# Source field > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Source_field > Markdown URL: https://mediated.wiki/source/Source_field.md > Source: https://en.wikipedia.org/wiki/Source_field > Source revision: 1353521910 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Type of field appearing in the Lagrangian}} In [theoretical physics](/source/theoretical_physics), a source is an abstract concept, developed by [Julian Schwinger](/source/Julian_Schwinger), motivated by the physical effects of surrounding particles involved in creating or destroying another [particle](/source/Particle_physics).{{Cite journal |last=Schwinger |first=Julian |date=1966-12-23 |title=Particles and Sources |url=https://link.aps.org/doi/10.1103/PhysRev.152.1219 |journal=Physical Review |language=en |volume=152 |issue=4 |pages=1219–1226 |doi=10.1103/PhysRev.152.1219 |issn=0031-899X|url-access=subscription }} 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 [probability amplitude](/source/probability_amplitude) 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 [tensorial](/source/Tensor_field){{Cite journal |last=Schwinger |first=Julian |date=1968-09-25 |title=Sources and Gravitons |url=https://link.aps.org/doi/10.1103/PhysRev.173.1264 |journal=Physical Review |language=en |volume=173 |issue=5 |pages=1264–1272 |doi=10.1103/PhysRev.173.1264 |issn=0031-899X|url-access=subscription }} and [spinorial](/source/Spin_representation){{Cite journal |last=Schwinger |first=Julian |date=1967-06-25 |title=Sources and Electrodynamics |url=https://link.aps.org/doi/10.1103/PhysRev.158.1391 |journal=Physical Review |language=en |volume=158 |issue=5 |pages=1391–1407 |doi=10.1103/PhysRev.158.1391 |issn=0031-899X|url-access=subscription }} nature of the source. An example that Julian Schwinger referred to is the creation of \eta^* meson due to the mass correlations among five \pi mesons.{{Cite journal |last=Kalbfleisch |first=George R. |last2=Alvarez |first2=Luis W. |last3=Barbaro-Galtieri |first3=Angela |last4=Dahl |first4=Orin I. |last5=Eberhard |first5=Philippe |last6=Humphrey |first6=William E. |last7=Lindsey |first7=James S. |last8=Merrill |first8=Deane W. |last9=Murray |first9=Joseph J. |last10=Rittenberg |first10=Alan |last11=Ross |first11=Ronald R. |last12=Shafer |first12=Janice B. |last13=Shively |first13=Frank T. |last14=Siegel |first14=Daniel M. |last15=Smith |first15=Gerald A. |date=1964-05-04 |title=Observation of a Nonstrange Meson of Mass 959 MeV |url=https://link.aps.org/doi/10.1103/PhysRevLett.12.527 |journal=Physical Review Letters |language=en |volume=12 |issue=18 |pages=527–530 |doi=10.1103/PhysRevLett.12.527 |issn=0031-9007}} The same idea can be used to define '''source fields'''. Mathematically, a source field is a ''background'' field J coupled to the original field \phi as S_\text{source} = J\phi. This term appears in the action in [Richard Feynman](/source/Richard_Feynman)'s [path integral formulation](/source/path_integral_formulation) and is responsible for the theory interactions. In a collision reaction a source could be other particles in the collision.{{Cite book |last=Schwinger |first=Julian |title=Particles, sources, and fields |date=1998 |publisher=Advanced Book Program, Perseus Books |isbn=0-7382-0053-0 |location=Reading, Mass. |pages= |oclc=40544377}} Therefore, the source appears in the vacuum amplitude acting from both sides on the [Green's function correlator](/source/Correlation_function_(quantum_field_theory)) of the theory. [Schwinger's source theory](/source/Quantum_field_theory) stems from [Schwinger's quantum action principle](/source/Schwinger's_quantum_action_principle) and can be related to the path integral formulation as the variation with respect to the source per se \delta J corresponds to the field \phi, i.e.{{Citation |last=Milton |first=Kimball A. |title=Quantum Action Principle |date=2015 |url=https://link.springer.com/10.1007/978-3-319-20128-3_4 |work=Schwinger's Quantum Action Principle |series=SpringerBriefs in Physics |pages=31–50 |access-date=2023-05-06 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-20128-3_4 |isbn=978-3-319-20127-6|url-access=subscription }} \delta J = \int \mathcal{D}\phi \, \exp\left(-i\!\int\! d^4x \, J(x,t) \phi(x,t)\right). Also, a source acts [effectively](/source/Effective_action){{Cite book |last=Toms |first=David J. |url=https://www.cambridge.org/core/product/identifier/9780511585913/type/book |title=The Schwinger Action Principle and Effective Action |date=2007-11-15 |publisher=Cambridge University Press |isbn=978-0-521-87676-6 |edition=1 |doi=10.1017/cbo9780511585913.008}} 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 [partial differential equation](/source/partial_differential_equation)s) for \phi. When the field \phi is the [electromagnetic potential](/source/electromagnetic_potential) or the [metric tensor](/source/metric_tensor), the source field is the [electric current](/source/electric_current) or the [stress–energy tensor](/source/stress%E2%80%93energy_tensor), respectively.{{Cite book |last=Zee |first=A. |title=Quantum field theory in a nutshell |date=2010 |publisher=Princeton University Press |isbn=978-0-691-14034-6 |edition=2nd |location=Princeton, N.J. |oclc=318585662}}{{Cite journal |last=Weinberg |first=Steven |date=1965-05-24 |title=Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations |url=https://link.aps.org/doi/10.1103/PhysRev.138.B988 |journal=Physical Review |language=en |volume=138 |issue=4B |pages=B988–B1002 |doi=10.1103/PhysRev.138.B988 |issn=0031-899X|url-access=subscription }} In terms of the statistical and non-relativistic applications, Schwinger's source formulation plays crucial rules in understanding many non-equilibrium systems.{{Cite journal |last=Schwinger |first=Julian |date=May 1961 |title=Brownian Motion of a Quantum Oscillator |url=https://pubs.aip.org/aip/jmp/article/2/3/407-432/224719 |journal=Journal of Mathematical Physics |language=en |volume=2 |issue=3 |pages=407–432 |doi=10.1063/1.1703727 |issn=0022-2488|url-access=subscription }}{{Cite book |last=Kamenev |first=Alex |title=Field theory of non-equilibrium systems |date=2011 |isbn=978-1-139-11485-1 |location=Cambridge |oclc=760413528}} Source theory is theoretically significant as it needs neither divergence regularizations nor renormalization. == Relation between path integral formulation and source formulation == In the Feynman's path integral formulation with normalization \mathcal{N}\equiv Z[J=0], the [partition function](/source/Partition_function_(quantum_field_theory)){{Cite book |last=Ryder |first=Lewis |title=Quantum Field Theory |publisher=Cambridge University Press |year=1996 |isbn=978-0-521-47814-4 |edition=2nd |pages=175}} is given by Z[J] = \mathcal{N} \int \mathcal{D}\phi \, \exp\left[-i\left(\int dt ~ \mathcal{L}(t;\phi,\dot{\phi})+ \int d^4x \, J(x,t) \phi(x,t)\right)\right]. One can expand the current term in the exponent \mathcal{N} \int \mathcal{D}\phi ~ \exp\left(-i \int d^4x \, J(x,t)\phi(x,t)\right) = \mathcal{N} \sum^{\infty}_{n=0} \frac{i^n}{n!} \int d^4x_1 \cdots \int d^4x_n J(x_1) \cdots J(x_1) \left\langle \phi(x_1) \cdots \phi(x_n) \right\rangle to generate [Green's functions](/source/Propagator) ([correlators](/source/Correlation_function_(quantum_field_theory))) 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}, where the fields inside the expectation function \langle\phi(x_1)\cdots\phi(x_n)\rangle are in their [Heisenberg picture](/source/Heisenberg_picture)s. On the other hand, one can define the correlation functions for higher order terms, e.g., for \frac{1}{2} m^2 \phi^2 term, the [coupling constant](/source/coupling_constant) like m is promoted to a spacetime-dependent source \mu(x) such that 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. One implements the quantum variational methodology to realize that J is an ''external driving source'' of \phi. From the perspectives of [probability theory](/source/probability_theory), Z[J] can be seen as the expectation value of the function e^{J\phi} . This motivates considering the Hamiltonian of forced harmonic oscillator as a [toy model](/source/toy_model) \mathcal{H} = E \hat{a}^{\dagger} \hat{a} - \frac{1}{\sqrt{2E}} \left(J\hat{a}^{\dagger} + J^{*}\hat{a}\right) where E^2 = m^2 + \mathbf{p}^2 . In fact, the current is real, that is J=J^{*}.{{Cite book |last=Nastase |first=Horatiu |url=https://www.cambridge.org/highereducation/product/9781108624992/book |title=Introduction to Quantum Field Theory |date=2019-10-17 |publisher=Cambridge University Press |isbn=978-1-108-62499-2 |edition=1 |doi=10.1017/9781108624992.009|s2cid=241983970 }} And the Lagrangian is \mathcal{L}=i\hat{a}^{\dagger}\partial_0(\hat{a})-\mathcal{H} . From now on we drop the hat and the asterisk. Remember that [canonical quantization](/source/Canonical_quantization) states \phi\sim (a^{\dagger}+a). In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives \delta_J\langle0,x'_0|0,x''_0\rangle_J = i \left\langle0,x'_0\right| \int^{x'_0}_{x''_0}dx_0 ~ \delta J{\left(a^{\dagger}+a\right)} {\left|0,x''_0\right\rangle}_J, where x_0'>x_0> x_0'' . As the integral is in the [time domain](/source/time_domain), one can [Fourier transform](/source/Fourier_transform) it, together with the creation/annihilation operators, such that the amplitude eventually becomes {\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)}. It is easy to notice that there is a singularity at f=E . Then, we can exploit the i\varepsilon-prescription and shift the pole f-E+i\varepsilon such that for x_0> x_0' the Green's function is revealed \begin{align} &{\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^{-i f \left(x_0 - x'_0\right)}}{f - E + i \varepsilon} \end{align} The last result is the Schwinger's source theory for interacting scalar fields and can be generalized to any spacetime regions. The discussed examples below follow the metric \eta_{\mu\nu}=\text{diag}(1,-1,-1,-1) . == Source theory for scalar fields == [Causal perturbation theory](/source/Causal_perturbation_theory) explains how sources weakly act. For a weak source emitting spin-0 particles J_e by acting on the [vacuum state](/source/Quantum_vacuum_state) with a probability amplitude \langle 0|0\rangle_{J_{e}}\sim1, a single particle with momentum p and amplitude \langle p|0\rangle_{J_{e}} is created within certain spacetime region x'. Then, another weak source J_a absorbs that single particle within another spacetime region x such that the amplitude becomes \langle 0|p\rangle_{J_{a}}. Thus, the full vacuum amplitude is given by {\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') where \Delta(x-x') is the propagator (correlator) of the sources. The second term of the last amplitude defines the [partition function of free scalar field theory](/source/Partition_function_(quantum_field_theory)). And for some interaction theory, the Lagrangian of a scalar field \phi coupled to a current J is given by{{Cite book |last=Ramond |first=Pierre |title=Field Theory: A Modern Primer |publisher=Routledge |year=2020 |isbn=978-0-367-15491-2 |edition=2nd}} \mathcal{L} = \tfrac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \tfrac{1}{2} m^2 \phi^2 + J\phi. If one adds -i\varepsilon to the mass term then Fourier transforms both J and \phi to the momentum space, the vacuum amplitude becomes \langle 0|0\rangle = \exp{\left(\frac{i}{2} \int \frac{d^4p}{{\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)}, where \tilde{\phi}(p) = \phi(p) + \frac{J(p)}{p_{\mu} p^{\mu} - m^2 + i \varepsilon}. It is easy to notice that the \tilde{\phi}(p) \left(p_{\mu} p^{\mu} - m^2 + i \varepsilon\right) \tilde{\phi}(-p) term in the amplitude above can be Fourier transformed into \tilde{\phi}(x) \left(\Box + m^2\right) \tilde{\phi}(x) = \tilde{\phi}(x) \, J(x) , i.e., the equation of motion \left(\Box + m^2\right) \tilde{\phi} = J . As the variation of the free action, that of the term \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^2 \phi^2 , yields the equation of motion, one can redefine the Green's function as the inverse of the operator G(x_1,x_2) \equiv {\left(\Box + m^2\right)}^{-1} such that \left(\Box_{x_1} + m^2\right) G(x_1,x_2) = \delta(x_1-x_2) [if and only if](/source/if_and_only_if) \left(p_{\mu} p^{\mu} - m^2\right) G(p) = 1, which is a direct application of the general role of functional derivative \frac{\delta J(x_2)}{\delta J(x_1)}=\delta(x_1-x_2). Thus, the [generating functional](/source/Partition_function_(quantum_field_theory)) is obtained from the partition function as follows. The last result allows us to read the partition function as Z[J] = Z[0] \exp\left(\tfrac{i}{2} \left\langle J(y) \Delta(y-y') J(y')\right\rangle\right) , where 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), and \langle J(y)\Delta(y-y')J(y')\rangle is the vacuum amplitude derived by the source \langle0|0\rangle_{J} . Consequently, the propagator is defined by varying the partition function as follows. \begin{align} {\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^4y' \, \Delta(x'-y') J(y') + \int d^4y \, 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{align} This motivates discussing the mean field approximation below. == Effective action, mean field approximation, and vertex functions == Based on Schwinger's source theory, [Steven Weinberg](/source/Steven_Weinberg) established the foundations of the effective field theory, which is widely appreciated among physicists. Despite the "[shoes incident](/source/Julian_Schwinger)", Weinberg gave the credit to Schwinger for catalyzing this theoretical framework.{{Cite journal |last=Weinberg |first=Steven |date=1979 |title=Phenomenological Lagrangians |url=https://linkinghub.elsevier.com/retrieve/pii/0378437179902231 |journal=Physica A: Statistical Mechanics and Its Applications |language= |volume=96 |issue=1–2 |pages=327–340 |doi=10.1016/0378-4371(79)90223-1|url-access=subscription }} All Green's functions may be formally found via [Taylor expansion](/source/Taylor_expansion) of the [partition sum](/source/partition_sum) considered as a function of the source fields. This method is commonly used in the [path integral formulation](/source/path_integral_formulation) of [quantum field theory](/source/quantum_field_theory). 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 [Wick-rotated](/source/Wick_rotation) amplitude W[J]=-i\ln(\langle 0|0 \rangle_{J}) , the partition function becomes Z[J]=e^{iW[J]} . One can introduce F[J]=iW[J] , which behaves as [Helmholtz free energy](/source/Helmholtz_free_energy) in [thermal field theories](/source/Thermal_quantum_field_theory),{{Cite book |last=Fradkin |first=Eduardo |title=Quantum Field Theory: An Integrated Approach |publisher=Princeton University Press |year=2021 |isbn=978-0-691-14908-0 |pages=331–341}} to absorb the [complex number](/source/complex_number), and hence \ln Z[J]=F[J] . The function F[J] is also called ''reduced quantum action''.{{Cite book |last=Zeidler |first=Eberhard |title=Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists |publisher=Springer |year=2006 |isbn=978-3-540-34762-0 |pages=455}} And with help of [Legendre transform](/source/Legendre_transformation), we can invent a "new" ''effective energy'' functional,{{Cite book |last1=Kleinert |first1=Hagen |title=Critical Properties of phi^4-Theories |last2=Schulte-Frohlinde |first2=Verena |publisher=World Scientific Publishing Co |year=2001 |isbn=978-981-279-994-4 |pages=68–70}} or [effective action](/source/effective_action), as \Gamma[\bar{\phi}] = W[J] - \int d^4x \, J(x) \bar{\phi}(x), with the transforms{{Cite journal |last=Jona-Lasinio |first=G. |date=1964-12-01 |title=Relativistic field theories with symmetry-breaking solutions |journal=Il Nuovo Cimento (1955-1965) |language=en |volume=34 |issue=6 |pages=1790–1795 |doi=10.1007/BF02750573 |s2cid=121276897 |issn=1827-6121}} \begin{align} &\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{align} The integration in the definition of the effective action is allowed to be replaced with sum over \phi, i.e., \Gamma[\bar{\phi}] = W[J] - J_a(x) \bar{\phi}^a(x) .{{Cite book |last1=Esposito |first1=Giampiero |url=http://link.springer.com/10.1007/978-94-011-5806-0 |title=Euclidean Quantum Gravity on Manifolds with Boundary |last2=Kamenshchik |first2=Alexander Yu. |last3=Pollifrone |first3=Giuseppe |date=1997 |publisher=Springer Netherlands |isbn=978-94-010-6452-1 |location=Dordrecht |language=en |doi=10.1007/978-94-011-5806-0}} The last equation resembles the thermodynamical relation F=E-TS between Helmholtz free energy and entropy. It is now clear that thermal and statistical field theories stem fundamentally from [functional integration](/source/functional_integration)s and [functional derivative](/source/functional_derivative)s. Back to the Legendre transforms, The \langle\phi\rangle is called ''[mean field](/source/Mean-field_theory)'' obviously because \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}}, while \bar{\phi} is a [background classical field](/source/Background_field_method). A field \phi is decomposed into a classical part \bar{\phi} and fluctuation part \eta, i.e., \phi=\bar{\phi}+\eta, so the vacuum amplitude can be reintroduced as 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, and any function \mathcal{F}[\phi] is defined as \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, where S[\phi] is the action of the free Lagrangian. The last two integrals are the pillars of any effective field theory. This construction is indispensable in studying scattering ([LSZ reduction formula](/source/LSZ_reduction_formula)), [spontaneous symmetry breaking](/source/spontaneous_symmetry_breaking),{{Citation |last1=Farhi |first1=E. |title=Dynamical Gauge Symmetry Breaking |date=January 1982 |url=https://www.worldscientific.com/doi/10.1142/9789814412698_0001 |work= |pages=1–14 |access-date=2023-05-17 |publisher=WORLD SCIENTIFIC |doi=10.1142/9789814412698_0001 |isbn=978-9971-950-24-8 |last2=Jackiw |first2=R.|url-access=subscription }} [Ward identities](/source/Ward_identities), [nonlinear sigma models](/source/Non-linear_sigma_model), and [low-energy effective theories](/source/Effective_field_theory). Additionally, this theoretical framework initiates line of thoughts, publicized mainly be [Bryce DeWitt](/source/Bryce_DeWitt) who was a PhD student of Schwinger, on developing a [canonical quantized](/source/Canonical_quantum_gravity) effective theory for [quantum gravity](/source/quantum_gravity).{{Cite book |title=Quantum theory of gravity: essays in honor of the 60. birthday of Bryce S. DeWitt |date=1984 |publisher=Hilger |isbn=978-0-85274-755-1 |editor-last=Christensen |editor-first=Steven M. |location=Bristol |editor-last2=DeWitt |editor-first2=Bryce S.}} Back to Green functions of the actions. Since \Gamma[\bar{\phi}] is the Legendre transform of F[J], and F[J] defines N-points ''[connected](/source/Ursell_function)'' correlator G^{N,~c}_{F[J]}=\frac{\delta F[J]}{\delta J(x_1)\cdots \delta J(x_N)}\Big|_{J=0}, then the corresponding correlator obtained from F[J], known as [vertex function](/source/vertex_function), is given by G^{N,~c}_{\Gamma[J]} = \left.\frac{\delta \Gamma[\bar{\phi}]}{\delta \bar{\phi}(x_1) \cdots \delta\bar{\phi}(x_N)}\right|_{\bar{\phi}=\langle\phi\rangle}. Consequently, in the one particle irreducible graphs (usually acronymized as '''1PI'''), the connected 2-point F -correlator is defined as the inverse of the 2-point \Gamma -correlator, i.e., the usual reduced correlation is G^{(2)}_{F[J]}=\frac{\delta \bar{\phi}(x_1)}{\delta J(x_2)}\Big|_{J=0}=\frac{1}{p_{\mu}p^{\mu}-m^2} , and the effective correlation is G^{(2)}_{\Gamma[\phi]}=\frac{\delta J(x_1)}{\delta \bar{\phi}(x_2)}\Big|_{\bar{\phi}=\langle\phi\rangle}=p_{\mu}p^{\mu}-m^2 . For J_i =J(x_i), the most general relations between the N-points connected F[J] and Z[J] are obtained from [Faà di Bruno's formula](/source/Fa%C3%A0_di_Bruno's_formula) \begin{align} \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{align} and \begin{align} \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{align} == Source theory for fields == === Vector fields === For a weak source producing a [massive spin-1 particle](/source/Proca_action) with a general current J=J_e+J_a acting on different causal spacetime points x_0> x_0', the vacuum amplitude is \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)} In momentum space, the spin-1 particle with rest mass m has a definite momentum p_{\mu}=(m,0,0,0) in its [rest frame](/source/rest_frame), i.e. p_{\mu}p^{\mu}=m^2 . Then, the amplitude gives \begin{alignat}{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{alignat} where \eta_{\mu\nu}=\text{diag}(1,-1,-1,-1) and (J_{\mu}(p))^T is the transpose of J_{\mu}(p) . The last result matches with the used propagator in the vacuum amplitude in the configuration space, that is, \left\langle 0\right| T A_{\mu}(x) A_{\nu}(x') \left|0\right\rangle = -i\int\frac{d^4p}{{\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^{i p^{\mu}\left(x_{\mu} - x'_{\mu}\right)}. When \xi = 1 , the chosen Feynman–'t Hooft [gauge-fixing](/source/Propagator) makes the spin-1 massless. And when \xi = 0 , the chosen Landau [gauge-fixing](/source/gauge_fixing) makes the spin-1 massive.{{Cite book |last=Bogoli︠u︡bov |first=N. N. |title=Quantum fields |date=1982 |publisher=Benjamin/Cummings Pub. Co., Advanced Book Program/World Science Division |others=D. V. Shirkov |isbn=0-8053-0983-7 |location=Reading, MA |oclc=8388186}} The massless case is obvious as studied in [quantum electrodynamics](/source/quantum_electrodynamics). 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 [Belinfante-Rosenfeld tensor](/source/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor) is improved so it ends up being conserved. And to get the equation of motion for the massive vector, one can define 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]. One can apply integration by part on the second term then single out \int dx J_{\mu}(x) to get a definition of the massive spin-1 field 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]. Additionally, the equation above says that \partial_{\mu}A^{\mu} = \tfrac{1}{m^2} \partial_{\mu}J^{\mu} . Thus, the equation of motion can be written in any of the following forms \begin{align} &\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{align} === Massive totally symmetric spin-2 fields === For a weak source in a [flat Minkowski background](/source/Minkowski_space), producing then absorbing a [massive spin-2 particle](/source/Massive_gravity) with a general redefined [energy-momentum tensor](/source/Stress%E2%80%93energy_tensor), acting as a current, \bar{T}^{\mu\nu} = T^{\mu\nu} - \tfrac{1}{3} \eta_{\mu\alpha} \bar{\eta}_{\nu\beta}T^{\alpha\beta}, where \bar{\eta}_{\mu\nu}(p) = \eta_{\mu\nu} - \tfrac{1}{m^2} p_{\mu}p_{\nu} is the [vacuum polarization tensor](/source/Vacuum_polarization), the vacuum amplitude in a compact form is \begin{align} \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{align} or \begin{align} \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{align} This amplitude in momentum space gives (transpose is imbedded) \begin{align} \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{align} \begin{align} \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{align} And with help of symmetric properties of the source, the last result can be written as T^{\mu\nu}(p)\Pi_{\mu\nu\kappa\lambda}(p)T^{\kappa\lambda}(p) , where the projection operator, or the Fourier transform of [Jacobi field](/source/Jacobi_field) operator obtained by applying [Peierls braket](/source/Peierls_bracket) on [Schwinger's variational principle](/source/Schwinger's_variational_principle),{{Cite book |last=DeWitt-Morette |first=Cecile |title=Quantum Field Theory: Perspective and Prospective |date=1999 |publisher=Springer Netherlands |others=Jean Bernard Zuber |isbn=978-94-011-4542-8 |location=Dordrecht |oclc=840310329}} is \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). In N-dimensional flat spacetime, 2/3 is replaced by 2/(N−1).{{Cite book |last=DeWitt |first=Bryce S. |title=The global approach to quantum field theory |date=2003 |publisher=Oxford University Press |isbn=0-19-851093-4 |location=Oxford |oclc=50323237}} And for [massless spin-2 fields](/source/Linearized_gravity), the [projection operator](/source/Propagator) is defined as \Pi^{m=0}_{\mu\nu\kappa\lambda} = \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) . Together with help of [Ward-Takahashi identity](/source/Ward%E2%80%93Takahashi_identity), 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 \bar{\eta}_{\nu\beta} and the improved energy momentum tensor \bar{T}^{\mu\nu} appear in the early versions of [massive gravity theories](/source/Massive_gravity).{{Cite journal |last1=Ogievetsky |first1=V.I |last2=Polubarinov |first2=I.V |date=November 1965 |title=Interacting field of spin 2 and the einstein equations |url=https://linkinghub.elsevier.com/retrieve/pii/0003491665900771 |journal=Annals of Physics |language=en |volume=35 |issue=2 |pages=167–208 |doi=10.1016/0003-4916(65)90077-1|url-access=subscription }}{{Cite journal |last1=Freund |first1=Peter G. O. |last2=Maheshwari |first2=Amar |last3=Schonberg |first3=Edmond |date=August 1969 |title=Finite-Range Gravitation |journal=The Astrophysical Journal |language=en |volume=157 |page=857 |doi=10.1086/150118 |issn=0004-637X|doi-access=free }} Interestingly, massive gravity theories have not been widely appreciated until recently due to apparent inconsistencies obtained in the early 1970s studies of the exchange of a single spin-2 field between two sources. But in 2010 the dRGT approach{{Cite journal |last1=de Rham |first1=Claudia |last2=Gabadadze |first2=Gregory |date=2010-08-10 |title=Generalization of the Fierz-Pauli action |url=https://link.aps.org/doi/10.1103/PhysRevD.82.044020 |journal=Physical Review D |volume=82 |issue=4 |article-number=044020 |doi=10.1103/PhysRevD.82.044020|arxiv=1007.0443 |s2cid=119289878 }} of exploiting [Stueckelberg field redefinition](/source/Stueckelberg_action) led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier. If one looks at \langle0|0\rangle_{T} and follows the same procedure used to define massive spin-1 fields, then it is easy to define massive spin-2 fields as \begin{align} 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{align} The corresponding divergence condition is read \partial^{\mu}h_{\mu\nu}-\partial_{\nu}h=\frac{1}{m^2}\partial^{\mu}T_{\mu\nu}, where the current \partial^{\mu}T_{\mu\nu} 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 \mathfrak{T}_{\mu\nu}=T_{\mu\nu}-\frac{1}{4}\eta_{\mu\nu}\mathfrak{T} such that \partial^{\mu}\mathfrak{T}_{\mu\nu}=0 according to [Belinfante-Rosenfeld](/source/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor) construction. Thus, the equation of motion \begin{align} \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{align} becomes \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}. One can use the divergence condition to decouple the non-physical fields \partial^{\mu}h_{\mu\nu} and h, so the equation of motion is simplified as{{Cite journal |last1=Van Kortryk |first1=Thomas |last2=Curtright |first2=Thomas |last3=Alshal |first3=Hassan |date=2021 |title=On Enceladian Fields |url=http://www.bjp-bg.com/paper1.php?id=1247 |journal=Bulgarian Journal of Physics |volume=48 |issue=2 |pages=138–145}} \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}. === Massive totally symmetric arbitrary integer spin fields === One can generalize T^{\mu\nu}(p) source to become S^{\mu_1\cdots\mu_{\ell}}(p) [higher-spin](/source/Higher-spin_theory) source such that T^{\mu\nu}(p)\Pi_{\mu\nu\kappa\lambda}(p)T^{\kappa\lambda}(p) becomes 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) . The generalized projection operator also helps generalizing the electromagnetic polarization vector e^{\mu}_{m}(p) of the [quantized electromagnetic vector potential](/source/Quantization_of_the_electromagnetic_field) as follows. For spacetime points x and x' , the [addition theorem of spherical harmonics](/source/Spherical_harmonics) states that 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^{\ell}_{m=-\ell}Y_{\ell,m}(x)Y_{\ell,m}^{*}(x'). Also, [the representation theory](/source/Spherical_harmonics) of the space of complex-valued [homogeneous polynomial](/source/homogeneous_polynomial)s of degree \ell on a unit (N-1)-sphere defines the polarization tensor as{{Citation |last1=Gallier |first1=Jean |title=Spherical Harmonics and Linear Representations of Lie Groups |date=2020 |url=http://link.springer.com/10.1007/978-3-030-46047-1_7 |work=Differential Geometry and Lie Groups |volume=13 |pages=265–360 |access-date=2023-05-08 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-46047-1_7 |isbn=978-3-030-46046-4 |last2=Quaintance |first2=Jocelyn|series=Geometry and Computing |s2cid=122806576 |url-access=subscription }}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}.Then, the generalized polarization vector ise^{\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). And the projection operator can be defined as \Pi^{\mu_1\cdots\mu_{\ell}\nu_1\cdots\nu_{\ell}}(p)=\sum\limits^{\ell}_{m=-\ell}[e^{\mu_1\cdots \mu_{\ell}}_{m}(p)]~[e^{\nu_1\cdots \nu_{\ell}}_{m}(p)]^*. 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 \Delta(x-x') in configuration space, we write \langle0|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]}. === Mixed symmetric arbitrary spin fields === Also, it is theoretically consistent to generalize the source theory to describe hypothetical gauge fields with [antisymmetric](/source/Kalb%E2%80%93Ramond_field) and mixed symmetric properties in arbitrary dimensions and [arbitrary spins](/source/Higher-spin_theory). 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 [Curtright field](/source/Curtright_field) T_{[\mu\nu]\lambda} and a source S_{[\mu\nu]\lambda}=\partial_{\alpha}\partial^{\alpha}T_{[\mu\nu]\lambda} , the vacuum amplitude is\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)} which for a theory in N=4 makes the source eventually reveal that it is a theory of a non physical field.{{Cite journal |last=Curtright |first=Thomas |date=1985-12-26 |title=Generalized gauge fields |journal=Physics Letters B |language=en |volume=165 |issue=4 |pages=304–308 |doi=10.1016/0370-2693(85)91235-3 |issn=0370-2693}} However, the [massive version](/source/Dual_graviton) survives in N≥5. === Arbitrary half-integer spin fields === For spin-{{1/2}} [fermion propagator](/source/Propagator) S(x-x')=(p \!\!\!/+m)\Delta(x-x') and current J=J_e+J_a as defined above, the vacuum amplitude is \begin{align} \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{align} In momentum space the reduced amplitude is given by W_{\frac{1}{2}}=-\frac{1}{3}\int \frac{d^4p}{(2\pi)^4}~J(-p)\left[\gamma^0\frac{p \!\!\!/+m}{p^2-m^2}\right]~J(p). For spin-{{Frac|3|2|}} [Rarita-Schwinger](/source/Rarita%E2%80%93Schwinger_equation) fermions, \Pi_{\mu\nu} = \bar{\eta}_{\mu\nu} - \tfrac{1}{3} \gamma^{\alpha} \bar{\eta}_{\alpha\mu} \gamma^{\beta} \bar{\eta}_{\beta\nu}. Then, one can use \gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu} and the on-shell p\!\!\!/=-m to get \begin{align} W_{\frac{3}{2}} &= - \frac{2}{5} \int \frac{d^4p}{{\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^4p}{{\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{align} One can replace the reduced metric \bar{\eta}_{\mu\nu} with the usual one \eta_{\mu\nu} if the source J_{\mu} is replaced with \bar{J}_{\mu}(p)=\frac{2}{5}\gamma^{\alpha}\Pi_{\mu\alpha\nu\beta}\gamma^{\beta}J^{\nu}(p). For spin-(j + \tfrac{1}{2}) , the above results can be generalized to W_{j+\frac{1}{2}} = - \frac{j+1}{2j+3} \int \frac{d^4p}{{\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). The factor \frac{j+1}{2j+3} 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. These conditions can be derived from the Fierz-Pauli{{Cite journal |date=1939-11-28 |title=On relativistic wave equations for particles of arbitrary spin in an electromagnetic field |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1939.0140 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |language=en |volume=173 |issue=953 |pages=211–232 |doi=10.1098/rspa.1939.0140 |s2cid=123189221 |issn=0080-4630|url-access=subscription }} and the Fang-Fronsdal{{Cite journal |last=Fronsdal |first=Christian |date=1978-11-15 |title=Massless fields with integer spin |url=https://link.aps.org/doi/10.1103/PhysRevD.18.3624 |journal=Physical Review D |volume=18 |issue=10 |pages=3624–3629 |doi=10.1103/PhysRevD.18.3624|url-access=subscription }}{{Cite journal |last1=Fang |first1=J. |last2=Fronsdal |first2=C. |date=1978-11-15 |title=Massless fields with half-integral spin |url=https://link.aps.org/doi/10.1103/PhysRevD.18.3630 |journal=Physical Review D |volume=18 |issue=10 |pages=3630–3633 |doi=10.1103/PhysRevD.18.3630|url-access=subscription }} conditions on the fields themselves. The Lagrangian formulations of massive fields and their conditions were studied by Lambodar Singh and [Carl Hagen](/source/C._R._Hagen).{{Cite journal |last1=Singh |first1=L. P. S. |last2=Hagen |first2=C. R. |date=1974-02-15 |title=Lagrangian formulation for arbitrary spin. I. The boson case |url=https://link.aps.org/doi/10.1103/PhysRevD.9.898 |journal=Physical Review D |language=en |volume=9 |issue=4 |pages=898–909 |doi=10.1103/PhysRevD.9.898 |issn=0556-2821|url-access=subscription }}{{Cite journal |last1=Singh |first1=L. P. S. |last2=Hagen |first2=C. R. |date=1974-02-15 |title=Lagrangian formulation for arbitrary spin. II. The fermion case |url=https://link.aps.org/doi/10.1103/PhysRevD.9.910 |journal=Physical Review D |language=en |volume=9 |issue=4 |pages=910–920 |doi=10.1103/PhysRevD.9.910 |issn=0556-2821|url-access=subscription }} The non-relativistic version of the projection operators, developed by Charles Zemach who is another student of Schwinger,{{Cite journal |last=Zemach |first=Charles |date=1965-10-11 |title=Use of Angular-Momentum Tensors |url=https://link.aps.org/doi/10.1103/PhysRev.140.B97 |journal=Physical Review |volume=140 |issue=1B |pages=B97–B108 |doi=10.1103/PhysRev.140.B97|url-access=subscription }} is used heavily in hadron spectroscopy. Zemach's method could be relativistically improved to render the covariant projection operators.{{Cite journal |last1=Filippini |first1=V. |last2=Fontana |first2=A. |last3=Rotondi |first3=A. |date=1995-03-01 |title=Covariant spin tensors in meson spectroscopy |url=https://link.aps.org/doi/10.1103/PhysRevD.51.2247 |journal=Physical Review D |volume=51 |issue=5 |pages=2247–2261 |doi=10.1103/PhysRevD.51.2247|pmid=10018695 |url-access=subscription }}{{Cite journal |last=Chung |first=S. U. |date=1998-01-01 |title=General formulation of covariant helicity-coupling amplitudes |url=https://link.aps.org/doi/10.1103/PhysRevD.57.431 |journal=Physical Review D |volume=57 |issue=1 |pages=431–442 |doi=10.1103/PhysRevD.57.431|url-access=subscription }} == See also == * [Keldysh-Schwinger formalism](/source/Keldysh_formalism) * [Schwinger function](/source/Schwinger_function) * [Wigner-Bargmann equations](/source/Bargmann%E2%80%93Wigner_equations) * [Joos–Weinberg equation](/source/Joos-Weinberg_equation) == References == {{reflist}} {{DEFAULTSORT:Source Field}} Category:Quantum field theory --- Adapted from the Wikipedia article [Source field](https://en.wikipedia.org/wiki/Source_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Source_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.