# 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.