PlanetPhysics/Representations of Canonical Anti Commutation Relations CAR

Thsi is a contributed topic in progress on representations of anti-commutation relations (CAR). (See also previous entries on the representations of canonical commutation and anti-commutation relations (CCR, CCAR)).

Representations of Canonical Anti-commutation Relations (CAR)

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CAR Representations in a Non-Abelian Gauge Theory

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One can also provide a representation of canonical anti-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]). Supersymmetry theories admit both CAR and CCR representations. Note also the connections of such representations to locally compact quantum groupoid representations.

All Sources

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[2] [3] [4] [5] [6] [1] [7] [8] [9] [10] [11]

References

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  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.
  11. Vainerman, L. 2003, Locally Compact Quantum Groups and Groupoids: Contributed Lectures., 247 pages; Walter de Gruyter Gmbh and Co, Berlin. (commutative and non-commutative quantum algebra, free download at this web link)