Pure Yang–Mills and the states we can observe

A gauge potential is a useful variable, but its components are not independent physical observables. The mass gap concerns the physical state space after gauge redundancy has been handled.

Prerequisite: the problem and the basic idea of a gauge transformation.

By the end:

• Distinguish gauge redundancy from physical states.
• Explain why a gluon propagator is not automatically a physical spectrum.

A redundancy is not an ordinary symmetry

An ordinary global symmetry can transform one physical state into another. A gauge transformation instead changes the description of the same local physical configuration. For the convention $D_\mu=\partial_\mu-iA_\mu$,

$$A_\mu\longmapsto U A_\mu U^{-1}+iU\partial_\mu U^{-1}, \qquad F_{\mu\nu}\longmapsto U F_{\mu\nu}U^{-1}.$$

The potential transforms inhomogeneously. The curvature transforms by conjugation, so traces of suitable products of curvature are gauge invariant. This gives us observables such as

$$\mathcal O_S(x)=\operatorname{tr}F_{\mu\nu}(x)F_{\mu\nu}(x).$$

In the quantum theory this is a renormalized composite operator, not an ordinary product of pointwise functions. We will usually smear observables with smooth test functions before treating them as operators on states. The distributions bridge explains why.

Gauss’s law constrains states

In canonical language the temporal gauge field enforces Gauss’s law. Schematically, pure Yang–Mills physical states obey

$$(D_iE_i)^a|\Psi\rangle=0.$$

The equation says that the redundant gauge directions do not create additional local physical degrees of freedom. A precise construction must implement this requirement, or use an equivalent formulation directly in gauge-invariant observables.

Gauge fixing is a way to organize calculations. It does not change the definition of the physical question. In covariant perturbation theory, ghosts and unphysical polarizations help implement consistency; they do not become particles whose energies must be included in the physical mass gap.

Thus a pole or an infrared mass parameter in a gauge-fixed propagator must be interpreted through its relation to physical observables. It cannot simply be inserted as the answer to the Clay problem.

What does a glueball operator create?

Let $\widetilde{\mathcal O}=\mathcal O-\langle\Omega,\mathcal O\Omega\rangle$ be a vacuum-subtracted, appropriately smeared gauge-invariant operator. It creates the vector

$$|\psi_{\mathcal O}\rangle=\widetilde{\mathcal O}|\Omega\rangle.$$

This is usually a superposition of many energy eigenstates or spectral components. Calling $\mathcal O$ a “glueball operator” does not mean it creates exactly one glueball.

Its transformation under spatial rotations, parity, and charge conjugation determines which channels it can probe. For example, a suitable scalar, parity-even, charge-conjugation-even operator probes the $0^{++}$ channel. A differently transforming operator probes different states. At finite lattice spacing, only the lattice rotation group is exact, so identifying continuum spin is another step.

The Athenodorou–Teper spectrum study illustrates how lattice operators in different symmetry channels are used to extract physical spectral information.

Why “massless gluon” does not settle the spectrum

At weak coupling, the expansion around a simple vacuum configuration gives massless gauge-field propagators. This is the correct starting point for short-distance perturbation theory.

The mass gap is a statement about the spectrum of the fully interacting physical theory at arbitrarily long distances. A perturbative field variable and a physical asymptotic excitation need not coincide. The distinction is already familiar in simpler systems: an interacting many-body Hamiltonian may be written in variables that do not directly create its lowest-energy excitations.

This observation explains how the two descriptions can be compatible. It is not a proof that the expected gapped spectrum exists. The construction and spectral estimate still have to be supplied.

Local and extended probes

Local curvature operators probe glueball channels. Wilson loops probe a different aspect of the theory:

$$W(C)=\frac{1}{N}\operatorname{tr}\,\mathcal P \exp\!\left(i\oint_C A_\mu\,dx^\mu\right),$$

where this convention transports column vectors along $C$. The lattice lesson uses inverse transporters; the oriented loop product must be chosen consistently. The trace removes dependence on the gauge frame at the base point.

A large rectangle with spatial extent $R$ and Euclidean time extent $T$ can probe the energy of external static sources. Under the appropriate spectral interpretation,

$$\langle W(R,T)\rangle\sim e^{-T V(R)} \quad(T\ \text{large}).$$

An area-law contribution corresponds to a linear term in $V(R)$. This is a confinement diagnostic, not the same observable as a local glueball correlator. Wilson’s original paper introduced the lattice formulation in this context.

Exercises

1. Why take a trace?

Show that $\operatorname{tr}(F_{\mu\nu}F_{\rho\sigma})$ is unchanged by a local gauge transformation.

Hint. Use conjugation of $F$ and cyclicity of the trace.

Solution. The transformed product is $UF_{\mu\nu}F_{\rho\sigma}U^{-1}$. Its trace equals the original trace. This classical identity motivates gauge-invariant composites; quantum renormalization remains necessary.

2. A symmetry selection rule

Suppose parity leaves the vacuum invariant and $\mathcal O$ is parity even. Can $\mathcal O|\Omega\rangle$ overlap with a parity-odd eigenstate?

Hint. Insert the parity operator on both sides of the matrix element.

Solution. The matrix element equals its own negative and therefore vanishes. A correlator can miss a state for an exact symmetry reason, even if that state has low energy.

Continue at your own depth

Gauge groups and physical representations develops this lesson without adding a prerequisite to the next lesson.

Read further

• David Tong, Gauge Theory, gauge invariance and physical observables.
• Wilson, Confinement of quarks.
• Athenodorou and Teper, glueball spectra in SU(N) theories.

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