Grassmann Numbers and Clifford-Jordan-Wigner Representation of Supersymmetry

The elementary particles of Physics are classified according to the behavior of the multi-particle states under exchange of identical particles: bosonic states are symmetric while fermionic states are antisymmetric. This manifests itself also in the commutation properties of the respective creation...

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Bibliographic Details
Published in:Journal of physics. Conference series 2013-01, Vol.411 (1), p.12009-19
Main Authors: Catto, Sultan, Choun, Yoon S, Gürcan, Yasemin, Khalfan, Amish, Kurt, Levent
Format: Article
Language:English
Subjects:
Bosons
Commutation
Complex variables
Contracts
Differential calculus
Elementary particles
Exchange
Fermions
Integral calculus
Operators
Operators (mathematics)
Physics
Representations
Supersymmetry
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Summary:The elementary particles of Physics are classified according to the behavior of the multi-particle states under exchange of identical particles: bosonic states are symmetric while fermionic states are antisymmetric. This manifests itself also in the commutation properties of the respective creation operators: bosonic creation operators commute while fermionic ones anticommute. It is natural therefore to study bosons using commuting entities (e.g. complex variables), whereas to describe fermions, anticommuting variables are more naturally suited. In this paper we introduce these anticommuting- and at first sight unfamiliar- variables (Grassmann numbers) and investigate their properties. In particular, we briefly discuss differential and integral calculus on Grassmann numbers. Work supported in part by DOE contracts No. DE-AC-0276-ER 03074 and 03075; NSF Grant No. DMS-8917754.
ISSN:1742-6596
1742-6588
1742-6596
DOI:10.1088/1742-6596/411/1/012009