Lattice Gauge Hamiltonians and Gauss’s Law

A Hamiltonian lattice gauge theory places a group-valued transporter and its conjugate electric field on each oriented spatial link. The electric term is kinetic energy on the group, plaquettes supply magnetic energy, and local Gauss generators implement gauge transformations. Gauge invariance is exact when every term commutes with every Gauss generator, including matter and boundary-charge contributions.

Required background. Regulated Hamiltonian field theory fixes the algebra–representation contract; links and plaquettes fix orientation; gauge orbits and Gauss constraints fix the physical meaning of the constraint.

Helpful background. Representations, intertwiners, and invariants organize non-Abelian link states and gauge-invariant vertices.

Link algebra and electric fields

Gauge-Hamiltonian convention card. Links are oriented from $x$ to $x+\hat i$ , reversal sends $U_\ell\to U_\ell^\dagger$ , and generators are Hermitian. Compact-group Haar measure is normalized. Left and right electric actions belong to opposite endpoints with the displayed signs; boundary flux and external charge are part of the declared Gauss sector.

For compact $U(1)$ , a link Hilbert space is $L^2(U(1))$ . In the angle basis,

U_\ell=e^{i\theta_\ell},\qquad E_\ell=-i\frac{\partial}{\partial\theta_\ell}, \qquad [E_\ell,U_{\ell'}]=\delta_{\ell\ell'}U_\ell.

The electric eigenstates $\lvert n\rangle$ , $n\in\mathbb Z$ , obey $E\lvert n\rangle=n\lvert n\rangle$ and $U\lvert n\rangle=\lvert n+1\rangle$ . Reversing a link sends $U\to U^\dagger$ and changes which endpoint sees outgoing rather than incoming electric flux.

For a non-Abelian group, left and right electric generators act on the two link endpoints. With Hermitian generators,

[L^a,U]=T^aU,\qquad [R^a,U]=-UT^a,

up to the declared sign convention. Their quadratic Casimirs agree on a link. The Peter–Weyl basis $\lvert r,m,n\rangle$ makes the representation label $r$ and endpoint indices explicit.

Kogut–Susskind Hamiltonian

In three spatial dimensions for $SU(N)$ , one common normalization is

H=\frac{g_0^2}{2a}\sum_{\ell,a}E_\ell^aE_\ell^a +\frac{1}{2g_0^2a}\sum_p \left[2N-\operatorname{tr}U_p-\operatorname{tr}U_p^\dagger\right] +H_{\mathrm{matter}}.

The first term is diagonal in electric representations; the second changes flux around a plaquette. Additive constants may be dropped. Other spatial dimensions and anisotropic conventions carry different powers of $a$ , so this displayed formula should not be transplanted without dimensional translation.

This electric-Casimir plus magnetic-plaquette structure is the canonical Hamiltonian construction of Kogut and Susskind 1975, pp. 395–408.

For staggered $U(1)$ matter, a representative hopping term is

H_{\mathrm{hop}}=-\kappa\sum_{x,i} \left[\psi_x^\dagger U_i(x)\psi_{x+\hat i}+\text{h.c.}\right].

It is invariant because the link transports the charge between the endpoint fields.

Gauss’s law including matter and boundaries

For oriented $U(1)$ links, define

G_x=\sum_i\left[E_i(x)-E_i(x-\hat i)\right]-\rho_x.

Physical states in a prescribed charge sector obey

G_x\lvert\psi\rangle=q_x^{\mathrm{ext}}\lvert\psi\rangle.

On a closed lattice, summing $G_x$ cancels every internal link flux, so total dynamical plus external charge must satisfy the global consistency condition. With open boundaries, the uncancelled flux equals the boundary charge; imposing zero at every boundary site would incorrectly discard charged sectors.

The electric energy commutes with $G_x$ because it is a function of flux. In a plaquette, raising the outgoing flux at a vertex is accompanied by lowering the adjacent incoming flux, so the divergence is unchanged. The matter hopping simultaneously moves charge and changes link flux. Hence

[H,G_x]=0

for all $x$ . This term-by-term proof is stronger than observing small drift in one state. The operator algebra and its strong- and weak-coupling checks are reviewed in Kogut 1979, pp. 659–713.

The shared diagram locates Gauss’s law between the regulated tensor product and all physical observables.

A regulated Hamiltonian and local Hilbert space feed exact constraints or penalty suppression, then a physical sector; a positive transfer-matrix branch and a direct real-time branch meet only at matched renormalized continuum observables, with leakage and positivity failures marked.

Gauss generators and boundary charges define the sector before either Euclidean or real-time observables are interpreted. Exact commutation preserves that sector; penalty enforcement remains approximate until leakage is quantified. The diagram is schematic.

Electric and magnetic regimes

At large $g_0^2$ , electric flux costs dominate and strong-coupling perturbation theory begins from local Casimir eigenstates. At small $g_0^2$ , plaquette alignment favors smooth magnetic fields, but many representations contribute and a local representation cutoff becomes demanding. These limits are useful checks, not two separate target theories.

For compact $U(1)$ , a one-link electric eigenstate gives $E^2=n^2$ . A plaquette operator shifts the four oriented link fluxes while preserving Gauss’s law at each vertex. Explicitly checking the four divergences is a compact implementation test.

Adversarial failure case

Replace $U_i(x)$ by $U_i(x)^\dagger$ in one matter-hopping term but retain its Hermitian conjugate. The resulting Hamiltonian is still Hermitian, preserves norm, and can even conserve the total charge, yet its commutator with the two endpoint Gauss generators is nonzero because charge and flux now move with incompatible orientations. Starting from a physical basis state therefore produces local leakage. Evaluate $[H,G_x]$ term by term—or evolve that state for a short time and measure $\sum_x\langle(G_x-q_x)^2\rangle$ —to expose the error.

Observable-level validation

Check the link commutators and the left/right transformation law on a complete one-link basis, including the cutoff boundary when present.
Evaluate $[H_{\rm electric},G_x]$ , $[H_{\rm magnetic},G_x]$ , and $[H_{\rm matter},G_x]$ separately at every affected vertex.
Reverse every link orientation and verify that a plaquette trace and the spectrum are unchanged after translating the basis.
Sum Gauss’s law over a closed lattice, or match the uncancelled flux to the declared open-boundary charge.
Reproduce a one-link Casimir energy and a one-plaquette flux transition before interpreting larger-volume observables.

Common pitfalls

Using one electric generator at both endpoints. Non-Abelian links carry left and right actions related by the link. Confusing them breaks the local transformation law.

Miscounting dependent constraints. On a closed connected Abelian lattice, the local Gauss constraints obey one global relation. Normalized group averaging remains an idempotent projector; independent-constraint counts and orbit-volume formulas must account for the relation. For a non-Abelian gauge action, identify its actual kernel or common-center action rather than assuming the same one-relation rule.

Calling a penalty term gauge invariant enforcement. A gauge-invariant penalty can energetically separate sectors, but finite-energy states can still contain unphysical weight. Measure it.

Learning outcomes

Starting from oriented link and matter transformation laws, derive every term in $G_x$ and verify $[H,G_x]=0$ including boundary and external-charge contributions.
For a stated gauge group and normalization, compute one electric and one plaquette benchmark and identify the lattice-spacing and representation factors needed for its continuum interpretation.

Exercises

Show directly that the $U(1)$ plaquette operator commutes with every $G_x$ .

Solution

At each plaquette vertex, the ordered product raises one oriented outgoing flux and raises the adjacent path flux that is incoming with the opposite sign in the divergence. Their changes cancel. Sites away from the plaquette are untouched.

On a closed lattice with no external charge, sum Gauss’s law over all sites.