1.7 Quantization of a Dirac field

The Lagrangian of the free Dirac field $\psi$ is

$${\mathcal{L}}_0=\bar{\psi }(\mathrm{i}/\partial -m)\psi ,\text{\hspace{1em}(1.233)}$$

where $\mathcal{J}={\gamma }^{\mu }{\partial }_{\mu }$ . In general $A\equiv {\gamma }^{\mu }{A}_{\mu }$ . Variation with respect to $\bar{\psi }$ yields the Dirac equation

$$\left(\mathrm{i}/\partial -m\right)\psi =0.\text{\hspace{1em}(1.234)}$$

The Dirac field, in canonical quantization, satisfies the anti-commutation relation

$$\{\bar{\psi }\left(x^0,x\right),\psi \left(x^0,y\right)\}=\delta (x-y).\text{\hspace{1em}(1.235)}$$

Accordingly, it is expressed as a Grassmann number function in path integrals. The generating functional is

$$Z_0[\bar{\eta },\eta ]=\int \mathcal{D}\bar{\psi }\mathcal{D}\psi \exp \left[i\int dx(\bar{\psi }(i/\partial -m)\psi +\bar{\psi }\eta +\bar{\eta }\psi )\right]\text{\hspace{1em}(1.236)}$$

where $\eta ,\bar{\eta }$ are Grassmannian sources.

The propagator is given by the functional derivative with respect to the sources,

$$\begin{array}{rl}S(x-y)& =-\frac{{\delta }^2{Z}_0[\bar{\eta },\eta ]}{\delta \bar{\eta }(x)\delta \eta (y)}\\ & =\frac{1}{(2\pi {)}^d}\int {\mathrm{d}}^{d}k\frac{{\mathrm{e}}^{\mathrm{i}kx}}{k-m-\mathrm{i}\epsilon }=(\mathrm{i}/\partial +m+\mathrm{i}\epsilon )\Delta (x-y)\end{array}\text{\hspace{1em}(1.237)}$$

where $\Delta (x-y)$ is the scalar field propagator.

By making use of the Dirac equations

$$\left(\mathrm{i}/\partial -m\right)\psi =-\eta \bar{\psi }(\mathrm{i}\overleftrightarrow{\partial }+m)=\bar{\eta }\text{\hspace{1em}(1.238)}$$

the generating functional is cast into the form

$$Z_0[\bar{\eta },\eta ]=Z_0[0,0]\exp \left[-\mathrm{i}\int \mathrm{d}x\mathrm{d}y\bar{\eta }(x)S(x-y)\eta (y)\right].\text{\hspace{1em}(1.239)}$$

After Wick rotation $\tau =\mathrm{i}{x}^0$ , the normalization factor is obtained as

$$Z_0[0,0]=\mathrm{Det}(\mathrm{i}/\partial -m)=\prod _i{\lambda }_i\text{\hspace{1em}(1.240)}$$

where ${\lambda }_i$ is the $i$ th eigenvalue of the Dirac operator $\mathrm{i}/\partial -m$

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

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