有限温度量子场论-第二章 虚时路径积分（2.5 狄拉克场）

前面两节中计算的中性标量场和复标量场都对应玻色子的情况。这一节计算的Dirac场对应费米子。

Non-interacting spin 1/2 fermions with mass could be described by Dirac field:

$$\mathcal{L}_{0}=\bar{\psi}(i \not \partial-m) \psi$$

Here , and are Dirac matrix:

$$\gamma^{0}=\left(\begin{array}{cc} I_{2} & 0 \\ 0 & -I_{2} \end{array}\right), \quad \gamma^{i}=\left(\begin{array}{cc} 0 & \sigma_{i} \\ -\sigma_{i} & 0 \end{array}\right)$$

The canonically conjugate fields are given by:

$$\Pi \equiv \frac{\partial \mathcal{L}_{0}}{\partial \dot{\psi}}=\bar{\psi} i \gamma^{0}=i \psi^{\dagger}, \quad \bar{\Pi} \equiv \frac{\partial \mathcal{L}_{0}}{\partial \dot{\bar{\psi}}}=0$$