Proca equation

With Minkowski metric signature $(+---)$ and source coupling $-{\mu }_0{J}_{\mu }{A}^{\mu }$ added to the Proca action, a massive vector field coupled to a conserved current obeysIn vacuum this gives $(\square +m^2)A^{\nu }=0$ after imposing the dynamically derived Lorenz constraint in Proca theory. The explicit mass term breaks the gauge invariance of the massless Maxwell theory.

A Proca field is a massive spin-one vector field. Its field equation enforces transversality and leaves three physical polarizations.

For a vertex $2i{g}^2{\eta }_{\mu \nu }$ and two identical outgoing real scalar particles, the longitudinal-pair amplitude is $g^2(s-2m_A^2)/m_A^2$. The high-energy cross section is the displayed expression, including the final-state factor $1/2!$. An unpolarized incoming pair averages over nine spin states and has leading cross section $g^{4}s/(288\pi m_A^4)$. A written interaction $g^2{A}_{\mu }{A}^{\mu }{h}^2$ instead gives vertex $4i{g}^2{\eta }_{\mu \nu }$ and four times these cross sections. The distinction separates a vertex normalization from polarization averaging and identical-particle phase-space counting.

For Minkowski metric $(+---)$ and $\mathcal{L}=-G_{\mu \nu }^{\ast }{G}^{\mu \nu }/2+M^2{B}_{\mu }^{\ast }{B}^{\mu }$, use coordinates $B^{\mu },B^{\ast \mu }$. The spatial canonical momenta are ${\pi }_i=G^{\ast 0i}$, ${\pi }_i^{\ast }=G^{0i}$; the temporal momenta vanish. Constraint preservation gives ${\partial }_i{\pi }_i=M^2{B}_0^{\ast }$. Eliminating $B_0$ and its conjugate gives the positive reduced Hamiltonian density displayed above, up to a spatial surface term. In particular the massive Gauss law has a mass-dependent right-hand side; setting it to zero removes a physical longitudinal particle polarization.

For metric signature $(+---)$, the momentum-space Feynman propagator of a Proca field isThe longitudinal numerator term inverts the massive vector kinetic operator without gauge fixing.

In mostly-plus signature, the unitary gauge limit $\xi \to \infty$ leaves the longitudinal Proca propagator term $-i{P}_L/M^2$. Its ultraviolet scaling is constant in the momentum magnitude, whereas finite $\xi$ gives $1/k^2$ scaling. Thus taking the gauge-parameter limit before the ultraviolet limit obscures ordinary power-counting renormalizability. A conserved external current annihilates the longitudinal term, so the exchange between two such currents still falls as $1/k^2$. The issue does not make a free massive-vector Gaussian integral inconsistent.

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.

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.

A vector field is transverse when ${\partial }_{\mu }{V}^{\mu }=0$ in spacetime, or $\mathbf{k}\cdot \mathbf{V}=0$ for a spatial Fourier mode.

Taking a divergence and using antisymmetry of $F^{\mu \nu }$ and current conservation gives $m^2{\partial }_{\mu }{A}^{\mu }=0$. For nonzero mass, the Lorenz condition follows from the field equation rather than from a gauge choice.

The static Green function of ${\nabla }^2-m^2$ in three dimensions is proportional to $e^{-mr}/r$. It describes an interaction screened beyond range $m^{-1}$ and tends to the Coulomb potential as $m\to 0$.