Pure gauge theories

Wilson proposed the investigation of non-Abelian gauge theories using a numerical approach. This approach, to be useful for computers must do two things.

• Approximate functional integrals by integrals over discrete spacetime points thereby creating an artificial lattice.
• Make the size of the lattice finite.

At the center of non-Abelian gauge theories are the gauge fields and gauge invariance. The problem is that gauge invariance is tied closely with spacetime dependence, involving derivatives. If one defines quantities on the lattice carelessly then one can use gauge invariance and one cannot expect that the theory would be a good approximation to continuum theories. Therefore Wilson proposed to use a compact version of the gauge group $SU(N)$. Let us start with the observation that in continuum theory the following object is gauge covariant unitary matrix:

$$U_\mu(x)=P\left[\exp\left(ig\int_x^{x+a\hat\mu}dx'_\mu A_\mu(x')\right)\right],$$ (3)

where $P$ is path ordering and $\hat \mu$ is a unit vector along the $\mu$th axis, and $A_\mu(x)$ is the matrix valued gauge field. $g$ is the bare gauge coupling.

$U_\mu(x)$ transforms very simply under gauge transformation:

$$U_\mu(x)\to U_\mu^V(x) =V(x)U_\mu(x)V^\dagger(x+a\hat\mu).$$

In lattice theory, rather than using the gauge fields themselves we only use the unitary matrices $U_\mu(x)$. One can imagine them as being defined on the links connecting latttice points of the four dimensional lattice of lattice parameter $a$. In particular $U_\mu(x)$ defined in () is located on the link connecting lattice points $x$ and $x+a\hat \mu$.

The action of the pure gauge theory is gauge invariant. Therefore one needs to construct gauge invariant quantities from the unitary operators $U_\mu(x)$. The simplest nontrivial gauge invariant is formed from the trace of the product of four unitary operators around a plaquette in the $(\mu,\nu)$ plane,

$${\rm Tr}\,U_{\mu\nu}^p(x)={\rm Tr}[U_\mu(x)U_\nu(x+a\hat \mu)U^\dagger_\mu(x+a\hat \nu)U^\dagger_\nu(x)].$$ (4)

This trace can be seen as the closed loop integral

$$[ {\rm Tr} \,U_{\mu\nu}^p(x)=P\left[\exp\left(ig\oint dx_\mu A_\mu(x)\right)\right],$$ (5)

where the loop is to be taken at the edge of the plaquette. Then, of course, this integral is gauge invariant.

Then the Euclidean action can be defined as

$$S=\frac{1}{g^2} \sum_{x,(\mu\nu)}\left[2N-{\rm Tr}\,[U_{\mu\nu}^p(x)+U_{\mu\nu}^{p\dagger}(x)]\right],$$ (6)

where $g$ is the gauge coupling and the summation over $(\mu\nu)$ refers to the six possible choices of the $(\mu\nu)$ pair.