Correlation function (quantum field theory): Difference between revisions

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{{other uses|Correlation function (disambiguation)}}

{{Quantum field theory}}

{{short description|Expectation value of time-ordered quantum operators}}

In [[quantum field theory]], '''correlation functions''', often referred to as '''correlators''' or '''[[GreensGreen's function (many-body theory)|Green's functions]]''', are [[vacuum expectation value|vacuum expectation values]] of [[time ordering|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 [[observable|observables]] such as [[S-matrix]] elements, although they are not themselves observables. This is because they need not be [[gauge theory|gauge invariant]], nor are they [[uniqueness quantification|unique]], with different correlation functions resulting in the same S-matrix and therefore describing the same [[physics]].<ref>{{cite arXiv|last1=Manohar|first1=A.V.|date=2018|title=Introduction to Effective Field Theories|class=hep-ph |eprint=1804.05863}}</ref> They are closely related to [[correlation function]]s between [[random variable]]s, although they are nonetheless different objects, being defined in [[Minkowski space]]time and on quantum operators.

== 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|''x)''}} in spacetime, the {{math|''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">

:<math>

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|edition=9|isbn=9781107034730}}</ref>

<math display="block">

:<math>

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 <blockquote>(sum over all bubble diagrams)<math>\times</math>(sum of all diagrams with no bubbles). </blockquote>The first term then cancels with the normalization factor in the denominator meaning that the {{math|''n''}}-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles

<math display="block">

:<math>

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">

:<math>

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]}}.

:<math>

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)}\biggright|_{J=0}.

:<math>

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>{{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)}\biggright|_{J=0}.

:<math>

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>

== 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 = \biggleft[i \int d^4 x_1 e^{-ip_1 x_1} \left(\partial^2_{x_1} + m^2\right)\biggright]\cdots \biggleft[i \int d^4 x_n e^{ip_n x_n} \left(\partial_{x_n}^2 + m^2\right)\biggright] \langle \Omega |T\{\phi(x_1)\dots \phi(x_n)\}|\Omega\rangle.

:<math>

\langle f|S|i\rangle = \bigg[i \int d^4 x_1 e^{-ip_1 x_1}(\partial^2_{x_1}+m^2)\bigg]\cdots \bigg[i \int d^4 x_n e^{ip_n x_n}(\partial_{x_n}^2 +m^2)\bigg]\langle \Omega |T\{\phi(x_1)\dots \phi(x_n)\}|\Omega\rangle.

</math>

Here the particles in the initial state <math>|i\rangle</math> have a <math>-i</math> sign in the exponential, while the particles in the final state <math>|f\rangle</math> have a <math>+i</math>. 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 <math>x_i</math> and the other at some internal vertex <math>x</math>. The significance of this formula becomes clear after the application of the [[Klein–Gordon equation|Klein–Gordon]] operators to these external legs using

<math display="block">

\left(\partial^2_{x_i} + m^2\right)\Delta_F(x_i,x) = -i\delta^4(x_i-x).

:<math>

(\partial^2_{x_i}+m^2)\Delta_F(x_i,x) = -i\delta^4(x_i-x).

</math>

This is said to amputate the diagrams by removing the external leg propagators and putting the external states [[on shell and off shell|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 transform|Fourier transformation]] operation where the integration is over the internal point positions <math>x</math> 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''' <math>\tilde G(q_1, \dots, q_n)</math>, defined through the Fourier transformation of the correlation function<ref>{{cite book|last=Năstase|first=H.|author-link=Horațiu Năstase|date=2019|title=Introduction to Quantum Field Theory|url=|doi=|___location=|publisher=Cambridge University Press|chapter=9|page=79|isbn=978-1108493994}}</ref>

<math display="block">

(2\pi)^4 \delta^{(4)}(q_1+\cdots + q_n) \tilde G_n(q_1, \dots, q_n) = \int d^4x_14 x_1 \dots d^4 x_n \biggleft(\prod^n_{i=1} e^{-iq_ix_ii q_i x_i}\biggright) G_n(x_1, \dots, x_n),

:<math>

(2\pi)^4 \delta^{(4)}(q_1+\cdots + q_n)\tilde G_n(q_1, \dots, q_n) = \int d^4x_1 \dots d^4 x_n \bigg(\prod^n_{i=1}e^{-iq_ix_i}\bigg)G_n(x_1, \dots, x_n),

</math>

where by convention the momenta are directed inwards into the diagram. A useful quantity to calculate when calculating scattering amplitudes is the matrix element <math>\mathcal M</math> which is defined from the S-matrix via

:<math display="block">\langle f| S - 1 |i\rangle = i(2\pi)^4 \delta^4{\Bigbigg(\sum_i p_i\Bigbigg)} \mathcal M</math>

where <math>p_i</math> 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<ref>{{cite book|last1=Mandl|first1=F.| last2=Shaw|first2=G.| date=2010| title=Quantum Field Theory|publisher=John Wiley & Sons| chapter=12|edition=2| page=254| isbn=9780471496847}}</ref>

:<math>\langle f| S - 1|i\rangle = i(2\pi)^4 \delta^4\Big(\sum_i p_i\Big) \mathcal M</math>

<math display="block">

where <math>p_i</math> 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<ref>{{cite book|last1=Mandl|first1=F.|last2=Shaw|first2=G.|date=2010|title=Quantum Field Theory|publisher=John Wiley & Sons|chapter=12|edition=2|page=254|isbn=9780471496847}}</ref>

:<math>

i \mathcal M = \tilde G_n^c(p_1, \dots, -p_n)_{\text{amputated}}.

</math>

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 theorem]] because scattering processes that occur at large separations do not interfere with each other so can be treated separately.<ref>{{cite book|first=S.|last=Weinberg| author1-link=Steven Weinberg|title=The Quantum Theory of Fields: Volume 1Foundations| publisher=Cambridge University Press| date=1995| chapter=166|volume=41| page=270| isbn=9780521670531}}</ref>

==See also==

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* [[Green's function (many-body theory)]]

* [[Partition function (mathematics)]]

* [[Source field]]

== Notes ==

{{reflist|group=note}}

==References==

Line 108 ⟶ 98:

==Further reading==

* Altland, A.; Simons, B. (2006). ''Condensed Matter Field Theory''. [[Cambridge University Press]].

* SchroederPeskin, D.VM.; PeskinSchroeder, MD.,V. (2018) ''An Introduction to Quantum Field Theory''. [[Addison-Wesley]].

{{DEFAULTSORT:Correlation Function (Quantum Field Theory)}}