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:

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