Correlation function (quantum field theory)

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

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

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

Correlation functions are also called simply correlators. 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.