Why a lattice makes the theory definable

A finite lattice replaces a formal field integral by finitely many ordinary integrals over compact groups. Gauge invariance can be preserved exactly throughout this regularization.

Prerequisite: physical observables. The Haar measure bridge explains the integration measure.

By the end:

• Write a finite-lattice expectation using normalized Haar measure.
• Check gauge invariance with the site conventions.

Links carry parallel transport

Take a four-dimensional hypercubic lattice with spacing $a$. To each positively oriented link from $x$ toward $x+a\hat\mu$, assign a matrix $U_{x,\mu}\in SU(N)$. This convention transports a column vector at the endpoint back to the base point. Reversing an orientation uses its inverse.

A lattice gauge transformation chooses $\Omega_x\in SU(N)$ at each site and acts by

$$U_{x,\mu}\longmapsto \Omega_xU_{x,\mu}\Omega_{x+a\hat\mu}^{-1}.$$

The elementary plaquette based at $x$ is

$$U_{x,\mu\nu} =U_{x,\mu}U_{x+a\hat\mu,\nu} U_{x+a\hat\nu,\mu}^{-1}U_{x,\nu}^{-1}.$$

The matrices at interior endpoints cancel under a gauge transformation, leaving conjugation at $x$. Therefore $\operatorname{tr}U_{x,\mu\nu}$ is gauge invariant. Larger closed loops work the same way.

Our link convention is compatible with $D_\mu=\partial_\mu-iA_\mu$ and $U_{x,\mu}\simeq e^{-iaA_\mu(x)}$ for a smooth connection. Reversing the link convention changes this sign without changing the real traced plaquette action.

The Wilson action and measure

For the fundamental representation, use

$$S_W[U]=\beta\sum_p \left(1-\frac1N\operatorname{Re}\operatorname{tr}U_p\right), \qquad \beta=\frac{2N}{g_0^2}.$$

The sum counts each elementary plaquette once with a chosen orientation. The normalized expectation is

$$\langle\mathcal O\rangle_{a,L} =\frac{1}{Z_{a,L}} \int\prod_{\ell}dU_\ell\, \mathcal O[U]\,e^{-S_W[U]}.$$

Each $dU_\ell$ is normalized Haar measure, invariant under left and right multiplication. On a finite lattice the configuration space is a finite product of compact groups. The action is continuous and bounded; its exponential is strictly positive. Consequently $0<Z_{a,L}<\infty$.

This is an actual finite-dimensional definition, not an instruction to interpret an undefined $\mathcal DA$. Gauge-invariant expectations can be evaluated without first fixing a gauge. Wilson’s original formulation is the foundational source.

Recover the classical continuum expression

For a smooth field, expanding a small plaquette gives

$$U_{x,\mu\nu}=1-ia^2F_{\mu\nu} -\frac{a^4}{2}F_{\mu\nu}^2 +\text{derivative and higher-order terms}.$$

The tracelessness of $F$ removes the linear trace. At leading order,

$$1-\frac1N\operatorname{Re}\operatorname{tr}U_{x,\mu\nu} =\frac{a^4}{2N}\operatorname{tr}F_{\mu\nu}^2+\cdots.$$

Substitute $\beta=2N/g_0^2$ and convert the site sum to an integral:

$$S_W\longrightarrow \frac{1}{g_0^2}\int d^4x\sum_{\mu<\nu} \operatorname{tr}F_{\mu\nu}^2 =\frac{1}{2g_0^2}\int d^4x\, \operatorname{tr}F_{\mu\nu}F_{\mu\nu}.$$

The factor of two in the final expression comes from counting both $(\mu,\nu)$ and $(\nu,\mu)$.

This calculation explains the choice of lattice action. It does not prove the quantum continuum limit: typical quantum fields are not smooth configurations to which a naive pointwise expansion can be uniformly applied.

Three scales that should not be confused

If the lattice has $N_s$ sites along each spatial direction, its physical side length is $L=N_sa$. A correlation length can be measured either in lattice sites, $\xi_{\mathrm{latt}}$, or in physical units, $\xi_{\mathrm{phys}}=a\xi_{\mathrm{latt}}$.

Decreasing $a$ with fixed $N_s$ shrinks the physical box. Increasing $N_s$ with fixed $a$ removes the volume cutoff but leaves the ultraviolet cutoff. A continuum, infinite-volume calculation has to control both.

Likewise, a Wilson-loop area in plaquette units differs from its physical area by a factor $a^2$. A string tension in lattice units must be divided by $a^2$ to obtain the corresponding physical quantity.

Whenever a paper quotes a lattice mass, length, or coupling, write its convention next to the number. Comparing two calculations before this translation can create an apparent disagreement that is only a difference of units or normalization.

What the lattice has accomplished

We have a finite regulated probability measure, exact gauge invariance, and well-defined bounded loop observables. These are substantial achievements. The remaining work includes the continuum limit, the infinite-volume limit, physical reconstruction, and spectral estimates.

For a construction of the limiting theory, regularized observables must also be chosen and renormalized appropriately. Lüscher’s Wilson-flow paper provides an important example of useful gauge-invariant quantities and scales.

Exercises

1. Check a transformation

Multiply the four transformed links around a plaquette. What happens to the gauge matrices?

Hint. Adjacent endpoint factors cancel.

Solution. The product becomes $\Omega_xU_{x,\mu\nu}\Omega_x^{-1}$. Cyclicity then makes its trace invariant.

2. Count the physical length

An initial lattice has $N_s=32$ and spacing $a$. Halve $a$ while keeping the same physical box. What must happen to $N_s$?

Hint. Keep $N_sa$ fixed.

Solution. Increase $N_s$ to $64$. In four dimensions an isotropic lattice then contains sixteen times as many sites, before considering any additional sampling cost.

Continue at your own depth

The transfer matrix and physical time develops this lesson without adding a prerequisite to the next lesson.

Read further

• Wilson, Confinement of quarks.
• Lüscher, Properties and uses of the Wilson flow in lattice QCD.

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