Correlation function (quantum field theory)

Definition

For a scalar field theory with a single field ${\displaystyle \varphi (x)}$ and a vacuum state ${\displaystyle |\mathrm{\Omega }⟩}$ at every event x in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of n field operators in the Heisenberg picture $${\displaystyle G_n(x_1,\dots ,x_n)=⟨\mathrm{\Omega }|T\{\varphi (x_1)\dots \varphi (x_n)\}|\mathrm{\Omega }⟩.}$$

Here ${\displaystyle T\{\cdots \}}$ is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as $${\displaystyle G_n(x_1,\dots ,x_n)=\frac{⟨0|T\{\varphi (x_1)\dots \varphi (x_n)e^{iS[\varphi ]}\}|0⟩}{⟨0|e^{iS[\varphi ]}|0⟩},}$$ where ${\displaystyle |0⟩}$ is the ground state of the free theory and ${\displaystyle S[\varphi ]}$ is the action. Expanding ${\displaystyle e^{iS[\varphi ]}}$ using its Taylor series, the n-point correlation function becomes a sum of interaction picture correlation functions which can be evaluated using Wick's theorem. A diagrammatic way to represent the resulting sum is via Feynman diagrams, where each term can be evaluated using the position space Feynman rules.

The series of diagrams arising from ${\displaystyle ⟨0|e^{iS[\varphi ]}|0⟩}$ is the set of all vacuum bubble diagrams, which are diagrams with no external legs. Meanwhile, ${\displaystyle ⟨0|\varphi (x_1)\dots \varphi (x_n)e^{iS[\varphi ]}|0⟩}$ is given by the set of all possible diagrams with exactly n external legs. Since this sum also includes disconnected diagrams with vacuum bubbles, the numerator factorizes into

(sum over all bubble diagrams)${\displaystyle \times }$(sum of all diagrams with no bubbles).

The first factor then cancels with the normalization factor in the denominator meaning that the n-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles

$${\displaystyle G_n(x_1,\dots ,x_n)=⟨0|T\{\varphi (x_1)\dots \varphi (x_n)e^{iS[\varphi ]}\}|0{⟩}_{\text{no bubbles}}.}$$

While not including any vacuum bubbles, the sum does include disconnected diagrams, which are diagrams where at least one external leg is not connected to all other external legs through some connected path. Excluding these disconnected diagrams instead defines connected n-point correlation functions $${\displaystyle G_n^c(x_1,\dots ,x_n)=⟨0|T\{\varphi (x_1)\dots \varphi (x_n)e^{iS[\varphi ]}\}|0{⟩}_{\text{connected, no bubbles}}}$$

It is often preferable to work directly with these as they contain all the information that the full correlation functions contain since any disconnected diagram is merely a product of connected diagrams. By excluding other sets of diagrams one can define other correlation functions such as one-particle irreducible correlation functions.

In the path integral formulation, n-point correlation functions are written as a functional average $${\displaystyle G_n(x_1,\dots ,x_n)=\frac{\int \mathcal{D}\varphi \varphi (x_1)\dots \varphi (x_n)e^{iS[\varphi ]}}{\int \mathcal{D}\varphi e^{iS[\varphi ]}}.}$$

They can be evaluated using the partition functional ${\displaystyle Z[J]}$ which acts as a generating functional, with ${\displaystyle J}$ being a source-term, for the correlation functions $${\displaystyle G_n(x_1,\dots ,x_n)=(-i{)}^n\frac{1}{Z[J]}{\frac{{\delta }^{n}Z[J]}{\delta J(x_1)\dots \delta J(x_n)}|}_{J=0}.}$$

Similarly, connected correlation functions can be generated using ${\displaystyle W[J]=-i\ln Z[J]}$ as $${\displaystyle G_n^c(x_1,\dots ,x_n)=(-i{)}^{n-1}{\frac{{\delta }^{n}W[J]}{\delta J(x_1)\dots \delta J(x_n)}|}_{J=0}.}$$