Correlation function (quantum field theory)

For other uses, see Correlation function (disambiguation).

In quantum field theory, the (real space) n-point correlation function is defined as the functional average (functional expectation value) of a product of $n$ field operators at different positions

$C_{n}\left(x_{1},x_{2},\ldots ,x_{n}\right):=\left\langle \phi (x_{1})\phi (x_{2})\cdots \phi (x_{n})\right\rangle ={\frac {\int {\mathcal {D}}\phi \;e^{-S[\phi ]}\phi (x_{1})\cdots \phi (x_{n})}{\int {\mathcal {D}}\phi \;e^{-S[\phi ]}}}$

For time-dependent correlation functions, the time-ordering operator $T$ is included.

Correlation functions are also called simply correlators.

The correlation function can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x.

See also

• Connected correlation function
• One particle irreducible correlation function
• Green's function (many-body theory)
• Partition function (mathematics)

Further reading

• Alexander Altland, Ben Simons (2006). Condensed Matter Field Theory. Cambridge University Press.
• Schroeder, Daniel V. and Michael Peskin, An Introduction to Quantum Field Theory. Addison-Wesley.