有限温度量子场论-第二章 虚时路径积分（2.4 复标量场）

复标量场是课程中第一个可以引入一个化学势的场。
由于复标量场的拉氏量具有对称性，它就对应一个守恒荷，进而可以引入一个化学势。

The Lagrangian of the complex field is:

$$\mathcal{L}=\partial_{\mu} \Phi^{*} \partial^{\mu} \Phi-m^{2} \Phi^{*} \Phi-V\left(2 \Phi^{*} \Phi\right)$$

The complex field can be decomposed into real and imaginary parts:

$$\Phi = \frac{1}{\sqrt{2}}(\phi_1+i\phi_2)$$

The Lagrangian in and is:

$$\mathcal{L}=\frac{1}{2}\left[\partial_{\mu} \phi_{1} \partial^{\mu} \phi_{1}+\partial_{\mu} \phi_{2} \partial^{\mu} \phi_{2}-m^{2}\left(\phi_{1}^{2}+\phi_{2}^{2}\right)\right]-V\left(\phi_{1}^{2}+\phi_{2}^{2}\right)$$

This Lagrangian density is invariant under global transformations:

$$\Phi \to \Phi' = e^{i\alpha}\Phi$$

This symmetry corresponds to a conserved charge:

$$Q=\int_{V} d^{3} \mathbf{x} J^{0}$$

where is the zeroth component of the Noether current. For the conserved we can introduce a chemical potential .