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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 (x) in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of <math>n</math> field operators in the [[Heisenberg picture]]
<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 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|edition=9|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},
</math>
where <math>|0\rangle</math> is the ground state of the free theory and <math>S[\phi]</math> is the [[action (physics)|action]]. Expanding <math>e^{iS[\phi]}</math> 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 diagram|Feynman diagrams]], where each term can be evaluated using the position space Feynman rules.
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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>n</math> external legs. Since this also includes disconnected diagrams with vacuum bubbles, the sum factorizes into (sum over all bubble diagrams)<math>\times</math>(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
<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 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}}
</math>
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In the [[path integral formulation]], n-point correlation functions are written as a functional average
<math display="block">
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]}}.▼
▲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]}}.
</math>
They can be evaluated using the [[partition function (quantum field theory)|partition functional]] <math>Z[J]</math> which acts as a [[generating function|generating functional]], with <math>J</math> being a source-term, for the correlation functions
<math display="block">
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)}\
▲G_n(x_1, \dots, x_n) = (-i)^n \frac{1}{Z[J]}\frac{\delta^n Z[J]}{\delta J(x_1) \dots \delta J(x_n)}\bigg|_{J=0}.
</math>
Similarly, connected correlation functions can be generated using <math>W[J] = -i \ln Z[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)}\
▲G_n^c(x_1, \dots, x_n) = (-i)^{n-1}\frac{\delta^n W[J]}{\delta J(x_1) \dots \delta J(x_n)}\bigg|_{J=0}.
</math>
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