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arXiv:cond-mat/9403088

Recent Developments in Supersymmetric Quantum Mechanics

Georg Junker

Published Mar 29, 1994• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/9403088

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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.
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Using a realization of the noncompact orthosymplectic superalgebra Osp(22,R)\mathrm{Osp}(\frac{2}{2}, R)Osp(22​,R), the two-level Jaynes-Cummings Hamiltonian is shown to be a supersymmetric quantum-mechanical system in the sense of Witten. The existence of the supersymmetry is shown to place a restriction on the physical parameters of the Hamiltonian.
In this paper, we develop the groundwork for a graph theoretic toy model of supersymmetric quantum mechanics. Using discrete Witten–Morse theory, we demonstrate that finite graphs have a natural supersymmetric structure and use this structure to incorporate supersymmetry into an existing model of graph quantum mechanics. We prove that although key characteristics of continuum supersymmetric systems are preserved on finite unweighted graphs, supersymmetry cannot be spontaneously broken. Finally, we prove new results about the behavior of supersymmetric graph quantum systems under edge rewiring.