi} and \textit{M. Skeide}, Math. Notes 68, No. 6, 683--694 (2000; Zbl 1029.46120); translation from Mat. Zametki 68, No. 6, 803--818 (2000) Full Text: DOI