Gauge Symmetry, Constraints, and Physical States

A gauge potential contains redundant descriptions of the same physical configuration, so quantizing all of its components as independent oscillators creates unphysical directions. A consistent quantum theory must impose the constraints, reduce by gauge transformations, or introduce gauge fixing and then select a physical state space. All routes must agree on gauge-invariant observables, including amplitudes between physical asymptotic states.

Required background. Quantum Mechanics supplies state spaces, inner products, and operator constraints; Classical Field Theory supplies gauge redundancy and canonical momenta; Special Relativity supplies Lorentz covariance and null polarizations. Helpful background. Calculus and Linear Algebra supplies kernels, quotients, and dimension counting; Waves, Green Functions, and Geometric Optics supplies transverse and longitudinal mode decompositions. The conventions for $F_{\mu\nu}$ and current conservation are fixed on Maxwell’s Equations in Covariant Form.

Constraints in canonical language

For electromagnetism, the transformation

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

leaves $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ unchanged. This is not an ordinary symmetry mapping one physical state to another when $\alpha$ is a gauge parameter that vanishes appropriately at the boundary. It is a many-to-one parametrization of the same physical state.

Boundary conditions matter. Transformations that remain nontrivial at a boundary can act as genuine symmetries with charges; calling every such transformation “pure gauge” would discard physics.

As anticipated on Classical Field Modes and Canonical Variables, the electromagnetic Lagrangian has no quadratic time derivative for $A_0$. Its conjugate momentum obeys the primary constraint

$$\pi^0\approx0.$$

Here $\approx$ means equality after restriction to the constraint surface, not numerical approximation.

With $\mathcal L=-F_{\mu\nu}F^{\mu\nu}/4$ and the site’s convention $F_{0i}=-E_i$,

$$\pi^i=F^{i0}=-E^i.$$

Preserving $\pi^0\approx0$ in time produces Gauss’s law. In vacuum,

$$\partial_i\pi^i\approx0.$$

$A_0$ acts as a Lagrange multiplier, not a propagating oscillator. Gauge transformations move the longitudinal part of $\mathbf A$, while the two transverse polarizations carry the radiative degrees of freedom.

A useful local canonical count for a massless vector in four dimensions begins with four fields $A_\mu$ and four conjugate momenta $\pi^\mu$, hence eight phase-space dimensions per spatial point. The primary constraint $\pi^0\approx0$ and the secondary Gauss constraint $\partial_i\pi^i\approx0$ are each first class: their Poisson brackets with every constraint vanish on the constraint surface, and their Hamiltonian flows generate gauge directions. Each first-class constraint removes one constraint direction and one gauge-orbit direction, so

$$8-2\times2=4$$

phase-space dimensions remain: two canonical pairs, or two configuration-space polarizations. This local count assumes regular, independent constraints and boundary conditions for which gauge transformations carry no boundary charge; boundary modes, large gauge transformations, and zero modes require a separate analysis.

Date 2010, Chapter 2, printed pp. 8–11, and Chapter 4, printed pp. 17–22, especially p. 22, PDF applies the constrained Hamiltonian framework to electromagnetism and derives the first-class reduction rule.

Gauss’s law on the quantum state space

For a gauge parameter $\alpha(\mathbf x)$ that vanishes at the boundary, define the smeared Gauss generator

$$\hat G[\alpha] =-\int d^3x\,\alpha(\mathbf x) \partial_i\hat\pi^i(\mathbf x).$$

Dirac quantization imposes

$$\hat G[\alpha]|\Psi_{\mathrm{phys}}\rangle=0 \qquad\text{for every allowed }\alpha.$$

Because $\hat G[\alpha]$ generates $A_i\mapsto A_i+\partial_i\alpha$, the condition says that a physical wavefunctional is constant along small gauge orbits:

$$\Psi_{\mathrm{phys}}[\mathbf A+\boldsymbol\nabla\alpha] =\Psi_{\mathrm{phys}}[\mathbf A].$$

With charged matter, the site’s Maxwell convention gives the Gauss constraint $\partial_i\pi^i+\rho\approx0$, so the generator becomes $-\int d^3x\,\alpha(\partial_i\pi^i+\rho)$. If $\alpha$ is nonzero at a boundary, integration by parts also exposes a surface term; the corresponding transformation can carry a charge and need not act trivially. The physical-state condition must therefore be stated with both its matter content and boundary class.

Two quantization strategies

Reduced quantization solves the constraints and fixes a gauge before quantizing. In Coulomb gauge,

$$\boldsymbol\nabla\cdot\mathbf A=0,$$

only transverse modes are promoted to independent oscillators. Physical content is transparent, but Lorentz covariance is not manifest and nonlocal projectors appear.

Covariant quantization adds a gauge-fixing term, for example

$$\mathcal L_{\mathrm{gf}} =-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$$

in the chapter’s mostly-plus action convention. The resulting propagator is manifestly covariant, but temporal and longitudinal polarizations enter intermediate expressions. The auxiliary state space can have an indefinite inner product, so an additional physical-state construction is essential.

For the free electromagnetic field, the Gupta–Bleuler condition is the representative covariant statement

$$\left(\partial_\mu A^\mu\right)^{(+)} |\Psi_{\mathrm{phys}}\rangle=0,$$

where $(+)$ denotes the positive-frequency part. It removes negative-norm excitations from physical matrix elements; residual zero-norm states are quotiented out. In BRST language, a nilpotent charge $Q_{\mathrm B}$ replaces the subsidiary condition:

$$Q_{\mathrm B}|\Psi_{\mathrm{phys}}\rangle=0, \qquad |\Psi_{\mathrm{phys}}\rangle \sim|\Psi_{\mathrm{phys}}\rangle +Q_{\mathrm B}|\chi\rangle.$$

Physical states are cohomology classes—closed states modulo exact states. Faddeev–Popov ghosts decouple from ordinary Abelian QED because the gauge-fixing determinant is field independent, but they are essential in interacting non-Abelian gauge theories.

Tong 2006–2007, §6.2, printed pp. 128–136, PDF develops Coulomb and Lorenz-gauge quantization and the Gupta–Bleuler state condition. Skinner, Advanced Quantum Field Theory, Chapter 8, §8.1 and §§8.2–8.2.2, printed pp. 154–169, PDF derives the ghost determinant and the BRST cohomology construction.

The two strategies must agree on gauge-invariant observables. Gauge-parameter dependence in an alleged observable signals an incomplete calculation, inconsistent truncation, or a quantity that was never physical.

Ward identities and observables

Gauge redundancy constrains correlation functions and amplitudes. For an external photon of momentum $k^\mu$, gauge invariance implies a Ward identity of the form

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

Replacing a physical polarization by $\epsilon^\mu\mapsto\epsilon^\mu+c\,k^\mu$ therefore leaves the amplitude unchanged. Current conservation and the cancellation of unphysical polarizations are two expressions of the same structure.

The same identity gives a direct gauge-choice check for an internal photon. Between two conserved momentum-space currents, any two covariant gauges differ only by a longitudinal term:

$$D_{\mu\nu}^{(\xi)}(k)-D_{\mu\nu}^{(1)}(k) =\mathcal N_\xi(k)\,k_\mu k_\nu,$$

where $\mathcal N_\xi(k)$ is proportional to $(\xi-1)/(k^2-i0)^2$ up to the overall propagator convention. Therefore

$$J^\mu \left[D_{\mu\nu}^{(\xi)}-D_{\mu\nu}^{(1)}\right] J'^\nu =\mathcal N_\xi(k) (k_\mu J^\mu)(k_\nu J'^\nu) =0.$$

This proves gauge-parameter independence for the representative current-exchange amplitude; it does not prove that an arbitrary off-shell Green function is gauge independent. Tong 2006–2007, §6.2, especially printed pp. 134–135, and §§6.3–6.4.1, printed pp. 136–143, PDF derives the covariant propagator, current conservation, and the equivalent Ward cancellation.

Local observables include $F_{\mu\nu}$ and gauge-invariant composites. Wilson loops,

$$W[C]=\exp\left(iq\oint_C A_\mu\,dx^\mu\right),$$

show that the potential can have gauge-invariant nonlocal information even when $F_{\mu\nu}$ vanishes in a neighborhood of the closed loop. For a spatial loop around an inaccessible magnetic-flux tube, Stokes’s theorem gives $W[C]=e^{iq\Phi_B}$, with $\Phi_B$ the enclosed magnetic flux, although the magnetic field vanishes along the loop itself. If the loop instead bounds a surface on which a smooth potential exists and $F=0$ everywhere, Stokes’s theorem gives $W[C]=1$. The topology and enclosed region matter. Tong 2018, §1.1.1, Eq. (1.2), printed pp. 4–5, PDF develops the closed-path phase and its flux interpretation. Gauge invariance does not mean that only pointwise field strengths are useful.

The gravity preview

Linearized gravity has the analogous redundancy

$$h_{\mu\nu}\longmapsto h_{\mu\nu}+\partial_\mu\zeta_\nu+\partial_\nu\zeta_\mu$$

up to a sign fixed by the coordinate-transformation convention. Constraints and gauge freedom reduce the ten components of $h_{\mu\nu}$ to two radiative helicities in four-dimensional vacuum. In full general relativity, diffeomorphism invariance also complicates locality: coordinate components at a named coordinate point are not automatically observables.

This analogy does not make gravity an ordinary Yang–Mills theory. Its gauge transformations act on spacetime geometry itself, its coupling is dimensionful, and boundaries carry gravitational charges. Donoghue 1995, §2, printed pp. 2–6, PDF gives the perturbative-gravity gauge structure and two-helicity count.

Exercises

Count photon polarizations

Take a null momentum $k^\mu=(\omega,0,0,\omega)$. A vector polarization obeying $k_\mu\epsilon^\mu=0$ has three independent components. Show that $\epsilon^\mu\sim\epsilon^\mu+c\,k^\mu$ leaves two.

Solution

The transversality condition relates the temporal and longitudinal components, leaving them as one shared direction plus the two transverse components $\epsilon^1,\epsilon^2$. Adding $c\,k^\mu$ changes only that shared temporal–longitudinal direction. Quotienting by it leaves the two transverse polarizations.

For a Proca field with no Stueckelberg or Higgs compensator, no gauge equivalence removes this direction, and three physical polarizations remain. Gauge-invariant massive-vector formulations retain redundancy while describing the same three physical polarizations. The massless limit can therefore require additional structure.

For the geometric constraint framework, continue to Constraints, Moment Maps, and Gauge Orbits. For fields and gravity, the exact next steps are Dirac and Maxwell Fields, Weak Metric Perturbations and Gauge Freedom, Diffeomorphism Gauge Freedom, and Covariant Gauge Fixing and Ghost Fields.

References

• Ghanashyam Date, “Lectures on Constrained Systems,” arXiv:1010.2062 [gr-qc] (2010), Chapter 2, printed pp. 8–11, and Chapter 4, printed pp. 17–22. Author-manuscript PDF.
• John F. Donoghue, “Introduction to the Effective Field Theory Description of Gravity,” lectures presented at the Advanced School on Effective Theories, Almuñécar, Spain (1995), arXiv:gr-qc/9512024, §2, printed pp. 2–6. Author-manuscript PDF.
• David Skinner, Advanced Quantum Field Theory, University of Cambridge lecture notes, Chapter 8, §8.1 and §§8.2–8.2.2, printed pp. 154–169. Official chapter PDF.
• David Tong, Gauge Theory, University of Cambridge lecture notes (2018), §1.1.1, Eq. (1.2), printed pp. 4–5. Official chapter PDF.
• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, Michaelmas Terms 2006 and 2007, §§6.1–6.4.1, printed pp. 124–143. Official author PDF.