Vector fields and gauge redundancy

A Lorentz vector has four components, yet a massive spin-one particle has three polarizations and a photon has two. The difference is not obtained by crossing out components: it comes from the constraint class and, in the massless theory, a gauge equivalence. This lesson develops that distinction far enough to count the states, invert the gauge-fixed kinetic operator, test the propagator against a conserved source, and recognize the Ward identity that later protects physical amplitudes.

The scope is deliberately controlled: a free real vector field in four-dimensional Minkowski space, with no dynamical charged matter and with boundary, harmonic, topological, and zero-momentum sectors excluded. Those qualifications matter—especially when deciding which transformations are redundancies rather than physical symmetries.

Required background. Classical fields, actions, and local dynamics supplies action variation and boundary terms; Quantum fields, states, and observables supplies the distinction between field components and physical states; and Lorentz field representations and Poincaré particle representations supplies the relation between a four-vector field and spin labels.

Helpful background. Functional integrals and correlators explains why a two-point function is an inverse kernel together with a state and pole prescription.

The mass term changes the vector theory

Use the site’s $(+---)$ metric, natural units, and

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

Assume that variations vanish on the boundary, or have compact support, so that the surface term in the action variation vanishes. The Proca and Maxwell Lagrangians are

$$\begin{aligned} \mathcal L_{\mathrm P} &=-\frac14F_{\mu\nu}F^{\mu\nu} +\frac12m^2A_\mu A^\mu, \qquad m>0,\\ \mathcal L_{\mathrm M} &=-\frac14F_{\mu\nu}F^{\mu\nu}. \end{aligned}$$

Varying $A_\nu$ gives

$$\partial_\mu F^{\mu\nu}+m^2A^\nu=0.$$

For Proca theory, taking a divergence and using the antisymmetry of $F$ yields

$$m^2\partial_\nu A^\nu=0.$$

Because $m>0$, transversality is a consequence of the dynamics:

$$\partial\cdot A=0, \qquad (\Box+m^2)A^\nu=0.$$

It is not a gauge condition. The mass term is not invariant under an arbitrary local shift $A_\mu\mapsto A_\mu+\partial_\mu\alpha$, so Proca theory has no gauge orbit to quotient. The detailed derivation and positivity check are on The Proca field; the action and three-polarization result are also developed in Schwartz 2014, § 8.2.2, pp. 114–117.

At $m=0$, the divergence of the field equation is instead the identity $0=0$. Maxwell theory has the off-shell invariance

$$A_\mu\longmapsto A_\mu+\partial_\mu\alpha, \qquad F_{\mu\nu}\longmapsto F_{\mu\nu}.$$

Here a crucial clause is hidden in the word “gauge”: the allowed $\alpha$ must preserve the field space and boundary conditions, and the transformation must be declared physically redundant. Compactly supported $\alpha$ give the clean local case used below. A transformation with nontrivial boundary action may carry a charge and need not be quotiented. See The free Maxwell field and gauge redundancy for the complete local classical construction.

First- and second-class constraints give different counts

Both actions contain no velocity $\dot A_0$, so both have the primary constraint

$$\pi^0\approx0, \qquad \pi^i=F^{i0}=F_{0i}.$$

The next step separates the theories. Requiring the primary constraint to be preserved in time produces

$$\begin{aligned} \text{Proca:}\qquad \chi_{\mathrm P} &=\partial_i\pi^i+m^2A_0\approx0,\\ \{\pi^0(\mathbf x),\chi_{\mathrm P}(\mathbf y)\} &=-m^2\delta^{(3)}(\mathbf x-\mathbf y), \end{aligned}$$

whereas

$$\begin{aligned} \text{Maxwell:}\qquad \chi_{\mathrm M} &=\partial_i\pi^i\approx0,\\ \{\pi^0(\mathbf x),\chi_{\mathrm M}(\mathbf y)\} &=0. \end{aligned}$$

The sign of the first bracket reverses if its arguments are reversed; its nonzero rank is the invariant point. Thus the Proca pair is second class, while the Maxwell primary constraint and Gauss constraint are first class. For a regular constrained system,

$$N_{\mathrm{physical\ pairs}} =\frac12 \left( N_{\mathrm{phase}} -2N_{\mathrm{first\ class}} -N_{\mathrm{second\ class}} \right).$$

Starting from eight phase-space variables $(A_\mu,\pi^\mu)$ per spatial point gives

$$\begin{aligned} N_{\mathrm P}&=\frac12(8-2)=3,\\ N_{\mathrm M}&=\frac12(8-2\times2)=2. \end{aligned}$$

Each first-class constraint removes one direction by restriction and one by the associated gauge quotient. Maxwell’s two first-class constraints do not mean that it has two independent spacetime gauge functions: the transformations they generate are linked by one function $\alpha(t,\mathbf x)$ and its time derivative. The massive and massless constraint analyses are developed in Weinberg 1995, § 7.6, pp. 326–330 and § 8.2, pp. 344–346. Weinberg uses the opposite metric signature; the displayed signs here follow the site convention, while the bracket rank and degree counts are unchanged. The Maxwell canonical chain is worked out on Maxwell constraints as a worked application, and the general counting rule is explained in Henneaux and Teitelboim 1992, ch. 1.

Keep the following objects separate:

Object	What it does in Maxwell theory
Primary constraint	Records the singular Legendre map: $\pi^0\approx0$
Gauss constraint	Restricts initial data: $\partial_i\pi^i\approx0$
Gauge orbit	Identifies admitted representatives related by $\partial_\mu\alpha$
Gauge condition	Selects representatives; a gauge-fixing term can implement the choice and make the quadratic kernel invertible
Residual transformation	Preserves the chosen gauge condition and still requires a boundary interpretation
Physical observable	Is well defined on the admitted gauge-equivalence classes
Physical state	Survives the constraint or subsidiary condition and the null-state quotient

Restriction to the constraint surface is not gauge fixing, and gauge fixing is not an additional physical law.

Plane waves expose the physical polarizations

Take a positive-frequency mode

$$A^\mu(x)=\varepsilon^\mu(p)e^{-ip\cdot x}.$$

The Proca equation becomes

$$(m^2-p^2)\varepsilon^\nu +p^\nu(p\cdot\varepsilon)=0.$$

Its divergence gives $p\cdot\varepsilon=0$, and a nonzero mode then obeys $p^2=m^2$. In the rest frame $p^\mu=(m,\mathbf0)$, transversality sets $\varepsilon^0=0$ and leaves three spatial polarizations. With $\varepsilon^{(\lambda)*}\!\cdot\varepsilon^{(\lambda')}=-\delta_{\lambda\lambda'}$, their on-shell completeness relation is

$$\sum_{\lambda=1}^{3} \varepsilon_\mu^{(\lambda)}(p) \varepsilon_\nu^{(\lambda)}(p)^* =-\eta_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}.$$

There is no identification $\varepsilon\sim\varepsilon+cp$ in Proca theory. The negative Lorentz norm of a spacelike polarization is not a negative Hilbert-space norm.

For Maxwell theory, the mode equation is

$$-p^2\varepsilon^\nu+p^\nu(p\cdot\varepsilon)=0.$$

A mode with nonzero field strength has $p^2=0$ and $p\cdot\varepsilon=0$. That condition leaves a three-dimensional subspace, but it contains the null vector $p^\mu$ itself. A plane-wave gauge transformation identifies

$$\varepsilon^\mu\sim\varepsilon^\mu+c\,p^\mu.$$

For $p^\mu=(\omega,0,0,\omega)$, transversality says $\varepsilon^0=\varepsilon^3$. The gauge shift can set both to zero, leaving

$$\varepsilon^\mu=(0,\varepsilon^1,\varepsilon^2,0).$$

These two classes become helicities $+1$ and $-1$ in a circular basis. The gauge-invariant plane-wave field strength,

$$F_{\mu\nu} =-i(p_\mu\varepsilon_\nu-p_\nu\varepsilon_\mu),$$

is unchanged by the shift. This is a quotient of solutions, not a statement that two preselected components of $A_\mu$ are observable in every frame. Massive and massless spin-one polarizations develops the explicit bases, completeness relations, and little-group interpretation.

The limit $m\to0$ is therefore structural, not a literal deletion of the third Proca polarization. The longitudinal polarization grows like $p^\mu/m$ at fixed nonzero momentum. Whether a quantity has a smooth limit depends on what it is contracted with and what is held fixed.

Covariant gauge fixing makes the kernel invertible

The Maxwell quadratic kernel has a null direction proportional to $p_\mu$, so it has no inverse on all four potential components. A linear covariant gauge adds

$$\mathcal L_{\mathrm{gf}} =-\frac1{2\xi}(\partial_\mu A^\mu)^2, \qquad \xi\ne0.$$

After an integration by parts under the stated falloff assumptions,

$$\mathcal L_{\mathrm M}+\mathcal L_{\mathrm{gf}} \doteq \frac12A_\mu \left[ \eta^{\mu\nu}\Box -\left(1-\frac1\xi\right) \partial^\mu\partial^\nu \right]A_\nu.$$

The gauge-fixing term does not impose $\partial\cdot A=0$ as a physical operator equation. It supplies an invertible quadratic operator. With the Minkowski-vacuum Feynman boundary prescription, its inverse is

$$D_{\mu\nu}^{(\xi)}(p) =\frac{-i}{p^2+i0} \left[ \eta_{\mu\nu} -(1-\xi)\frac{p_\mu p_\nu}{p^2+i0} \right].$$

For $\xi\ne1$, the longitudinal double pole is a prescribed distribution; it must not be treated as an ordinary rational function with the $i0$ discarded. In Feynman gauge, $\xi=1$,

$$D_{\mu\nu}^{(1)}(p) =\frac{-i\eta_{\mu\nu}}{p^2+i0}.$$

The numerator $-\eta_{\mu\nu}$ is not a sum over two positive physical photon polarizations. In Feynman gauge, covariant canonical quantization first constructs an indefinite auxiliary space. In the free theory, the Gupta–Bleuler condition

$$(\partial\cdot A)^{(+)}|\psi\rangle=0$$

selects a positive-semidefinite subspace, after which null states are quotiented. The resulting one-photon space has exactly the two transverse classes. A direct transverse quantization reaches the same physical Fock space without introducing four covariant oscillator families, at the cost of losing manifest covariance. The two constructions and their limitations are compared in Covariant free-photon quantization and propagator and Physical-mode quantization of the free electromagnetic field. The subspace-then-null-quotient structure is emphasized by Steinmann 1989, pp. 299–302.

For this free Abelian field, the Faddeev–Popov determinant in a linear gauge is field independent, so no interacting ghost is needed. That statement does not extend to Yang–Mills theory, and Gupta–Bleuler is not a replacement for the BRST construction in a general interacting gauge theory.

A conserved source removes the auxiliary directions

Couple the potential to a prescribed test current by

$$S_{\mathrm{int}}=-\int\mathrm d^4x\,J^\mu A_\mu.$$

Under $A_\mu\mapsto A_\mu+\partial_\mu\alpha$,

$$\delta S_{\mathrm{int}} =\int\mathrm d^4x\,\alpha\,\partial_\mu J^\mu -\int_{\partial\Omega}\mathrm d\Sigma_\mu\,\alpha J^\mu.$$

Thus current conservation and the boundary condition are exactly what make the coupling invariant under the admitted local redundancy. For two conserved momentum-space currents,

$$p_\mu J^\mu=0, \qquad p_\nu J'^{\nu}=0,$$

the gauge-dependent term in the propagator drops out:

$$J^\mu(-p)D_{\mu\nu}^{(\xi)}(p)J'^{\nu}(p) =-\frac{i\,J(-p)\cdot J'(p)}{p^2+i0}.$$

Equivalently, for any two gauge parameters,

$$J^\mu \left(D_{\mu\nu}^{(\xi)}-D_{\mu\nu}^{(\xi')}\right) J'^{\nu} =0.$$

This is a powerful but bounded check. It shows gauge-parameter independence of this free conserved-source response; it does not make $\langle A_\mu A_\nu\rangle$ gauge independent or prove the consistency of an interacting gauge theory. The inverse-kernel and source tests are given in Schwartz 2014, § 8.5, pp. 128–130.

The same contraction explains one smooth part of the Proca massless limit. The free Proca propagator is

$$D_{\mu\nu}^{\mathrm P}(p) =\frac{-i}{p^2-m^2+i0} \left(\eta_{\mu\nu}-\frac{p_\mu p_\nu}{m^2}\right).$$

For conserved currents, the apparently singular longitudinal numerator vanishes upon contraction, and the exchange tends smoothly to the massless conserved-current result. For a nonconserved source, the $p_\mu p_\nu/m^2$ term generally diverges. A smooth source response therefore does not mean that the Proca constraint structure or longitudinal polarization has a componentwise smooth limit.

The pole check is now unambiguous: physical Proca propagation has a pole at $p^2=m^2$, while physical photon propagation has a pole at $p^2=0$. A gauge-dependent longitudinal double pole in a covariant representative is not evidence for an extra particle; it disappears from the conserved-source contraction.

Polarization equivalence becomes a Ward identity

An amplitude with one external photon can be written

$$\mathcal A=\varepsilon_\mu(k)\,\mathcal M^\mu(k,\ldots), \qquad k^2=0.$$

Physical polarization representatives obey $\varepsilon_\mu\sim\varepsilon_\mu+c\,k_\mu$. Independence of the chosen representative requires

$$k_\mu\mathcal M^\mu=0.$$

At this level, the equation is the on-shell Ward check: replacing an external polarization by its momentum must give zero. In an interacting quantum theory, the Ward–Takahashi identity is stronger. It relates correlation functions and includes contact terms from charged insertions; gauge fixing, regularization, and possible breaking terms must be controlled. That development belongs to Symmetry, currents, and Ward identities.

Do not call the local gauge redundancy an ordinary global symmetry acting on distinct physical states. Global symmetries can carry charges and act nontrivially on the physical Hilbert space; an admitted gauge redundancy identifies descriptions. Boundary conditions can turn what looks locally like the same formula into a nontrivial transformation, which is why the admissible transformation group must always be stated.

Common pitfalls

Calling Proca transversality a gauge choice. For $m>0$, $\partial\cdot A=0$ follows from the field equation. Proca has a second-class constraint pair and no local gauge quotient.

Calling Gauss law a gauge condition. Gauss law restricts allowed initial data. A condition such as Coulomb gauge or Lorenz gauge selects a representative and must be paired with a residual-gauge and boundary analysis.

Counting the propagator numerator as photon states. The Feynman-gauge tensor $-\eta_{\mu\nu}$ belongs to an auxiliary covariant description. The physical state space is obtained only after the subsidiary condition and null quotient, or by quantizing transverse modes directly.

Dropping longitudinal terms without checking conservation. A factor of $p_\mu$ vanishes only when it contracts a current or amplitude that obeys the appropriate Ward identity. Gauge-variant Green functions may retain explicit $\xi$ dependence.

Treating every $\partial_\mu\alpha$ shift as redundant. Allowed fields, falloff, topology, boundaries, and charged observables determine whether a transformation is quotiented or acts physically.

Exercises

1. Starting with eight phase-space variables per point, reproduce the Proca and Maxwell degree counts. Explain why two Maxwell first-class constraints correspond to one spacetime gauge function rather than two independent functions.

   Solution

   Proca has the primary constraint $\pi^0\approx0$ and the secondary constraint $\partial_i\pi^i+m^2A_0\approx0$. Their mutual bracket is proportional to $m^2$ and is nonsingular for $m>0$, so both are second class. They remove two phase-space dimensions, leaving $8-2=6$, or three canonical pairs.

   Maxwell has $\pi^0\approx0$ and $\partial_i\pi^i\approx0$. Both are first class, so each removes one dimension by restriction and one by quotient. The result is $8-2\times2=4$ phase-space dimensions, or two canonical pairs. The primary constraint generates the change of $A_0$, whereas the Gauss constraint generates the longitudinal spatial shift. Matching them to $\delta A_\mu=\partial_\mu\alpha$ ties their descriptors as the value of one function $\alpha$ and its time derivative $\dot\alpha$.

2. Let $p^\mu=(\omega,0,0,\omega)$ and let a Maxwell polarization satisfy $p\cdot\varepsilon=0$. Find an explicit representative of every physical class and verify that a shift by $c p^\mu$ leaves $F_{\mu\nu}$ unchanged.

   Solution

   Transversality gives $p\cdot\varepsilon=\omega(\varepsilon^0-\varepsilon^3)=0$, so $\varepsilon^0=\varepsilon^3$. Under $\varepsilon^\mu\mapsto\varepsilon^\mu+c p^\mu$, both entries change by $c\omega$. Choosing $c=-\varepsilon^0/\omega$ sets them both to zero, leaving the representative $(0,\varepsilon^1,\varepsilon^2,0)$. Hence each physical class is specified by two transverse numbers.

   The field-strength change is

   $$\delta F_{\mu\nu} =-ic(p_\mu p_\nu-p_\nu p_\mu)=0.$$

   Thus the shift changes the potential representative but not the local gauge-invariant plane-wave field strength.

3. Contract both the covariant Maxwell propagator and the Proca propagator with conserved currents. Determine which terms survive and explain what this proves—and what it does not prove—about the limit $m\to0$.

   Solution

   Every gauge-dependent Maxwell term is proportional to $p_\mu p_\nu$. Since $p\cdot J=p\cdot J'=0$,

   $$J^\mu D_{\mu\nu}^{(\xi)}J'^{\nu} =-\frac{iJ\cdot J'}{p^2+i0},$$

   independent of $\xi$. The $p_\mu p_\nu/m^2$ part of the Proca propagator vanishes for the same reason, giving

   $$J^\mu D_{\mu\nu}^{\mathrm P}J'^{\nu} =-\frac{iJ\cdot J'}{p^2-m^2+i0}.$$

   This conserved-source response tends smoothly to the Maxwell expression as $m\to0$. It does not show that the normalized longitudinal polarization, the constraint classification, a nonconserved-source response, or every observable has a smooth limit. Those objects change discontinuously or can diverge.

Where to go next

You are ready to continue when you can obtain the three-versus-two count in both constraint and polarization language; explain why gauge fixing is not a physical state condition; invert the covariant Maxwell kernel; and make the gauge-dependent propagator term vanish using a stated conservation law.

Continue to Symmetry, currents, and Ward identities to turn the external-photon replacement test into identities for correlation functions. Before the branches rejoin in perturbation theory, also complete Fermions, spin, and anticommutation if you have not already done so.

References

• Henneaux, Marc, and Claudio Teitelboim. Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992. Publisher.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Steinmann, Othmar. “On the Characterization of Physical States in Gauge Theories.” Annales de l’Institut Henri Poincaré. Physique Théorique 51, no. 3 (1989): 299–321. NUMDAM.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.