Lattice U(1) Gauge Theory: Link Variables and Wilson Action

1. The Challenge of Gauge Invariance on a Lattice

Let's recall how we discretized the scalar field theory. We replaced the continuous field $\varphi (x)$ with values ${\varphi }_n$ on the lattice sites $x=na$, and the derivative ${\partial }_{\mu }\varphi (x)$ with a finite difference:

This worked perfectly. It's tempting to do the same for the electromagnetic field $A_{\mu }(x)$. However, this naive approach fails spectacularly because it violates gauge invariance.

A U(1) gauge transformation acts on the continuum fields as:

The covariant derivative $D_{\mu }\psi =({\partial }_{\mu }+ig{A}_{\mu })\psi$ was constructed precisely so that it transforms simply: $D_{\mu }\psi \to e^{i\alpha (x)}(D_{\mu }\psi )$.

If we try to discretize this on the lattice, the finite difference of a matter field, ${\Delta }_{\mu }{\psi }_n=({\psi }_{n+\hat{\mu }}-{\psi }_n)/a$, does not transform covariantly. The gauge transformation introduces non-local terms involving ${\alpha }_{n+\hat{\mu }}$ and ${\alpha }_n$. The entire structure of gauge theory, which relies on these careful cancellations, falls apart. We need a more sophisticated approach.

2. From Fields to Connections: The Link Variable

The key insight, developed by Kenneth Wilson, is to rethink what a gauge field is. In physics, a gauge field is a connection: it tells us how to relate, or "connect," the phase of a field at one point to the phase at a nearby point. It facilitates parallel transport.

On a lattice, the most fundamental connection is between two adjacent sites. This suggests that our fundamental degree of freedom should not live on the sites, but on the links connecting them.

We define a new variable, the link variable $U_{\mu }(x)$, which lives on the link starting at site $x$ and pointing in the $\mu$ direction. For a U(1) gauge theory, this variable is an element of the U(1) group itself—a phase factor. It's related to the continuum gauge field $A_{\mu }$ by:

where $a$ is the lattice spacing and $g$ is the coupling constant (related to the electric charge).

With this new variable, how do gauge transformations work?
A matter field $\psi (x)$ still lives on the sites and transforms as $\psi (x)\to \Omega (x)\psi (x)$, where $\Omega (x)=e^{i\alpha (x)}$ is a U(1) phase at each site $x$.

We want to build a "covariant finite difference" that has a simple transformation property. Consider the term ${\psi }^{†}(x)U_{\mu }(x)\psi (x+a\hat{\mu })$. If the link variable $U_{\mu }(x)$ transforms as:

then the whole object is gauge invariant:

This is exactly what we need. The link variable $U_{\mu }(x)$ acts as a transporter, rotating the field from site $x+a\hat{\mu }$ back to site $x$ so their phases can be compared in a gauge-invariant way.

3. The Wilson Plaquette Action

We have successfully defined gauge fields ($U_{\mu }$) and matter fields ($\psi$) on the lattice and their interactions. Now, we need an action for the gauge field itself—a lattice equivalent of the Maxwell action, $\frac{1}{4}\int d^{4}x{F}_{\mu \nu }{F}^{\mu \nu }$. This action must be built purely from the link variables and must be locally gauge invariant.

How can we combine link variables to form a gauge-invariant object? Look at the transformation rule: $U_{\mu }(x)\to \Omega (x)U_{\mu }(x){\Omega }^{†}(x+a\hat{\mu })$. The gauge factors $\Omega$ appear at the start and end points of the link. If we form a closed loop of links, the $\Omega$ factors at the internal vertices will cancel out.

The smallest, non-trivial closed loop on a hypercubic lattice is a 1x1 square called a plaquette. We define the plaquette variable $U_{\mu \nu }(x)$ as the ordered product of link variables around a square in the $\mu$-$\nu$ plane starting at site $x$:

Notice the dagger on the links that go in the negative direction, corresponding to the inverse group element. Since U(1) is an Abelian group, the trace is just the complex number itself. Let's check its gauge invariance:
The factors transform as:

• $U_{\mu }(x)\to \Omega (x)U_{\mu }(x){\Omega }^{†}(x+a\hat{\mu })$
• $U_{\nu }(x+a\hat{\mu })\to \Omega (x+a\hat{\mu })U_{\nu }(x+a\hat{\mu }){\Omega }^{†}(x+a\hat{\mu }+a\hat{\nu })$
• $U_{\mu }^{†}(x+a\hat{\nu })\to \Omega (x+a\hat{\mu }+a\hat{\nu })U_{\mu }^{†}(x+a\hat{\nu }){\Omega }^{†}(x+a\hat{\nu })$ (using $(ABC{)}^{†}=C^{†}{B}^{†}{A}^{†}$)
• $U_{\nu }^{†}(x)\to \Omega (x+a\hat{\nu })U_{\nu }^{†}(x){\Omega }^{†}(x)$

Multiplying these together, you can see that the internal $\Omega$ factors cancel beautifully, leaving the product invariant. The plaquette is the elementary building block of gauge actions on the lattice.

The simplest gauge-invariant action is the Wilson action, formed by summing the real part of the plaquette variables over all plaquettes $p$ on the lattice:

Here, $U_p$ is the plaquette variable $U_{\mu \nu }(x)$ for a specific plaquette, and $\beta$ is a constant related to the coupling.

4. The Continuum Limit: Recovering Maxwell's Equations

The Wilson action might seem abstract, but it has the correct physics embedded in it. To see this, we can show that in the continuum limit ($a\to 0$), it becomes the familiar Maxwell action.

Let's sketch the key steps of the derivation for the U(1) case.

1. Expand the links: For small $a$, we Taylor expand the link variables.
   $U_{\mu }(x)=e^{iag{A}_{\mu }(x)}\approx 1+iag{A}_{\mu }(x)-\frac{1}{2}(ag{A}_{\mu }(x){)}^2+\dots$
2. Expand the plaquette: We substitute these into the plaquette product $U_{\mu \nu }=U_{\mu }(x)U_{\nu }(x+a\hat{\mu })U_{\mu }^{†}(x+a\hat{\nu })U_{\nu }^{†}(x)$. Since U(1) is Abelian, the exponentials combine easily: $$U_{\mu \nu }(x)=\exp \left\{iag\left(A_{\mu }(x)+A_{\nu }(x+a\hat{\mu })-A_{\mu }(x+a\hat{\nu })-A_{\nu }(x)\right)\right\}$$
3. Recognize the curl: The term in the exponent is a finite-difference representation of the curl of $A$. $$(A_{\nu }(x+a\hat{\mu })-A_{\nu }(x))-(A_{\mu }(x+a\hat{\nu })-A_{\mu }(x))\approx a({\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu })=a{F}_{\mu \nu }$$ So, the plaquette variable is approximately the exponential of the field strength tensor: $$U_{\mu \nu }(x)\approx e^{i{a}^{2}g{F}_{\mu \nu }(x)}$$
4. Expand the action: Now substitute this into the Wilson action term: $$1-\text{Re }{U}_{\mu \nu }=1-\cos (a^{2}g{F}_{\mu \nu })\approx 1-\left(1-\frac{(a^{2}g{F}_{\mu \nu }{)}^2}{2}+\dots \right)\approx \frac{1}{2}{a}^4{g}^2{F}_{\mu \nu }^2$$
5. From Sum to Integral: The final step is to replace the sum over all plaquettes with an integral. There are 6 orientations for a plaquette ($xy,yz,zx,xt,yt,zt$). A sum over all sites and these orientations corresponds to $\frac{1}{2}{\sum }_{\mu ,\nu }$. So, ${\sum }_p\to \frac{1}{2}{\sum }_x{\sum }_{\mu \ne \nu }\to \frac{1}{2a^4}\int d^{4}x{\sum }_{\mu \ne \nu }$.
   Putting it all together: $$S_W=\beta \sum _p(1-\text{Re }{U}_p)\approx \beta \sum _p\frac{1}{2}{a}^4{g}^2{F}_{\mu \nu }^2\to \frac{\beta g^2}{4}\int d^{4}x\sum _{\mu ,\nu }{F}_{\mu \nu }{F}_{\mu \nu }$$

By setting the overall constant $\beta =1/g^2$, we recover the standard Euclidean action for electromagnetism:

This confirms that the Wilson action, built from these seemingly strange link variables, correctly reproduces continuum physics at long distances (low energies).

Conclusion

Today, we have made the crucial transition from scalar fields to gauge fields on the lattice. You have learned the fundamental concepts that underpin all of modern lattice gauge theory.

Key Takeaways:

• Naively discretizing the continuum gauge field $A_{\mu }$ on lattice sites breaks local gauge invariance.
• The correct fundamental degree of freedom for a lattice gauge theory is the link variable $U_{\mu }(x)$, a group element that lives on the links between sites. It acts as a parallel transporter.
• The link variable transforms as $U_{\mu }(x)\to \Omega (x)U_{\mu }(x){\Omega }^{†}(x+a\hat{\mu })$ under a gauge transformation $\psi (x)\to \Omega (x)\psi (x)$.
• The smallest gauge-invariant, non-trivial object is the plaquette, a closed 1x1 loop of link variables.
• The Wilson plaquette action, formed by summing plaquettes over the lattice, is gauge invariant by construction and correctly reproduces the continuum Maxwell (or Yang-Mills) action in the $a\to 0$ limit.

Preview of the Next Lesson:

We now have a valid lattice action for U(1) gauge theory. What's next? We will put it to use! In our next lesson, we will implement a Metropolis-Hastings algorithm to simulate this pure U(1) gauge theory. This will involve generating configurations of link variables $\{U_{\mu }(x)\}$ according to the Boltzmann weight $e^{-S_W[U]}$, setting the stage for computing physical observables like the force between static charges.