functorial quantum field theory
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Types of quantum field thories
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.
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.
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.
For the moment see below.
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).
A fairly comprehensive survey and discussion of supersymmetric quantum mechanics as such, with emphasis on its relation to spectral geometry (“noncommutative geometry”) is in
and with more emphasis on the relation to the superstring (2-spectral triples) in section 7 of:
Another survey is
On supersymmetric quantum mechanics in the perspective of supergeometry (integration over supermanifolds, picture changing operators):
Supersymmetric quantum mechanics gained attention with the work
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
and then in
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
The relation of the partition function of supersymmetric quantum mechanics to index theory was suggested in unpublished work of Edward Witten and formulated in
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)
The superstring‘s supersymmetric quantum mechanics on smooth loop space (the string’s Wheeler superspace) motivated the subject in
Edward Witten, from p. 92 (32 of 39) on in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.). Symposium on Anomalies, Geometry, Topology_, pp. 61–99. World Scientific, 1985 (pdf, spire:214913)
Edward Witten, The index of the Dirac operator in loop space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire)
and is further explored in:
For supersymmetric quantum mechanics of D0-branes see:
Discussion of M-theory on Calabi-Yau 5-folds in terms of supersymmetric quantum mechanics:
Last revised on February 16, 2021 at 05:58:59. See the history of this page for a list of all contributions to it.