nLab supersymmetric quantum mechanics

Idea

Where a system of quantum mechanics is specified by

• a Hilbert space $\mathcal{H}$;

• a hermitean operator $H$ on $\mathcal{H}$ – the Hamiltonian;

a system of supersymmetric quantum mechanics has

• a super Hilbert space $\mathcal{H}$;

• an odd linear operator $D$ in $\mathcal{H}$, the supercharge

• such that $D \circ D = H$ is the Hamiltonian.

If we regard the Hamiltonian as the generator of the Poincare Lie algebra in one dimension – the super translation Lie algebra –, then the graded commutator $[D,D] = 2 H$ is the supersymmetry extension to the super Poincaré Lie algebra in super-dimension $(1|1)$.

The data of a system of supersymmetric quantum mechanics may also be formalized in terms of a spectral triple.

Properties

Relation to spinning particles

A simple but often underappreciated fact is that the worldline theory of any spinning particle is supersymmetric, and hence is supersymmetric quantum mechanics, on the worldline. In this sense relativistic supersymmetric quantum mechanics is not the exception but the rule, it is something exhibited by every fermion in the world. See at spinning particle – Worldline supersymmetryfor more on this.

Of course the bulk of the literature considers non-relativistic supersymmetric quantum mechanics. That is much less relevant in nature.

Relation to index theory

Another fairly simple but very deep fact is that the partition function of a supersymmetric quantum mechanical system, namely the supertrace of its propagator, is equivalently what in mathematics (in index theory) is called the index of the supercharge regarded as a Fredholm operator. See the references below for more on this.

This relation is at the heart of a deep and ubiquituous role that supersymmetric quantum mechanics plays in the mathematics of K-theory and related topics (and vice versa). For a general abstract discussion of why there is such a relation see also at super algebra – Abstract idea and at super line 2-bundle.

Relation to Morse theory

For the moment see below.

Relation to superstrings

Supersymmetric quantum mechanics was introduced or at least became famous with (Witten 82). As explained at the end of (Witten 85), Witten had come to consider this while looking at the point particle limit of the superstring sigma-model. The superstring sigma-model is a kind of supersymmetric quantum mechanics on loop space (see also at 2-spectral triple) and ordinary supersymmetric quantum mechanics is obtained from this in the limit of vanishing loop size (see e.g Schreiber 04). Under this identification the above discussion of index theory translates to Witten’s interpretation of the universal elliptic genus as what is now known as the Witten genus (see there for more).

One way to make this rigorously precise would be to realize the Dirac-Ramond operator of the superstring as an actual Dirac operator on smooth loop space (the string’s Wheeler superspace), as originally suggested in (Witten 87b).

Related entries

• hadron supersymmetry

• Morse theory

• index

• (1,1)-dimensional Euclidean field theories and K-theory

References

General

Lecture notes:

• David Tong: Supersymmetric Quantum Mechanics, lecture notes [webpage, pdf]

A fairly comprehensive survey and discussion of supersymmetric quantum mechanics as such, with emphasis on its relation to spectral geometry (“noncommutative geometry”) is in

• Jürg Fröhlich, Oiver Grandjean, Andreas Recknagel, Supersymmetric quantum theory and (non-commutative) differential geometry, Commun.Math.Phys. 193 (1998) 527-594 (arXiv:hep-th/9612205)

and with more emphasis on the relation to the superstring (2-spectral triples):

• Jürg Fröhlich, Oliver Grandjean, Andreas Recknagel, section 7 of: Supersymmetric quantum theory, non-commutative geometry, and gravitation Lecture Notes Les Houches (1995) (arXiv:hep-th/9706132).

Another survey is

• Fred Cooper, Avinash Khare, Uday Sukhatme, Supersymmetry and Quantum Mechanics, Physics Reports 251 (1995) 267-385 [arXiv:hep-th/9405029]

On supersymmetric quantum mechanics in the perspective of supergeometry (integration over supermanifolds, picture changing operators):

• Leonardo Castellani, Roberto Catenacci, Pietro Grassi, Super Quantum Mechanics in the Integral Form Formalism, Ann. Henri Poincaré (2018) 19: 1385 (arXiv:1706.04704)

See also:

• Nick Dorey, Boan Zhao, Supersymmetric quantum mechanics and growth of sheaf cohomology [arXiv:2209.11834]

• Vyacheslav P. Spiridonov, Variations of supersymmetric quantum mechanics and superconformal indices [arXiv:2404.10609]

Relation to semiclassical approximation:

• Asim Gangopadhaya, Jonathan Bougie, Constantin Rasinariu: Recent Advances in Semiclassical Methods Inspired by Supersymmetric Quantum Mechanics [arXiv:2408.15424]

Relation to Morse theory and string theory

Supersymmetric quantum mechanics gained attention with the work

• Edward Witten, Supersymmetry and Morse theory J. Differential Geom. 17 4 (1982) 661-692. [euclid:jdg/1214437492, pdf, spire:176416]

which showed that Morse theory may be equivalently interpreted as the study of supersymmetric vacua in supersymmetric quantum mechanics, and which was part of what gained Witten the Fields medal 1990. In this article a certain family of deformations of superparticles on a Riemannian manifold are considered and the supersymmetric ground states are shown to be given by the Morse theory of the deformation function.

For a survey of the relation to Morse theory see for instance

• Gábor Pete, section 2 of Morse theory, lecture notes 1999-2001 (pdf)

• Rohit Jain, Supersymmetric Schrödinger operators with applications to Morse theory (pdf)

This deformed supersymmetric quantum mechanics arises as the point-particle limit of the type II superstring regarded as quantum mechanics on the smooth loop space (the string’s Wheeler superspace), a relation that is stated more explicitly in

• Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913]

and then in

• Edward Witten, The Index Of The Dirac Operator In Loop Space, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326, Springer (1988) 161-181 [doi:10.1007/BFb0078045, spire]

The relation between the 2d SCFT describing the type II superstring and this deformed supersymmetric quantum mechanics on smooth loop space has been further explored in

• Urs Schreiber, On deformations of 2d SCFTs, JHEP 0406:058,2004 (arXiv:hep-th/0401175)

Further discussion of supersymmetric quantum mechanics in string theory:

On supersymmetric quantum mechanics of D0-branes:

• BFSS matrix model, BMN matrix model

Discussion of M-theory on Calabi-Yau 5-folds in terms of supersymmetric quantum mechanics:

• Alexander Haupt, Andre Lukas, Kellogg Stelle, M-theory on Calabi-Yau Five-Folds, JHEP 0905:069, 2009 (arXiv:0810.2685)

Relation to index theory

The relation of the partition function of supersymmetric quantum mechanics to index theory was suggested in unpublished work of Edward Witten and formulated in

• Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 16-173. (Euclid)

• Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)

• D. Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89–95, (doi);

• Varghese Mathai, Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms. Topology 25 (1986), no. 1, 85–110;

• Ezra Getzler, A short proof of the Atiyah-Singer index theorem, Topology 25 (1986), 111-117 (pdf)

• D. Quillen, Superconnection character forms and the Cayley transform, Topology 27 (1988), no. 2, 211–238