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

• 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.