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\section*{supersymmetric quantum mechanics}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToSpinningParticles}{Relation to spinning particles}\dotfill \pageref*{RelationToSpinningParticles} \linebreak
\noindent\hyperlink{RelationToIndexTheory}{Relation to index theory}\dotfill \pageref*{RelationToIndexTheory} \linebreak
\noindent\hyperlink{relation_to_morse_theory}{Relation to Morse theory}\dotfill \pageref*{relation_to_morse_theory} \linebreak
\noindent\hyperlink{RelationToSuperstrings}{Relation to superstrings}\dotfill \pageref*{RelationToSuperstrings} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{ReferencesGeneral}{General}\dotfill \pageref*{ReferencesGeneral} \linebreak
\noindent\hyperlink{ReferencesRelationToMorseTheory}{Relation to Morse theory}\dotfill \pageref*{ReferencesRelationToMorseTheory} \linebreak
\noindent\hyperlink{ReferencesRelationToIndexTheory}{Relation to index theory}\dotfill \pageref*{ReferencesRelationToIndexTheory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Where a system of [[quantum mechanics]] is specified by

\begin{itemize}%
\item a [[Hilbert space]] $\mathcal{H}$;


\item a [[hermitean operator]] $H$ on $\mathcal{H}$ -- the [[Hamiltonian]];



\end{itemize}
a system of \emph{supersymmetric quantum mechanics} has

\begin{itemize}%
\item a [[super Hilbert space]] $\mathcal{H}$;


\item an odd [[linear operator]] $D$ in $\mathcal{H}$, the [[supercharge]]


\item such that $D \circ D = H$ is the [[Hamiltonian]].



\end{itemize}
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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToSpinningParticles}{}\subsubsection*{{Relation to spinning particles}}\label{RelationToSpinningParticles}

A simple but often underappreciated fact is that the [[worldline theory]] of any [[spinning particle]] is supersymmetric, and hence is supersymmetric quantum mechanics, \emph{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 \emph{\href{spinning+particle#WorldlineSupersymmetry}{spinning particle -- Worldline supersymmetry}}for more on this.

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

\hypertarget{RelationToIndexTheory}{}\subsubsection*{{Relation to index theory}}\label{RelationToIndexTheory}

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 \hyperlink{ReferencesRelationToIndexTheory}{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 \emph{\href{super+algebra#AbstractIdea}{super algebra -- Abstract idea}} and at \emph{[[super line 2-bundle]]}.

\hypertarget{relation_to_morse_theory}{}\subsubsection*{{Relation to Morse theory}}\label{relation_to_morse_theory}

For the moment see \hyperlink{ReferencesRelationToMorseTheory}{below}.

\hypertarget{RelationToSuperstrings}{}\subsubsection*{{Relation to superstrings}}\label{RelationToSuperstrings}

Supersymmetric quantum mechanics was introduced or at least became famous with (\hyperlink{Wittem82}{Witten 82}). As explained at the end of (\hyperlink{Witten85}{Witten 85}), [[Edward Witten|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 \hyperlink{Schreiber04}{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 \emph{[[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]], as originally suggested in (\hyperlink{Witten87b}{Witten 87b}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Morse theory]]


\item [[index]]


\item [[(1,1)-dimensional Euclidean field theories and K-theory]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{ReferencesGeneral}{}\subsubsection*{{General}}\label{ReferencesGeneral}

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

\begin{itemize}%
\item [[Jürg Fröhlich]], Oiver Grandjean, [[Andreas Recknagel]], \emph{Supersymmetric quantum theory and (non-commutative) differential geometry}, Commun.Math.Phys. 193 (1998) 527-594 (\href{http://arxiv.org/abs/hep-th/9612205}{arXiv:hep-th/9612205})

\end{itemize}
and with more emphasis on the relation to the [[superstring]] in section 7 of

\begin{itemize}%
\item [[Jürg Fröhlich]], Oliver Grandjean, [[Andreas Recknagel]], \emph{Supersymmetric quantum theory, non-commutative geometry, and gravitation} Lecture Notes Les Houches (1995) (\href{http://arxiv.org/abs/hep-th/9706132}{arXiv:hep-th/9706132}).

\end{itemize}
Another survey is

\begin{itemize}%
\item Fred Cooper, Avinash Khare, Uday Sukhatme, \emph{Supersymmetry and Quantum Mechanics} Physics Reports Volume 251 (1995), 267-385. (\href{http://arxiv.org/abs/hep-th/9405029}{arXiv:hep-th/9405029})

\end{itemize}
\hypertarget{ReferencesRelationToMorseTheory}{}\subsubsection*{{Relation to Morse theory}}\label{ReferencesRelationToMorseTheory}

Supersymmetric quantum mechanics gained attention with the work

\begin{itemize}%
\item [[Edward Witten]], \emph{Supersymmetry and Morse theory} J. Differential Geom. Volume 17, Number 4 (1982), 661-692. (\href{http://projecteuclid.org/euclid.jdg/1214437492}{Euclid}, \href{http://www.cimat.mx/~gil/docencia/2012/teoria_de_morse/witten_supersymmetry_and_morse_theory.pdf}{pdf}, \href{http://inspirehep.net/record/176416?ln=de}{spire})

\end{itemize}
which showed that [[Morse theory]] may be equivalently interpreted as the study of [[supersymmetry|supersymmetric]] [[vacua]] in supersymmetric quantum mechanics, and which was part of what gained Witten the \href{http://159.226.47.99:8080/general/prize/medal/1990.htm}{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

\begin{itemize}%
\item [[Gábor Pete]], section 2 of \emph{Morse theory}, lecture notes 1999-2001 (\href{http://www.math.bme.hu/~gabor/morse.pdf}{pdf})


\item Rohit Jain, \emph{Supersymmetric Schr\"o{}dinger operators with applications to Morse theory} (\href{http://www.ma.utexas.edu/users/rjain/SUSY.pdf}{pdf})



\end{itemize}
This deformed supersymmetric quantum mechanics arises as the point-particle limit of the [[type II superstring]] regarded as [[quantum mechanics]] on the [[loop space]], a relation that is stated more explicitly in

\begin{itemize}%
\item [[Edward Witten]], from p. 92 (32 of 39) on in \emph{Global anomalies in string theory}, in W. Bardeen and A. White (eds.) \emph{Symposium on Anomalies}, Geometry, Topology, pp. 61--99. World Scientific, 1985 ([[WittenGlobalAnomaliesInStringTheory.pdf:file]])

\end{itemize}
and then in

\begin{itemize}%
\item [[Edward Witten]], \emph{The Index Of The Dirac Operator In Loop Space} Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (\href{http://inspirehep.net/record/245523}{spire})

\end{itemize}
The relationon between the 2d [[SCFT]] describing the [[type II superstring]] and this deformed supersymmetric quantum mechanics on loop space has been further explored in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{On deformations of 2d SCFTs}, JHEP 0406:058,2004 (\href{http://arxiv.org/abs/hep-th/0401175}{arXiv:hep-th/0401175})

\end{itemize}
\hypertarget{ReferencesRelationToIndexTheory}{}\subsubsection*{{Relation to index theory}}\label{ReferencesRelationToIndexTheory}

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

\begin{itemize}%
\item [[Luis Alvarez-Gaumé]], \emph{Supersymmetry and the Atiyah-Singer index theorem}, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (\href{http://projecteuclid.org/euclid.cmp/1103940278}{Euclid})

\end{itemize}
\begin{itemize}%
\item [[Ezra Getzler]], \emph{Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem}, Comm. Math. Phys. 92 (1983), 163-178. (\href{http://math.northwestern.edu/~getzler/Papers/1103940796.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[D. Quillen]], \emph{Superconnections and the Chern character}, Topology 24 (1985), no. 1, 89--95, (\href{https://doi.org/10.1016/0040-9383%2885%2990047-3}{doi});


\item [[Varghese Mathai]], [[Daniel Quillen]], \emph{Superconnections, Thom classes, and equivariant differential forms}. Topology 25 (1986), no. 1, 85--110;


\item [[Ezra Getzler]], \emph{A short proof of the Atiyah-Singer index theorem}, Topology 25 (1986), 111-117 (\href{http://math.northwestern.edu/~getzler/Papers/local.pdf}{pdf})



\end{itemize}
\begin{itemize}%
\item [[D. Quillen]], \emph{Superconnection character forms and the Cayley transform}, Topology 27 (1988), no. 2, 211--238

\end{itemize}


\end{document}