Source field#Source theory for massive totally symmetric spin-2 fields

In the Feynman's path integral formulation with normalization $\backslash mathcal\{N\}\backslash equiv\; Z[J=0]$, the partition function{{Cite book |last=Ryder |first=Lewis |title=Quantum Field Theory |publisher=Cambridge University Press |year=1996 |isbn=9780521478144 |edition=2nd |pages=175}} is given by

$$Z[J]\; =\; \backslash mathcal\{N\}\; \backslash int\; \backslash mathcal\{D\}\backslash phi\; \backslash ,\; \backslash exp\backslash left[-i\backslash left(\backslash int\; dt\; ~\; \backslash mathcal\{L\}(t;\backslash phi,\backslash dot\{\backslash phi\})+\; \backslash int\; d^4x\; \backslash ,\; J(x,t)\; \backslash phi(x,t)\backslash right)\backslash right].$$

One can expand the current term in the exponent $$\backslash mathcal\{N\}\; \backslash int\; \backslash mathcal\{D\}\backslash phi\; ~\; \backslash exp\backslash left(-i\; \backslash int\; d^4x\; \backslash ,\; J(x,t)\backslash phi(x,t)\backslash right)$$

= \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 (correlators) $$G(t\_1,\backslash cdots,t\_n)\; =\; \{\backslash left(-i\backslash right)\}^n\; \backslash left.\backslash frac\{\backslash delta^n\; Z[J]\}\{\backslash delta\; J(t\_1)\; \backslash cdots\; \backslash delta\; J(t\_n)\}\backslash right|\_\{J=0\},$$ where the fields inside the expectation function $\backslash langle\backslash phi(x\_1)\backslash cdots\backslash phi(x\_n)\backslash rangle$ are in their Heisenberg pictures. On the other hand, one can define the correlation functions for higher order terms, e.g., for $\backslash frac\{1\}\{2\}\; m^2\; \backslash phi^2$ term, the coupling constant like $m$ is promoted to a spacetime-dependent source $\backslash mu(x)$ such that $$i\; \backslash frac\{1\}\{\backslash mathcal\{N\}\}\; \backslash left.\backslash frac\{\backslash delta\; \}\{\backslash delta\; \backslash mu^2\}\; Z[J,\backslash mu]\; \backslash right|\_\{m^2=\backslash mu^2\}\; =\; \backslash left\backslash langle\; \backslash tfrac\{1\}\{2\}\; \backslash phi^2\; \backslash right\backslash rangle.$$

One implements the quantum variational methodology to realize that $J$ is an external driving source of $\backslash phi$. From the perspectives of probability theory, $Z[J]$ can be seen as the expectation value of the function $e^\{J\backslash phi\}$. This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model

$$\backslash mathcal\{H\}\; =\; E\; \backslash hat\{a\}^\{\backslash dagger\}\; \backslash hat\{a\}\; -\; \backslash frac\{1\}\{\backslash sqrt\{2E\}\}\; \backslash left(J\backslash hat\{a\}^\{\backslash dagger\}\; +\; J^\{*\}\backslash hat\{a\}\backslash right)$$ where $E^2\; =\; m^2\; +\; \backslash mathbf\{p\}^2$.

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 $\backslash mathcal\{L\}=i\backslash hat\{a\}^\{\backslash dagger\}\backslash partial\_0(\backslash hat\{a\})-\backslash mathcal\{H\}$ . From now on we drop the hat and the asterisk. Remember that canonical quantization states $\backslash phi\backslash sim\; (a^\{\backslash dagger\}+a)$. In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives

$$\backslash delta\_J\backslash langle0,x\text{'}\_0|0,x$$_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 $x\_0\text{'}>x\_0>\; x\_0$ .

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

$$\{\backslash left\backslash langle\; 0,\; x\text{'}\_0\; |\; 0,\; x\text{'}\text{'}\_0\; \backslash right\backslash rangle\}\_J\; =\; \backslash exp\{\backslash left(\backslash frac\{i\}\{2\backslash pi\}\backslash int\; df\; ~\; J(f)\; \backslash frac\{1\}\{f-E\}\; J(-f)\backslash right)\}.$$

It is easy to notice that there is a singularity at $f=E$ . Then, we can exploit the $i\backslash varepsilon$-prescription and shift the pole $f-E+i\backslash varepsilon$ such that for $x\_0>\; x\_0\text{'}$ the Green's function is revealed

$$\backslash begin\{align\}$$

&{\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 $\backslash eta\_\{\backslash mu\backslash nu\}=\backslash text\{diag\}(1,-1,-1,-1)$.

Causal perturbation theory explains how sources weakly act. For a weak source emitting spin-0 particles $J\_e$ by acting on the vacuum state with a probability amplitude $\backslash langle\; 0|0\backslash rangle\_\{J\_\{e\}\}\backslash sim1$, a single particle with momentum $p$ and amplitude $\backslash langle\; p|0\backslash rangle\_\{J\_\{e\}\}$ is created within certain spacetime region $x\text{'}$. Then, another weak source $J\_a$ absorbs that single particle within another spacetime region $x$ such that the amplitude becomes $\backslash langle\; 0|p\backslash rangle\_\{J\_\{a\}\}$. Thus, the full vacuum amplitude is given by

$$\{\backslash left\backslash langle\; 0\; |\; 0\; \backslash right\backslash rangle\}\_\{J\_e\; +\; J\_a\}\; \backslash sim\; 1\; +\; \backslash frac\{i\}\{2\}\; \backslash int\; dx\; \backslash ,\; dx\text{'}\; \backslash ,\; J\_a(x)\; \backslash Delta(x-x\text{'})\; J\_e(x\text{'})$$

where $\backslash Delta(x-x\text{'})$ is the propagator (correlator) of the sources. The second term of the last amplitude defines the partition function of free scalar field theory. And for some interaction theory, the Lagrangian of a scalar field $\backslash phi$ 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=978-0367154912 |edition=2nd}}

$$\backslash mathcal\{L\}\; =\; \backslash tfrac\{1\}\{2\}\; \backslash partial\_\{\backslash mu\}\; \backslash phi\; \backslash partial^\{\backslash mu\}\; \backslash phi\; -\; \backslash tfrac\{1\}\{2\}\; m^2\; \backslash phi^2\; +\; J\backslash phi.$$

If one adds $-i\backslash varepsilon$ to the mass term then Fourier transforms both $J$ and $\backslash phi$ to the momentum space, the vacuum amplitude becomes

$$\backslash langle\; 0|0\backslash rangle\; =\; \backslash exp\{\backslash left(\backslash frac\{i\}\{2\}\; \backslash int\; \backslash frac\{d^4p\}\{\{\backslash left(2\backslash pi\backslash right)\}^4\}\; \backslash left[$$

\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 $$\backslash tilde\{\backslash phi\}(p)\; =\; \backslash phi(p)\; +\; \backslash frac\{J(p)\}\{p\_\{\backslash mu\}\; p^\{\backslash mu\}\; -\; m^2\; +\; i\; \backslash varepsilon\}.$$ It is easy to notice that the $\backslash tilde\{\backslash phi\}(p)\; \backslash left(p\_\{\backslash mu\}\; p^\{\backslash mu\}\; -\; m^2\; +\; i\; \backslash varepsilon\backslash right)\; \backslash tilde\{\backslash phi\}(-p)$ term in the amplitude above can be Fourier transformed into $\backslash tilde\{\backslash phi\}(x)\; \backslash left(\backslash Box\; +\; m^2\backslash right)\; \backslash tilde\{\backslash phi\}(x)\; =\; \backslash tilde\{\backslash phi\}(x)\; \backslash ,\; J(x)$, i.e., the equation of motion $\backslash left(\backslash Box\; +\; m^2\backslash right)\; \backslash tilde\{\backslash phi\}\; =\; J$. As the variation of the free action, that of the term $\backslash frac\{1\}\{2\}\; \backslash partial\_\{\backslash mu\}\; \backslash phi\; \backslash partial^\{\backslash mu\}\; \backslash phi\; -\; \backslash frac\{1\}\{2\}\; m^2\; \backslash phi^2$, yields the equation of motion, one can redefine the Green's function as the inverse of the operator $G(x\_1,x\_2)\; \backslash equiv\; \{\backslash left(\backslash Box\; +\; m^2\backslash right)\}^\{-1\}$ such that $\backslash left(\backslash Box\_\{x\_1\}\; +\; m^2\backslash right)\; G(x\_1,x\_2)\; =\; \backslash delta(x\_1-x\_2)$ if and only if $\backslash left(p\_\{\backslash mu\}\; p^\{\backslash mu\}\; -\; m^2\backslash right)\; G(p)\; =\; 1$, which is a direct application of the general role of functional derivative $\backslash frac\{\backslash delta\; J(x\_2)\}\{\backslash delta\; J(x\_1)\}=\backslash delta(x\_1-x\_2)$. Thus, the generating functional is obtained from the partition function as follows. The last result allows us to read the partition function as $Z[J]\; =\; Z[0]\; \backslash exp\backslash left(\backslash tfrac\{i\}\{2\}\; \backslash left\backslash langle\; J(y)\; \backslash Delta(y-y\text{'})\; J(y\text{'})\backslash right\backslash rangle\backslash right)$, where $$Z[0]\; =\; \backslash int\; \backslash mathcal\{D\}\backslash tilde\{\backslash phi\}\; \backslash ,\; \backslash exp\backslash left(-i\; \backslash int\; dt\; \backslash left[\backslash tfrac\{1\}\{2\}\; \backslash partial\_\{\backslash mu\}\; \backslash tilde\{\backslash phi\}\; \backslash partial^\{\backslash mu\}\; \backslash tilde\{\backslash phi\}\; -\; \backslash tfrac\{1\}\{2\}\; \backslash left(m^2\; -\; i\; \backslash varepsilon\backslash right)\; \backslash tilde\{\backslash phi\}^2\backslash right]\backslash right),$$ and $\backslash langle\; J(y)\backslash Delta(y-y\text{'})J(y\text{'})\backslash rangle$ is the vacuum amplitude derived by the source $\backslash langle0|0\backslash rangle\_\{J\}$. Consequently, the propagator is defined by varying the partition function as follows.

$$\backslash begin\{align\}$$

{\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.

Based on Schwinger's source theory, Steven Weinberg established the foundations of the effective field theory, which is widely appreciated among physicists. Despite the "shoes incident", 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 of the partition sum considered as a function of the source fields. This method is commonly used in the path integral formulation of 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 amplitude $W[J]=-i\backslash ln(\backslash langle\; 0|0\; \backslash rangle\_\{J\})$, the partition function becomes $Z[J]=e^\{iW[J]\}$. One can introduce $F[J]=iW[J]$, which behaves as Helmholtz free energy in thermal field theories,{{Cite book |last=Fradkin |first=Eduardo |title=Quantum Field Theory: An Integrated Approach |publisher=Princeton University Press |year=2021 |isbn=9780691149080 |pages=331–341}} to absorb the complex number, and hence $\backslash ln\; Z[J]=F[J]$. 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=9783540347620 |pages=455}} And with help of Legendre transform, 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=9789812799944 |pages=68–70}} or effective action, as

$$\backslash Gamma[\backslash bar\{\backslash phi\}]\; =\; W[J]\; -\; \backslash int\; d^4x\; \backslash ,\; J(x)\; \backslash bar\{\backslash phi\}(x),$$ with the transforms{{Cite journal |last=Jona-Lasinio |first=G. |date=1964-12-01 |title=Relativistic field theories with symmetry-breaking solutions |url=https://doi.org/10.1007/BF02750573 |journal=Il Nuovo Cimento (1955-1965) |language=en |volume=34 |issue=6 |pages=1790–1795 |doi=10.1007/BF02750573 |s2cid=121276897 |issn=1827-6121}} $$\backslash begin\{align\}$$

&\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 $\backslash phi$, i.e., $\backslash Gamma[\backslash bar\{\backslash phi\}]\; =\; W[J]\; -\; J\_a(x)\; \backslash bar\{\backslash phi\}^a(x)$.{{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 integrations and functional derivatives. Back to the Legendre transforms,

The $\backslash langle\backslash phi\backslash rangle$ is called mean field obviously because $\backslash langle\backslash phi\backslash rangle=\backslash frac\{\backslash int\; \backslash mathcal\{D\}\backslash phi\; ~\; e^\{-i\; [\; \backslash int\; dt\; ~\; \backslash mathcal\{L\}(t;\backslash phi,\backslash dot\{\backslash phi\})+\backslash int\; dx^4\; J(x,t)\backslash phi(x,t)]\}~\backslash phi~\}\{Z[J]/\backslash mathcal\{N\}\}$, while $\backslash bar\{\backslash phi\}$ is a background classical field. A field $\backslash phi$ is decomposed into a classical part $\backslash bar\{\backslash phi\}$ and fluctuation part $\backslash eta$, i.e., $\backslash phi=\backslash bar\{\backslash phi\}+\backslash eta$, so the vacuum amplitude can be reintroduced as

$$e^\{i\backslash Gamma[\backslash bar\{\backslash phi\}]\}\; =\; \backslash mathcal\{N\}\; \backslash int\; \backslash exp\backslash left[i\; \backslash left(\; S[\backslash phi]\; -\; \backslash frac\{\backslash delta\backslash Gamma[\backslash bar\{\backslash phi\}]\}\{\backslash delta\backslash bar\{\backslash phi\}\}\; \backslash eta\; \backslash right)\; \backslash right]\; d\backslash phi,$$

and any function $\backslash mathcal\{F\}[\backslash phi]$ is defined as

$$\backslash langle\backslash mathcal\{F\}[\backslash phi]\backslash rangle\; =\; e^\{-i\backslash Gamma[\backslash bar\{\backslash phi\}]\}\; ~\; \backslash mathcal\{N\}\; \backslash int\; \backslash mathcal\{F\}[\backslash phi]\; \backslash exp\; \backslash left[i\; \backslash left(S[\backslash phi]\; -\; \backslash frac\{\backslash delta\backslash Gamma[\backslash bar\{\backslash phi\}]\}\{\backslash delta\backslash bar\{\backslash phi\}\}\; \backslash eta\backslash right)\backslash right]\; d\backslash phi,$$

where $S[\backslash phi]$ 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), spontaneous symmetry breaking,{{Cite journal |last=Jona-Lasinio |first=G. |date=1964-12-01 |title=Relativistic field theories with symmetry-breaking solutions |url=https://doi.org/10.1007/BF02750573 |journal=Il Nuovo Cimento (1955-1965) |language=en |volume=34 |issue=6 |pages=1790–1795 |doi=10.1007/BF02750573 |s2cid=121276897 |issn=1827-6121}}{{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, nonlinear sigma models, and low-energy effective theories. Additionally, this theoretical framework initiates line of thoughts, publicized mainly be Bryce DeWitt who was a PhD student of Schwinger, on developing a canonical quantized effective theory for 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 $\backslash Gamma[\backslash bar\{\backslash phi\}]$ is the Legendre transform of $F[J]$, and $F[J]$ defines N-points connected correlator $G^\{N,~c\}\_\{F[J]\}=\backslash frac\{\backslash delta\; F[J]\}\{\backslash delta\; J(x\_1)\backslash cdots\; \backslash delta\; J(x\_N)\}\backslash Big|\_\{J=0\}$, then the corresponding correlator obtained from $F[J]$, known as vertex function, is given by $G^\{N,~c\}\_\{\backslash Gamma[J]\}\; =\; \backslash left.\backslash frac\{\backslash delta\; \backslash Gamma[\backslash bar\{\backslash phi\}]\}\{\backslash delta\; \backslash bar\{\backslash phi\}(x\_1)\; \backslash cdots\; \backslash delta\backslash bar\{\backslash phi\}(x\_N)\}\backslash right|\_\{\backslash bar\{\backslash phi\}=\backslash langle\backslash phi\backslash rangle\}$. 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 $\backslash Gamma$-correlator, i.e., the usual reduced correlation is $G^\{(2)\}\_\{F[J]\}=\backslash frac\{\backslash delta\; \backslash bar\{\backslash phi\}(x\_1)\}\{\backslash delta\; J(x\_2)\}\backslash Big|\_\{J=0\}=\backslash frac\{1\}\{p\_\{\backslash mu\}p^\{\backslash mu\}-m^2\}$, and the effective correlation is $G^\{(2)\}\_\{\backslash Gamma[\backslash phi]\}=\backslash frac\{\backslash delta\; J(x\_1)\}\{\backslash delta\; \backslash bar\{\backslash phi\}(x\_2)\}\backslash Big|\_\{\backslash bar\{\backslash phi\}=\backslash langle\backslash phi\backslash rangle\}=p\_\{\backslash mu\}p^\{\backslash mu\}-m^2$. For $J\_i\; =J(x\_i)$, the most general relations between the N-points connected $F[J]$ and $Z[J]$ are

$$\backslash begin\{align\}$$

\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

$$\backslash begin\{align\}$$

\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}

For a weak source producing a missive spin-1 particle with a general current $J=J\_e+J\_a$ acting on different causal spacetime points $x\_0>\; x\_0\text{'}$, the vacuum amplitude is

$$\backslash langle\; 0|0\backslash rangle\_\{J\}=\backslash exp\{\backslash left(\backslash frac\{i\}\{2\}\backslash int\; dx~dx\text{'}\backslash left[J\_\{\backslash mu\}(x)\backslash Delta(x-x\text{'})J^\{\backslash mu\}(x\text{'})+\backslash frac\{1\}\{m^2\}\backslash partial\_\{\backslash mu$$

}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 $p\_\{\backslash mu\}=(m,0,0,0)$ in its rest frame, i.e. $p\_\{\backslash mu\}p^\{\backslash mu\}=m^2$. Then, the amplitude gives

$$\backslash begin\{alignat\}\{2\}$$

(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 $\backslash eta\_\{\backslash mu\backslash nu\}=\backslash text\{diag\}(1,-1,-1,-1)$ and $(J\_\{\backslash mu\}(p))^T$ is the transpose of $J\_\{\backslash mu\}(p)$. The last result matches with the used propagator in the vacuum amplitude in the configuration space, that is,

$$\backslash left\backslash langle\; 0\backslash right|\; T\; A\_\{\backslash mu\}(x)\; A\_\{\backslash nu\}(x\text{'})\; \backslash left|0\backslash right\backslash rangle$$

= -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 $\backslash xi\; =\; 1$, the chosen Feynman–'t Hooft gauge-fixing makes the spin-1 massless. And when $\backslash xi\; =\; 0$, the chosen Landau 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. 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 is improved so it ends up being conserved. And to get the equation of motion for the massive vector, one can define

$$W[J]=-i\backslash ln(\backslash langle\; 0|0\backslash rangle\_\{J\})=\backslash frac\{1\}\{2\}\backslash int\; dx~dx\text{'}\backslash left[J\_\{\backslash mu\}(x)\backslash Delta(x-x\text{'})J^\{\backslash mu\}(x\text{'})+\backslash frac\{1\}\{m^2\}\backslash partial\_\{\backslash mu$$

}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 $\backslash int\; dx\; J\_\{\backslash mu\}(x)$ to get a definition of the massive spin-1 field

$$A\_\{\backslash mu\}(x)\backslash equiv\backslash int\; dx\text{'}\backslash Delta(x-x\text{'})J^\{\backslash mu\}(x\text{'})-\backslash frac\{1\}\{m^2\}\backslash partial\_\{\backslash mu$$

}\left[\int dx'\Delta(x-x')\partial'_{\nu}J^{\nu}(x')\right].

Additionally, the equation above says that $\backslash partial\_\{\backslash mu\}A^\{\backslash mu\}\; =\; \backslash tfrac\{1\}\{m^2\}\; \backslash partial\_\{\backslash mu\}J^\{\backslash mu\}$. Thus, the equation of motion can be written in any of the following forms

$$\backslash begin\{align\}$$

&\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}

For a weak source in a flat Minkowski background, producing then absorbing a missive spin-2 particle with a general redefined energy-momentum tensor, acting as a current, $\backslash bar\{T\}^\{\backslash mu\backslash nu\}\; =\; T^\{\backslash mu\backslash nu\}\; -\; \backslash tfrac\{1\}\{3\}\; \backslash eta\_\{\backslash mu\backslash alpha\}\; \backslash bar\{\backslash eta\}\_\{\backslash nu\backslash beta\}T^\{\backslash alpha\backslash beta\}$, where $\backslash bar\{\backslash eta\}\_\{\backslash mu\backslash nu\}(p)\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; -\; \backslash tfrac\{1\}\{m^2\}\; p\_\{\backslash mu\}p\_\{\backslash nu\}$ is the vacuum polarization tensor, the vacuum amplitude in a compact form is

$$\backslash begin\{align\}$$

\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

$$\backslash begin\{align\}$$

\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)

$$\backslash begin\{align\}$$

\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}

$$\backslash begin\{align\}$$

\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 $T^\{\backslash mu\backslash nu\}(p)\backslash Pi\_\{\backslash mu\backslash nu\backslash kappa\backslash lambda\}(p)T^\{\backslash kappa\backslash lambda\}(p)$, where the projection operator, or the Fourier transform of Jacobi field operator obtained by applying Peierls braket on 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 $\backslash Pi\_\{\backslash mu\backslash nu\backslash kappa\backslash lambda\}(p)\; =\; \backslash tfrac\{1\}\{2\}\; \backslash left(\backslash bar\{\backslash eta\}\_\{\backslash mu\backslash kappa\}(p)\; \backslash bar\{\backslash eta\}\_\{\backslash nu\backslash lambda\}(p)\; +\; \backslash bar\{\backslash eta\}\_\{\backslash mu\backslash lambda\}(p)\; \backslash bar\{\backslash eta\}\_\{\backslash nu\backslash kappa\}(p)\; -\; \backslash tfrac\{2\}\{3\}\; \backslash bar\{\backslash eta\}\_\{\backslash mu\backslash nu\}(p)\; \backslash bar\{\backslash eta\}\_\{\backslash kappa\backslash lambda\}(p)\backslash right)$.

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, the projection operator is defined as $\backslash Pi^\{m=0\}\_\{\backslash mu\backslash nu\backslash kappa\backslash lambda\}\; =\; \backslash tfrac\{1\}\{2\}\; \backslash left(\backslash eta\_\{\backslash mu\backslash kappa\}\; \backslash eta\_\{\backslash nu\backslash lambda\}\; +\; \backslash eta\_\{\backslash mu\backslash lambda\}\; \backslash eta\_\{\backslash nu\backslash kappa\}\; -\; \backslash tfrac\{1\}\{2\}\; \backslash eta\_\{\backslash mu\backslash nu\}\; \backslash eta\_\{\backslash kappa\backslash lambda\}\backslash right)$.

Together with help of Ward-Takahashi 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 $\backslash bar\{\backslash eta\}\_\{\backslash nu\backslash beta\}$ and the improved energy momentum tensor $\backslash bar\{T\}^\{\backslash mu\backslash nu\}$ appear in the early versions of massive gravity theories.{{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 |pages=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 1970's 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 |pages=044020 |doi=10.1103/PhysRevD.82.044020|arxiv=1007.0443 |s2cid=119289878 }} of exploiting Stueckelberg field redefinition led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier.

If one looks at $\backslash langle0|0\backslash rangle\_\{T\}$ and follows the same procedure used to define massive spin-1 fields, then it is easy to define massive spin-2 fields as

$$\backslash begin\{align\}$$

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 $\backslash partial^\{\backslash mu\}h\_\{\backslash mu\backslash nu\}-\backslash partial\_\{\backslash nu\}h=\backslash frac\{1\}\{m^2\}\backslash partial^\{\backslash mu\}T\_\{\backslash mu\backslash nu\}$, where the current $\backslash partial^\{\backslash mu\}T\_\{\backslash mu\backslash nu\}$ 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 $\backslash mathfrak\{T\}\_\{\backslash mu\backslash nu\}=T\_\{\backslash mu\backslash nu\}-\backslash frac\{1\}\{4\}\backslash eta\_\{\backslash mu\backslash nu\}\backslash mathfrak\{T\}$ such that $\backslash partial^\{\backslash mu\}\backslash mathfrak\{T\}\_\{\backslash mu\backslash nu\}=0$ according to Belinfante-Rosenfeld construction. Thus, the equation of motion

$$\backslash begin\{align\}$$

\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

$$\backslash left(\; \backslash square+m^\{2\}\backslash right)\; h\_\{\backslash mu\backslash nu\}=\backslash mathfrak\{T\}\_\{\backslash mu\backslash nu\}-\backslash frac\{1\}\{4\}$$

~\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 $\backslash partial^\{\backslash mu\}h\_\{\backslash mu\backslash nu\}$ 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}}

$$\backslash left(\; \backslash square+m^\{2\}\backslash right)\; h\_\{\backslash mu\backslash nu\}=\backslash mathfrak\{T\}\_\{\backslash mu\backslash nu\}-\backslash frac\{1\}\{3\}$$

~\eta_{\mu\nu}\mathfrak{T}-\frac{1}{3m^{2}}~\partial_{\mu}\partial_{\nu}

\mathfrak{T}.

One can generalize $T^\{\backslash mu\backslash nu\}(p)$ source to become $S^\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\}(p)$ higher-spin source such that $T^\{\backslash mu\backslash nu\}(p)\backslash Pi\_\{\backslash mu\backslash nu\backslash kappa\backslash lambda\}(p)T^\{\backslash kappa\backslash lambda\}(p)$ becomes $S^\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\}(p)\; \backslash Pi\_\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)\; S^\{\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)$ . The generalized projection operator also helps generalizing the electromagnetic polarization vector $e^\{\backslash mu\}\_\{m\}(p)$ of the quantized electromagnetic vector potential as follows. For spacetime points $x$ and $x\text{'}$, the addition theorem of spherical harmonics states that

$$x^\{\backslash mu\_1\}\backslash cdots\; x^\{\backslash mu\_\{\backslash ell\}\}\; \backslash Pi\_\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)\; x\text{'}^\{\backslash nu\_1\}\backslash cdots\; x\text{'}^\{\backslash nu\_\{\backslash ell\}\}=\backslash frac\{2^\backslash ell(\backslash ell!)^2\}\{(2\backslash ell)\; !\}\backslash frac\{4\backslash pi\}\{2\backslash ell+\; 1\}\backslash sum\backslash limits^\{\backslash ell\}\_\{m=-\backslash ell\}Y\_\{\backslash ell,m\}(x)Y\_\{\backslash ell,m\}^\{*\}(x\text{'}).$$

Also, the representation theory of the space of complex-valued homogeneous polynomials of degree $\backslash ell$ 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 }}$$e\_\{(m)\}(x\_1,\backslash dots,x\_n)\; =\; \backslash sum\_\{i\_1\backslash dots\; i\_\backslash ell\}\; e\_\{(m)i\_1\backslash dots\; i\_\backslash ell\}x\_\{i\_1\}\backslash cdots\; x\_\{i\_\backslash ell\},~\; \backslash forall\; x\_i\backslash in\; S^\{N-1\}.$$Then, the generalized polarization vector is$$e^\{\backslash mu\_\{1\}\backslash cdots\backslash mu\_\{\backslash ell\}\}(p)~\; x\_\{\backslash mu\_\{1\}\}\backslash cdots\; x\_\{\backslash mu\_\{\backslash ell\}\}=\backslash sqrt\{\backslash frac\{2^\backslash ell(\backslash ell!)^2\}\{(2\backslash ell)\; !\}\backslash frac\{4\backslash pi\}\{2\backslash ell+\; 1\}\}~~Y\_\{\backslash ell,m\}(x).$$

And the projection operator can be defined as $$\backslash Pi^\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)=\backslash sum\backslash limits^\{\backslash ell\}\_\{m=-\backslash ell\}[e^\{\backslash mu\_1\backslash cdots\; \backslash mu\_\{\backslash ell\}\}\_\{m\}(p)]~[e^\{\backslash nu\_1\backslash cdots\; \backslash nu\_\{\backslash ell\}\}\_\{m\}(p)]^*.$$

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 $\backslash Delta(x-x\text{'})$ in configuration space, we write

$$\backslash langle0|0\backslash rangle\_S=\backslash exp\{\backslash left[\backslash frac\{i\}\{2\}\backslash int\backslash frac\{dp^4\}\{(2\backslash pi)^4\}S^\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\}(-p)\; \backslash frac\{\backslash Pi\_\{\backslash mu\_1\backslash cdots\backslash mu\_\{\backslash ell\}\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)\}\{p\_\{\backslash sigma\}p^\{\backslash sigma\}-m^2+i\backslash varepsilon\}\; S^\{\backslash nu\_1\backslash cdots\backslash nu\_\{\backslash ell\}\}(p)\backslash right]\}.$$

Also, it is theoretically consistent to generalize the source theory to describe hypothetical gauge fields with antisymmetric and mixed symmetric properties in arbitrary dimensions and arbitrary spins. 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 $T\_\{[\backslash mu\backslash nu]\backslash lambda\}$ and a source $S\_\{[\backslash mu\backslash nu]\backslash lambda\}=\backslash partial\_\{\backslash alpha\}\backslash partial^\{\backslash alpha\}T\_\{[\backslash mu\backslash nu]\backslash lambda\}$ , the vacuum amplitude is$$\backslash langle\; 0|0\backslash rangle\_\{S\}=\backslash exp\{\backslash left(-\backslash frac\{1\}\{2\}\backslash int\; dx~dx\text{'}\backslash left[S\_\{[\backslash mu\backslash nu]\backslash lambda\}(x)\backslash Delta(x-x\text{'})S\_\{[\backslash mu\backslash nu]\backslash lambda\}(x\text{'})+\backslash frac\{2\}\{3-N\}S\_\{[\backslash mu\backslash alpha]\backslash alpha\}(x)\backslash Delta(x-x\text{'})S\_\{[\backslash mu\backslash beta]\backslash beta\}(x\text{'})\backslash right]\backslash right)\}$$ 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 |url=https://dx.doi.org/10.1016/0370-2693%2885%2991235-3 |journal=Physics Letters B |language=en |volume=165 |issue=4 |pages=304–308 |doi=10.1016/0370-2693(85)91235-3 |issn=0370-2693|url-access=subscription }} However, the massive version survives in N≥5.

For spin-{{1/2}} fermion propagator $S(x-x\text{'})=(p\; \backslash !\backslash !\backslash !/+m)\backslash Delta(x-x\text{'})$ and current $J=J\_e+J\_a$ as defined above, the vacuum amplitude is

$$\backslash begin\{align\}$$

\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

$$W\_\{\backslash frac\{1\}\{2\}\}=-\backslash frac\{1\}\{3\}\backslash int\; \backslash frac\{d^4p\}\{(2\backslash pi)^4\}~J(-p)\backslash left[\backslash gamma^0\backslash frac\{p\; \backslash !\backslash !\backslash !/+m\}\{p^2-m^2\}\backslash right]~J(p).$$

For spin-{{Frac|3|2|}} Rarita-Schwinger fermions, $\backslash Pi\_\{\backslash mu\backslash nu\}\; =\; \backslash bar\{\backslash eta\}\_\{\backslash mu\backslash nu\}\; -\; \backslash tfrac\{1\}\{3\}\; \backslash gamma^\{\backslash alpha\}\; \backslash bar\{\backslash eta\}\_\{\backslash alpha\backslash mu\}\; \backslash gamma^\{\backslash beta\}\; \backslash bar\{\backslash eta\}\_\{\backslash beta\backslash nu\}.$ Then, one can use $\backslash gamma\_\{\backslash mu\}=\backslash eta\_\{\backslash mu\backslash nu\}\backslash gamma^\{\backslash nu\}$ and the on-shell $p\backslash !\backslash !\backslash !/=-m$ to get

$$\backslash begin\{align\}$$

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 $\backslash bar\{\backslash eta\}\_\{\backslash mu\backslash nu\}$ with the usual one $\backslash eta\_\{\backslash mu\backslash nu\}$ if the source $J\_\{\backslash mu\}$ is replaced with $\backslash bar\{J\}\_\{\backslash mu\}(p)=\backslash frac\{2\}\{5\}\backslash gamma^\{\backslash alpha\}\backslash Pi\_\{\backslash mu\backslash alpha\backslash nu\backslash beta\}\backslash gamma^\{\backslash beta\}J^\{\backslash nu\}(p).$

For spin-$(j\; +\; \backslash tfrac\{1\}\{2\})$, the above results can be generalized to

$$W\_\{j+\backslash frac\{1\}\{2\}\}\; =\; -\; \backslash frac\{j+1\}\{2j+3\}\; \backslash int\; \backslash frac\{d^4p\}\{\{\backslash left(2\backslash pi\backslash right)\}^4\}\; \backslash ,\; J^\{\backslash mu\_1\; \backslash cdots\; \backslash mu\_j\}(-p)\; ~\; \backslash left[\backslash gamma^0\; \backslash frac\{~\backslash gamma^\{\backslash alpha\}\; ~\; \backslash Pi\_\{\backslash mu\_1\; \backslash cdots\; \backslash mu\_j\; \backslash alpha\; \backslash nu\_1\; \backslash cdots\; \backslash nu\_j\; \backslash beta\}\; ~\; \backslash gamma^\backslash beta\}\{p^2-m^2\}\backslash right]\; J^\{\backslash nu\_1\backslash cdots\backslash nu\_j\}(p).$$

The factor $\backslash frac\{j+1\}\{2j+3\}$ 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 form 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.{{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 }}

• Keldysh-Schwinger formalism
• Schwinger function
• Wigner-Bargmann equations
• Joos–Weinberg equation

{{reflist}}

{{DEFAULTSORT:Source Field}}

Category:Quantum field theory