Source field

In the Feynman's path integral formulation with normalization ${\displaystyle {\mathcal {N}}\equiv Z[J=0]}$, partition function[8]

${\displaystyle Z[J]={\mathcal {N}}\int {\mathcal {D}}\phi ~e^{-i\int dt~[{\mathcal {L}}(t;\phi ,{\dot {\phi }})+J(t)\phi (t)]}}$

generates Green's functions (correlators)

${\displaystyle G(t_{1},\cdots ,t_{N})={\frac {\delta }{i\delta J(t_{1})}}\cdots {\frac {\delta }{i\delta J(t_{N})}}Z[J]{\Bigg |}_{J=0}}$ .

Upon studying the field equation using the variational process, one can see that ${\displaystyle J}$ is an external driving source of ${\displaystyle \phi }$. From the perspectives of probability theory, ${\displaystyle Z[J]}$ can be seen as the expectation value of the function ${\displaystyle e^{J\phi }}$. This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model

${\displaystyle {\mathcal {H}}=E{\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{\sqrt {2E}}}(J{\hat {a}}^{\dagger }+J^{*}a)}$ where ${\displaystyle E^{2}=m^{2}+{\vec {p}}^{2}}$.

In fact, the current is real, that is ${\displaystyle J=J^{*}}$.[9] And the Lagrangian is ${\displaystyle {\mathcal {L}}=i{\hat {a}}^{\dagger }\partial _{0}({\hat {a}})-{\mathcal {H}}}$ .

From now on we drop the hat and the asterisk. Remember that canonical quantization states ${\displaystyle \phi \sim (a^{\dagger }+a)}$. In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives

${\displaystyle \delta _{J}\langle 0,x'_{0}|0,x''_{0}\rangle _{J}=i{\Big \langle }0,x'_{0}{\Big |}\int _{x''_{0}}^{x'_{0}}dx_{0}~\delta J{\Big (}a^{\dagger }+a{\Big )}{\Big |}0,x''_{0}~{\Big \rangle }_{J}}$, where ${\displaystyle x_{0}'>x_{0}>x_{0}''}$ .

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[2]

${\displaystyle \langle 0,x'_{0}|0,x''_{0}\rangle _{J}=\exp {{\Big [}{\frac {i}{2\pi }}\int df~J(f){\frac {1}{f-E}}J(-f){\Big ]}}}$.

It is easy to notice that there is a singularity at ${\displaystyle f=E}$ . Then, we can exploit the ${\displaystyle i\epsilon }$-prescription and shift the pole ${\displaystyle f-E+i\epsilon }$ such that for ${\displaystyle x_{0}>x_{0}'}$ the Green's function is revealed

${\displaystyle {\begin{aligned}\langle 0|0\rangle _{J}&=\exp {{\Big [}{\frac {i}{2}}\int dx_{0}~dx'_{0}J(x_{0})\Delta (x_{0}-x'_{0})J(x'_{0}){\Big ]}}\\&\Delta (x_{0}-x'_{0})=\int {\frac {df}{2\pi }}{\frac {e^{-if(x_{0}-x'_{0})}}{f-E+i\epsilon }}\end{aligned}}}$

The last result is the Schwinger's source theory for scalar fields and can be generalized to any spacetime regions.[3] The discussed examples below follow the metric ${\displaystyle \eta _{\mu \nu }={\text{diag}}(1,-1,-1,-1)}$.

Causal perturbation theory explains how sources weakly act. For a weak source emits spin-0 particles ${\displaystyle J_{e}}$ by acting on the vacuum state with a probability amplitude ${\displaystyle \langle 0|0\rangle _{J_{e}}\sim 1}$, a single particle with momentum ${\displaystyle p}$ and amplitude ${\displaystyle \langle p|0\rangle _{J_{e}}}$ is created within certain spacetime region ${\displaystyle x'}$. Then, another weak source ${\displaystyle J_{a}}$ absorbs that single particle within another spacetime region ${\displaystyle x}$ such that the amplitude becomes ${\displaystyle \langle 0|p\rangle _{J_{a}}}$.[1] Thus the full vacuum amplitude is given by

${\displaystyle \langle 0|0\rangle _{J_{e}+J_{a}}\sim 1+{\frac {i}{2}}\int dx~dx'J_{a}(x)\Delta (x-x')J_{e}(x')}$

where ${\displaystyle \Delta (x-x')}$ is the propagator (correlator) of the sources. The second term of the last amplitude defines the partition function of free scalar field theory. And for some interaction theory, the Lagrangian of a scalar field ${\displaystyle \phi }$ coupled to a current ${\displaystyle J}$ is given by[10]

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}+J\phi .}$

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

${\displaystyle \langle 0|0\rangle =\exp {\left({\frac {i}{2}}\int {\frac {d^{4}p}{(2\pi )^{4}}}\left[{\tilde {\phi }}(p)(p_{\mu }p^{\mu }-m^{2}+i\epsilon ){\tilde {\phi }}(-p)+J(p){\frac {1}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}J(-p)\right]\right)}}$,

where ${\displaystyle {\tilde {\phi }}(p)=\phi (p)+{\frac {J(p)}{p_{\mu }p^{\mu }-m^{2}+i\epsilon }}.}$ It is easy to notice that the ${\displaystyle {\tilde {\phi }}(p)(p_{\mu }p^{\mu }-m^{2}+i\epsilon ){\tilde {\phi }}(-p)}$ term in the amplitude above can be Fourier transform into ${\displaystyle {\tilde {\phi }}(x)(\Box +m^{2}){\tilde {\phi }}(x)={\tilde {\phi }}(x)(J{\tilde {\phi }}(x))}$, i.e. ${\displaystyle (\Box +m^{2}){\tilde {\phi }}=J}$.

Thus, the generating functional is obtained from the partition function as follows.[4] The last result allows us to read the partition function as

${\displaystyle Z[J]=Z[0]e^{{\frac {i}{2}}\langle J(y)\Delta (y-y')J(y')\rangle }}$

where ${\displaystyle Z[0]=\int {\mathcal {D}}{\tilde {\phi }}~e^{-i\int dt~[{\frac {1}{2}}\partial _{\mu }{\tilde {\phi }}\partial ^{\mu }{\tilde {\phi }}-{\frac {1}{2}}(m^{2}-i\epsilon ){\tilde {\phi }}^{2}]}}$, and ${\displaystyle \langle J(y)\Delta (y-y')J(y')\rangle }$ is the vacuum amplitude derived by the source ${\displaystyle \langle 0|0\rangle _{J}}$. Consequently, the propagator is defined by varying the partition function as follows.

${\displaystyle {\begin{aligned}{\frac {-1}{Z[0]}}{\frac {\delta ^{2}Z[J]}{\delta J(x)\delta J(x')}}{\Bigg \vert }_{J=0}&={\frac {-1}{2Z[0]}}{\frac {\delta }{\delta J(x)}}{\Bigg \{}Z[J]\left(\int d^{4}y'\Delta (x'-y')J(y')+\int d^{4}yJ(y)\Delta (y-x')\right){\Bigg \}}{\Bigg \vert }_{J=0}={\frac {Z[J]}{Z[0]}}\Delta (x-x'){\Bigg \vert }_{J=0}\\\quad \\&=\Delta (x-x').\end{aligned}}}$This means motivates the discussing the mean field approximation below.

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 can be 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 ${\displaystyle W[J]=i\langle 0|0\rangle _{J}}$, the partition function becomes ${\displaystyle Z[J]=e^{iW[J]}}$. One can introduce ${\displaystyle F[J]=iW[J]}$, which behaves as a free energy in thermal field theories,[11] to absorb the complex number, and hence ${\displaystyle \ln Z[J]=F[J]}$. The function ${\displaystyle F[J]}$ is also called reduced quantum action.[12] And with help of Legendre transform, we can invent a "new" effective energy functional[13] ${\displaystyle \Gamma [{\bar {\phi }}]=W[J]-\int d^{4}xJ(x){\bar {\phi }}(x)}$ with the transforms ${\displaystyle {\frac {\delta W}{\delta J}}{\Bigg |}_{\bar {\phi }}={\bar {\phi }}~~~,~~~{\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}}}{\Bigg |}_{J}=-J}$ . The ${\displaystyle {\bar {\phi }}}$ is called mean field obviously because ${\displaystyle {\bar {\phi }}={\frac {\int {\mathcal {D}}\phi ~e^{-i\int dt~[{\mathcal {L}}(t;\phi ,{\dot {\phi }})+J(t)\phi (t)]}~\phi ~}{Z[J]/{\mathcal {N}}}}}$.[12] Since ${\displaystyle \Gamma [{\bar {\phi }}]}$ is the Legendre transform of ${\displaystyle F[J]}$, and ${\displaystyle F[J]}$ defines N-points connected correlator ${\displaystyle G_{F[J]}^{N,~c}={\frac {\delta F[J]}{\delta J(x_{1})\cdots \delta J(x_{N})}}{\Big |}_{J=0}}$, then the corresponding correlator obtained from ${\displaystyle F[J]}$, known as vertex function, is given by ${\displaystyle G_{\Gamma [J]}^{N,~c}={\frac {\delta \Gamma [{\bar {\phi }}]}{\delta {\bar {\phi }}(x_{1})\cdots \delta {\bar {\phi }}(x_{N})}}{\Big |}_{{\bar {\phi }}=0}}$. Consequently in the one particle irreducible graphs, the connected 2-point ${\displaystyle F}$-correlator is the inverse of the 2-point ${\displaystyle \Gamma }$-correlator; e.g. the usual reduced correlation is ${\displaystyle G_{F[J]}^{(2)}={\frac {\delta {\bar {\phi }}(x_{1})}{\delta J(x_{2})}}{\Big |}_{J=0}={\frac {1}{p_{\mu }p^{\mu }-m^{2}}}}$, and the effective correlation is ${\displaystyle G_{\Gamma [\phi ]}^{(2)}={\frac {\delta J(x_{1})}{\delta {\bar {\phi }}(x_{2})}}{\Big |}_{{\bar {\phi }}=0}=p_{\mu }p^{\mu }-m^{2}}$.

This construction is indispensable in studying scattering (LSZ reduction formula), spontaneous symmetry breaking,[14][15] Ward identities, nonlinear sigma models, and low-energy effective theories.[11]

For a weak source producing a missive spin-1 particle with general current ${\displaystyle J=J_{e}+J_{a}}$ acting on different causal spacetime points ${\displaystyle x_{0}>x_{0}'}$, the vacuum amplitude is

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

In momentum space, the spin-1 particle with rest mass ${\displaystyle m}$ has a definite momentum ${\displaystyle p_{\mu }=(m,0,0,0)}$ in its rest frame, i.e. ${\displaystyle p_{\mu }p^{\mu }=m^{2}}$. Then, the amplitude gives[1]

${\displaystyle {\begin{alignedat}{2}(J_{\mu }(p))^{T}~J^{\mu }(p)-{\frac {1}{m^{2}}}(p_{\mu }J^{\mu }(p))^{T}~p_{\nu }J^{\nu }(p)&=(J_{\mu }(p))^{T}~J^{\mu }(p)-(J^{\mu }(p))^{T}~{\frac {p_{\mu }p_{\nu }}{p_{\sigma }p^{\sigma }}}~J^{\nu }(p)\\&=(J^{\mu }(p))^{T}~\left[\eta _{\mu \nu }-{\frac {p_{\mu }p_{\nu }}{p_{\sigma }p^{\sigma }}}\right]~J^{\nu }(p)\end{alignedat}}}$

where ${\displaystyle \eta _{\mu \nu }={\text{diag}}(1,-1,-1,-1)}$ and ${\displaystyle (J_{\mu }(p))^{T}}$ is the transpose of ${\displaystyle J_{\mu }(p)}$. The last result matches with the propagator in configuration space

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

When ${\displaystyle \xi =1}$, the chosen Feynman-'t Hooft gauge-fixing makes the spin-1 massless. And when ${\displaystyle \xi =0}$, the chosen Landau gauge-fixing makes the spin-1 massive.[16]

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, ${\displaystyle {\bar {T}}^{\mu \nu }=T^{\mu \nu }-{\frac {1}{3}}\eta _{\mu \alpha }{\bar {\eta }}_{\nu \beta }T^{\alpha \beta }}$, where ${\displaystyle {\bar {\eta }}_{\mu \nu }(p)=\eta _{\mu \nu }-{\frac {1}{m^{2}}}p_{\mu }p_{\nu }}$ is the vacuum polarization tensor, the vacuum amplitude in a compact form is[1]

${\displaystyle {\begin{aligned}\langle 0|0\rangle _{\bar {T}}=\exp {\Big (}-{\frac {i}{2}}\int {\Big [}{\bar {T}}_{\mu \nu }(x)\Delta (x-x'){\bar {T}}^{\mu \nu }(x')&+{\frac {2}{m^{2}}}\eta _{\lambda \nu }\partial _{\mu }{\bar {T}}^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }{\bar {T}}^{\kappa \lambda }(x')\\&+{\frac {1}{m^{4}}}\partial _{\mu }\partial _{\mu }{\bar {T}}^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }\partial '_{\lambda }{\bar {T}}^{\kappa \lambda }(x'){\Big ]}dx~dx'{\Big )}\end{aligned}}}$

or

${\displaystyle {\begin{aligned}\langle 0|0\rangle _{T}=\exp {\Bigg (}-{\frac {i}{2}}\int &{\Bigg [}T_{\mu \nu }(x)\Delta (x-x')T^{\mu \nu }(x')+{\frac {2}{m^{2}}}\eta _{\lambda \nu }\partial _{\mu }T^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }T^{\kappa \lambda }(x')\\&+{\frac {1}{m^{4}}}\partial _{\mu }\partial _{\mu }T^{\mu \nu }(x)\Delta (x-x')\partial '_{\kappa }\partial '_{\lambda }T^{\kappa \lambda }(x')\\&-{\frac {1}{3}}\left(\eta _{\mu \nu }T^{\mu \nu }(x)-{\frac {1}{m^{2}}}\partial _{\mu }\partial _{\mu }T^{\mu \nu }(x)\right)\Delta (x-x')\left(\eta _{\mu \nu }T^{\mu \nu }(x')-{\frac {1}{m^{2}}}\partial '_{\mu }\partial '_{\mu }T^{\mu \nu }(x')\right){\Bigg ]}dx~dx'{\Bigg )}.\end{aligned}}}$

This amplitude in momentum space gives (transpose is imbedded)

${\displaystyle {\begin{aligned}{\bar {T}}_{\mu \nu }(p)\eta ^{\mu \kappa }\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)\eta ^{\mu \kappa }p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)\\&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }p^{\mu }p^{\kappa }{\bar {T}}_{\kappa \lambda }(p)+{\frac {1}{m^{4}}}{\bar {T}}_{\mu \nu }(p)p^{\mu }p^{\nu }p^{\kappa }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p)=\end{aligned}}}$

${\displaystyle {\begin{aligned}\eta ^{\mu \kappa }{\Big (}{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)&-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p){\Big )}\\&-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }{\Big (}{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p){\Big )}=\\{\Big (}\eta ^{\mu \kappa }-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }{\Big )}{\Big (}&{\bar {T}}_{\mu \nu }(p)\eta ^{\nu \lambda }{\bar {T}}_{\kappa \lambda }(p)-{\frac {1}{m^{2}}}{\bar {T}}_{\mu \nu }(p)p^{\nu }p^{\lambda }{\bar {T}}_{\kappa \lambda }(p){\Big )}=\\&{\bar {T}}_{\mu \nu }(p){\Big (}\eta ^{\mu \kappa }-{\frac {1}{m^{2}}}p^{\mu }p^{\kappa }{\Big )}{\Big (}\eta ^{\nu \lambda }-{\frac {1}{m^{2}}}p^{\nu }p^{\lambda }{\Big )}{\bar {T}}_{\kappa \lambda }(p)\end{aligned}}}$

And with help of symmetric properties of the source, the last result can be written as ${\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}$, where the projection operator (the Fourier transform of Jacobi field operator obtained by applying Peierls braket on Schwinger's variational principle[17]) is ${\displaystyle \Pi _{\mu \nu \kappa \lambda }(p)={\frac {1}{2}}{\Big (}{\bar {\eta }}_{\mu \kappa }(p){\bar {\eta }}_{\nu \lambda }(p)+{\bar {\eta }}_{\mu \lambda }(p){\bar {\eta }}_{\nu \kappa }(p)-{\frac {2}{3}}{\bar {\eta }}_{\mu \nu }(p){\bar {\eta }}_{\kappa \lambda }(p){\Big )}}$.

In N-dimensional flat spacetime, 2/3 is replaced by 2/(N-1).[18] And for massless spin-2 fields, the projection operator is[1] ${\displaystyle \Pi _{\mu \nu \kappa \lambda }^{m=0}={\frac {1}{2}}{\Big (}\eta _{\mu \kappa }\eta _{\nu \lambda }+\eta _{\mu \lambda }\eta _{\nu \kappa }-{\frac {1}{2}}\eta _{\mu \nu }\eta _{\kappa \lambda }{\Big )}}$.

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 ${\displaystyle {\bar {\eta }}_{\nu \beta }}$ and the improved energy momentum tensor ${\displaystyle {\bar {T}}^{\mu \nu }}$ appear in the early versions of massive gravity theories.[19][20] 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[21] of exploiting Stueckelberg field redefinition led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier.

One can generalize ${\displaystyle T^{\mu \nu }(p)}$ source to become ${\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)}$ higher-spin source such that ${\displaystyle T^{\mu \nu }(p)\Pi _{\mu \nu \kappa \lambda }(p)T^{\kappa \lambda }(p)}$ becomes ${\displaystyle S^{\mu _{1}\cdots \mu _{\ell }}(p)\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)S^{\nu _{1}\cdots \nu _{\ell }}(p)}$ .[1] The generalized projection operator also helps generalizing the electromagnetic polarization vector ${\displaystyle e_{m}^{\mu }(p)}$ of the quantized electromagnetic vector potential as follows. For spacetime points ${\displaystyle x~{\text{and}}~x'}$, the addition theorem of spherical harmonics states that

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

Also, the representation theory of the space of complex-valued homogeneous polynomials of degree ${\displaystyle \ell }$ on a unit (N-1)-sphere defines the polarization tensor as[22]

$${\displaystyle e_{(m)}(x_{1},\dots ,x_{n})=\sum _{i_{1}\dots i_{\ell }}e_{(m)i_{1}\dots i_{\ell }}x_{i_{1}}\cdots x_{i_{\ell }},~\forall x_{i}\in S^{N-1}.}$$

Then, the generalized polarization vector is

${\displaystyle e^{\mu _{1}\cdots \mu _{\ell }}(p)~x_{\mu _{1}}\cdots x_{\mu _{\ell }}={\sqrt {{\frac {2^{\ell }(\ell$ !)^{2}}{(2\ell )!}}{\frac {4\pi }{2\ell +1}}}}~~Y_{\ell ,m}(x)} .

And the projection operator can be defined as

${\displaystyle \Pi ^{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)=\sum \limits _{m=-\ell }^{\ell }[e_{m}^{\mu _{1}\cdots \mu _{\ell }}(p)]~[e_{m}^{\nu _{1}\cdots \nu _{\ell }}(p)]^{*}}$ .

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

${\displaystyle \langle 0|0\rangle _{S}=\exp {{\Big [}{\frac {i}{2}}\int {\frac {dp^{4}}{(2\pi )^{4}}}S^{\mu _{1}\cdots \mu _{\ell }}(-p){\frac {\Pi _{\mu _{1}\cdots \mu _{\ell }\nu _{1}\cdots \nu _{\ell }}(p)}{p_{\sigma }p^{\sigma }-m^{2}+i\epsilon }}S^{\nu _{1}\cdots \nu _{\ell }}(p){\Big ]}}}$.

The source theory can be linked to Wigner-Bargmann and Joos–Weinberg systems of higher-spin fields.

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 ${\displaystyle T_{[\mu \nu ]\lambda }}$ and a source ${\displaystyle S_{[\mu \nu ]\lambda }=\partial _{\alpha }\partial ^{\alpha }T_{[\mu \nu ]\lambda }}$ , the vacuum amplitude is${\displaystyle \langle 0|0\rangle _{S}=\exp {\left(-{\frac {1}{2}}\int dx~dx'\left[S_{[\mu \nu ]\lambda }(x)\Delta (x-x')S_{[\mu \nu ]\lambda }(x')+{\frac {2}{3-N}}S_{[\mu \alpha ]\alpha }(x)\Delta (x-x')S_{[\mu \beta ]\beta }(x')\right]\right)}}$ which for a theory in N=4 makes the source eventually reveal that it is a theory of a non physical field.[23] However, the massive version survives in N≥5.

• Keldysh-Schwinger formalism
• Schwinger function