对易与反对易公式(update 09/July/2020)

在运动方程中, 经常要计算各种产生湮灭算符之间的对易关系. 而对于 Fermion 来说, 它满足的是反对易关系, 因此在最后一步中, 需要用下述关系, 写成反对易子的形式

$$\begin{split} [AB, C] =& A\{B, C\} - \{A,C\}B\\ [A,BC] =&\{A, B\} C -B\{A,C\} \end{split}$$

Boson

Boson 满足对易关系

$$[a_k, a^{\dagger}_{k'}] = \delta_{kk'}$$ $$\sum_{k'}[a^{\dagger}_{k3}a_{k3},a^{\dagger}_{k'2}a_{k'3}] = -a^{\dagger}_{k2}a_{k3}$$ $$\sum_{k'}[a^{\dagger}_{k3}a_{k3},a^{\dagger}_{k'3}a_{k'2}] = a^{\dagger}_{k3}a_{k2}$$

Fermion

Fermion 满足反对易关系

$$\{a_k, a^{\dagger}_{k'}\} = \delta_{kk'}$$ $$\sum_{k'}[a^{\dagger}_{k3}a_{k3},a^{\dagger}_{k'2}a_{k'3}] = -a^{\dagger}_{k2}a_{k3}$$ $$\sum_{k'}[a^{\dagger}_{k3}a_{k3},a^{\dagger}_{k'3}a_{k'2}] = a^{\dagger}_{k3}a_{k2}$$