Supersymmetric quantum mechanics

introduction

Consider a quantum mechanical system consisting of a Hilbert (Fock) space \(\it F\) and Hamiltonian \(H\). The system is said to be supersymmetric quantum mechanical (SQM) if

1.\(\it F\) has a decomposition \({\it F}={\it F}^B \oplus {\it F}^F\) and states in \({\it F}^B\) and \({\it F}^F\) are called bosonic and fermionic states respectively. There is an operator \((-1)^F\) such that \begin{eqnarray} &&(-1)^F \Psi =\Psi \ \ if \ \Psi \in {\it F}^B \\ &&(-1)^F \Psi =-\Psi \ \ if \ \Psi \in {\it F}^F \end{eqnarray} \(F\) and \((-1)^F\) are called fermion number operator and chirality operator.

2.There are N operators \(Q^I\), \(I=1,\cdots,N\), such that \begin{eqnarray} Q^I,{Q^I}^\dagger &:&{\it F}^B \rightarrow {\it F}^F ,\\ Q^I,{Q^I}^\dagger &:&{\it F}^F \rightarrow {\it F}^B ,\\ \left\{ (-1)^F,Q^I\right\}&=&\left\{ (-1)^F,{Q^I}^\dagger\right\}=0 \end{eqnarray} \(Q^I\) are called supersymmetry (SUSY) charges or generators.

3.The SUSY generators satisfy the general superalgebra condition: \begin{eqnarray} \left\{ Q^I,{Q^J}^\dagger \right\}&=&2 \delta^{IJ} H\\ \left\{ Q^I,{Q^J}\right\}&=&\left\{ Q^I,{Q^J}\right\}=0 \end{eqnarray} where \(I,J=1,\cdots,N\).

A quantum system satisfying the above conditions is said to have a type N supersymmetry.

expositions

Muhammad Abdul Wasay, Supersymmetric quantum mechanics and topology, http://arxiv.org/abs/1603.07691v1
van Loon, Mark. “Path Integral Methods in Index Theorems.” arXiv:1509.03063 [math-Ph, Physics:quant-Ph], September 10, 2015. http://arxiv.org/abs/1509.03063.
Li, Si. “Supersymmetric Quantum Mechanics and Lefschetz Fixed-Point Formula.” arXiv:hep-th/0511101, November 8, 2005. http://arxiv.org/abs/hep-th/0511101.
Cooper, Fred, Avinash Khare, and Uday Sukhatme. “Supersymmetry and Quantum Mechanics.” Physics Reports 251, no. 5–6 (January 1995): 267–385. doi:10.1016/0370-1573(94)00080-M.

articles

Dana Fine, Stephen Sawin, Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac, arXiv:1605.06982 [math-ph], May 23 2016, http://arxiv.org/abs/1605.06982

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