1.6 Quantization of a scalar field

1.6.1 Free scalar field

The analysis made in the previous sections may be easily generalized to a case with many degrees of freedom. We are interested, in particular, in a system with infinitely many degrees of freedom; the quantum field theory (QFT). Let us start our exposition with the simplest case, that is, the scalar field theory. Let $\varphi (x)$ be a real scalar field at the spacetime coordinates $x=(x,x^0)$ where $x$ is the space coordinate while $x^0$ is the time coordinate. The action depends on $\varphi$ and its derivatives ${\partial }_{\mu }\varphi (x)=\partial \varphi (x)/\partial x^{\mu }$ :

$$S=\int \mathrm{d}x\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi ).\text{\hspace{1em}(1.207)}$$

Here $\mathcal{L}$ is the Lagrangian density. The Euler-Lagrange equation now takes the form

$$\frac{\partial }{\partial x^{\mu }}\left(\frac{\partial \mathcal{L}}{\partial \left({\partial }_{\mu }\varphi \right)}\right)-\frac{\partial \mathcal{L}}{\partial \varphi }=0.\text{\hspace{1em}(1.208)}$$

The Lagrangian density of a free scalar field is

$${\mathcal{L}}_0(\varphi ,{\partial }_{\mu }\varphi )=-\frac{1}{2}\left({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi +m^2{\varphi }^2\right).\text{\hspace{1em}(1.209)}$$

The Euler-Lagrange equation derived from this Lagrangian density is the Klein-Gordon equation

$$(\square -m^2)\varphi =0,\text{\hspace{1em}(1.210)}$$

where $\square ={\partial }^{\mu }{\partial }_{\mu }=-{\partial }_0^2+{\nabla }^2$

The vacuum-to-vacuum amplitude in the presence of a source $J$ has the path integral representation $⟨0,\infty |0,-\infty {⟩}^J\propto Z_0[J]$ , where

$$Z_0[J]=\int \mathcal{D}\varphi \exp \left[i\int dx\left({\mathcal{L}}_0+J\varphi +\frac{\mathrm{i}}{2}\epsilon {\varphi }^2\right)\right]\text{\hspace{1em}(1.211)}$$

where the $\mathrm{i}\epsilon$ term has been added to regularize the path integral. ¹ Integration by parts yields

$$Z_0[J]=\int \mathcal{D}\varphi \exp \left[\mathrm{i}\int \mathrm{d}x\left(\frac{1}{2}\left\{\varphi \left(\square -m^2\right)\varphi +\mathrm{i}\epsilon {\varphi }^2\right\}+J\varphi \right)\right].\text{\hspace{1em}(1.212)}$$

Let ${\varphi }_{\mathrm{c}}$ be the classical solution to the Klein-Gordon equation in the presence of the source,

$$\left(\square -m^2+\mathrm{i}\epsilon \right){\varphi }_{\mathrm{c}}=-J.\text{\hspace{1em}(1.213)}$$

The solution is easily found to be

$${\varphi }_{\mathrm{c}}(x)=-\int \mathrm{d}y\Delta (x-y)J(y)\text{\hspace{1em}(1.214)}$$

where $\Delta (x-y)$ is the Feynman propagator

$$\Delta (x-y)=\frac{-1}{(2\pi {)}^d}\int {\mathrm{d}}^{d}k\frac{{\mathrm{e}}^{\mathrm{i}k(x-y)}}{k^2+m^2-\mathrm{i}\epsilon }.\text{\hspace{1em}(1.215)}$$

Here $d$ denotes the spacetime dimension. Note that $\Delta (x-y)$ satisfies

$$(\square -m^2+\mathrm{i}\epsilon )\Delta (x-y)={\delta }^d(x-y).$$

It is easy to show that (exercise) the functional $Z_0[J]$ is now written as

$$Z_0[J]=Z_0[0]\exp \left[-\frac{\mathrm{i}}{2}\int \mathrm{d}x\mathrm{d}yJ(x)\Delta (x-y)J(y)\right].\text{\hspace{1em}(1.216)}$$

It is instructive to note that the propagator is conversely obtained by the functional derivative of $Z_0[J]$ ,

$$\Delta (x-y)=\frac{\mathrm{i}}{Z_0[0]}{\frac{{\delta }^2{Z}_0[J]}{\delta J(x)\delta J(y)}|}_{J=0}.\text{\hspace{1em}(1.217)}$$

The amplitude $Z_0[0]$ is the vacuum-to-vacuum amplitude in the absence of the source and may be evaluated as follows. Let us introduce the imaginary time $x^4=\tau =\mathrm{i}{x}^0$ . Then, we obtain

$$\begin{array}{rl}{Z}_0[0]& =\int \bar{\mathcal{D}}\varphi \exp \left[\frac{1}{2}\int \mathrm{d}x\varphi (\bar{\square }-m^2)\varphi \right]\\ & ={\left[\mathrm{Det}\left(\bar{\square }-m^2\right)\right]}^{-1/2},\end{array}\text{\hspace{1em}(1.218)}$$

where $\bar{\square }={\partial }_{\tau }^2+{\nabla }^2$ and the determinant is understood in the sense of section 1.4, namely it is the product of eigenvalues with a relevant boundary condition.

A free complex scalar field theory has a Lagrangian density

$${\mathcal{L}}_0=-{\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2|\varphi {|}^2+J{\varphi }^{\ast }+J^{\ast }\varphi \text{\hspace{1em}(1.219)}$$

where the source terms have been included. The generating functional is now given by

$$\begin{array}{rl}{Z}_0[J,J^{\ast }]& =\int \mathcal{D}\varphi \mathcal{D}{\varphi }^{\ast }\exp \left[\mathrm{i}\int \mathrm{d}x({\mathcal{L}}_0-\mathrm{i}\epsilon |\varphi {|}^2)\right]\\ & =\int \mathcal{D}\varphi \mathcal{D}{\varphi }^{\ast }\exp \left[\mathrm{i}\int \mathrm{d}x\left\{{\varphi }^{\ast }(\square -m^2+\mathrm{i}\epsilon )\varphi +J^{\ast }\varphi +J{\varphi }^{\ast }\right\}\right].\end{array}\text{\hspace{1em}(1.220)}$$

The propagator is now given by

$$\Delta (x-y)=\frac{\mathrm{i}}{Z_0[0,0]}\frac{{\delta }^2{Z}_0[J,J^{\ast }]}{\delta J^{\ast }(x)\delta J(y)}{|}_{J=J^{\ast }=0}.\text{\hspace{1em}(1.221)}$$

By substituting the Klein-Gordon equations

$$\left(\square -m^2\right){\varphi }_{\mathrm{c}}=-J\left(\square -m^2\right){\varphi }_{\mathrm{c}}^{\ast }=-J^{\ast }\text{\hspace{1em}(1.222)}$$

we separate the generating functional as

$$Z_0[J,J^{\ast }]=Z_0[0,0]\exp \left[-\mathrm{i}\int \mathrm{d}x\mathrm{d}y{J}^{\ast }(x)\Delta (x-y)J(y)\right]\text{\hspace{1em}(1.223)}$$

where

$$\begin{array}{rl}{Z}_0[0,0]& =\int \mathcal{D}\varphi \mathcal{D}{\varphi }^{\ast }\exp \left[-\mathrm{i}\int \mathrm{d}x{\varphi }^{\ast }(\square -m^2-\mathrm{i}\epsilon )\varphi \right]\\ & ={\left[\mathrm{Det}\left(\bar{\square }-m^2\right)\right]}^{-1}.\end{array}\text{\hspace{1em}(1.224)}$$

Wick rotation has been made to occur at the last line.

1.6.2 Interacting scalar field

It is possible to add interaction terms to the free field Lagrangian (1.209),

$$\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi )={\mathcal{L}}_0(\varphi ,{\partial }_{\mu }\varphi )-V(\varphi ).\text{\hspace{1em}(1.225)}$$

The possible form of $V(\varphi )$ is restricted by the symmetry and renormalizability of the theory. A typical form of $V$ is a polynomial

$$V(\varphi )=\frac{g}{n!}{\varphi }^n(n\ge 3,n\in \mathbb{N})$$

where the constant $g\in \mathbb{R}$ controls the strength of the interaction. The generating functional is defined similarly to the free theory as

$$Z[J]=\int \mathcal{D}\varphi \exp \left[\mathrm{i}\int \mathrm{d}x\left\{\frac{1}{2}\varphi \left(\square -m^2\right)\varphi -V(\varphi )+J\varphi \right\}\right].\text{\hspace{1em}(1.226)}$$

The presence of $V(\varphi )$ makes things slightly more complicated. It can be handled at least perturbatively as

$$\begin{array}{rl}& Z[J]=\int \mathcal{D}\varphi \exp \left[-\mathrm{i}\int \mathrm{d}xV(\varphi )\right]\exp \left[\mathrm{i}\int \mathrm{d}x\{{\mathcal{L}}_0+J\varphi \}\right]\\ & =\exp \left[-\mathrm{i}\int \mathrm{d}xV\left(\frac{1}{\mathrm{i}}\frac{\delta }{\delta J(x)}\right)\right]\int \mathcal{D}\varphi \exp \left[\mathrm{i}\int \mathrm{d}x\{{\mathcal{L}}_0+J\varphi \}\right]\\ & =\exp \left[-\mathrm{i}\int \mathrm{d}xV\left(\frac{1}{\mathrm{i}}\frac{\delta }{\delta J(x)}\right)\right]Z_0[J]\\ & =\sum _{k=0}^{\infty }\int \mathrm{d}{x}_1\cdots \int \mathrm{d}{x}_k\frac{(-\mathrm{i}{)}^k}{k!}\\ & \times V\left(\frac{1}{\mathrm{i}}\frac{\delta }{\delta J\left(x_1\right)}\right)\dots V\left(\frac{1}{\mathrm{i}}\frac{\delta }{\delta J\left(x_k\right)}\right)Z_0[J].\end{array}\text{\hspace{1em}(1.227)}$$

The generating functional $Z[J]$ generates the vacuum expectation value of the $T$ -product of field operators, also known as the Green function $G_n(x_1,\dots ,x_n)$ , as

$$\begin{array}{rl}& G_n\left(x_1,\dots ,x_n\right)\equiv ⟨0|T[\varphi (x_1)\dots \varphi (x_n)]|0⟩\\ & ={\frac{(-\mathrm{i}{)}^n{\delta }^n}{\delta J(x_1)\dots \delta J(x_n)}Z[J]|}_{J=0}.\end{array}\text{\hspace{1em}(1.228)}$$

Since this is the $n$ th functional derivative of $Z[J]$ around $J=0$ , we obtain the functional Taylor expansion of $Z[J]$ as

$$\begin{array}{rl}Z[J]& =\sum _{n=1}^{\infty }\frac{1}{n!}\left[\prod _{i=1}^n\int \mathrm{d}{x}_{i}J(x_i)\right]⟨0|T[\varphi (x_1)\dots \varphi (x_n)]|0⟩\\ & =⟨0|T{\mathrm{e}}^{\int \mathrm{d}xJ(x)\varphi (x)}|0⟩.\end{array}\text{\hspace{1em}(1.229)}$$

The connected $n$ -point functions are generated by $W[J]$ defined by

$$Z[J]={\mathrm{e}}^{-W[J]}.\text{\hspace{1em}(1.230)}$$

The effective action $\Gamma [{\varphi }_{\mathrm{cl}}]$ is defined by the Legendre transformation

$$\Gamma \left[{\varphi }_{\mathrm{cl}}\right]\equiv W[J]-\int \mathrm{d}\tau \mathrm{d}xJ{\varphi }_{\mathrm{cl}}\text{\hspace{1em}(1.231)}$$

where

$${\varphi }_{\mathrm{cl}}\equiv ⟨\varphi {⟩}^J=\frac{\delta W[J]}{\delta J}.\text{\hspace{1em}(1.232)}$$

The functional $\Gamma [{\varphi }_{\mathrm{cl}}]$ generates one-particle irreducible diagrams.

Endnotes

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

OCR 来源 SHA-256：0b6324f01172a3548cd34382e82fd8bfae574602b6b2cfaa357ba96bc2c35478

OCR 产物 SHA-256：1385a70831e1d2b8520e753c932a4e7bb566b7484989df204432c698fe609f6a

Footnotes

1. Alternatively, we can introduce the imaginary time $\tau =\mathrm{i}{x}^0$ to Wick rotate the time axis. ↩