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Correlation function (quantum field theory)

Summary

In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements.

Definition

For a scalar field theory with a single field ${\displaystyle \phi (x)}$  and a vacuum state ${\displaystyle |\Omega \rangle }$  at every event (x) in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of ${\displaystyle n}$  field operators in the Heisenberg picture

${\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle \Omega |T\{{\mathcal {\phi }}(x_{1})\dots {\mathcal {\phi }}(x_{n})\}|\Omega \rangle .}$

Here ${\displaystyle T\{\cdots \}}$  is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as[1]

${\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle }{\langle 0|e^{iS[\phi ]}|0\rangle }},}$

where ${\displaystyle |0\rangle }$  is the ground state of the free theory and ${\displaystyle S[\phi ]}$  is the action. Expanding ${\displaystyle e^{iS[\phi ]}}$  using its Taylor series, the n-point correlation function becomes a sum of interaction picture correlation functions which can be evaluated using Wick's theorem. A diagrammatic way to represent the resulting sum is via Feynman diagrams, where each term can be evaluated using the position space Feynman rules.

A connected Feynman diagram which contributes to the connected six-point correlation function.

A disconnected Feynman diagram which does not contribute to the connected six-point correlation function.

The series of diagrams arising from ${\displaystyle \langle 0|e^{iS[\phi ]}|0\rangle }$  is the set of all vacuum bubble diagrams, which are diagrams with no external legs. Meanwhile, ${\displaystyle \langle 0|\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}|0\rangle }$  is given by the set of all possible diagrams with exactly ${\displaystyle n}$  external legs. Since this also includes disconnected diagrams with vacuum bubbles, the sum factorizes into (sum over all bubble diagrams)${\displaystyle \times }$ (sum of all diagrams with no bubbles). The first term then cancels with the normalization factor in the denominator meaning that the n-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles

${\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{no bubbles}}.}$

While not including any vacuum bubbles, the sum does include disconnected diagrams, which are diagrams where at least one external leg is not connected to all other external legs through some connected path. Excluding these disconnected diagrams instead defines connected n-point correlation functions

${\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{connected, no bubbles}}}$

It is often preferable to work directly with these as they contain all the information that the full correlation functions contain since any disconnected diagram is merely a product of connected diagrams. By excluding other sets of diagrams one can define other correlation functions such as one-particle irreducible correlation functions.

In the path integral formulation, n-point correlation functions are written as a functional average

${\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\int {\mathcal {D}}\phi \ \phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}}{\int {\mathcal {D}}\phi \ e^{iS[\phi ]}}}.}$

They can be evaluated using the partition functional ${\displaystyle Z[J]}$  which acts as a generating functional, with ${\displaystyle J}$  being a source-term, for the correlation functions

${\displaystyle G_{n}(x_{1},\dots ,x_{n})=(-i)^{n}{\frac {1}{Z[J]}}\left.{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}$

Similarly, connected correlation functions can be generated using ${\displaystyle W[J]=-i\ln Z[J]}$  as

${\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=(-i)^{n-1}\left.{\frac {\delta ^{n}W[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}$

Relation to the S-matrix

Scattering amplitudes can be calculated using correlation functions by relating them to the S-matrix through the LSZ reduction formula

${\displaystyle \langle f|S|i\rangle =\left[i\int d^{4}x_{1}e^{-ip_{1}x_{1}}\left(\partial _{x_{1}}^{2}+m^{2}\right)\right]\cdots \left[i\int d^{4}x_{n}e^{ip_{n}x_{n}}\left(\partial _{x_{n}}^{2}+m^{2}\right)\right]\langle \Omega |T\{\phi (x_{1})\dots \phi (x_{n})\}|\Omega \rangle .}$

Here the particles in the initial state ${\displaystyle |i\rangle }$  have a ${\displaystyle -i}$  sign in the exponential, while the particles in the final state ${\displaystyle |f\rangle }$  have a ${\displaystyle +i}$ . All terms in the Feynman diagram expansion of the correlation function will have one propagator for each external leg, that is a propagators with one end at ${\displaystyle x_{i}}$  and the other at some internal vertex ${\displaystyle x}$ . The significance of this formula becomes clear after the application of the Klein–Gordon operators to these external legs using

${\displaystyle \left(\partial _{x_{i}}^{2}+m^{2}\right)\Delta _{F}(x_{i},x)=-i\delta ^{4}(x_{i}-x).}$

This is said to amputate the diagrams by removing the external leg propagators and putting the external states on-shell. All other off-shell contributions from the correlation function vanish. After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a Fourier transformation operation where the integration is over the internal point positions ${\displaystyle x}$  that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the amputated correlation functions with on-shell external states.

It is common to directly deal with the momentum space correlation function ${\displaystyle {\tilde {G}}(q_{1},\dots ,q_{n})}$ , defined through the Fourier transformation of the correlation function[2]

${\displaystyle (2\pi )^{4}\delta ^{(4)}(q_{1}+\cdots +q_{n}){\tilde {G}}_{n}(q_{1},\dots ,q_{n})=\int d^{4}x_{1}\dots d^{4}x_{n}\left(\prod _{i=1}^{n}e^{-iq_{i}x_{i}}\right)G_{n}(x_{1},\dots ,x_{n}),}$

where by convention the momenta are directed inwards into the diagram. A useful quantity to calculate when calculating scattering amplitudes is the matrix element ${\displaystyle {\mathcal {M}}}$  which is defined from the S-matrix via
${\displaystyle \langle f|S-1|i\rangle =i(2\pi )^{4}\delta ^{4}{{\bigg (}\sum _{i}p_{i}{\bigg )}}{\mathcal {M}}}$

where ${\displaystyle p_{i}}$  are the external momenta. From the LSZ reduction formula it then follows that the matrix element is equivalent to the amputated connected momentum space correlation function with properly orientated external momenta[3]
${\displaystyle i{\mathcal {M}}={\tilde {G}}_{n}^{c}(p_{1},\dots ,-p_{n})_{\text{amputated}}.}$

For non-scalar theories the reduction formula also introduces external state terms such as polarization vectors for photons or spinor states for fermions. The requirement of using the connected correlation functions arises from the cluster decomposition because scattering processes that occur at large separations do not interfere with each other so can be treated separately.[4]