The Free Vector Field

Spin-1 quantization differs from the scalar and Dirac cases in one decisive way: a four-component vector field $A_{\mu }(x)$ carries more components than physical polarizations, and the surplus is removed either by a mass constraint (Proca) or by gauge redundancy (Maxwell). This page quantizes both, derives the photon propagator and its gauge dependence, and records the polarization sums used in QED calculations.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Degrees of freedom: the core problem

A Lorentz vector $A_{\mu }$ has four components, but a massive spin-1 particle has three physical polarizations and a massless one only two (helicities $\pm 1$). The quantization must project out the unphysical components without spoiling Lorentz covariance — the tension that makes gauge fields subtle and ultimately forces the Faddeev–Popov construction in the non-abelian case.

The massive (Proca) field

The Proca Lagrangian for a massive vector is

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{m}^2{A}_{\mu }{A}^{\mu },F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$$

The equation of motion ${\partial }_{\mu }{F}^{\mu \nu }+m^2{A}^{\nu }=0$, upon taking ${\partial }_{\nu }$, forces the constraint

$${\partial }_{\mu }{A}^{\mu }=0(m\ne 0),$$

which removes one of the four components, leaving the three polarizations of a massive spin-1 particle. Each surviving component then obeys Klein–Gordon, $(\square +m^2)A^{\nu }=0$. Quantization proceeds as for three scalars, with mode expansion over three polarization vectors ${ϵ}_{\mu }^{(\lambda )}(p)$, $\lambda =1,2,3$, satisfying $p^{\mu }{ϵ}_{\mu }^{(\lambda )}=0$. The propagator is

$$D_{\mu \nu }(p)=\frac{-i\left({\eta }_{\mu \nu }-p_{\mu }{p}_{\nu }/m^2\right)}{p^2-m^2+iϵ},$$

with the completeness relation ${\sum }_{\lambda }{ϵ}_{\mu }^{(\lambda )}{ϵ}_{\nu }^{(\lambda )\ast }=-{\eta }_{\mu \nu }+p_{\mu }{p}_{\nu }/m^2$. The $p_{\mu }{p}_{\nu }/m^2$ term is the source of the notorious high-energy growth of massive-vector amplitudes, resolved only when the mass comes from the Higgs mechanism.

The massless (Maxwell) field and gauge redundancy

Setting $m=0$ gives the Maxwell Lagrangian $\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$, but now the constraint ${\partial }_{\mu }{A}^{\mu }=0$ is not forced — instead the theory has a gauge symmetry

$$A_{\mu }(x)\to A_{\mu }(x)+{\partial }_{\mu }\alpha (x),$$

leaving $F_{\mu \nu }$ and $\mathcal{L}$ invariant. This is a redundancy: $A_{\mu }$ and $A_{\mu }+{\partial }_{\mu }\alpha$ describe the same physics. Two obstructions to naive canonical quantization follow:

1. The conjugate momentum to $A^0$ vanishes, ${\pi }^0=\partial \mathcal{L}/\partial {\dot{A}}_0=0$ — a primary constraint; $A^0$ is not dynamical.
2. Without fixing the gauge the kinetic operator is non-invertible, so no propagator exists.

The gauge freedom must be fixed before quantization. There is no way to do this while keeping manifest Lorentz covariance and a positive-definite Hilbert space and locality all at once — one always sacrifices one of the three.

Gauge fixing and the photon propagator

Covariant ($R_{\xi }$) gauges — Gupta–Bleuler

Add a gauge-fixing term to make the kinetic operator invertible while keeping Lorentz covariance manifest:

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2.$$

The equation of motion becomes $\square A^{\nu }-(1-1/\xi ){\partial }^{\nu }({\partial }_{\mu }{A}^{\mu })=0$, now invertible, giving the photon propagator in $R_{\xi }$ gauge:

$$\boxed{D_{\mu \nu }(p)=\frac{-i}{p^2+iϵ}\left[{\eta }_{\mu \nu }-(1-\xi )\frac{p_{\mu }{p}_{\nu }}{p^2}\right].}$$

• $\xi =1$ (Feynman gauge): $D_{\mu \nu }=-i{\eta }_{\mu \nu }/(p^2+iϵ)$ — the simplest form, standard for QED.
• $\xi =0$ (Landau gauge): transverse, $p^{\mu }{D}_{\mu \nu }=0$.

Physical amplitudes are independent of $\xi$ — the $p_{\mu }{p}_{\nu }$ pieces drop out against the Ward identity (conservation of the external current $p^{\mu }{\mathcal{M}}_{\mu }=0$). Checking $\xi$-independence is a standard consistency test of a QED calculation.

The price of covariant gauge fixing is a Hilbert space with negative-norm timelike photons and zero-norm longitudinal photons. The Gupta–Bleuler condition ${\partial }_{\mu }{A}^{\mu (+)}|\text{phys}⟩=0$ (annihilation part of the Lorenz condition) selects the physical subspace, in which the unphysical polarizations pair up into zero-norm states that decouple from all observables. In the non-abelian case this bookkeeping requires the Faddeev–Popov ghosts.

Physical polarizations and the polarization sum

A real (on-shell, $p^2=0$) photon has only the two transverse polarizations ${ϵ}_{\mu }^{(\pm )}$, with $p\cdot ϵ=0$. In amplitude calculations the gauge-dependent completeness relation is replaced, thanks to the Ward identity, by the simple substitution

$$\sum _{\lambda =\pm }{ϵ}_{\mu }^{(\lambda )}{ϵ}_{\nu }^{(\lambda )\ast }⟶-{\eta }_{\mu \nu }$$

whenever the photon attaches to a conserved current. This is the rule used in the Compton and other QED cross-section computations.

Coupling to matter: the gauge principle

The reason $A_{\mu }$ exists at all is the gauge principle: promoting the global $U(1)$ symmetry of a charged field to a local one, $\psi \to e^{-i\alpha (x)}\psi$, spoils invariance through ${\partial }_{\mu }\psi \to e^{-i\alpha }({\partial }_{\mu }-i{\partial }_{\mu }\alpha )\psi$. Invariance is restored by introducing $A_{\mu }$ with transformation $A_{\mu }\to A_{\mu }+\frac{1}{e}{\partial }_{\mu }\alpha$ and replacing ${\partial }_{\mu }\to D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ (the covariant derivative of preliminaries § Gauge Fields). This is Step 2 of the QED derivation; its non-abelian generalization is Yang–Mills theory.

Summary

Where this leads

• QED — coupling the photon to the Dirac field: QED from postulates.
• Non-abelian gauge theory, where the gauge field self-interacts and gauge fixing needs ghosts: Yang–Mills, Faddeev–Popov.
• Massive gauge bosons done consistently via spontaneous breaking: the Higgs mechanism.
• Discrete symmetries acting on $A_{\mu }$: C, P, T on fields.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 4.8, 9.4.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 8.
• Srednicki, Quantum Field Theory, Ch. 54–57.