Definition 0.1   In connection with the Schrödinger representation, one defines a Schrödinger d-system as a set of self-adjoint operators on a Hilbert space (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces of such that with the following two properties:

• (i) each reduces all and all ;
• (ii) the set is, in each , unitarily equivalent to the Schrödinger representation [8].

Definition 0.2   A set of self-adjoint operators on a Hilbert space is called a Weyl representation with degrees of freedom if and satisfy the Weyl relations:

1. 

2. 

3. 

with .

The Schrödinger representation is a Weyl representation of CCR.

Von Neumann established a uniqueness theorem: if the Hilbert space is separable, then every Weyl representation of CCR with degrees of freedom is a Schrödinger -system ([6]). Since the pioneering work of von Neumann [6] there have been numerous reports published concerning representation theory of CCR (viz. ref. [8] and references cited therein).

Bibliography

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Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th., 1993, v.16, 38–63.

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Arai A., Analysis on anticommuting self–adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1–15.

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Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139–173.

5

Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys., 1987, V.28, 472–476.

6

von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann., 1931, v.104, 570–578.

7

Pedersen S., Anticommuting self–adjoint operators, J. Funct. Anal., 1990, V.89, 428–443.

8

Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.

9

Reed M. and Simon B., Methods of Modern Mathematical Physics., vol.I, Academic Press, New York, 1972.