# Lattice Field Theory: Gauge Fields

My in-progress, constantly changing notes on lattice field theory. This post is about Gauge fields on the lattice.

## Wilson’s Action

For a lattice gauge field (which consists of the set of links between all lattice sites) $$\{U(b)\}$$, the Wilson action is given by

$S(\{U(b)\}) = \sum_p S_p(U_p)$

where the sum is over all plaquettes in all orientations, $$U_p$$ is the plaquette $$p$$, and

$S_p(U_p) = -\beta\left(\frac{1}{2\ \textup{Tr}\ \Bbb{I}}\left( \textup{Tr}\ U + \textup{Tr}\ U^{-1}\right) - 1\right)$

for an arbitrary compact (matrix) gauge group $$G$$. (How would this change for a non matrix Lie group, such as an exceptional?)

For the case of $$G = \textup{SU}(N)$$ this reduces to

$S_p(U_p) = \beta\left( 1 - \frac{1}{N} \textup{Re Tr } U \right).$

The most important feature of any lattice action is that it should reproduce the correct continuum action on taking the lattice spacing to zero.

Consider a Lie algebra valued vector field $$A_\mu(x) = -igA^a_\mu(x)T_a$$ where $$T_a$$ are the generators of the Lie algebra, such that the links can be written as $$U(x,\mu) = U(x, x+a\mu) = e^{-aA_\mu(x)}$$ (in analogy to the path ordered exponential in the continuum case).

We have that

$U(x, x+a\mu) = e^{-aA_\mu(x)} = \Bbb{I} - aA_\mu(x) + \frac{a^2}{2}A_\mu(x)^2 + O(a^3)$

Using the fact that $$U(x,y) = U(y,x)^{-1}$$, and $$A_\mu(x + a\nu) = A_\mu(x) + a\Delta^f_\nu A_\mu(x)$$, the plaquettes are (always working to order $$a^2$$)

\begin{aligned} U_{x,\mu\nu} &= U(x,\mu)U(x+a\mu, \nu)U(x+a\nu,\mu)^{-1} U(x,\nu)^{-1}\\ &= \left[\Bbb{I} - aA_\mu(x) + \frac{a^2}{2}A_\mu(x)^2\right] \left[\Bbb{I} - aA_\nu(x+a\mu) + \frac{a^2}{2}A_\nu(x+a\mu)^2\right]\\ &\quad \times \left[\Bbb{I} + aA_\mu(x + a\nu) + \frac{a^2}{2}A_\mu(x+a\nu)^2\right] \left[\Bbb{I} + aA_\nu(x) + \frac{a^2}{2}A_\nu(x)^2\right]\\ &= \Bbb{I} + a\left( A_\nu(x) + A_\mu(x+a\nu) - A_\nu(x+a\mu) - A_\mu(x) \right)\\ &\quad + a^2\left( A_\mu(x)A_\nu(x+a\mu) - A_\mu(x)A_\mu(x+a\nu) - A_\mu(x)A_\nu(x) - A_\nu(x+a\mu)A_\mu(x+a\nu) \right)\\ &\quad + a^2\left( -A_\nu(x+a\mu)A_\nu(x) + A_\mu(x+a\nu)A_\nu(x) + A_\mu(x+a\nu)A_\nu(x) \right)\\ &\quad + \frac{a^2}{2} \left(A_\mu(x)^2 + A_\nu(x+a\mu)^2 + A_\mu(x+a\nu)^2 + A_\nu(x)^2\right) + O(a^3)\\ &= \Bbb{I} - a^2\left(\Delta^f_\mu A_\nu(x) - \Delta^f_\nu A_\mu(x) + \textcolor{red}{[A_\nu(x), A_\mu(x)]} \right) + O(a^3). \end{aligned}

## Group Characters

Let $$r$$ be a unitary representation of the compact Lie group $$G$$. The corresponding matrices are denoted $$D^{(r)}(U)$$ for all $$U \in G$$

A matter field in the representation $$r$$ transforms as

$\phi'(x) = D^{(r)}(\Lambda^{-1}(x))\phi(x)$

under a local gauge transformation $$\Lambda(x) \in G$$.

The gauge invariant coupling term is given by $$\overline{\phi(x)} D^{(r)}(U(x,y)) \phi(y)$$, because under a gauge transformation

\begin{aligned} \overline{\phi(x)} D^{(r)}(U(x,y)) \phi(y) & \to \overline{D^{(r)}(\Lambda^{-1}(x))\phi(x)} D^{(r)}(\Lambda^{-1}(x)U(x,y)\Lambda(y))D^{(r)}(\Lambda^{-1}(y))\phi(y)\\ & = \overline{\phi(x)} D^{(r)}(\Lambda(x)) D^{(r)}(\Lambda^{-1}(x)) D^{(r)}(U(x,y)) D^{(r)}(\Lambda(y)) D^{(r)}(\Lambda^{-1}(y)) \phi(y)\\ &= \overline{\phi(x)} D^{(r)}(U(x,y)) \phi(y). \end{aligned}

The character of $$r$$ is the function $$\mathcal{X}_r(U) := \textup{Tr}\ D^{r}(U)$$ which associates a number to each group element $$U$$ in the representation $$r$$, given by the trace of its representative matrix.

In particular, the character of the identity is $$\mathcal{X}_r(\Bbb{I}) = d_r$$, the dimension of the representation. Characters are invariant functions or class functions on the group, meaning that for any two group elements $$V, U \in G$$, we have

$\mathcal{X}_r(V^{-1}UV) = \textup{Tr}\ D^{(r)}(V^{-1}UV) = \textup{Tr}\ D^{(r)}(V^{-1}) D^{(r)}(U) D^{(r)}(V) = \textup{Tr}\ D^{(r)}(U) = \mathcal{X}_r(U).$

Particularly important are the unitary irreducible representations ‘irreps’ of $$G$$ (representations which cannot be decomposed into proper subrepresentations). Denote the set of all inequivalent unitary irreps of $$G$$ as $$\hat{G}$$. Given two such representations $$r, s \in \hat{G}$$, we have a crucial fact - the characters of the different representations are orthogonal with respect to the Haar measure:

$\int dU \mathcal{X}_r(U)\mathcal{X}_s(U) = \delta_{rs}$