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

$C_n(x_1, x_2,\ldots,x_n) := \left\langle \phi(x_1) \phi(x_2) \ldots \phi(x_n)\right\rangle =\frac{\int D \phi \; e^{-S[\phi]}\phi(x_1)\ldots \phi(x_n)}{\int D \phi \; e^{-S[\phi]}}$

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

Correlation functions are also called simply correlators. Sometimes, the phrase Green's function is used not only for two-point functions, but for any correlators.

• The correlation function can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x. In the free theory, it is simply the Feynman propagator(for n=2). [For more information see 'An Introduction to Quantum Field Theory' by Peskin & Schroeder, Section 4.2 : Perturbation Expansion of Correlation Functions]

See also

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

References

Books

• Alexander Altland, Ben Simons (2006): Condensed Matter Field Theory Cambridge University Press