Polarization vector (${\epsilon }^{\mu }(p,\lambda )$)

A Lorentz four-vector coefficient selecting a spin-one polarization in a mode expansion of a free field. Its allowed components depend on the field equations and gauge fixing.

The sum over the two physical transverse photon polarization vectors is not the full tensor $-{\eta }_{\mu \nu }$. It differs by gauge-dependent terms proportional to the photon four-momentum. When the amplitude satisfies the Ward identity, those terms vanish on contraction and one may use $-{\eta }_{\mu \nu }$ as a replacement inside the squared amplitude. This replacement must not be mistaken for a free-standing equality of physical polarization tensors.

A physical polarization of a massive spin-one gauge boson whose spatial polarization points along its spatial momentum. Together with two transverse polarizations it supplies the three physical spin states. In the Higgs mechanism, a broken-direction Goldstone boson supplies this extra state as the gauge boson acquires mass.

Two orthonormal transverse polarization vectors span the plane perpendicular to a nonzero momentum. Summing their dyadic products gives the spatial transverse projector. It has trace and rank two and acts as the identity on physical photon polarizations.

For $m>0$, the three physical polarization vectors are transverse to $k$ and have norm $-1$ in metric $(+---)$. In the rest frame their sum is $\mathrm{diag}(0,1,1,1)$; Lorentz covariance then gives the displayed expression for $k^2=m^2$. This includes the longitudinal mode. It cannot be specialized to $m=0$ by setting its denominator to zero.

A timelike basis polarization in covariant photon quantization has positive Minkowski metric squared length, conventionally ${ϵ}^{0\mu }=(1,0)$ with signature $(+,-,-,-)$. It is distinct from the spatial longitudinal polarization ${ϵ}^{3\mu }=(0,\hat{\mathbf{p}})$. The extra minus sign in the covariant oscillator commutator makes its one-photon state a negative-norm photon state. Neither this temporal polarization nor an isolated longitudinal photon survives the Gupta-Bleuler null-state quotient.