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Journal of Physics A Mathematical and General

Supersymmetric quantum mechanical models with continuous spectrum and the Witten index

Harry Whiting Braden, Alan Macfarlane

Published Nov 11, 1985• 27 citations
Publisher

Abstract

The authors discuss supersymmetric regularisations of soluble supersymmetric quantum mechanical models with continuous parts to the energy spectra. Results include a demonstration that the Witten index changes discontinuously from zero to one upon removal of the regularisation.
DOI
10.1088/0305-4470/18/16/017

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Some recent results in supersymmetric quantum mechanics are presented. New semi-classical approximation formulas for Witten's realization of supersymmetric quantum mechanics are discussed. Implications of the supersymmetric structure of Pauli's Hamiltonian are also considered. In particular, the paramagnetisation of a non-interacting electron gas is related to a modified version of Witten's index. We also show that the supersymmetry in this system provides a new counter example for the paramagnetic conjecture.
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of N≤2\mathcal N \leq 2N≤2 quantum mechanical systems is #P\#P#P-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.