infinitesimal exponential power series Hamiltonian function field
where ε is an infinitesimal number and ϕI is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in 




ϕ

4




{\displaystyle \phi ^{4}}

-theory, the interacting term of the Hamiltonian is 




H

I


(
t
)
=
∫

d

3


x



λ

4
!




ϕ

I


(
x

)

4




{\textstyle H_{I}(t)=\int d^{3}x\,{\frac {\lambda }{4!}}\phi _{I}(x)^{4}}

, and the expansion of the two-point correlator in terms of 



λ


{\displaystyle \lambda }

 becomes



⟨
Ω

|

T
{
ϕ
(
x
)
ϕ
(
y
)
}

|

Ω
⟩
=




∑

n
=
0


∞





(
−
i
λ

)

n




(
4
!

)

n


n
!



∫

d

4



z

1


⋯
∫

d

4



z

n


⟨
0

|

T
{

ϕ

I


(
x
)

ϕ

I


(
y
)

ϕ

I


(

z

1



)

4


⋯

ϕ

I


(

z

n



)

4


}

|

0
⟩



∑

n
=
0


∞





(
−
i
λ

)

n




(
4
!

)

n


n
!



∫

d

4



z

1


⋯
∫

d

4



z

n


⟨
0

|

T
{

ϕ

I


(

z

1



)

4


⋯

ϕ

I


(

z

n



)

4


}

|

0
⟩



.


{\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle ={\frac {\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(x)\phi _{I}(y)\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }{\displaystyle \sum _{n=0}^{\infty }{\frac {(-i\lambda )^{n}}{(4!)^{n}n!}}\int d^{4}z_{1}\cdots \int d^{4}z_{n}\langle 0|T\{\phi _{I}(z_{1})^{4}\cdots \phi _{I}(z_{n})^{4}\}|0\rangle }}.}

This perturbation expansion expresses the interacting two-point function in terms of quantities 



⟨
0

|

⋯

|

0
⟩


{\displaystyle \langle 0|\cdots |0\rangle }

 that are evaluated in the free theory.
