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 {http://planetphysics.us/encyclopedia/Formula.html}: 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).

All SourcesEdit

[3][4][5][6][7][2][8][1][9]

ReferencesEdit

  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.