Lattice gauge theory

Remarks

A lattice gauge transformation is a map $g \colon \Lambda \to G$; it acts by $U_e \mapsto g(x) U_e \, g(y)^{-1}$ for an edge $e$ from $x$ to $y$. It changes $U_p$ into $g(x) U_p \, g(x)^{-1}$, where $x$ is the starting corner of $p$, so $\operatorname{tr} U_p$ and the action do not change. The Haar measure is invariant too, so $\mu$ is gauge invariant; this is the lattice version of the gauge group symmetry.

Example: $G = U(1)$, with $U_e = e^{i\theta_e}$. Then $U_p = e^{i\theta_p}$, where $\theta_p$ is the signed sum of the four angles around $p$, and $S(U) = \beta \sum_p (1 - \cos\theta_p)$. Let $a > 0$ be the lattice spacing and $e > 0$ a coupling, and let $\theta_e = a e A_e$ for a smooth potential $A$, where $A_e$ is the component of $A$ along $e$. Then $\theta_p \approx a^2 e F_p$, where $F_p$ is the component of $F = dA$ on $p$, and $S \approx \frac{\beta e^2}{2} \sum_p a^4 F_p^2$. For $d = 4$ and $\beta = 1/e^2$ this is a Riemann sum for the Euclidean Maxwell action $\frac14 \int F_{\mu\nu} F_{\mu\nu} \, d^4x$.

Non-example: the function $\sum_{e} \operatorname{Re}\operatorname{tr} U_e$ is not gauge invariant, because $\operatorname{tr}(g(x) U_e \, g(y)^{-1})$ differs from $\operatorname{tr} U_e$ when $g(x) \ne g(y)$.

The natural observables are traces of products of edge variables around closed paths, the Wilson loops. For $G = SU(3)$ and $d = 4$, with quark fields added, the theory is lattice quantum chromodynamics.