PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger representation, one defines a Schr\"odinger 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 Schr\"odinger representation [1].

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. Failed to parse (syntax error): {\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{−ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},}

with .

The Schr\"odinger representation is a Weyl representation of CCR.

Von Neumann established a uniqueness \htmladdnormallink{theorem {}: if the Hilbert space is separable, then every Weyl representation of CCR with degrees of freedom is a Schr\"odinger -system} ([2]). Since the pioneering work of von Neumann [2] there have been numerous reports published concerning representation theory of CCR (viz. ref. [1] and references cited therein).

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  1. 1.0 1.1 1.2 Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.
  2. 2.0 2.1 2.2 von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren, Math. Ann. , 1931, v.104, 570--578.
  3. 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.
  4. Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th. , 1993, v.16, 38--63.
  5. Arai A., Analysis on anticommuting self--adjoint operators, Adv. Stud. Pure Math. , 1994, v.23, 1--15.
  6. Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139--173.
  7. Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys. , 1987, V.28, 472--476.
  8. Pedersen S., Anticommuting self--adjoint operators, J. Funct. Anal., 1990, V.89, 428--443.
  9. Reed M. and Simon B., Methods of Modern Mathematical Physics ., vol.I, Academic Press, New York, 1972.