Generating Functional for Free Bosonic Lagrangian

Generating functional for scalar fields in the presence of source J is defined by

Vacuum-to-vacuum transition amplitude can be written as

For free bosonic particle, the lagrangian is

Then, the transition amplitude is

(1)

Let us now simplify the expression by evaluate this

where can be written as D'Alembertian. Exchange the position, so we have

The first term of right side can be vanished by taking at infinity. Then,

Substitute this result to Eq. (1). Thus

To evaluate it more simplify, let us change to

This is the semi-classical approximation that will separate the classical aspect and the quantum aspect. 

Using the fact that

,
then we have

(2)

(3)

This equation have a solution in Fourier representation, that is

(4)

where is called Feynman propagator and must obey

The Fourier representation of this Green's function is

(5)

Subtituting Eq. (3) to Eq. (2)

(6)

Furthermore, substituting Eq. (4) to Eq. (6)

In the case of abbreviation the notation, let

'
thus

The result of the path integral on the kinetic term must be a number, then just called it as N

Therefore, we have generating functional of vacuum in simpler form.

Generating functional for interacting field can be obtained as functional derivatives of this vacuum transition in respect to the external current.