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Source field

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In <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a>, a source is an abstract concept, developed by <a href="Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>, motivated by the physical effects of surrounding particles involved in creating or destroying another <a href="Particle_physics" title="Particle physics">particle</a>.<a href="#cite_note-:7-1">[1]</a> So, one can perceive sources as the origin of the physical properties carried by the created or destroyed particle, and thus one can use this concept to study all quantum processes including the spacetime localized properties and the energy forms, i.e., mass and momentum, of the phenomena. The <a href="Probability_amplitude" title="Probability amplitude">probability amplitude</a> of the created or the decaying particle is defined by the effect of the source on a localized spacetime region such that the affected particle captures its physics depending on the <a href="Tensor_field" title="Tensor field">tensorial</a><a href="#cite_note-2">[2]</a> and <a href="Spin_representation" title="Spin representation">spinorial</a><a href="#cite_note-3">[3]</a> nature of the source. An example that Julian Schwinger referred to is the creation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta ^{*}}</annotation>

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</math><img src="./235cbf4e713cd159112727685db969e2fdf249cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.228ex; height:2.843ex;" alt="{\displaystyle \eta ^{*}}" loading="lazy"> meson due to the mass correlations among five <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> mesons.<a href="#cite_note-4">[4]</a>

Same idea can be used to define source fields. Mathematically, a source field is a background field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

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</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> coupled to the original field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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<mtext>source</mtext>

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<annotation encoding="application/x-tex">{\displaystyle S_{\text{source}}=J\phi .}</annotation>

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This term appears in the action in <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a>'s <a href="Path_integral_formulation" title="Path integral formulation">path integral formulation</a> and responsible for the theory interactions. In a collision reaction a source could be other particles in the collision.<a href="#cite_note-:0-5">[5]</a> Therefore, the source appears in the vacuum amplitude acting from both sides on the <a href="Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">Green's function correlator</a> of the theory.<a href="#cite_note-:7-1">[1]</a>

<a href="Quantum_field_theory#Source_theory" title="Quantum field theory">Schwinger's source theory</a> stems from <a href="Schwinger's_quantum_action_principle" title="Schwinger's quantum action principle">Schwinger's quantum action principle</a> and can be related to the path integral formulation as the variation with respect to the source per se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta J}">

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<annotation encoding="application/x-tex">{\displaystyle \delta J}</annotation>

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</math><img src="./5f5e94fa1378844785f31ae2b1fc681ace9d2728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.52ex; height:2.343ex;" alt="{\displaystyle \delta J}" loading="lazy"> corresponds to the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \delta J=\int {\mathcal {D}}\phi \,\exp \left(-i\!\int \!d^{4}x\,J(x,t)\phi (x,t)\right).}</annotation>

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Also, a source acts <a href="Effective_action" title="Effective action">effectively</a><a href="#cite_note-:2-7">[7]</a> in a region of the spacetime. As one sees in the examples below, the source field appears on the right-hand side of the equations of motion (usually second-order <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>) for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> is the <a href="Electromagnetic_potential" class="mw-redirect" title="Electromagnetic potential">electromagnetic potential</a> or the <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, the source field is the <a href="Electric_current" title="Electric current">electric current</a> or the <a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>, respectively.<a href="#cite_note-:3-8">[8]</a><a href="#cite_note-9">[9]</a>

In terms of the statistical and non-relativistic applications, Schwinger's source formulation plays crucial rules in understanding many non-equilibrium systems.<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a> Source theory is theoretically significant as it needs neither divergence regularizations nor renormalization.<a href="#cite_note-:0-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Relation_between_path_integral_formulation_and_source_formulation">Relation between path integral formulation and source formulation</h2></div>

In the Feynman's path integral formulation with normalization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}\equiv Z[J=0]}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}\equiv Z[J=0]}</annotation>

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</math><img src="./2658cc961c4014392e41b538b34a00bf5804e192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:14.141ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}\equiv Z[J=0]}" loading="lazy">, the <a href="Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">partition function</a><a href="#cite_note-12">[12]</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle Z[J]={\mathcal {N}}\int {\mathcal {D}}\phi \,\exp \left[-i\left(\int dt~{\mathcal {L}}(t;\phi ,{\dot {\phi }})+\int d^{4}x\,J(x,t)\phi (x,t)\right)\right].}</annotation>

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One can expand the current term in the exponent <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}\int {\mathcal {D}}\phi ~\exp \left(-i\int d^{4}x\,J(x,t)\phi (x,t)\right)={\mathcal {N}}\sum _{n=0}^{\infty }{\frac {i^{n}}{n!}}\int d^{4}x_{1}\cdots \int d^{4}x_{n}J(x_{1})\cdots J(x_{1})\left\langle \phi (x_{1})\cdots \phi (x_{n})\right\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}\int {\mathcal {D}}\phi ~\exp \left(-i\int d^{4}x\,J(x,t)\phi (x,t)\right)={\mathcal {N}}\sum _{n=0}^{\infty }{\frac {i^{n}}{n!}}\int d^{4}x_{1}\cdots \int d^{4}x_{n}J(x_{1})\cdots J(x_{1})\left\langle \phi (x_{1})\cdots \phi (x_{n})\right\rangle }</annotation>

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to generate <a href="Propagator" title="Propagator">Green's functions</a> (<a href="Correlation_function_(quantum_field_theory)" title="Correlation function (quantum field theory)">correlators</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t_{1},\cdots ,t_{n})={\left(-i\right)}^{n}\left.{\frac {\delta ^{n}Z[J]}{\delta J(t_{1})\cdots \delta J(t_{n})}}\right|_{J=0},}">

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<annotation encoding="application/x-tex">{\displaystyle G(t_{1},\cdots ,t_{n})={\left(-i\right)}^{n}\left.{\frac {\delta ^{n}Z[J]}{\delta J(t_{1})\cdots \delta J(t_{n})}}\right|_{J=0},}</annotation>

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</math> where the fields inside the expectation function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi (x_{1})\cdots \phi (x_{n})\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle \phi (x_{1})\cdots \phi (x_{n})\rangle }</annotation>

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</math><img src="./f1140d91541cdbeefba7b5ed739d745ae5fb5477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.628ex; height:2.843ex;" alt="{\displaystyle \langle \phi (x_{1})\cdots \phi (x_{n})\rangle }" loading="lazy"> are in their <a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg pictures</a>. On the other hand, one can define the correlation functions for higher order terms, e.g., for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}m^{2}\phi ^{2}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}m^{2}\phi ^{2}}</annotation>

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</math><img src="./b116af39081722eabc7d284105006dba892bfadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.192ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}m^{2}\phi ^{2}}" loading="lazy"> term, the coupling constant like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is promoted to a spacetime-dependent source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (x)}">

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</math><img src="./f339251a09ebf15dd50bb751d27b02820f68c545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \mu (x)}" loading="lazy"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\frac {1}{\mathcal {N}}}\left.{\frac {\delta }{\delta \mu ^{2}}}Z[J,\mu ]\right|_{m^{2}=\mu ^{2}}=\left\langle {\tfrac {1}{2}}\phi ^{2}\right\rangle .}">

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<annotation encoding="application/x-tex">{\displaystyle i{\frac {1}{\mathcal {N}}}\left.{\frac {\delta }{\delta \mu ^{2}}}Z[J,\mu ]\right|_{m^{2}=\mu ^{2}}=\left\langle {\tfrac {1}{2}}\phi ^{2}\right\rangle .}</annotation>

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One implements the quantum variational methodology to realize that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is an external driving source of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle Z[J]}</annotation>

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</math><img src="./82f5d3b05047c46140ec4c32564aac5465f34692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.445ex; height:2.843ex;" alt="{\displaystyle Z[J]}" loading="lazy"> can be seen as the expectation value of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{J\phi }}">

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<annotation encoding="application/x-tex">{\displaystyle e^{J\phi }}</annotation>

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</math><img src="./51fd0090316bc8f6cd7f17e8f4074b524a2a6c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.336ex; height:2.676ex;" alt="{\displaystyle e^{J\phi }}" loading="lazy">. This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=E{\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{\sqrt {2E}}}\left(J{\hat {a}}^{\dagger }+J^{*}{\hat {a}}\right)}</annotation>

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</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}=m^{2}+\mathbf {p} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle E^{2}=m^{2}+\mathbf {p} ^{2}}</annotation>

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In fact, the current is real, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=J^{*}}">

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</math><img src="./39a2f1a9e9040a9be5c11269b7052fac80d9d33e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.15ex; height:2.343ex;" alt="{\displaystyle J=J^{*}}" loading="lazy">.<a href="#cite_note-13">[13]</a> And the Lagrangian is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=i{\hat {a}}^{\dagger }\partial _{0}({\hat {a}})-{\mathcal {H}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=i{\hat {a}}^{\dagger }\partial _{0}({\hat {a}})-{\mathcal {H}}}</annotation>

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</math><img src="./91ffb59d81350e5e681f006c157a01bd790d3ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.829ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}=i{\hat {a}}^{\dagger }\partial _{0}({\hat {a}})-{\mathcal {H}}}" loading="lazy"> . From now on we drop the hat and the asterisk. Remember that <a href="Canonical_quantization#Real_scalar_field" title="Canonical quantization">canonical quantization</a> states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \sim (a^{\dagger }+a)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi \sim (a^{\dagger }+a)}</annotation>

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</math><img src="./78db0d1503380518e93bc7fe33e581734207395b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.555ex; height:3.176ex;" alt="{\displaystyle \phi \sim (a^{\dagger }+a)}" loading="lazy">. In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives

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<annotation encoding="application/x-tex">{\displaystyle \delta _{J}\langle 0,x'_{0}|0,x''_{0}\rangle _{J}=i\left\langle 0,x'_{0}\right|\int _{x''_{0}}^{x'_{0}}dx_{0}~\delta J{\left(a^{\dagger }+a\right)}{\left|0,x''_{0}\right\rangle }_{J},}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}'>x_{0}>x_{0}''}</annotation>

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As the integral is in the time domain, one can Fourier transform it, together with the creation/annihilation operators, such that the amplitude eventually becomes<a href="#cite_note-:1-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\left\langle 0,x'_{0}|0,x''_{0}\right\rangle }_{J}=\exp {\left({\frac {i}{2\pi }}\int df~J(f){\frac {1}{f-E}}J(-f)\right)}.}</annotation>

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It is easy to notice that there is a singularity at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=E}">

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</math><img src="./76b863e668b0c72831f559d33243e488437e9fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.621ex; height:2.509ex;" alt="{\displaystyle f-E+i\varepsilon }" loading="lazy"> such that for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}>x_{0}'}">

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</math><img src="./fc60b1892c8bd2d51092bb7c40abc83ad2eedc81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.866ex; height:2.843ex;" alt="{\displaystyle x_{0}>x_{0}'}" loading="lazy"> the Green's function is revealed

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\left\langle 0|0\right\rangle }_{J}=\exp {\left({\frac {i}{2}}\int dx_{0}\,dx'_{0}\,J(x_{0})\Delta (x_{0}-x'_{0})J(x'_{0})\right)}\\[1ex]&\Delta (x_{0}-x'_{0})=\int {\frac {df}{2\pi }}{\frac {e^{-if\left(x_{0}-x'_{0}\right)}}{f-E+i\varepsilon }}\end{aligned}}}</annotation>

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The last result is the Schwinger's source theory for interacting scalar fields and can be generalized to any spacetime regions.<a href="#cite_note-:2-7">[7]</a> The discussed examples below follow the metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }={\text{diag}}(1,-1,-1,-1)}">

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<div class="mw-heading mw-heading2"><h2 id="Source_theory_for_scalar_fields">Source theory for scalar fields</h2></div>

<a href="Causal_perturbation_theory" title="Causal perturbation theory">Causal perturbation theory</a> explains how sources weakly act. For a weak source emitting spin-0 particles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{e}}">

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</math><img src="./c85589239dbfeece1b7f2a7784ae8db73a8ec1f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.289ex; height:2.509ex;" alt="{\displaystyle J_{e}}" loading="lazy"> by acting on the <a href="Quantum_vacuum_state" title="Quantum vacuum state">vacuum state</a> with a probability amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|0\rangle _{J_{e}}\sim 1}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{J_{e}}\sim 1}</annotation>

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</math><img src="./2372be07a09604da82ac19e25342ea76c6b2703c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.973ex; height:2.843ex;" alt="{\displaystyle \langle 0|0\rangle _{J_{e}}\sim 1}" loading="lazy">, a single particle with momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle p|0\rangle _{J_{e}}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle p|0\rangle _{J_{e}}}</annotation>

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</math><img src="./cd89381c115d86bca622348fb685140494598209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.719ex; height:2.843ex;" alt="{\displaystyle \langle p|0\rangle _{J_{e}}}" loading="lazy"> is created within certain spacetime region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">

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</math><img src="./0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy">. Then, another weak source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{a}}">

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</math><img src="./6247391885abd321eb94cbe9f0aa17858af7d530.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.392ex; height:2.509ex;" alt="{\displaystyle J_{a}}" loading="lazy"> absorbs that single particle within another spacetime region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> such that the amplitude becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|p\rangle _{J_{a}}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|p\rangle _{J_{a}}}</annotation>

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</math><img src="./005798ad55c4e953ff5341624ef63743984d698f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.803ex; height:2.843ex;" alt="{\displaystyle \langle 0|p\rangle _{J_{a}}}" loading="lazy">.<a href="#cite_note-:0-5">[5]</a> Thus, the full vacuum amplitude is given by

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<annotation encoding="application/x-tex">{\displaystyle {\left\langle 0|0\right\rangle }_{J_{e}+J_{a}}\sim 1+{\frac {i}{2}}\int dx\,dx'\,J_{a}(x)\Delta (x-x')J_{e}(x')}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x-x')}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta (x-x')}</annotation>

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</math><img src="./5b0e16ca756dd4390a65bc1a73afd870ee9a465c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.93ex; height:3.009ex;" alt="{\displaystyle \Delta (x-x')}" loading="lazy"> is the propagator (correlator) of the sources. The second term of the last amplitude defines the <a href="Partition_function_(quantum_field_theory)#Free_theories" title="Partition function (quantum field theory)">partition function of free scalar field theory</a>. And for some interaction theory, the Lagrangian of a scalar field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

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</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> is given by<a href="#cite_note-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\tfrac {1}{2}}m^{2}\phi ^{2}+J\phi .}</annotation>

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If one adds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\varepsilon }">

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</math><img src="./a6aed6bc7f333f25fe37a9bc56ffe31ee2c3299c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.694ex; height:2.343ex;" alt="{\displaystyle -i\varepsilon }" loading="lazy"> to the mass term then Fourier transforms both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle =\exp {\left({\frac {i}{2}}\int {\frac {d^{4}p}{{\left(2\pi \right)}^{4}}}\left[{\tilde {\phi }}(p)\left(p_{\mu }p^{\mu }-m^{2}+i\varepsilon \right){\tilde {\phi }}(-p)+J(p){\frac {1}{p_{\mu }p^{\mu }-m^{2}+i\varepsilon }}J(-p)\right]\right)},}</annotation>

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where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\phi }}(p)=\phi (p)+{\frac {J(p)}{p_{\mu }p^{\mu }-m^{2}+i\varepsilon }}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\phi }}(p)=\phi (p)+{\frac {J(p)}{p_{\mu }p^{\mu }-m^{2}+i\varepsilon }}.}</annotation>

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</math> It is easy to notice that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\phi }}(p)\left(p_{\mu }p^{\mu }-m^{2}+i\varepsilon \right){\tilde {\phi }}(-p)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\phi }}(p)\left(p_{\mu }p^{\mu }-m^{2}+i\varepsilon \right){\tilde {\phi }}(-p)}</annotation>

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</math><img src="./b5c745a14742163ece76ae66f84694705cb74cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.051ex; height:3.343ex;" alt="{\displaystyle {\tilde {\phi }}(p)\left(p_{\mu }p^{\mu }-m^{2}+i\varepsilon \right){\tilde {\phi }}(-p)}" loading="lazy"> term in the amplitude above can be Fourier transformed into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\phi }}(x)\left(\Box +m^{2}\right){\tilde {\phi }}(x)={\tilde {\phi }}(x)\,J(x)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\phi }}(x)\left(\Box +m^{2}\right){\tilde {\phi }}(x)={\tilde {\phi }}(x)\,J(x)}</annotation>

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</math><img src="./65315566bcb9a0b3a108338c585517ae50731a6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.562ex; height:3.343ex;" alt="{\displaystyle {\tilde {\phi }}(x)\left(\Box +m^{2}\right){\tilde {\phi }}(x)={\tilde {\phi }}(x)\,J(x)}" loading="lazy">, i.e., the equation of motion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\Box +m^{2}\right){\tilde {\phi }}=J}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\Box +m^{2}\right){\tilde {\phi }}=J}</annotation>

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</math><img src="./dea5df928779c634a99ae2b4abe5b8068a0a485f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.297ex; height:3.343ex;" alt="{\displaystyle \left(\Box +m^{2}\right){\tilde {\phi }}=J}" loading="lazy">. As the variation of the free action, that of the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}}">

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</math><img src="./bfd7e29c085907109c26cebb2e2eec374202a3f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.486ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}}" loading="lazy">, yields the equation of motion, one can redefine the Green's function as the inverse of the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle G(x_{1},x_{2})\equiv {\left(\Box +m^{2}\right)}^{-1}}">

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<annotation encoding="application/x-tex">{\textstyle G(x_{1},x_{2})\equiv {\left(\Box +m^{2}\right)}^{-1}}</annotation>

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</math><img src="./134d11092a9dc7cbd0cea6f6382a64e6550e9e89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.742ex; height:3.676ex;" alt="{\textstyle G(x_{1},x_{2})\equiv {\left(\Box +m^{2}\right)}^{-1}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\Box _{x_{1}}+m^{2}\right)G(x_{1},x_{2})=\delta (x_{1}-x_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\Box _{x_{1}}+m^{2}\right)G(x_{1},x_{2})=\delta (x_{1}-x_{2})}</annotation>

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</math><img src="./1ff19a805e4949656e77987838378d7e3bec9227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.267ex; height:3.343ex;" alt="{\displaystyle \left(\Box _{x_{1}}+m^{2}\right)G(x_{1},x_{2})=\delta (x_{1}-x_{2})}" loading="lazy"> <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left(p_{\mu }p^{\mu }-m^{2}\right)G(p)=1}">

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<annotation encoding="application/x-tex">{\textstyle \left(p_{\mu }p^{\mu }-m^{2}\right)G(p)=1}</annotation>

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</math><img src="./98ee88215b6cc23e7d1adcb091787977abe7d9c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.304ex; height:3.176ex;" alt="{\textstyle \left(p_{\mu }p^{\mu }-m^{2}\right)G(p)=1}" loading="lazy">, which is a direct application of the general role of functional derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\delta J(x_{2})}{\delta J(x_{1})}}=\delta (x_{1}-x_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta J(x_{2})}{\delta J(x_{1})}}=\delta (x_{1}-x_{2})}</annotation>

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</math><img src="./b3dbd9f3034e25e93c3feeeac79ebb4a94ba0911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.114ex; height:6.509ex;" alt="{\displaystyle {\frac {\delta J(x_{2})}{\delta J(x_{1})}}=\delta (x_{1}-x_{2})}" loading="lazy">. Thus, the <a href="Partition_function_(quantum_field_theory)#Scalar_theories" title="Partition function (quantum field theory)">generating functional</a> is obtained from the partition function as follows.<a href="#cite_note-:3-8">[8]</a> The last result allows us to read the partition function as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z[J]=Z[0]\exp \left({\tfrac {i}{2}}\left\langle J(y)\Delta (y-y')J(y')\right\rangle \right)}">

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<annotation encoding="application/x-tex">{\textstyle Z[J]=Z[0]\exp \left({\tfrac {i}{2}}\left\langle J(y)\Delta (y-y')J(y')\right\rangle \right)}</annotation>

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</math><img src="./5a909b21c9db38630f00ec5820bc55d9bfdbd7b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:40.753ex; height:3.509ex;" alt="{\textstyle Z[J]=Z[0]\exp \left({\tfrac {i}{2}}\left\langle J(y)\Delta (y-y')J(y')\right\rangle \right)}" loading="lazy">, where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z[0]=\int {\mathcal {D}}{\tilde {\phi }}\,\exp \left(-i\int dt\left[{\tfrac {1}{2}}\partial _{\mu }{\tilde {\phi }}\partial ^{\mu }{\tilde {\phi }}-{\tfrac {1}{2}}\left(m^{2}-i\varepsilon \right){\tilde {\phi }}^{2}\right]\right),}">

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<annotation encoding="application/x-tex">{\displaystyle Z[0]=\int {\mathcal {D}}{\tilde {\phi }}\,\exp \left(-i\int dt\left[{\tfrac {1}{2}}\partial _{\mu }{\tilde {\phi }}\partial ^{\mu }{\tilde {\phi }}-{\tfrac {1}{2}}\left(m^{2}-i\varepsilon \right){\tilde {\phi }}^{2}\right]\right),}</annotation>

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</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle J(y)\Delta (y-y')J(y')\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle J(y)\Delta (y-y')J(y')\rangle }</annotation>

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</math><img src="./37f41d8071d375044dae93a2771017c9c623fd7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.958ex; height:3.009ex;" alt="{\displaystyle \langle J(y)\Delta (y-y')J(y')\rangle }" loading="lazy"> is the vacuum amplitude derived by the source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|0\rangle _{J}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{J}}</annotation>

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</math><img src="./381de4aef342adcf040ea13de5f70f5e27e36150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.054ex; height:2.843ex;" alt="{\displaystyle \langle 0|0\rangle _{J}}" loading="lazy">. Consequently, the propagator is defined by varying the partition function as follows.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\left.{\frac {-1}{Z[0]}}{\frac {\delta ^{2}Z[J]}{\delta J(x)\delta J(x')}}\right\vert }_{J=0}&={\frac {-1}{2Z[0]}}{\frac {\delta }{\delta J(x)}}{\left[Z[J]\left(\int d^{4}y'\,\Delta (x'-y')J(y')+\int d^{4}y\,J(y)\Delta (y-x')\right)\right]}_{J=0}\\[1.5ex]&={\left.{\frac {Z[J]}{Z[0]}}\Delta (x-x')\right\vert }_{J=0}\\[1.5ex]&=\Delta (x-x').\end{aligned}}}</annotation>

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This motivates discussing the mean field approximation below.

<div class="mw-heading mw-heading2"><h2 id="Effective_action,_mean_field_approximation,_and_vertex_functions">Effective action, mean field approximation, and vertex functions</h2></div>

Based on Schwinger's source theory, <a href="Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a> established the foundations of the effective field theory, which is widely appreciated among physicists. Despite the "<a href="Julian_Schwinger#Career" title="Julian Schwinger">shoes incident</a>", Weinberg gave the credit to Schwinger for catalyzing this theoretical framework.<a href="#cite_note-15">[15]</a>

All Green's functions may be formally found via <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> of the <a href="Partition_sum" class="mw-redirect" title="Partition sum">partition sum</a> considered as a function of the source fields. This method is commonly used in the <a href="Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. The general method by which such source fields are utilized to obtain propagators in both quantum, statistical-mechanics and other systems is outlined as follows. Upon redefining the partition function in terms of <a href="Wick_rotation" title="Wick rotation">Wick-rotated</a> amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W[J]=-i\ln(\langle 0|0\rangle _{J})}">

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<annotation encoding="application/x-tex">{\displaystyle W[J]=-i\ln(\langle 0|0\rangle _{J})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle Z[J]=e^{iW[J]}}</annotation>

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</math><img src="./dc4b0b664e9498c999d20f44d7b26b684a797fda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.104ex; height:3.343ex;" alt="{\displaystyle Z[J]=e^{iW[J]}}" loading="lazy">. One can introduce <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[J]=iW[J]}">

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<annotation encoding="application/x-tex">{\displaystyle F[J]=iW[J]}</annotation>

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</math><img src="./f86d2bf7b8cc0b272e372a776296da3ca187fc2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.607ex; height:2.843ex;" alt="{\displaystyle F[J]=iW[J]}" loading="lazy">, which behaves as <a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> in <a href="Thermal_quantum_field_theory" title="Thermal quantum field theory">thermal field theories</a>,<a href="#cite_note-:4-16">[16]</a> to absorb the complex number, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln Z[J]=F[J]}">

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<annotation encoding="application/x-tex">{\displaystyle \ln Z[J]=F[J]}</annotation>

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</math><img src="./5692e01cc7ca90056b58f047b638fb843955ca8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.376ex; height:2.843ex;" alt="{\displaystyle \ln Z[J]=F[J]}" loading="lazy">. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[J]}">

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<annotation encoding="application/x-tex">{\displaystyle F[J]}</annotation>

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</math><img src="./c65a0fdde138719f3fdf5d4c2be422b5e7791722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.506ex; height:2.843ex;" alt="{\displaystyle F[J]}" loading="lazy"> is also called reduced quantum action.<a href="#cite_note-:5-17">[17]</a> And with help of <a href="Legendre_transformation" title="Legendre transformation">Legendre transform</a>, we can invent a "new" effective energy functional,<a href="#cite_note-18">[18]</a> or <a href="Effective_action" title="Effective action">effective action</a>, as

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<annotation encoding="application/x-tex">{\displaystyle \Gamma [{\bar {\phi }}]=W[J]-\int d^{4}x\,J(x){\bar {\phi }}(x),}</annotation>

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</math> with the transforms<a href="#cite_note-19">[19]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\frac {\delta W}{\delta J}}={\bar {\phi }}~,&&{\frac {\delta W}{\delta J}}{\Bigg |}_{J=0}=\langle \phi \rangle ~,\\[1.2ex]&{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}{\Bigg |}_{J}=-J~,&&{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}{\Bigg |}_{{\bar {\phi }}=\langle \phi \rangle }=0.\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {\delta W}{\delta J}}={\bar {\phi }}~,&&{\frac {\delta W}{\delta J}}{\Bigg |}_{J=0}=\langle \phi \rangle ~,\\[1.2ex]&{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}{\Bigg |}_{J}=-J~,&&{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}{\Bigg |}_{{\bar {\phi }}=\langle \phi \rangle }=0.\end{aligned}}}</annotation>

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The integration in the definition of the effective action is allowed to be replaced with sum over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma [{\bar {\phi }}]=W[J]-J_{a}(x){\bar {\phi }}^{a}(x)}</annotation>

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</math><img src="./cac589c2b4708d4790b89d6684b99df939159537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.592ex; height:3.176ex;" alt="{\displaystyle \Gamma [{\bar {\phi }}]=W[J]-J_{a}(x){\bar {\phi }}^{a}(x)}" loading="lazy">.<a href="#cite_note-:6-20">[20]</a> The last equation resembles the thermodynamical relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=E-TS}">

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</math><img src="./795f17f6bcbb99cc55572b0b75275c3bdad9986b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.591ex; height:2.343ex;" alt="{\displaystyle F=E-TS}" loading="lazy"> between Helmholtz free energy and entropy. It is now clear that thermal and statistical field theories stem fundamentally from <a href="Functional_integration" title="Functional integration">functional integrations</a> and <a href="Functional_derivative" title="Functional derivative">functional derivatives</a>. Back to the Legendre transforms,

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi \rangle }">

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</math><img src="./2c1c908b5b57f8fe50e6c90dd43bb12a9979312f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.195ex; height:2.843ex;" alt="{\displaystyle \langle \phi \rangle }" loading="lazy"> is called <a href="Mean-field_theory" title="Mean-field theory">mean field</a> obviously because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \phi \rangle ={\frac {\int {\mathcal {D}}\phi ~e^{-i[\int dt~{\mathcal {L}}(t;\phi ,{\dot {\phi }})+\int dx^{4}J(x,t)\phi (x,t)]}~\phi ~}{Z[J]/{\mathcal {N}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \phi \rangle ={\frac {\int {\mathcal {D}}\phi ~e^{-i[\int dt~{\mathcal {L}}(t;\phi ,{\dot {\phi }})+\int dx^{4}J(x,t)\phi (x,t)]}~\phi ~}{Z[J]/{\mathcal {N}}}}}</annotation>

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</math><img src="./6d58982754a0de68a98d106e5b7ab52b5a831538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.596ex; height:7.343ex;" alt="{\displaystyle \langle \phi \rangle ={\frac {\int {\mathcal {D}}\phi ~e^{-i[\int dt~{\mathcal {L}}(t;\phi ,{\dot {\phi }})+\int dx^{4}J(x,t)\phi (x,t)]}~\phi ~}{Z[J]/{\mathcal {N}}}}}" loading="lazy">, while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\phi }}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{i\Gamma [{\bar {\phi }}]}={\mathcal {N}}\int \exp \left[i\left(S[\phi ]-{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}\eta \right)\right]d\phi ,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \langle {\mathcal {F}}[\phi ]\rangle =e^{-i\Gamma [{\bar {\phi }}]}~{\mathcal {N}}\int {\mathcal {F}}[\phi ]\exp \left[i\left(S[\phi ]-{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}\eta \right)\right]d\phi ,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle S[\phi ]}</annotation>

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</math><img src="./e8f21c5301f7d31b5b5862c5691f37eb6938b665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.178ex; height:2.843ex;" alt="{\displaystyle S[\phi ]}" loading="lazy"> is the action of the free Lagrangian. The last two integrals are the pillars of any effective field theory.<a href="#cite_note-:6-20">[20]</a> This construction is indispensable in studying scattering (<a href="LSZ_reduction_formula" title="LSZ reduction formula">LSZ reduction formula</a>), <a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a>,<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> <a href="Ward_identities" class="mw-redirect" title="Ward identities">Ward identities</a>, <a href="Non-linear_sigma_model" title="Non-linear sigma model">nonlinear sigma models</a>, and <a href="Effective_field_theory#Effective_field_theories_in_gravity" title="Effective field theory">low-energy effective theories</a>.<a href="#cite_note-:4-16">[16]</a> Additionally, this theoretical framework initiates line of thoughts, publicized mainly be <a href="Bryce_DeWitt" title="Bryce DeWitt">Bryce DeWitt</a> who was a PhD student of Schwinger, on developing a <a href="Canonical_quantum_gravity" title="Canonical quantum gravity">canonical quantized</a> effective theory for quantum gravity.<a href="#cite_note-23">[23]</a>

Back to Green functions of the actions. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma [{\bar {\phi }}]}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma [{\bar {\phi }}]}</annotation>

</semantics>

</math><img src="./c3edd19858346d860cc67cda8df126c3b0f3759e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.214ex; height:3.009ex;" alt="{\displaystyle \Gamma [{\bar {\phi }}]}" loading="lazy"> is the Legendre transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[J]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F[J]}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F[J]}</annotation>

</semantics>

</math><img src="./c65a0fdde138719f3fdf5d4c2be422b5e7791722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.506ex; height:2.843ex;" alt="{\displaystyle F[J]}" loading="lazy"> defines N-points <a href="Ursell_function" title="Ursell function">connected</a> correlator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{F[J]}^{N,~c}={\frac {\delta F[J]}{\delta J(x_{1})\cdots \delta J(x_{N})}}{\Big |}_{J=0}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{F[J]}^{N,~c}={\frac {\delta F[J]}{\delta J(x_{1})\cdots \delta J(x_{N})}}{\Big |}_{J=0}}</annotation>

</semantics>

</math><img src="./fbddf4f5418b41aa282f85817accafce846347f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.784ex; height:6.509ex;" alt="{\displaystyle G_{F[J]}^{N,~c}={\frac {\delta F[J]}{\delta J(x_{1})\cdots \delta J(x_{N})}}{\Big |}_{J=0}}" loading="lazy">, then the corresponding correlator obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[J]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F[J]}</annotation>

</semantics>

</math><img src="./c65a0fdde138719f3fdf5d4c2be422b5e7791722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.506ex; height:2.843ex;" alt="{\displaystyle F[J]}" loading="lazy">, known as <a href="Vertex_function" title="Vertex function">vertex function</a>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\Gamma [J]}^{N,~c}=\left.{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}(x_{1})\cdots \delta {\bar {\phi }}(x_{N})}}\right|_{{\bar {\phi }}=\langle \phi \rangle }}">

</mrow>

</mrow>

</msubsup>

<msub>

<mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

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<msub>

</mrow>

</msub>

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</mrow>

<mover>

</mover>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\Gamma [J]}^{N,~c}=\left.{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}(x_{1})\cdots \delta {\bar {\phi }}(x_{N})}}\right|_{{\bar {\phi }}=\langle \phi \rangle }}</annotation>

</semantics>

</math><img src="./402017e5c910e46d004f01493be9ce8dbb35f111.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.267ex; height:7.343ex;" alt="{\displaystyle G_{\Gamma [J]}^{N,~c}=\left.{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}(x_{1})\cdots \delta {\bar {\phi }}(x_{N})}}\right|_{{\bar {\phi }}=\langle \phi \rangle }}" loading="lazy">. Consequently in the one particle irreducible graphs (usually acronymized as 1PI), the connected 2-point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">-correlator is defined as the inverse of the 2-point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">-correlator, i.e., the usual reduced correlation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{F[J]}^{(2)}={\frac {\delta {\bar {\phi }}(x_{1})}{\delta J(x_{2})}}{\Big |}_{J=0}={\frac {1}{p_{\mu }p^{\mu }-m^{2}}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

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</msub>

</mrow>

</mfrac>

</mrow>

<msub>

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</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{F[J]}^{(2)}={\frac {\delta {\bar {\phi }}(x_{1})}{\delta J(x_{2})}}{\Big |}_{J=0}={\frac {1}{p_{\mu }p^{\mu }-m^{2}}}}</annotation>

</semantics>

</math><img src="./bc229336a7dcec80f4492feeab61e63ceac8d45b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.568ex; height:6.843ex;" alt="{\displaystyle G_{F[J]}^{(2)}={\frac {\delta {\bar {\phi }}(x_{1})}{\delta J(x_{2})}}{\Big |}_{J=0}={\frac {1}{p_{\mu }p^{\mu }-m^{2}}}}" loading="lazy">, and the effective correlation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\Gamma [\phi ]}^{(2)}={\frac {\delta J(x_{1})}{\delta {\bar {\phi }}(x_{2})}}{\Big |}_{{\bar {\phi }}=\langle \phi \rangle }=p_{\mu }p^{\mu }-m^{2}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

<msub>

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</mfrac>

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<msub>

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<mover>

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<msub>

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<msup>

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</msup>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\Gamma [\phi ]}^{(2)}={\frac {\delta J(x_{1})}{\delta {\bar {\phi }}(x_{2})}}{\Big |}_{{\bar {\phi }}=\langle \phi \rangle }=p_{\mu }p^{\mu }-m^{2}}</annotation>

</semantics>

</math><img src="./1eaac7a5e3323d3e856c69242c5fc7af9f4498bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.959ex; height:6.676ex;" alt="{\displaystyle G_{\Gamma [\phi ]}^{(2)}={\frac {\delta J(x_{1})}{\delta {\bar {\phi }}(x_{2})}}{\Big |}_{{\bar {\phi }}=\langle \phi \rangle }=p_{\mu }p^{\mu }-m^{2}}" loading="lazy">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}=J(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle J_{i}=J(x_{i})}</annotation>

</semantics>

</math><img src="./5b8ffcb2f832b42a1d3bd22675555e2e944bf3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.598ex; height:2.843ex;" alt="{\displaystyle J_{i}=J(x_{i})}" loading="lazy">, the most general relations between the N-points connected <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[J]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F[J]}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z[J]}</annotation>

</semantics>

<mtr>

<mtd>

<mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

<mrow>

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<msub>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\delta ^{N}F}{\delta J_{1}\cdots \delta J_{N}}}=&{\frac {1}{Z[J]}}{\frac {\delta ^{N}Z[J]}{\delta J_{1}\cdots \delta J_{N}}}-{\Big \{}{\frac {1}{Z^{2}[J]}}{\frac {\delta Z[J]}{\delta J_{1}}}{\frac {\delta ^{N-1}Z[J]}{\delta J_{2}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+{\big \{}{\frac {1}{Z^{3}[J]}}{\frac {\delta Z[J]}{\delta J_{1}}}{\frac {\delta Z[J]}{\delta J_{2}}}{\frac {\delta ^{N-2}Z[J]}{\delta J_{3}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+\cdots \\&-{\Big \{}{\frac {1}{Z^{2}[J]}}{\frac {\delta ^{2}Z[J]}{\delta J_{1}\delta J_{2}}}{\frac {\delta ^{N-2}Z[J]}{\delta J_{3}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+{\Big \{}{\frac {1}{Z^{3}[J]}}{\frac {\delta ^{3}Z[J]}{\delta J_{1}\delta J_{2}\delta J_{3}}}{\frac {\delta ^{N-3}Z[J]}{\delta J_{4}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}-\cdots \end{aligned}}}</annotation>

</semantics>

</math>

and

<mtr>

<mtd>

<mfrac>

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</mfrac>

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<mfrac>

<mrow>

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<mfrac>

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<mtr>

<mtd>

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<mfrac>

<mrow>

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<mrow>

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<mfrac>

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<mfrac>

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<msup>

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<mrow>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{Z[J]}}{\frac {\delta ^{N}Z[J]}{\delta J_{1}\cdots \delta J_{N}}}=&{\frac {\delta ^{N}F[J]}{\delta J_{1}\cdots \delta J_{N}}}+{\Big \{}{\frac {\delta F[J]}{\delta J_{1}}}{\frac {\delta ^{N-1}F[J]}{\delta J_{2}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+{\Big \{}{\frac {\delta F[J]}{\delta J_{1}}}{\frac {\delta F[J]}{\delta J_{2}}}{\frac {\delta ^{N-2}F[J]}{\delta J_{3}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+\cdots \\&+{\Big \{}{\frac {\delta ^{2}F[J]}{\delta J_{1}\delta J_{2}}}{\frac {\delta ^{N-2}F[J]}{\delta J_{3}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+{\Big \{}{\frac {\delta ^{3}F[J]}{\delta J_{1}\delta J_{2}\delta J_{3}}}{\frac {\delta ^{N-3}F[J]}{\delta J_{4}\cdots \delta J_{N}}}+{\text{perm}}{\Big \}}+\cdots \end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Source_theory_for_fields">Source theory for fields</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Vector_fields">Vector fields</h3></div>

For a weak source producing a <a href="Proca_action" title="Proca action">missive spin-1 particle</a> with a general current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=J_{e}+J_{a}}">

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle J=J_{e}+J_{a}}</annotation>

</semantics>

</math><img src="./ed2caabaf16bfcc0c701ff204ce4806a1aa438cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.091ex; height:2.509ex;" alt="{\displaystyle J=J_{e}+J_{a}}" loading="lazy"> acting on different causal spacetime points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}>x_{0}'}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}>x_{0}'}</annotation>

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<msup>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{J}=\exp {\left({\frac {i}{2}}\int dx~dx'\left[J_{\mu }(x)\Delta (x-x')J^{\mu }(x')+{\frac {1}{m^{2}}}\partial _{\mu }J^{\mu }(x)\Delta (x-x')\partial '_{\nu }J^{\nu }(x')\right]\right)}}</annotation>

</semantics>

</math>

In momentum space, the spin-1 particle with rest mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> has a definite momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mu }=(m,0,0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle p_{\mu }=(m,0,0,0)}</annotation>

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</math><img src="./17cc0eb2d36ba4545c55a5e010f65ca0842000af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:16.02ex; height:3.009ex;" alt="{\displaystyle p_{\mu }=(m,0,0,0)}" loading="lazy"> in its rest frame, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mu }p^{\mu }=m^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{\mu }p^{\mu }=m^{2}}</annotation>

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</math><img src="./b05f0595d5cf4943afa03bd17745850a8a10a282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:11.068ex; height:3.343ex;" alt="{\displaystyle p_{\mu }p^{\mu }=m^{2}}" loading="lazy">. Then, the amplitude gives<a href="#cite_note-:0-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}(J_{\mu }(p))^{T}~J^{\mu }(p)-{\frac {1}{m^{2}}}(p_{\mu }J^{\mu }(p))^{T}~p_{\nu }J^{\nu }(p)&=(J_{\mu }(p))^{T}~J^{\mu }(p)-(J^{\mu }(p))^{T}~{\frac {p_{\mu }p_{\nu }}{p_{\sigma }p^{\sigma }}}{\bigg |}_{\text{on-shell}}~J^{\nu }(p)\\&=(J^{\mu }(p))^{T}~\left[\eta _{\mu \nu }-{\frac {p_{\mu }p_{\nu }}{m^{2}}}\right]~J^{\nu }(p)\end{alignedat}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }={\text{diag}}(1,-1,-1,-1)}</annotation>

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</math><img src="./b11db1d914af281eca5145c2e4e1626ae18e12c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.598ex; height:3.009ex;" alt="{\displaystyle \eta _{\mu \nu }={\text{diag}}(1,-1,-1,-1)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (J_{\mu }(p))^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle (J_{\mu }(p))^{T}}</annotation>

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</math><img src="./36f175dddd5b85a5e597d2926600d3ac76c5e20e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.691ex; height:3.343ex;" alt="{\displaystyle (J_{\mu }(p))^{T}}" loading="lazy"> is the transpose of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu }(p)}">

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<annotation encoding="application/x-tex">{\displaystyle J_{\mu }(p)}</annotation>

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</math><img src="./2355240e8d1a670bac721e38d81107818991c2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.492ex; height:3.009ex;" alt="{\displaystyle J_{\mu }(p)}" loading="lazy">. The last result matches with the used propagator in the vacuum amplitude in the configuration space, that is,

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<annotation encoding="application/x-tex">{\displaystyle \left\langle 0\right|TA_{\mu }(x)A_{\nu }(x')\left|0\right\rangle =-i\int {\frac {d^{4}p}{{\left(2\pi \right)}^{4}}}{\frac {1}{p_{\alpha }p^{\alpha }+i\varepsilon }}\left[\eta _{\mu \nu }-\left(1-\xi \right){\frac {p_{\mu }p_{\nu }}{p_{\sigma }p^{\sigma }-\xi m^{2}}}\right]e^{ip^{\mu }\left(x_{\mu }-x'_{\mu }\right)}.}</annotation>

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When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =1}">

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</math><img src="./b58b7a875fef17a64b2cc7710225b16f72950024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.291ex; height:2.509ex;" alt="{\displaystyle \xi =1}" loading="lazy">, the chosen Feynman–'t Hooft <a href="Propagator#Spin_1" title="Propagator">gauge-fixing</a> makes the spin-1 massless. And when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =0}">

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</math><img src="./da5354e193004a0e2f16e7d4a76ea499ffcca225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.291ex; height:2.509ex;" alt="{\displaystyle \xi =0}" loading="lazy">, the chosen Landau <a href="Gauge_fixing" title="Gauge fixing">gauge-fixing</a> makes the spin-1 massive.<a href="#cite_note-24">[24]</a> The massless case is obvious as studied in <a href="Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. The massive case is more interesting as the current is not demanded to conserved. However, the current can be improved in a way similar to how the <a href="Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor" title="Belinfante–Rosenfeld stress–energy tensor">Belinfante-Rosenfeld tensor</a> is improved so it ends up being conserved. And to get the equation of motion for the massive vector, one can define<a href="#cite_note-:0-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle W[J]=-i\ln(\langle 0|0\rangle _{J})={\frac {1}{2}}\int dx~dx'\left[J_{\mu }(x)\Delta (x-x')J^{\mu }(x')+{\frac {1}{m^{2}}}\partial _{\mu }J^{\mu }(x)\Delta (x-x')\partial '_{\nu }J^{\nu }(x')\right].}</annotation>

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One can apply integration by part on the second term then single out <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int dxJ_{\mu }(x)}">

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</math><img src="./0681a3dc2b092a234dc077827559a650a622a9b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.003ex; height:3.176ex;" alt="{\textstyle \int dxJ_{\mu }(x)}" loading="lazy"> to get a definition of the massive spin-1 field

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }(x)\equiv \int dx'\Delta (x-x')J^{\mu }(x')-{\frac {1}{m^{2}}}\partial _{\mu }\left[\int dx'\Delta (x-x')\partial '_{\nu }J^{\nu }(x')\right].}</annotation>

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Additionally, the equation above says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \partial _{\mu }A^{\mu }={\tfrac {1}{m^{2}}}\partial _{\mu }J^{\mu }}">

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<annotation encoding="application/x-tex">{\textstyle \partial _{\mu }A^{\mu }={\tfrac {1}{m^{2}}}\partial _{\mu }J^{\mu }}</annotation>

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</math><img src="./a0534dec6d5da2542081fc13b67e763ad5a5421e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:16.84ex; height:3.843ex;" alt="{\textstyle \partial _{\mu }A^{\mu }={\tfrac {1}{m^{2}}}\partial _{\mu }J^{\mu }}" loading="lazy">. Thus, the equation of motion can be written in any of the following forms

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left(\Box +m^{2}\right)A_{\mu }=J_{\mu }+{\tfrac {1}{m^{2}}}\partial _{\nu }\partial _{\mu }J^{\nu },\\[1ex]&\left(\Box +m^{2}\right)A_{\mu }+\partial _{\nu }\partial _{\mu }A^{\nu }=J_{\mu }.\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Massive_totally_symmetric_spin-2_fields">Massive totally symmetric spin-2 fields</h3></div>

For a weak source in a <a href="Minkowski_space" title="Minkowski space">flat Minkowski background</a>, producing then absorbing a <a href="Massive_gravity" title="Massive gravity">massive spin-2 particle</a> with a general redefined <a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">energy-momentum tensor</a>, acting as a current, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {T}}^{\mu \nu }=T^{\mu \nu }-{\tfrac {1}{3}}\eta _{\mu \alpha }{\bar {\eta }}_{\nu \beta }T^{\alpha \beta }}">

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<annotation encoding="application/x-tex">{\textstyle {\bar {T}}^{\mu \nu }=T^{\mu \nu }-{\tfrac {1}{3}}\eta _{\mu \alpha }{\bar {\eta }}_{\nu \beta }T^{\alpha \beta }}</annotation>

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</math><img src="./a0d81f1f1d90242662c3072e04471827ce983aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.007ex; height:3.676ex;" alt="{\textstyle {\bar {T}}^{\mu \nu }=T^{\mu \nu }-{\tfrac {1}{3}}\eta _{\mu \alpha }{\bar {\eta }}_{\nu \beta }T^{\alpha \beta }}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {\eta }}_{\mu \nu }(p)=\eta _{\mu \nu }-{\tfrac {1}{m^{2}}}p_{\mu }p_{\nu }}">

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<annotation encoding="application/x-tex">{\textstyle {\bar {\eta }}_{\mu \nu }(p)=\eta _{\mu \nu }-{\tfrac {1}{m^{2}}}p_{\mu }p_{\nu }}</annotation>

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</math><img src="./938af222eb902d5cfce45f0e371db2e5fd11cad0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:23.343ex; height:3.843ex;" alt="{\textstyle {\bar {\eta }}_{\mu \nu }(p)=\eta _{\mu \nu }-{\tfrac {1}{m^{2}}}p_{\mu }p_{\nu }}" loading="lazy"> is the <a href="Vacuum_polarization#Vacuum_polarization_tensor" title="Vacuum polarization">vacuum polarization tensor</a>, the vacuum amplitude in a compact form is<a href="#cite_note-:0-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle 0|0\rangle _{\bar {T}}=\exp {\Biggl (}-{\frac {i}{2}}\int {\biggl [}&{\bar {T}}_{\mu \nu }(x)\Delta (x-x'){\bar {T}}^{\mu \nu }(x')\\&+{\frac {2}{m^{2}}}\eta _{\lambda \nu }\partial _{\mu }{\bar {T}}^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }{\bar {T}}^{\kappa \lambda }(x')\\&+{\frac {1}{m^{4}}}\partial _{\mu }\partial _{\nu }{\bar {T}}^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }\partial '_{\lambda }{\bar {T}}^{\kappa \lambda }(x'){\biggr ]}dx\,dx'{\Biggr )},\end{aligned}}}</annotation>

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or

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle 0|0\rangle _{T}=\exp {\Biggl (}-{\frac {i}{2}}\int {\biggl [}&T_{\mu \nu }(x)\Delta (x-x')T^{\mu \nu }(x')\\&+{\frac {2}{m^{2}}}\eta _{\lambda \nu }\partial _{\mu }T^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }T^{\kappa \lambda }(x')\\&+{\frac {1}{m^{4}}}\partial _{\mu }\partial _{\mu }T^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }\partial '_{\lambda }T^{\kappa \lambda }(x')\\&-{\frac {1}{3}}\left(\eta _{\mu \nu }T^{\mu \nu }(x)-{\frac {1}{m^{2}}}\partial _{\mu }\partial _{\nu }T^{\mu \nu }(x)\right)\Delta (x-x')\left(\eta _{\kappa \lambda }T^{\kappa \lambda }(x')-{\frac {1}{m^{2}}}\partial '_{\kappa }\partial '_{\lambda }T^{\kappa \lambda }(x')\right){\biggr ]}dx~dx'{\Biggr )}.\end{aligned}}}</annotation>

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This amplitude in momentum space gives (transpose is imbedded)

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {T}}_{\mu \nu }(p)\eta ^{\mu \kappa }\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)\eta ^{\mu \kappa }p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)\\&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }p^{\mu }p^{\kappa }{\bar {T}}_{\kappa \lambda }(p)+{\frac {1}{m^{4}}}{\bar {T}}_{\mu \nu }(p)p^{\mu }p^{\nu }p^{\kappa }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)=\end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta ^{\mu \kappa }{\biggl (}{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p){\biggr )}\\&-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }\left({\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)\right)\\=\left(\eta ^{\mu \kappa }-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }\right)&\left({\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)\right)\\=&{\bar {T}}_{\mu \nu }(p)\left(\eta ^{\mu \kappa }-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }\right)\left(\eta ^{\nu \lambda }-{\frac {1}{m^{2}}}p^{\nu }p^{\lambda }\right){\bar {T}}_{\kappa \lambda }(p).\end{aligned}}}</annotation>

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And with help of symmetric properties of the source, the last result can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}">

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<annotation encoding="application/x-tex">{\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}</annotation>

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</math><img src="./58fc7b17328da13281eedaf767a2fdfd81e4bf90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.351ex; height:3.343ex;" alt="{\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}" loading="lazy">, where the projection operator, or the Fourier transform of Jacobi field operator obtained by applying <a href="Peierls_bracket" title="Peierls bracket">Peierls braket</a> on <a href="Schwinger's_variational_principle" class="mw-redirect" title="Schwinger's variational principle">Schwinger's variational principle</a>,<a href="#cite_note-25">[25]</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Pi _{\mu \nu \kappa \lambda }(p)={\tfrac {1}{2}}\left({\bar {\eta }}_{\mu \kappa }(p){\bar {\eta }}_{\nu \lambda }(p)+{\bar {\eta }}_{\mu \lambda }(p){\bar {\eta }}_{\nu \kappa }(p)-{\tfrac {2}{3}}{\bar {\eta }}_{\mu \nu }(p){\bar {\eta }}_{\kappa \lambda }(p)\right)}">

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<annotation encoding="application/x-tex">{\textstyle \Pi _{\mu \nu \kappa \lambda }(p)={\tfrac {1}{2}}\left({\bar {\eta }}_{\mu \kappa }(p){\bar {\eta }}_{\nu \lambda }(p)+{\bar {\eta }}_{\mu \lambda }(p){\bar {\eta }}_{\nu \kappa }(p)-{\tfrac {2}{3}}{\bar {\eta }}_{\mu \nu }(p){\bar {\eta }}_{\kappa \lambda }(p)\right)}</annotation>

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</math><img src="./d8ef29b85463b50b032641fbf6097f3cc68eafe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.373ex; height:4.843ex;" alt="{\textstyle \Pi _{\mu \nu \kappa \lambda }(p)={\tfrac {1}{2}}\left({\bar {\eta }}_{\mu \kappa }(p){\bar {\eta }}_{\nu \lambda }(p)+{\bar {\eta }}_{\mu \lambda }(p){\bar {\eta }}_{\nu \kappa }(p)-{\tfrac {2}{3}}{\bar {\eta }}_{\mu \nu }(p){\bar {\eta }}_{\kappa \lambda }(p)\right)}" loading="lazy">.

In N-dimensional flat spacetime, 2/3 is replaced by 2/(N−1).<a href="#cite_note-26">[26]</a> And for <a href="Linearized_gravity" title="Linearized gravity">massless spin-2 fields</a>, the <a href="Propagator#Graviton_propagator" title="Propagator">projection operator</a> is defined as<a href="#cite_note-:0-5">[5]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{\mu \nu \kappa \lambda }^{m=0}={\tfrac {1}{2}}\left(\eta _{\mu \kappa }\eta _{\nu \lambda }+\eta _{\mu \lambda }\eta _{\nu \kappa }-{\tfrac {1}{2}}\eta _{\mu \nu }\eta _{\kappa \lambda }\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \Pi _{\mu \nu \kappa \lambda }^{m=0}={\tfrac {1}{2}}\left(\eta _{\mu \kappa }\eta _{\nu \lambda }+\eta _{\mu \lambda }\eta _{\nu \kappa }-{\tfrac {1}{2}}\eta _{\mu \nu }\eta _{\kappa \lambda }\right)}</annotation>

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</math><img src="./eaabe6c4346d5c69a61db7c7d4c0c10c62e8affd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:39.984ex; height:3.676ex;" alt="{\displaystyle \Pi _{\mu \nu \kappa \lambda }^{m=0}={\tfrac {1}{2}}\left(\eta _{\mu \kappa }\eta _{\nu \lambda }+\eta _{\mu \lambda }\eta _{\nu \kappa }-{\tfrac {1}{2}}\eta _{\mu \nu }\eta _{\kappa \lambda }\right)}" loading="lazy">.

Together with help of <a href="Ward%E2%80%93Takahashi_identity" title="Ward–Takahashi identity">Ward-Takahashi identity</a>, the projector operator is crucial to check the symmetric properties of the field, the conservation law of the current, and the allowed physical degrees of freedom.

It is worth noting that the vacuum polarization tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\eta }}_{\nu \beta }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\eta }}_{\nu \beta }}</annotation>

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</math><img src="./afa5c7e0cb5e1d879671d3d1bde7bc0a533e8259.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.35ex; height:2.843ex;" alt="{\displaystyle {\bar {\eta }}_{\nu \beta }}" loading="lazy"> and the improved energy momentum tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {T}}^{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {T}}^{\mu \nu }}</annotation>

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</math><img src="./3dba0375144a90a70cdeaa1ab43d9f5dc3d6c58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.869ex; height:2.676ex;" alt="{\displaystyle {\bar {T}}^{\mu \nu }}" loading="lazy"> appear in the early versions of <a href="Massive_gravity" title="Massive gravity">massive gravity theories</a>.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> Interestingly, massive gravity theories have not been widely appreciated until recently due to apparent inconsistencies obtained in the early 1970's studies of the exchange of a single spin-2 field between two sources. But in 2010 the dRGT approach<a href="#cite_note-29">[29]</a> of exploiting <a href="Stueckelberg_action" title="Stueckelberg action">Stueckelberg field redefinition</a> led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier.

If one looks at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|0\rangle _{T}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{T}}</annotation>

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</math><img src="./6d8fe577a76cdb2ba3c61de9e44127c06f8d9844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.17ex; height:2.843ex;" alt="{\displaystyle \langle 0|0\rangle _{T}}" loading="lazy"> and follows the same procedure used to define massive spin-1 fields, then it is easy to define massive spin-2 fields as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h_{\mu \nu }(x)=&\int \Delta (x-x')T_{\mu \nu }(x')dx'\\&-{\frac {1}{m^{2}}}\partial _{\mu }\int \Delta (x-x')\partial '^{\kappa }T_{\kappa \nu }(x')dx'\\&-{\frac {1}{m^{2}}}\partial _{\nu }\int \Delta (x-x')\partial '^{\kappa }T_{\kappa \mu }(x')dx'\\&+{\frac {1}{m^{4}}}\partial _{\mu }\partial _{\nu }\int \Delta (x-x')\partial '_{\kappa }\partial '_{\lambda }T^{\kappa \lambda }(x')dx'\\&-{\frac {1}{3}}\left(\eta _{\mu \nu }-{\frac {1}{m^{2}}}\partial _{\mu }\partial _{\nu }\right)\int \Delta (x-x')\left[\eta _{\kappa \lambda }T^{\kappa \lambda }(x')-{\frac {1}{m^{2}}}\partial '_{\kappa }\partial '_{\lambda }T^{\kappa \lambda }(x')\right]dx'.\end{aligned}}}</annotation>

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The corresponding divergence condition is read <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{\mu }h_{\mu \nu }-\partial _{\nu }h={\frac {1}{m^{2}}}\partial ^{\mu }T_{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle \partial ^{\mu }h_{\mu \nu }-\partial _{\nu }h={\frac {1}{m^{2}}}\partial ^{\mu }T_{\mu \nu }}</annotation>

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</math><img src="./95ad50d8263409e62363b412c46096a99341718f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.566ex; height:5.509ex;" alt="{\displaystyle \partial ^{\mu }h_{\mu \nu }-\partial _{\nu }h={\frac {1}{m^{2}}}\partial ^{\mu }T_{\mu \nu }}" loading="lazy">, where the current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{\mu }T_{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle \partial ^{\mu }T_{\mu \nu }}</annotation>

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</math><img src="./aa9703a7ef8f6af2f358dacd6316ba01d67adba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.019ex; height:3.009ex;" alt="{\displaystyle \partial ^{\mu }T_{\mu \nu }}" loading="lazy"> is not necessarily conserved (it is not a gauge condition as that of the massless case). But the energy-momentum tensor can be improved as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathfrak {T}}_{\mu \nu }=T_{\mu \nu }-{\frac {1}{4}}\eta _{\mu \nu }{\mathfrak {T}}}">

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<annotation encoding="application/x-tex">{\textstyle {\mathfrak {T}}_{\mu \nu }=T_{\mu \nu }-{\frac {1}{4}}\eta _{\mu \nu }{\mathfrak {T}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \partial ^{\mu }{\mathfrak {T}}_{\mu \nu }=0}</annotation>

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</math><img src="./6147e5681af37456f94771880e84b834d76237ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.477ex; height:3.009ex;" alt="{\displaystyle \partial ^{\mu }{\mathfrak {T}}_{\mu \nu }=0}" loading="lazy"> according to <a href="Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor" title="Belinfante–Rosenfeld stress–energy tensor">Belinfante-Rosenfeld</a> construction. Thus, the equation of motion

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(\square +m^{2}\right)h_{\mu \nu }=T_{\mu \nu }&+{\dfrac {1}{m^{2}}}\left(\partial _{\mu }\partial ^{\rho }T_{\rho \nu }+\partial _{\nu }\partial ^{\rho }T_{\rho \mu }-{\frac {1}{2}}\eta _{\mu \nu }\partial ^{\rho }\partial ^{\sigma }T_{\rho \sigma }\right)\\&+{\frac {2}{3m^{4}}}\left(\partial _{\mu }\partial _{\nu }-{\frac {1}{4}}\eta _{\mu \nu }\square \right)\partial ^{\rho }\partial ^{\sigma }T_{\rho \sigma }\end{aligned}}}</annotation>

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becomes

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<annotation encoding="application/x-tex">{\displaystyle \left(\square +m^{2}\right)h_{\mu \nu }={\mathfrak {T}}_{\mu \nu }-{\frac {1}{4}}~\eta _{\mu \nu }{\mathfrak {T}}-{\dfrac {1}{6m^{4}}}\left(\partial _{\mu }\partial _{\nu }-{\frac {1}{4}}~\eta _{\mu \nu }\square \right)\left(\square +3m^{2}\right){\mathfrak {T}}.}</annotation>

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One can use the divergence condition to decouple the non-physical fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{\mu }h_{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle \partial ^{\mu }h_{\mu \nu }}</annotation>

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</math><img src="./e3229b7ab35d1faabae525a5fcf3d6705bcf9449.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6ex; height:3.009ex;" alt="{\displaystyle \partial ^{\mu }h_{\mu \nu }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, so the equation of motion is simplified as<a href="#cite_note-30">[30]</a>

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<annotation encoding="application/x-tex">{\displaystyle \left(\square +m^{2}\right)h_{\mu \nu }={\mathfrak {T}}_{\mu \nu }-{\frac {1}{3}}~\eta _{\mu \nu }{\mathfrak {T}}-{\frac {1}{3m^{2}}}~\partial _{\mu }\partial _{\nu }{\mathfrak {T}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Massive_totally_symmetric_arbitrary_integer_spin_fields">Massive totally symmetric arbitrary integer spin fields</h3></div>

One can generalize <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\mu \nu }(p)}">

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<annotation encoding="application/x-tex">{\displaystyle T^{\mu \nu }(p)}</annotation>

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</math><img src="./cc335e18716733a4dc17648d8ee9b2994d7ccf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.793ex; height:2.843ex;" alt="{\displaystyle T^{\mu \nu }(p)}" loading="lazy"> source to become <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)}">

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<annotation encoding="application/x-tex">{\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)}</annotation>

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</math><img src="./92947d2faff4637deed758a831eb76e6aeefae8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.193ex; height:2.843ex;" alt="{\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)}" loading="lazy"> <a href="Higher-spin_theory" title="Higher-spin theory">higher-spin</a> source such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}">

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<annotation encoding="application/x-tex">{\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)S^{\nu _{1}\cdots \nu _{\ell }}(p)}</annotation>

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</math><img src="./ca454c34bd3ec656ace2d7093ebcbc47623e8641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.545ex; height:3.009ex;" alt="{\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)S^{\nu _{1}\cdots \nu _{\ell }}(p)}" loading="lazy"> .<a href="#cite_note-:0-5">[5]</a> The generalized projection operator also helps generalizing the electromagnetic polarization vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{m}^{\mu }(p)}">

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<annotation encoding="application/x-tex">{\displaystyle e_{m}^{\mu }(p)}</annotation>

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</math><img src="./5a9952687723ee2f8c9d9784de69d62d18ee16a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.737ex; height:3.009ex;" alt="{\displaystyle e_{m}^{\mu }(p)}" loading="lazy"> of the <a href="Quantization_of_the_electromagnetic_field#Electromagnetic_field_and_vector_potential" title="Quantization of the electromagnetic field">quantized electromagnetic vector potential</a> as follows. For spacetime points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">

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</math><img src="./0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy">, the <a href="Spherical_harmonics#Addition_theorem" title="Spherical harmonics">addition theorem of spherical harmonics</a> states that

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<annotation encoding="application/x-tex">{\displaystyle x^{\mu _{1}}\cdots x^{\mu _{\ell }}\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)x'^{\nu _{1}}\cdots x'^{\nu _{\ell }}={\frac {2^{\ell }(\ell !)^{2}}{(2\ell )!}}{\frac {4\pi }{2\ell +1}}\sum \limits _{m=-\ell }^{\ell }Y_{\ell ,m}(x)Y_{\ell ,m}^{*}(x').}</annotation>

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Also, <a href="Spherical_harmonics#Connection_with_representation_theory" title="Spherical harmonics">the representation theory</a> of the space of complex-valued <a href="Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

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</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> on a unit (N-1)-sphere defines the polarization tensor as<a href="#cite_note-31">[31]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{(m)}(x_{1},\dots ,x_{n})=\sum _{i_{1}\dots i_{\ell }}e_{(m)i_{1}\dots i_{\ell }}x_{i_{1}}\cdots x_{i_{\ell }},~\forall x_{i}\in S^{N-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle e_{(m)}(x_{1},\dots ,x_{n})=\sum _{i_{1}\dots i_{\ell }}e_{(m)i_{1}\dots i_{\ell }}x_{i_{1}}\cdots x_{i_{\ell }},~\forall x_{i}\in S^{N-1}.}</annotation>

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</math>Then, the generalized polarization vector is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\mu _{1}\cdots \mu _{\ell }}(p)~x_{\mu _{1}}\cdots x_{\mu _{\ell }}={\sqrt {{\frac {2^{\ell }(\ell !)^{2}}{(2\ell )!}}{\frac {4\pi }{2\ell +1}}}}~~Y_{\ell ,m}(x).}">

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<annotation encoding="application/x-tex">{\displaystyle e^{\mu _{1}\cdots \mu _{\ell }}(p)~x_{\mu _{1}}\cdots x_{\mu _{\ell }}={\sqrt {{\frac {2^{\ell }(\ell !)^{2}}{(2\ell )!}}{\frac {4\pi }{2\ell +1}}}}~~Y_{\ell ,m}(x).}</annotation>

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And the projection operator can be defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)=\sum \limits _{m=-\ell }^{\ell }[e_{m}^{\mu _{1}\cdots \mu _{\ell }}(p)]~[e_{m}^{\nu _{1}\cdots \nu _{\ell }}(p)]^{*}.}">

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<annotation encoding="application/x-tex">{\displaystyle \Pi ^{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)=\sum \limits _{m=-\ell }^{\ell }[e_{m}^{\mu _{1}\cdots \mu _{\ell }}(p)]~[e_{m}^{\nu _{1}\cdots \nu _{\ell }}(p)]^{*}.}</annotation>

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The symmetric properties of the projection operator make it easier to deal with the vacuum amplitude in the momentum space. Therefore rather that we express it in terms of the correlator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x-x')}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{S}=\exp {\left[{\frac {i}{2}}\int {\frac {dp^{4}}{(2\pi )^{4}}}S^{\mu _{1}\cdots \mu _{\ell }}(-p){\frac {\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)}{p_{\sigma }p^{\sigma }-m^{2}+i\varepsilon }}S^{\nu _{1}\cdots \nu _{\ell }}(p)\right]}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Mixed_symmetric_arbitrary_spin_fields">Mixed symmetric arbitrary spin fields</h3></div>

Also, it is theoretically consistent to generalize the source theory to describe hypothetical gauge fields with <a href="Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">antisymmetric</a> and mixed symmetric properties in arbitrary dimensions and <a href="Higher-spin_theory" title="Higher-spin theory">arbitrary spins</a>. But one should take care of the unphysical degrees of freedom in the theory. For example in N-dimensions and for a mixed symmetric massless version of <a href="Curtright_field" title="Curtright field">Curtright field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{[\mu \nu ]\lambda }}">

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<annotation encoding="application/x-tex">{\displaystyle S_{[\mu \nu ]\lambda }=\partial _{\alpha }\partial ^{\alpha }T_{[\mu \nu ]\lambda }}</annotation>

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</math><img src="./e66bff662120bae66e328922175dc40851f72833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.962ex; height:3.176ex;" alt="{\displaystyle S_{[\mu \nu ]\lambda }=\partial _{\alpha }\partial ^{\alpha }T_{[\mu \nu ]\lambda }}" loading="lazy"> , the vacuum amplitude is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 0|0\rangle _{S}=\exp {\left(-{\frac {1}{2}}\int dx~dx'\left[S_{[\mu \nu ]\lambda }(x)\Delta (x-x')S_{[\mu \nu ]\lambda }(x')+{\frac {2}{3-N}}S_{[\mu \alpha ]\alpha }(x)\Delta (x-x')S_{[\mu \beta ]\beta }(x')\right]\right)}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle 0|0\rangle _{S}=\exp {\left(-{\frac {1}{2}}\int dx~dx'\left[S_{[\mu \nu ]\lambda }(x)\Delta (x-x')S_{[\mu \nu ]\lambda }(x')+{\frac {2}{3-N}}S_{[\mu \alpha ]\alpha }(x)\Delta (x-x')S_{[\mu \beta ]\beta }(x')\right]\right)}}</annotation>

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</math> which for a theory in N=4 makes the source eventually reveal that it is a theory of a non physical field.<a href="#cite_note-32">[32]</a> However, the <a href="Dual_graviton" title="Dual graviton">massive version</a> survives in N≥5.

<div class="mw-heading mw-heading3"><h3 id="Arbitrary_half-integer_spin_fields">Arbitrary half-integer spin fields</h3></div>

For spin-<style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>1⁄2 <a href="Propagator#Spin_1⁄2" title="Propagator">fermion propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x-x')=(p\!\!\!/+m)\Delta (x-x')}">

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<annotation encoding="application/x-tex">{\displaystyle S(x-x')=(p\!\!\!/+m)\Delta (x-x')}</annotation>

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</math><img src="./053f7c0c571b638bebd7896fcc10766488df7425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.382ex; height:3.009ex;" alt="{\displaystyle S(x-x')=(p\!\!\!/+m)\Delta (x-x')}" loading="lazy"> and current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=J_{e}+J_{a}}">

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<annotation encoding="application/x-tex">{\displaystyle J=J_{e}+J_{a}}</annotation>

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</math><img src="./ed2caabaf16bfcc0c701ff204ce4806a1aa438cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.091ex; height:2.509ex;" alt="{\displaystyle J=J_{e}+J_{a}}" loading="lazy"> as defined above, the vacuum amplitude is<a href="#cite_note-:0-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle 0|0\rangle _{J}&=\exp {\left[{\frac {i}{2}}\int dxdx'~J(x)~\left(\gamma ^{0}S(x-x')\right)~J(x')\right]}\\&=\langle 0|0\rangle _{J_{e}}\exp {\left[i\int dxdx'~J_{e}(x)~\left(\gamma ^{0}S(x-x')~\right)~J_{a}(x')\right]}\langle 0|0\rangle _{J_{a}}.\end{aligned}}}</annotation>

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</math>

In momentum space the reduced amplitude is given by

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<annotation encoding="application/x-tex">{\displaystyle W_{\frac {1}{2}}=-{\frac {1}{3}}\int {\frac {d^{4}p}{(2\pi )^{4}}}~J(-p)\left[\gamma ^{0}{\frac {p\!\!\!/+m}{p^{2}-m^{2}}}\right]~J(p).}</annotation>

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For spin-3⁄2 <a href="Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita-Schwinger</a> fermions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Pi _{\mu \nu }={\bar {\eta }}_{\mu \nu }-{\tfrac {1}{3}}\gamma ^{\alpha }{\bar {\eta }}_{\alpha \mu }\gamma ^{\beta }{\bar {\eta }}_{\beta \nu }.}">

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<annotation encoding="application/x-tex">{\textstyle \Pi _{\mu \nu }={\bar {\eta }}_{\mu \nu }-{\tfrac {1}{3}}\gamma ^{\alpha }{\bar {\eta }}_{\alpha \mu }\gamma ^{\beta }{\bar {\eta }}_{\beta \nu }.}</annotation>

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</math><img src="./d90a17dbab584eaeecb39cdd012bb1149d1ac65f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.427ex; height:3.676ex;" alt="{\textstyle \Pi _{\mu \nu }={\bar {\eta }}_{\mu \nu }-{\tfrac {1}{3}}\gamma ^{\alpha }{\bar {\eta }}_{\alpha \mu }\gamma ^{\beta }{\bar {\eta }}_{\beta \nu }.}" loading="lazy"> Then, one can use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mu }=\eta _{\mu \nu }\gamma ^{\nu }}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{\mu }=\eta _{\mu \nu }\gamma ^{\nu }}</annotation>

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</math><img src="./da8e056ff2dcecb0539aa43f67f55effc72813f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.16ex; height:3.009ex;" alt="{\displaystyle \gamma _{\mu }=\eta _{\mu \nu }\gamma ^{\nu }}" loading="lazy"> and the on-shell <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\!\!\!/=-m}">

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<annotation encoding="application/x-tex">{\displaystyle p\!\!\!/=-m}</annotation>

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</math><img src="./49d05d7e675dd738fb9ea049577c139e486e99d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.207ex; height:2.843ex;" alt="{\displaystyle p\!\!\!/=-m}" loading="lazy"> to get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{\frac {3}{2}}&=-{\frac {2}{5}}\int {\frac {d^{4}p}{{\left(2\pi \right)}^{4}}}\,J^{\mu }(-p)\left[\gamma ^{0}{\frac {(p\!\!\!/+m)\left({\bar {\eta }}_{\mu \nu }|_{\text{on-shell}}-{\frac {1}{3}}\gamma ^{\alpha }{\bar {\eta }}_{\alpha \mu }|_{\text{on-shell}}\gamma ^{\beta }{\bar {\eta }}_{\beta \nu }|_{\text{on-shell}}\right)}{p^{2}-m^{2}}}\right]~J^{\nu }(p)\\&=-{\frac {2}{5}}\int {\frac {d^{4}p}{{\left(2\pi \right)}^{4}}}\,J^{\mu }(-p)\left[\gamma ^{0}{\frac {\left(\eta _{\mu \nu }-{\frac {p_{\mu }p_{\nu }}{m^{2}}}\right)(p\!\!\!/+m)-{\frac {1}{3}}\left(\gamma _{\mu }+{\frac {1}{m}}p_{\mu }\right)\left(p\!\!\!/+m\right)\left(\gamma _{\nu }+{\frac {1}{m}}p_{\nu }\right)}{p^{2}-m^{2}}}\right]~J^{\nu }(p).\end{aligned}}}</annotation>

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One can replace the reduced metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\eta }}_{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\eta }}_{\mu \nu }}</annotation>

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</math><img src="./e686509cbbcc4a70346ed43566e7dcfc8aafcede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.399ex; height:2.843ex;" alt="{\displaystyle {\bar {\eta }}_{\mu \nu }}" loading="lazy"> with the usual one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }}">

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<annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }}</annotation>

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</math><img src="./9a8fb25ea6ee5d591c2d819519401a871ee04bc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.25ex; height:2.343ex;" alt="{\displaystyle \eta _{\mu \nu }}" loading="lazy"> if the source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{\mu }}">

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<annotation encoding="application/x-tex">{\displaystyle J_{\mu }}</annotation>

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</math><img src="./fa316ed147312de05f2ea5c7fcb26847c2e6d709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.843ex;" alt="{\displaystyle J_{\mu }}" loading="lazy"> is replaced with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {J}}_{\mu }(p)={\frac {2}{5}}\gamma ^{\alpha }\Pi _{\mu \alpha \nu \beta }\gamma ^{\beta }J^{\nu }(p).}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {J}}_{\mu }(p)={\frac {2}{5}}\gamma ^{\alpha }\Pi _{\mu \alpha \nu \beta }\gamma ^{\beta }J^{\nu }(p).}</annotation>

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</math><img src="./f1ca4697732e3c9d3390b104283f569bedcce6df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.226ex; height:5.176ex;" alt="{\displaystyle {\bar {J}}_{\mu }(p)={\frac {2}{5}}\gamma ^{\alpha }\Pi _{\mu \alpha \nu \beta }\gamma ^{\beta }J^{\nu }(p).}" loading="lazy">

For spin-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (j+{\tfrac {1}{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle (j+{\tfrac {1}{2}})}</annotation>

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</math><img src="./1ae8bfcc2de1e0b63dea918450bd50c40fefbdba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.266ex; height:3.509ex;" alt="{\displaystyle (j+{\tfrac {1}{2}})}" loading="lazy">, the above results can be generalized to

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<annotation encoding="application/x-tex">{\displaystyle W_{j+{\frac {1}{2}}}=-{\frac {j+1}{2j+3}}\int {\frac {d^{4}p}{{\left(2\pi \right)}^{4}}}\,J^{\mu _{1}\cdots \mu _{j}}(-p)~\left[\gamma ^{0}{\frac {~\gamma ^{\alpha }~\Pi _{\mu _{1}\cdots \mu _{j}\alpha \nu _{1}\cdots \nu _{j}\beta }~\gamma ^{\beta }}{p^{2}-m^{2}}}\right]J^{\nu _{1}\cdots \nu _{j}}(p).}</annotation>

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The factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {j+1}{2j+3}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {j+1}{2j+3}}}</annotation>

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</math><img src="./75646228565fab5eeed855ae831533268eaf4025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.96ex; height:5.843ex;" alt="{\displaystyle {\frac {j+1}{2j+3}}}" loading="lazy"> is obtained from the properties of the projection operator, the tracelessness of the current, and the conservation of the current after being projected by the operator.<a href="#cite_note-:0-5">[5]</a> These conditions can be derived form the Fierz-Pauli<a href="#cite_note-33">[33]</a> and the Fang-Fronsdal<a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a> conditions on the fields themselves. The Lagrangian formulations of massive fields and their conditions were studied by Lambodar Singh and <a href="C._R._Hagen" title="C. R. Hagen">Carl Hagen</a>.<a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> The non-relativistic version of the projection operators, developed by Charles Zemach who is another student of Schwinger,<a href="#cite_note-38">[38]</a> is used heavily in hadron spectroscopy. Zemach's method could be relativistically improved to render the covariant projection operators.<a href="#cite_note-39">[39]</a><a href="#cite_note-40">[40]</a>

<ul><li><a href="Keldysh_formalism" title="Keldysh formalism">Keldysh-Schwinger formalism</a></li>

<li><a href="Schwinger_function" title="Schwinger function">Schwinger function</a></li>

<li><a href="Bargmann%E2%80%93Wigner_equations" title="Bargmann–Wigner equations">Wigner-Bargmann equations</a></li>

<li><a href="Joos-Weinberg_equation" class="mw-redirect" title="Joos-Weinberg equation">Joos–Weinberg equation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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