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A free Proca field is consistent and has three physical particle polarizations. The difficulty for generic interactions is the ultraviolet behavior of its Proca propagator: the longitudinal numerator makes part of it approach order $1/m^2$, rather than decay as $1/k^2$. Longitudinal external polarization vectors also behave as ${\epsilon }_L^{\mu }=k^{\mu }/m+O(m/E)$. Without suitable identities and cancellations, amplitudes grow with energy and perturbative counterterms of increasing dimension are needed. Conserved-current Abelian models and the Stueckelberg mechanism are important exceptions; a massive interacting vector is not intrinsically inconsistent as an effective field theory.

Use a flat five-dimensional metric of signature $(+----)$ and a circle coordinate $y\sim y+2\pi r$. Start with the canonically normalized Maxwell actionExpand the real Maxwell field in a Fourier series:The circle integral makes different modes orthogonal. Writing $a^{(n)}=A_5^{(n)}$ and $k_n=n/r$, the resulting four-dimensional action isThe scalar sign is positive because the fifth direction is spacelike in the chosen mostly-minus metric. This expression follows directly by separating the $\mu \nu$ and the two identical $\mu 5$ contributions in $F_{MN}{F}^{MN}$.

A five-dimensional gauge transformation $A_M\mapsto A_M+{\partial }_M\Lambda$ acts on its modes byFor $n\ne 0$, define the gauge-invariant vectorThen $F_{\mu 5}^{(n)}=-i{k}_n{B}_{\mu }^{(n)}$, so its term is a Proca field mass term $+\frac{1}{2}{k}_n^2{B}_{\mu }^{(n)}{B}^{\mu (-n)}$. Equivalently, a periodic mode-dependent gauge transformation sets $a^{(n)}=0$. The nonzero modes are massive vectors whose longitudinal polarization is supplied by $a^{(n)}$ through the Stueckelberg mechanism:For each positive $n$ there is one complex vector, equivalently the two real sine/cosine vectors, each with three physical polarizations. The negative mode is its conjugate, not another independent complex field. This is the infinite Kaluza-Klein tower of the Maxwell reduction on a circle.

At $n=0$, $\delta a^{(0)}=0$ under periodic infinitesimal gauge transformations. The zero modes are a massless vector with two polarizations and a real massless scalar with one. The scalar is the circle component, or Wilson line degree of freedom; a generic constant value cannot be removed by a periodic gauge parameter. Both have $m_0=0$. At energies $E\ll 1/r$, the free theory's low-energy effective action is thereforeThere is no scalar potential in this free Maxwell theory. If the original action instead has coefficient $1/g_5^2$ and unrescaled constant zero-mode fields are used, integrating the circle gives $g_4^2=g_5^2/(2\pi r)$; canonical field rescaling recovers the normalization displayed above.

For the gravitational part, allow a circle radius field $R(x)$ with vacuum value $r$, and use a dimensionless angle $\chi \sim \chi +2\pi$ in the Kaluza-Klein theory ansatzThe vector comes from off-diagonal metric components; $R$ is the radion. Perform the coordinate transformation ${\chi }^{\prime }=\chi -\kappa \lambda (x)$, leaving $x$ unchanged. Since $d\chi =d{\chi }^{\prime }+\kappa {\partial }_{\mu }\lambda d{x}^{\mu }$, the metric has the same form withThus a higher-dimensional diffeomorphism becomes an Abelian gauge transformation in the reduced theory. Changing the sign convention for the internal coordinate shift reverses the displayed sign without changing its content.

More generally, continuous internal isometries give lower-dimensional gauge symmetries, with vector fields from metric components along the corresponding Killing vector fields. The circle translation group gives $U(1)$. A scalar mode ${\psi }_n(x)e^{in\chi }$ transforms as ${\psi }_n^{\prime }=e^{in\kappa \lambda }{\psi }_n$, so $D_{\mu }{\psi }_n=({\partial }_{\mu }-in\kappa A_{\mu }){\psi }_n$ transforms covariantly. This derives Kaluza-Klein charge quantization in the chosen vector normalization. The metric vector is a graviphoton; if a five-dimensional Maxwell field is present as well, its zero-mode photon is a separate field originating from that Maxwell gauge symmetry.

Use the Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$, $\hslash =c=1$, and $\square ={\partial }_a{\partial }^a=-{\partial }_t^2+{\nabla }^2$. The Proca field has nonzero mass $m$. Vary its action, using the fact that the gauge field strength is an antisymmetric second-rank tensor:The integration by parts discards a boundary term, with the variation fixed at the boundary. Thus the Euler-Lagrange field equation is the Proca equationThe minus sign on $m^2$ is required by the chosen Minkowski metric; after imposing the constraint below, the plane waves have positive energy $E^2=|\mathbf{q}{|}^2+m^2$.

For the plane wave $A^{\mu }(x)={ϵ}^{\mu }(k)e^{-ik\cdot x}$, the equations from part a becomeIn the rest frame $k^{\mu }=(m,0)$, transversality sets ${ϵ}^0=0$ while the three spatial components remain independent. Hence a Proca field has three polarization states, as required for a massive spin-one particle. A covariant orthonormal choice obeys

Positivity selectsfor which the kinetic terms combine, up to normalization and a total derivative, into the Proca field form $-F_{\mu \nu }{F}^{\mu \nu }/4$. Taking the divergence of the equation of motion givesFor the selected relation and $m\ne 0$, this implies the transverse vector field condition ${\partial }_{\mu }{V}^{\mu }=0$. The remaining three massive polarizations have positive on-shell energy when $\tilde{a}>0$.

For a free massive Proca field in the mostly-plus Minkowski metric, the covariant inverse of the quadratic action and the canonical time-ordered product of vector potentials differ locally. With Fourier transform $e^{iq\cdot (x-y)}$, the covariant Proca propagator is $-i({\eta }_{ab}+q_a{q}_b/m^2)/(q^2+m^2-i0)$, while the literal canonical two-point correlation function subtracts $i{\delta }_a^0{\delta }_b^0/m^2$ in momentum space. The time component is nondynamical; its physical-mode numerator is $|\mathbf{q}{|}^2/m^2$, rather than $-1+(q^0{)}^2/m^2$. The difference is an instantaneous Dirac delta function, not another propagating state. Both expressions agree away from coincidence. This distinction matters when identifying a covariant Green function with a canonical correlator in a constrained field theory.