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PlanetPhysics/Representations of Canonical Anti Commutation Relations CAR

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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.

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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)
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