Photon propagator

In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

At non-null four-momentum, these complementary projectors satisfy $P_L^2=P_L$, $P_T^2=P_T$ and $P_L{P}_T=0$. The gauge-fixed Maxwell kinetic operator is $-k^2(P_T+{\xi }^{-1}{P}_L)$; its inverse gives propagator $-i(P_T+\xi P_L)/k^2$. Transversality here removes one four-dimensional longitudinal direction, leaving rank three; it does not by itself project onto the two physical photon polarizations. The pole requires the Feynman i-epsilon prescription.

Define the inclusive photon-field tensor for hadronic final states by $W_A^{\mu \nu }=2\pi {\rho }_h(s)(-g^{\mu \nu }+q^{\mu }{q}^{\nu }/s)$ for timelike $q$ with $s=q^2$ and $q^0>0$. Here each final state has relativistic normalization and the sum includes the four-momentum delta function. This hadronic photon spectral density has mass dimension $-2$. At leading electromagnetic order, ${\rho }_h=e^2{\rho }_J/s$, with ${\rho }_J$ the dimensionless hadronic electromagnetic-current spectral density. Stating the tensor convention prevents mismatched powers of $s$ or $e$ in an inclusive cross-section.

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

For $q\cdot J=q\cdot K=0$, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:The identity follows from $\mathbf{q}\cdot \mathbf{J}=q^0{J}^0$ and $q^2=(q^0{)}^2-|\mathbf{q}{|}^2$. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.

Time-ordering the canonical transverse photon field gives a propagator with numerator equal to the spatial transverse projector and denominator $k^2+i0$. Its two energy poles encode forward propagation of the positive-frequency term and backward propagation of the negative-frequency term. This is the reduced free-field propagator; it is distinct from the covariant four-potential propagator.