= Proca time-ordering contact term
{c}
{title2=$\Delta^{\rm cov}_{ab}-\Delta^{T}_{ab}=i\delta_a^0\delta_b^0\delta^{(4)}/m^2$}
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 $-i(\eta_{ab}+q_aq_b/m^2)/(q^2+m^2-i0)$, 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 $|\boldsymbol q|^2/m^2$, 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.
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