On the Classification of N-extended Supersymmetric Quantum Mechanical Systems

TL;DR

This paper classifies irreducible multiplets of N-extended supersymmetry in one dimension, revealing a unique equivalence property and linking their classification to real-valued Clifford Gamma-matrices, with implications for spinning particle theories.

Contribution

It establishes a one-to-one correspondence between supersymmetric multiplets and real Clifford Gamma-matrices, providing a novel classification framework for supersymmetric systems in one dimension.

Findings

Any multiplet with 2d particles in M spin states is equivalent to a (d,d) multiplet with 2 spin states.

Classification of multiplets corresponds to real Clifford Gamma-matrices of Weyl type.

Results have implications for the theory of spinning particles.

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.