Perturbative Expansion of Correlation Functions

Explains the perturbative expansion of correlation functions.

Reference: (Peskin & Schroeder, 1995)

Unlike free field theories, no exactly solvable nonlinear interacting field theories are known in more than two dimensions. Therefore, we generally have to rely on perturbation theory. Thus, we want to be able to compute perturbative series easily. Ultimately, we aim to calculate observables, but first, we must consider the perturbative expansion of correlation functions. To achieve this, we need to rewrite the correlation functions into a form suitable for a perturbative expansion. Here, we provide an explanation within the context of canonical quantization.

If we understand how to rewrite the 2-point correlation function, the approach for the $n$-point correlation function is easy to deduce. Therefore, below we explain the rewriting for the 2-point correlation function:

Here, $\ket{\Omega}$ is the ground state of the interacting field theory. Note that this ground state differs from the ground state $\ket{0}$ of the free field theory. In addition, $\phi_{\mathrm{H}}$ is the field in the Heisenberg picture, which we will simply denote as $\phi$ below.

Our goal is to express the 2-point correlation function as a power series in $\lambda$, given a Hamiltonian such as:

The effect of $H_{\mathrm{int}}$ appears in the following two parts of the 2-point correlation function:

1. The definition of the Heisenberg field: $\phi(x) = e^{iHt}\phi(\mathbf{x})e^{-iHt}$
2. The definition of the ground state: $H\ket{\Omega} = 0$

Thus, if we can express these two parts in the language of free field theory, we can express the 2-point correlation function entirely in the language of free field theory. This is specifically what we aim to do.

Rewriting the Heisenberg Field

Consider the time evolution of a field at a certain time $t_0$ under $H$:

If $\lambda$ is small, we can assume that the time evolution of the field is dominated by $H_{0}$, which is expected to form the core of the perturbative expansion. Therefore, we define:

We call this the field in the interaction picture. This is a free field. Its relationship with the original field $\phi$ is given by:

where $U(t,t_0)$ satisfies the differential equation:

This precisely corresponds to the time evolution operator of the interacting field. The solution to this differential equation can formally be written as:

This is known as Dyson’s formula. Using this and equation (1), we successfully rewrote the original field $\phi$ in terms of the free field $\phi_{\mathrm{I}}$.

Rewriting the Ground State

Consider the time evolution of the free field theory ground state $\ket{0}$ under $H$:

Here, $\ket{n}$ are the unknown eigenstates of $H$, and $E_n$ are its unknown energy eigenvalues.

Assuming $\braket{\Omega|0}\ne 0$, since $E_0 < E_n$, the $n=0$ term becomes dominant in the limit $T\to\infty(1-i\epsilon)$. Therefore:

In this limit, we can shift the value of $T$ by a finite amount, yielding:

Thus, excluding an overall normalization factor, we successfully rewrote the ground state $\ket{\Omega}$ in terms of free field theory.

Rewriting the Correlation Function

Combining the above steps, for $x^0 > y^0$, we get:

Dividing the right-hand side by the normalization condition:

allows us to express the correlation function purely in the language of free field theory. This reasoning holds equally well for $x^0 < y^0$. In summary, we have:

For $n$-point functions, we simply add more $\phi_{\mathrm{I}}$ fields. We are now in a position to perform a series expansion in $\lambda$ up to any necessary order.

References