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

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

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).

• Very important note : [Mohamed 15/08/2018]

These (n) field operators at different positions , can be discribe according to Smarr's formula, the thermodynamics of black holes connect the mass M of a black hole to the surface A of its horizon and its possible kinetic moment L and electric charge Q:

M=M(A,L,Q).

It is nevertheless possible, using the appropriate variable change, to use Euler's theorem, which then gives:

M= kA/4[phi] +2ΩL+VQ.

with κ: surface gravity; 2Ω: velocity of the nuclear vector of particle transport proportional to the speed of light wave = f. λ where f is the frequency of the wave and λ is its length; V is and the electric potential of the black hole. L is the quantum kinetic moment and plays a fundamental role in atomic and molecular physics, in the classification of electronic terms.

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.