Two Significant Identities

In this section, you are going to meet two significant identities which play central roles in the following constructions. These identities are

\[\begin{split}\frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}W_0[J] = \int\mathcal{D}\varphi\ \varphi(x)\exp\{\mathrm{i}S\}\\ (\partial^2\!\!-\!m^2)\frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}W_0[J] = J(x)W_0[J]\end{split}\]

Note

To understand this section, you will find Functional Derivative helpful.

First Identity

The first identity can be verified through a calculation of functional derivative of \(W_0[J]\) with respect to \(J(x)\). Specifically, we have

\[\begin{split}\frac{\delta}{\delta J(x)}W_0[J] &= \int\mathcal{D}\varphi\frac{\delta}{\delta J(x)}\left[\exp\{-\frac{1}{2}\mathrm{i}\int\varphi(\partial^2\!\!-\!m^2)\varphi + \mathrm{i}\int J\varphi\}\right]\\ &= \int\mathcal{D}\varphi\frac{\delta}{\delta J(x)}\left[\mathrm{i}\int J\varphi\}\right]\exp\{\mathrm{i}S\} = \int\mathcal{D}\varphi\ \mathrm{i}\varphi(x)\exp\{\mathrm{i}S\}\end{split}\]

The right hand side of the identity is sometimes interpreted as the average value of the field \(\varphi(x)\) due to its formal similarities with the definition of average value in statistical mechanics.

Second Identity

The second identity can be gained through a direct calculation. Specifically, there is

\[\frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}W_0[J] = \frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}\exp\{-\frac{1}{2}\mathrm{i}\int J\Delta_F J\} = -\left[\int\Delta_F J(x)\right]\ W_0[J]\]

Therefore

\[(\partial^2\!\!-\!m^2)\frac{1}{\mathrm{i}}\frac{\delta}{\delta J(x)}W_0[J] = -\left[\int(\partial^2\!\!-\!m^2)\Delta_F J(x)\right]\ W_0[J] = J(x)W_0[J]\]

You are strongly suggested to remember the two identities since they are the very foundations of the following constructions of interaction theory and S-matrix.