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Representations of canonical anti-commutation relations (CAR) (Topic)

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)).

0.1 Representations of Canonical Anti-commutation Relations (CAR)

0.1.1 CAR Representations in a Non-Abelian Gauge Theory

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 ([6]). Supersymmetry theories admit both CAR and CCR representations. Note also the connections of such representations to locally compact quantum groupoid representations.

References

[1]   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.

[2]   Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th., 1993, v.16, 38–63.

[3]   Arai A., Analysis on anticommuting self–adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1–15.

[4]   Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139–173.

[5]   Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys., 1987, V.28, 472–476.

[6]   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.

[7]   von Neumann J., Die Eindeutigkeit der Schrödingerschen 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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"Representations of canonical anti-commutation relations (CAR)" is owned by bci1.
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See Also: canonical commutation and anti-commutation relations: their representations

Other names:  CAR, CCAR
Also defines:  CAR, CCAR
Keywords:  CAR, representations

Cross-references: supersymmetry, non-Abelian, two-dimensional, CCR, anti-commutation relations, representations
There are 9 references to this object.

This is version 4 of Representations of canonical anti-commutation relations (CAR), born on 2009-04-07, modified 2009-04-07.
Object id is 636, canonical name is RepresentationsOfCanonicalAntiCommutationRelationsCAR.
Accessed 4636 times total.

Classification:
Physics Classification00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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