| 一次量子化 | Boson | Fermion | Spin | |
| 共轭对Conjugate pair | \(\{x, p\}\) | \(\{a, a^{\dagger}\}\) | \(\{C, C^{\dagger}\}\) | \(\{\phi, \cos \theta\}\) |
| 相干态Coherent state | \( \) | \( \left | \alpha \right \rangle = e^{-\frac{1}{2}\alpha^{*}\alpha} \sum_n \frac{\alpha^n}{\sqrt{n!}} \left | n \right \rangle\) | \( \left | \eta \right \rangle = e^{-\eta C^{\dagger}} \left | 0 \right \rangle\) | \( \left | \vec{n} \right \rangle = (\cos \frac{\theta}{2}, \sin \frac{\theta}{2} e^{i\phi})^T\) |
| 单位"I" | \(\int dx \left | x \right \rangle \left \langle x \right | = \int dp \left | p \right \rangle \left \langle p \right | =I \) | \(\int d\alpha^{*}d\alpha \frac{1}{\pi} \left | \alpha \right \rangle \left \langle \alpha \right | = 1\) | \(\int d\overline{\eta} d\eta e^{-\overline{\eta}\eta} \left | \eta \right \rangle \left \langle \eta \right | = 1\) | \(\int d\Omega \left | \vec{n} \right \rangle \left \langle \vec{n} \right | = 1\) |
| 路径积分形式 | \(\int Dx e^{S}\) | \(\int D\overline{\phi}D\phi e^{iS(\overline{\phi}, \phi)} \) | \(\int D\overline{\eta}D\eta e^{iS(\overline{\eta}, \eta)} \) | \(\int D\vec{n} e^{i\int\left(\sin^2\frac{\theta}{2}\dot{\phi}-H\right)}\) |
| 结果 | \(\frac{1}{\sqrt{det M}}\) | \(\frac{1}{det(M)}\) | \(det(M)\) | \( \) |