Free Spin-One Fields and Constraints

A Lorentz four-vector has four components, but a free spin-one theory need not propagate four physical modes. In four-dimensional Minkowski spacetime the massive Proca field has three polarizations, the Maxwell field has two, and a covariant photon description may temporarily retain four potential components without making all four independent physical modes. This chapter develops the constraints, gauge equivalence, polarization sums, and quantization choices that make those statements compatible.

The comparison is deliberately bounded. It treats free Abelian fields and the physical-mode questions already visible there. General constrained reduction belongs to Mathematical Methods; the primary theory of gauge symmetry, gauge fixing, BRST/BV, and global structure belongs to Symmetry and Gauge Structure; interacting QED and Yang–Mills theory belong downstream.

Choose an entry route · See every page · Check your entry point

Enter the Proca–Maxwell comparison

Let $A_\mu$ be a real vector field and

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

With the site’s $(+---)$ convention, the free Proca action is

$$S_{\mathrm P}[A] =\int d^4x\left( -\frac14 F_{\mu\nu}F^{\mu\nu} +\frac12 m^2A_\mu A^\mu \right), \qquad m>0.$$

Its field equation and divergence are

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

Thus $\partial\cdot A=0$ is an equation-derived subsidiary condition when $m>0$; it is not a gauge choice. It removes one of the four apparent polarizations, leaving the three states of a massive spin-one representation. The Hamiltonian description reaches the same count differently: $A_0$ has no independent velocity, and the resulting pair of second-class constraints leaves six physical phase-space dimensions, or three configuration-space modes, per momentum. Schwartz 2014, §§ 8.2.2–8.2.4, pp. 114–120 gives the polarization construction, while Zinn-Justin 2021, § 21.1.1, pp. 508–509 gives the reduced degree count and positive Hamiltonian.

Setting $m=0$ changes the logical structure. Maxwell’s equation

$$\partial_\mu F^{\mu\nu}=0$$

is invariant under

$$A_\mu\longmapsto A_\mu+\partial_\mu\alpha,$$

and taking its divergence produces the identity $0=0$, not a new condition on $A_\mu$. Gauss law and the primary momentum constraint are first class; after imposing them and identifying gauge-related data, four physical phase-space dimensions remain, corresponding to two transverse photon polarizations. The reduction from three to two physical modes is therefore a change in constraint type and equivalence relation, not the deletion of a vector component by decree.

Description	Variables retained during the calculation	Constraint or equivalence structure	Physical one-particle modes in four dimensions
Proca, $m>0$	Four components of $A_\mu$	One nondynamical component and a second-class constraint pair; no gauge equivalence	Three massive polarizations
Maxwell, reduced description	Transverse data after solving Gauss law	First-class constraints plus quotient by gauge transformations	Two helicities
Maxwell, covariant description	Four components of a gauge-fixed potential	An indefinite auxiliary state space and a subsidiary condition; the positive-semidefinite physical subspace is quotiented by its null vectors	The resulting physical space contains the same two helicities

The third row previews only the free Abelian Gupta–Bleuler construction. Positivity belongs to the physical quotient, not to every component of the covariant potential or to the auxiliary inner product. Steinmann 1989, p. 300 states this indefinite-space, subsidiary-condition, and null-state quotient structure explicitly.

This table is a route map, not a substitute for the constraint calculation. The formula

$$N_{\mathrm{phys}} =N_{\mathrm{configuration}} -N_{\mathrm{first\ class}} -\frac12N_{\mathrm{second\ class}}$$

applies under the regularity hypotheses of constrained Hamiltonian mechanics. Constraints, Dirac brackets, and reduction develops that general framework; this chapter works out its Maxwell application.

Choose a route

The arrows below are recommended dependency paths. They answer different questions and meet again at the physical photon sector.

Reader goal	Suggested route	What you should be able to check at the exit
Compare a massive vector with a photon	Action principle plus one-particle states → the Proca field → the free Maxwell field → spin-one polarizations	Derive the three-versus-two count and state precisely what becomes singular or gauge-equivalent as $m\to0$
Follow the constraints into the physical Fock space	Hamiltonian initial data → the free Maxwell field → Maxwell constraints plus spin-one polarizations and canonical quantization → physical-mode quantization	Identify the primary and Gauss constraints, isolate transverse canonical data, and obtain two positive photon oscillators
Understand the covariant photon propagator	Maxwell constraints plus spin-one polarizations and canonical quantization → physical-mode quantization → covariant quantization and the propagator	Compare the covariant gauge-fixed potential description with the transverse two-mode description and test gauge-parameter independence of conserved-current propagator contractions

Check your entry point

The overview has no prerequisite. Individual leaves do. Use these diagnostics to find the shortest repair route.

Try this	Ready when	Repair if unsure
Take $\partial_\nu$ of $\partial_\mu F^{\mu\nu}+m^2A^\nu=0$	You use antisymmetry of $F^{\mu\nu}$ to obtain $m^2\partial\cdot A=0$ and explain why the conclusion changes at $m=0$	Review the action principle
Count a system with four configuration variables and two first-class constraints	You obtain $4-2=2$ physical configuration-space modes and can explain why each first-class constraint removes a constraint direction and a gauge direction	Review Hamiltonian initial data and constraints and reduction
Normalize a spacelike polarization vector	You distinguish the Lorentz norm $\varepsilon^*\cdot\varepsilon=-1$ from a negative Hilbert-space norm and can form a transverse projector	Review forms, adjoints, and isometries and spectral projectors
Compare $A_\mu$ with $F_{\mu\nu}$	You identify $F_{\mu\nu}$ as gauge invariant, avoid calling every potential component observable, and keep locality claims tied to the stated field and state space	Review spacelike compatibility and local observables

The six pages in order

The sidebar order gives one coherent traversal, but the chapter branches after the free Maxwell model.

1. The Proca Field. Derive the massive-vector equation, the subsidiary condition, three-mode polarization expansion, and positive physical Hamiltonian. The page also identifies why the longitudinal mode makes the massless limit conditional. The Higgs mechanism and interacting massive vectors remain downstream.

2. The Free Maxwell Field and Gauge Redundancy. Derive Maxwell’s equations, Gauss law, and the gauge equivalence of potentials while identifying the electric and magnetic fields as free gauge-invariant observables. It is a worked free model, not the primary general definition of gauge theory.

3. Massive and Massless Spin-One Polarizations. Construct explicit bases and completeness relations, connect the three-versus-two count to the relevant little groups, and test the longitudinal $m\to0$ behavior. Spinor-helicity methods and interacting longitudinal scattering are outside its scope.

4. Maxwell Constraints as a Worked Application. Follow the degenerate Legendre transform through the primary constraint, Gauss law, gauge directions, and reduced initial data. General Dirac–Bergmann theory remains with Mathematical Methods, while gauge generators and BRST/BV remain with Symmetry and Gauge Structure.

5. Physical-Mode Quantization of the Free Electromagnetic Field. Quantize the transverse canonical pair in radiation or Coulomb gauge, obtain the two-polarization photon Fock space, and make the nonlocal transverse projector visible. Charged matter and non-Abelian fields are not introduced.

6. Covariant Free-Photon Quantization and Propagator. Construct the free Abelian Gupta–Bleuler auxiliary space, impose its subsidiary physical-state condition, and pass from the positive-semidefinite subspace to the physical quotient by null states; derive the covariant photon propagator and relate the quotient to the two transverse modes. This page does not assign componentwise positivity to the covariant potential. General gauge fixing, ghosts, BRST/BV, interacting QED, and Yang–Mills dynamics hand off elsewhere. Steinmann 1989, p. 300 gives the state-space pattern summarized here.

Why the massless limit is conditional

For momentum $p^\mu=(E,\mathbf p)$, a convenient massive longitudinal polarization is

$$\varepsilon_L^\mu(p) =\frac1m\left( |\mathbf p|,E\widehat{\mathbf p} \right), \qquad p\cdot\varepsilon_L=0, \qquad \varepsilon_L^2=-1.$$

Along an on-shell family with fixed nonzero spatial momentum $\mathbf p$ and $E(m)=\sqrt{|\mathbf p|^2+m^2}$,

$$\varepsilon_L^\mu(p) =\frac{p^\mu}{m} +O\!\left(\frac{m}{E}\right),$$

so the vector itself has no finite componentwise limit. If it couples to a conserved source, $p_\mu J^\mu=0$, the leading term in $J\cdot\varepsilon_L$ cancels. Some source-dependent observables can consequently have a smooth limit even though the free Hilbert-space mode count, the constraint classification, and generic nonconserved-source correlators do not. Schwartz 2014, §§ 8.2.3–8.2.4, pp. 118–120 shows the longitudinal polarization becoming proportional to $p^\mu$, and Zinn-Justin 2021, §§ 21.2–21.2.1, pp. 510–511 isolates the role of current conservation in the propagator limit.

Three questions must therefore accompany any claim that “Proca becomes Maxwell”:

• Which variables or observables are being compared: $A_\mu$, $F_{\mu\nu}$, a conserved-current amplitude, or the state space?
• Which couplings are held fixed, and are all external sources conserved?
• Is the claim about a limit of correlation functions, a constraint surface, a representation of the Poincaré group, or an interacting theory?

The answer can be smooth for one of these objects and discontinuous for another. The chapter never treats the massless limit as erasing one oscillator by hand.

Physical modes and covariant fields answer different questions

Reduced and covariant quantization describe the same physical free photons, but they distribute covariance, locality, and positivity differently.

Formulation	Main advantage	Price paid before the physical sector is selected	Safe statement
Transverse or reduced modes	Positive Fock space with exactly two oscillators is explicit	The transverse projector is spatially nonlocal and Lorentz covariance is not manifest	The reduced photon states have positive norm and positive energy
Covariant gauge-fixed potential	A local covariant gauge-fixed action and a simple covariant propagator are manifest	Additional gauge-fixed components remain before claims are restricted to physical observables	Conserved-current and gauge-invariant quantities agree with reduced gauges under compatible prescriptions
Gauge-invariant field strength	Observable content and local causal relations can be stated without choosing a potential representative	Reconstructing a potential introduces gauge or global choices	Positivity and observable locality are asserted on the physical space for gauge-invariant quantities

The apparent tradeoff is not a contradiction. A covariant gauge-fixed $A_\mu$ can be a useful local field variable without each component representing an independent photon observable. Conversely, the transverse potential exposes positive physical oscillators but contains the nonlocal projector

$$P_{ij}^{\mathrm T}(\mathbf p) =\delta_{ij}-\frac{p_ip_j}{|\mathbf p|^2}.$$

Gauge-invariant matrix elements agree when both formulations are constructed with compatible boundary, state, and pole prescriptions. Schwartz 2014, §§ 8.4.2–8.6, pp. 126–132 develops the two physical polarizations, photon propagator, and gauge-equivalence argument; Zinn-Justin 2021, §§ 21.5–21.6, pp. 516–520 compares reduced gauges with covariant quantization for gauge-invariant observables.

Checks that travel with every spin-one calculation

Datum	Chapter declaration	Quick check
Field strength	$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$	$F$ is unchanged by $A\mapsto A+d\alpha$, and $\partial_{[\rho}F_{\mu\nu]}=0$
Proca sign package	$\mathcal L=-\tfrac14F^2+\tfrac12m^2A^2$ with $(+---)$	The equation is $\partial_\mu F^{\mu\nu}+m^2A^\nu=0$ and the three physical oscillator energies are positive
Fourier phase	Positive frequency uses $e^{-ip\cdot x}$ with $p^0>0$	$\partial_\mu\mapsto-ip_\mu$ and $p^2=m^2$ for Proca modes
Massive polarizations	$p\cdot\varepsilon^{(\lambda)}=0$ and $\varepsilon^{(\lambda)*}\cdot\varepsilon^{(\lambda')}=-\delta_{\lambda\lambda'}$	The completeness sum is $-\eta_{\mu\nu}+p_\mu p_\nu/m^2$ on shell
Massless polarizations	Two transverse representatives are chosen modulo shifts proportional to $p^\mu$	A conserved-current contraction is unchanged by $\varepsilon^\mu\mapsto\varepsilon^\mu+c\,p^\mu$
Hamiltonian constraints	Primary, secondary, first-class, and second-class labels are stated before counting	The phase-space count agrees with the polarization count
Covariant propagator	Gauge parameter, Fourier transform, metric, and $i0$ prescription are fixed together	Acting with the gauge-fixed quadratic operator gives the declared identity distribution
Physical-observable claim	The gauge-fixed potential, conserved sources, transverse modes, and field strength are distinguished	Gauge-dependent components receive no independent physical interpretation; conserved-current and gauge-invariant quantities carry the comparison

The Lorentz norm $\varepsilon^2=-1$ of a physical spacelike polarization is not a negative quantum probability. Hilbert-space positivity is a statement about the state inner product after quantization; it cannot be read from one Lorentz index contraction.

Four errors the chapter is designed to prevent

Counting components instead of modes. Four entries of $A_\mu$ do not imply four physical oscillators. Equations, constraints, and gauge equivalence must be included before the count is meaningful.

Describing the massless limit as deleting a state. The longitudinal polarization grows like $p^\mu/m$, while the constraint algebra changes and a gauge equivalence appears. A conserved source may remove the leading singularity, but that does not make every observable or state-space statement continuous.

Treating a gauge condition as the definition of gauge redundancy. Coulomb, Lorenz, temporal, and related conditions select or weight representatives; they do not create the equivalence $A\sim A+d\alpha$. Residual transformations and boundary data must still be checked.

Assigning gauge-fixed components physical probabilities. A covariant gauge-fixed description retains additional potential components to preserve a simple covariant action and propagator. No physical inner product follows from that propagator alone; the evidence developed here supports physical claims for transverse modes and for conserved-current or gauge-invariant quantities.

What the chapter establishes

By the end of the chapter, the reader can carry out the following controlled chain.

• Derive the Proca equation and subsidiary condition, construct its three polarizations, and verify the positive physical Hamiltonian.
• Derive Maxwell’s equations and Gauss law while separating gauge-equivalent potentials from gauge-invariant field strengths.
• Reconcile the polarization and Hamiltonian degree counts for both massive and massless spin one.
• State which Proca quantities have a smooth $m\to0$ limit and why conserved sources change the answer.
• Quantize the two transverse photon modes and compare that positive Fock description with a covariant gauge-fixed propagator description on conserved-current and gauge-invariant quantities.
• Derive and test the free photon propagator without assigning physical positivity or observability to unphysical components.

The chapter stops before charged matter, interacting QED, non-Abelian gauge fields, the Higgs mechanism, ghosts, BRST/BV cohomology, global gauge structure, generalized symmetries, infrared-dressed charged states, and interacting longitudinal-mode scattering. Its equivalence claims concern the free Abelian physical sector under compatible choices, not every gauge-fixed field component.

Review the chapter

Use the stated success criteria to diagnose which step needs revision, then follow the corresponding repair route.

Review mode	Prompt	A successful response	Repair route
Degree count	Explain why Proca has three physical polarizations while Maxwell has two	Gives both the polarization argument and the first-class versus second-class phase-space count	The Proca Field → Maxwell Constraints
Limit analysis	Decide whether a stated Proca observable has a smooth $m\to0$ limit	Identifies the observable, source conservation, longitudinal scaling, held-fixed coupling, and changed constraint structure	Spin-One Polarizations
Constraint derivation	Start from the Maxwell Lagrangian and identify the canonical data	Finds the vanishing momentum conjugate to $A_0$, derives Gauss law, and explains the associated gauge direction	The Free Maxwell Field → Maxwell Constraints
Formulation comparison	Compare transverse and covariant free-photon quantization	Separates manifest covariance, projector locality, additional gauge-fixed components, and the shared two-helicity gauge-invariant content without inferring Hilbert-space positivity from the propagator	Physical-Mode Quantization → Covariant Quantization
Boundary diagnosis	Decide where a question about ghosts, charged matter, or global gauge transformations belongs	Keeps this chapter’s free Abelian result and names the exact downstream treatment for the developed generalization	Where to continue

Where to continue

• Develop gauge structure: Symmetry and Gauge Structure develops gauge redundancy as a general organizing principle, gauge generators, global form, gauge fixing, ghosts, BRST/BV, and generalized symmetries.
• Add charged or non-Abelian dynamics: Gauge Theories and the Standard Model develops interacting QED, Yang–Mills theory, matter representations, renormalization, and physical scattering questions.
• Use external photon states: Perturbative QFT and Scattering develops interacting polarization sums, Ward identities in amplitudes, infrared structure, and cross sections.
• Strengthen the reduction machinery: Variational, Symplectic, and Constraint Methods develops general presymplectic reduction, first- and second-class constraints, Dirac brackets, and moment maps.
• Continue the free-field spine: Correlators, Sources, and Effective Actions organizes the common propagator and generating-functional language used by scalar, fermion, and vector fields.

References

• 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’I.H.P. Physique théorique 51, no. 3 (1989): 299–321. NUMDAM.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.