Talk:PlanetPhysics/Representations of Canonical Anti Commutation Relations CAR

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Thsi is a contributed topic in progress on \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of \htmladdnormallink{anti-commutation relations}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} (CAR). (See also previous entries on the representations of \htmladdnormallink{canonical commutation and anti-commutation relations}{http://planetphysics.us/encyclopedia/CanonicalCommutationAndAntiCommutationRepresentations.html} (\htmladdnormallink{CCR}{http://planetphysics.us/encyclopedia/SchwartzSpaceOfRapidlyDecreasingC_inftyFunctions.html}, CCAR)).

\subsection{Representations of Canonical Anti-commutation Relations (CAR)}

\subsubsection{CAR Representations in a Non-Abelian Gauge Theory}

One can also provide a representation of canonical anti-commutation relations in a \htmladdnormallink{non-Abelian gauge theory}{http://planetphysics.org/?op=getobj&from=lec&id=124} defined on a non-simply connected region in the \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} Euclidean space. Such representations were shown to provide also a mathematical expression for the \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}, Aharonov-Bohm effect (\cite{GMS81}). \htmladdnormallink{Supersymmetry}{http://planetphysics.us/encyclopedia/Supersymmetry.html} theories admit both CAR and CCR representations. Note also the connections of such representations to \htmladdnormallink{locally compact quantum groupoid representations.}{http://planetphysics.org/?op=getobj&from=lec&id=5}

\begin{thebibliography}{99}

\bibitem{AA93a} Arai A., Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications, {\em Integr. Equat. Oper. Th.}, 1993, v.17, 451--463.

\bibitem{AA93b} Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, {\em Integr. Equat. Oper. Th.}, 1993, v.16, 38--63.

\bibitem{AA94} Arai A., Analysis on anticommuting self--adjoint operators, {\em Adv. Stud. Pure Math.}, 1994, v.23, 1--15.

\bibitem{AA95} Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139--173.

\bibitem{AA87} Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, {\em J. Math. Phys.}, 1987, V.28, 472--476.

\bibitem{GMS81} 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.

\bibitem{JVN31} von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren, {\em Math. Ann.}, 1931, v.104, 570--578.

\bibitem{PS90} Pedersen S., Anticommuting self--adjoint operators, J. Funct. Anal., 1990, V.89, 428--443.

\bibitem{PCR67} Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.

\bibitem{RM-SB72} Reed M. and Simon B., {\em Methods of Modern Mathematical Physics}., vol.I, Academic Press, New York, 1972.

\bibitem{Vainerman} Vainerman, L. 2003, Locally Compact Quantum Groups and Groupoids: Contributed Lectures., 247 pages; Walter de Gruyter Gmbh and Co, Berlin. \htmladdnormallink{(commutative and non-commutative quantum algebra, free download at this web link)}{http://planetphysics.org/?op=getobj&from=lec&id=5}

\end{thebibliography}

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