# 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

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

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

Therefore

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.