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{{short description|Expectation value of time-ordered quantum operators}}
In [[quantum field theory]], '''correlation functions''', often referred to as '''correlators''' or '''[[
== Definition ==
For a [[scalar field theory]] with a single field <math>\phi(x)</math> and a [[quantum vacuum state|vacuum state]] <math>|\Omega\rangle</math> at every event
<math display="block">
G_n(x_1,\dots, x_n) = \langle \Omega|T\{\mathcal \phi(x_1)\dots \mathcal \phi(x_n)\}|\Omega\rangle.
</math>
Here <math>T\{\cdots \}</math> is the [[Path-ordering#Time_ordering|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<ref>{{cite book|first=M.D.|last=Schwartz|title=Quantum Field Theory and the Standard Model| publisher=Cambridge University Press| chapter=7|isbn=9781107034730}}</ref>
<math display="block">
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^{i S[\phi]}|0\rangle},
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}}
The series of diagrams arising from <math>\langle 0|e^{iS[\phi]}|0\rangle</math> is the set of all [[Feynman diagram#Vacuum bubbles|vacuum bubble]] diagrams, which are diagrams with no external legs. Meanwhile, <math>\langle 0|\phi(x_1)\dots \phi(x_n)e^{iS[\phi]}|0\rangle</math> is given by the set of all possible diagrams with exactly
<math display="block">
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}}.
</math>
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 {{math|''n''}}-point correlation functions'''
<math display="block">
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}}
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</math>
Similarly, connected correlation functions can be generated using <math>W[J] = -i \ln Z[J]</math>{{refn|group=note|The <math>-i</math> factor in the definition of <math>W[J]</math> is a matter of convention, with the sum of all connected Feynman diagrams instead given by <math>W'[J]=iW[J]</math>.}} as
<math display="block">
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}.
</math>
== 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]]
<math display="block">
\langle f|S|i\rangle = \left[i \int d^4 x_1 e^{-ip_1 x_1} \left(\partial^2_{x_1} + 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.
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* [[Green's function (many-body theory)]]
* [[Partition function (mathematics)]]
* [[Source field]]
== Notes ==
{{reflist|group=note}}
==References==
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