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 ${\displaystyle n}$ field operators at different positions

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

Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Space translation symmetry Time translation symmetry Rotation symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Yang–Mills theory Noether charge Topological charge
Tools Anomaly Crossing Effective field theory Expectation value Ghost fields Faddeev–Popov ghosts Feynman diagram Lattice gauge theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms Feynman parametrization
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Joos–Weinberg equation Weyl equation
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories Topological quantum field theory String theory Superstring theory M-Theory Supersymmetry Supergravity Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Bethe Bogoliubov Coleman Callan DeWitt Dirac Dyson Fermi Feynman Fierz Fock Fröhlich Gell-Mann Goldstone Gross Heisenberg 't Hooft Jackiw Jordan Landau Lee Lehmann Mandelstam Majorana Nambu Parisi Polyakov Salam Schwinger Skyrme Stueckelberg Symanzik Tomonaga Veltman Weinberg Weisskopf Weyl Wilczek Wilson Witten Yang Yukawa Zimmermann Zinn-Justin

For time-dependent correlation functions, the time-ordering operator ${\displaystyle T}$ 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.