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)).
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.
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- 1
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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.
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Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th., 1993, v.16, 38-63.
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Arai A., Analysis on anticommuting self-adjoint operators, Adv. Stud. Pure Math., 1994, v.23, 1-15.
- 4
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Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139-173.
- 5
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Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys., 1987, V.28, 472-476.
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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.
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von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren,
Math. Ann., 1931, v.104, 570-578.
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Pedersen S., Anticommuting self-adjoint operators, J. Funct. Anal., 1990, V.89, 428-443.
- 9
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Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.
- 10
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Reed M. and Simon B., Methods of Modern Mathematical Physics., vol.I, Academic Press, New York, 1972.
- 11
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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)
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.