CCR representation theory

Definition 0.1   In connection with the Schrödinger representation, one defines a Schrödinger d-system as a set $ \left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on a Hilbert space $ \mathcal{H}$ (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces $ \mathcal{H}_{\alpha}$ of $ \mathcal{H}$ such that $ \mathcal{H} = \oplus_{\alpha} \mathcal{H}_{\alpha}$ with the following two properties:

Definition 0.2   A set $ \left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on a Hilbert space $ \mathcal{H}$ is called a Weyl representation with $ d$ degrees of freedom if $ Q_j$ and $ P_j$ satisfy the Weyl relations:
  1. $\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{−ist} \hbar_{jk} e^{isP_k} \dot
e^{itQ_j},$

  2. $\displaystyle e^{itQ_j} \dot e^{isQ_k} = e^{isQ_k} \dot e^{itQ_j},$

  3. $\displaystyle e^{itP_j} \dot e{isP_k} = e^{isP_k} \dot e^{itP_j} ,$

with $ j, k = 1,..., d, s, t \in \mathbb{R}$ .

The Schrödinger representation $ \left\{Q_j,P_j\right\} ^d_{j =1}$ is a Weyl representation of CCR.

Von Neumann established a uniqueness theorem: if the Hilbert space $ \mathcal{H}$ is separable, then every Weyl representation of CCR with $ d$ degrees of freedom is a Schrödinger $ d$ -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

1
Arai A., Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications, Integr. Equat. Oper. Th., 1993, v.17, 451-463.

2
Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th., 1993, v.16, 38-63.

3
Arai A., Analysis on anticommuting self-adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1-15.

4
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.



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