On the Classification of N-extended Supersymmetric Quantum Mechanical Systems

Abstract

In this paper some properties of the irreducible multiplets of representation for the N = (p, q) - extended supersymmetry in one dimension are discussed. Essentially two results are here presented. At first a peculiar property of the one dimension is exhibited, namely that any multiplet containing 2d (d bosonic and d fermionic) particles in M different spin states, is equivalent to a (d,d) multiplet of just 2 spin states (all bosons and all fermions being grouped in the same spin). Later, it is shown that the classification of all multiplets of this kind carrying an irreducible representation of the N - extended supersymmetry is in one-to-one correspondence with the classification of real-valued Clifford Gamma-matrices of Weyl type. In particular, p+q is mapped into D, the space-time dimensionality, while 2d is determined to be the dimensionality of the corresponding Gamma-matrices. The implications of these results to the theory of spinning particles are analyzed.

15 pages, LaTex

References in corpus (4)

• N=4 Supersymmetric Multidimensional Quantum Mechanics, Partial SUSY Breaking and Superconformal Quantum Mechanics
• Supersymmetric Quantum Mechanics from Light Cone Quantization
• The Signature Triality of Majorana-Weyl Spacetimes
• Real Structures in Clifford Algebras and Majorana Conditions in Any Space-time

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