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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.

MSC:

46N50 Applications of functional analysis in quantum physics
46L60 Applications of selfadjoint operator algebras to physics
81S25 Quantum stochastic calculus
39A99 Difference equations
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