"Supersymmetric quantum mechanics"의 두 판 사이의 차이
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| + | ==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 | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 07:15 판
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