PlanetPhysics/Canonical Commutation and Anti Commutation Representations

This is a contributed topic on representations of canonical commutation and anti-commutation relations.

Representations of Canonical Commutation Relations (CCR) edit

Canonical Commutation Relations: edit

Consider a Hilbert space . For a linear operator {\mathbf O} on , we denote its domain by {\mathbf } With Arai's notation, a set of self-adjoint operators on (such as the position and momentum operators, for example) is called a representation of the canonical commutation relations (CCR) with degrees of freedom if there exists a dense subspace of such that:

  • (i) and
  • (ii) and satisfy the CCR relations: on , where is the Planck constant divided by .

A standard representation of the CCR is the well-known Schr\"odinger representation which is given by:

the multiplication operator by the j-th coordinate , with , with being the generalized partial differential operator in , and with being the Schwartz space of rapidly decreasing functions on , or , that is the space of functions on with compact support.

CCR Representations in a Non-Abelian Gauge Theory edit

One can provide a representation of canonical commutation relations in a non-Abelian gauge theory defined on a non-simply connected region in the two-dimensional Euclidean space. Such representations were shown to provide also a mathematical expression for the non-Abelian, Aharonov-Bohm effect ([1]).

Canonical Anticommutation Relations (CAR) edit

All Sources edit

[2][3][4][5][6][1][7][8][9][10]

References edit

  1. 1.0 1.1 Goldin G.A., Menikoff R. and Sharp D.H., Representations of a local current algebra in nonsimply connected space and the Aharonov--Bohm effect, J. Math. Phys., 1981, v.22, 1664--1668.
  2. 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.
  3. Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th. , 1993, v.16, 38--63.
  4. Arai A., Analysis on anticommuting self--adjoint operators, Adv. Stud. Pure Math. , 1994, v.23, 1--15.
  5. Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139--173.
  6. Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys. , 1987, V.28, 472--476.
  7. von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren, Math. Ann. , 1931, v.104, 570--578.
  8. Pedersen S., Anticommuting self--adjoint operators, J. Funct. Anal., 1990, V.89, 428--443.
  9. Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.
  10. Reed M. and Simon B., Methods of Modern Mathematical Physics ., vol.I, Academic Press, New York, 1972.