5 SU(N) Gauge Theory with Scalar Matter
5.1 From U(1) to SU(N)
The construction of scalar QED on the lattice generalises directly from the gauge group \(U(1)\) to \(SU(N)\). The two theories are compared in the table below.
| U(1) — Scalar QED | SU(N) — Yang–Mills + Scalar | |
|---|---|---|
| Scalar field | \(\varphi \in \mathbb{C}\) | \(\varphi \in \mathbb{C}^N\) |
| Gauge field | \(U_\mu(x) \in U(1)\) | \(U_\mu(x) \in SU(N)\) |
| Gauge transf. (scalar) | \(\varphi(x) \to e^{i\lambda(x)}\varphi(x)\) | \(\varphi(x) \to \Omega(x)\varphi(x)\) |
| Gauge transf. (link) | \(U_\mu(x) \to e^{i\lambda(x)}U_\mu(x)e^{-i\lambda(x+ae_\mu)}\) | \(U_\mu(x) \to \Omega(x)\,U_\mu(x)\,\Omega^\dagger(x+ae_\mu)\) |
| Covariant derivative | \(D_\mu^f\varphi = \tfrac{1}{a}\{U_\mu(x)\varphi(x+ae_\mu)-\varphi(x)\}\) | same |
| Gauge action \(S_g\) | \(\sum_{\mu<\nu,x}\operatorname{Re}[1-W_{\mu\nu}(x)]\) | \(\sum_{\mu<\nu,x}\operatorname{Re}\operatorname{tr}[1-W_{\mu\nu}(x)]\) |
| Matter action \(S_M\) | \(a^4\sum_{\mu,x}\|D_\mu^f\varphi(x)\|^2 + \cdots\) | same |
| Path-integral measure | \(\prod_{x,\mu}d\theta_\mu(x)\) (rotation-invariant) | \(\prod_{x,\mu}dU_\mu(x)\) (Haar measure) |
The structural parallel is exact: the only differences are that \(\varphi\) becomes a column vector, \(U_\mu\) becomes a matrix, the trace appears in the gauge action, and the link measure becomes the Haar measure on \(SU(N)\).
5.2 ① The Link Variable
The link variable \(U_\mu(x)\) is the non-Abelian parallel transporter from \(x\) to \(x+ae_\mu\), \[ U_\mu(x) \stackrel{\mathrm{def}}{=} W(x\to x+ae_\mu) = \mathcal{P}\exp\!\left\{i\int_x^{x+ae_\mu} dx_\beta^\mu\, A_\beta(x)\right\} \in SU(N), \] where \(A_\mu(x) = A_\mu^a(x)\,T^a\) is the \(\mathfrak{su}(N)\)-valued gauge field and \(\mathcal{P}\) denotes path ordering.
Under a gauge transformation \(\Omega(x)\in SU(N)\) and \(A_\mu \to \Omega A_\mu \Omega^\dagger + i(\partial_\mu\Omega)\Omega^\dagger\), the link transforms as \[ U_\mu(x) \;\to\; \Omega(x)\,U_\mu(x)\,\Omega^\dagger(x+ae_\mu). \]
5.3 ② Gauge Covariance of the Covariant Derivative
The forward lattice covariant derivative is defined as in the \(U(1)\) case: \[ D_\mu^f\varphi(x) \stackrel{\mathrm{def}}{=} \frac{U_\mu(x)\,\varphi(x+ae_\mu) - \varphi(x)}{a}. \]
Gauge covariance. Using the transformation laws \(U_\mu(x)\to\Omega(x)U_\mu(x)\Omega^\dagger(x+ae_\mu)\) and \(\varphi(x)\to\Omega(x)\varphi(x)\): \[ D_\mu^{f\,\prime}\varphi'(x) = \frac{\Omega(x)\,U_\mu(x)\,\Omega^\dagger(x+ae_\mu)\cdot\Omega(x+ae_\mu)\,\varphi(x+ae_\mu) - \Omega(x)\,\varphi(x)}{a} = \Omega(x)\,D_\mu^f\varphi(x). \] Therefore \(\|D_\mu^f\varphi\|^2 = (D_\mu^f\varphi)^\dagger(D_\mu^f\varphi)\) is gauge invariant.
Continuum limit. Writing \(U_\mu(x) = e^{iaA_\mu(x)+O(a^2)} = 1 + iaA_\mu(x) + O(a^2)\), one finds \[ D_\mu^f\varphi(x) \xrightarrow{a\to 0} \frac{(1+iaA_\mu)\varphi(x+ae_\mu) - \varphi(x)}{a} = (\partial_\mu + iA_\mu)\varphi(x) + O(a) = D_\mu\varphi(x) + O(a). \]
5.4 ③ The Plaquette
The plaquette \(W_{\mu\nu}(x)\) is the ordered product of link variables around the elementary square in the \((\mu,\nu)\)-plane with corner at \(x\): \[ W_{\mu\nu}(x) \stackrel{\mathrm{def}}{=} U_\mu(x)\,U_\nu(x+ae_\mu)\,U_\mu^\dagger(x+ae_\nu)\,U_\nu^\dagger(x). \]
It satisfies the following properties:
(a) \(W(x+ae_\mu\to x) = U_\mu^\dagger(x)\). (See proof below.)
(b) The link variable is the exponential of a Hermitian traceless matrix: \[ U_\mu(x) = e^{iB_\mu(x)}, \qquad B_\mu = aA_\mu + \frac{a^2}{2}\partial_\mu A_\mu + O(a^3). \] (See proof below.)
(c) \(W_{\mu\nu}(x) \in SU(N)\).
(d) Gauge invariance. Under a gauge transformation, consecutive factors cancel: \[ W_{\mu\nu}(x) \;\to\; \Omega(x)\,U_\mu(x)\,\Omega^\dagger(x+ae_\mu)\cdot\Omega(x+ae_\mu)\,U_\nu(x+ae_\mu)\,\cdots\,\Omega^\dagger(x) = \Omega(x)\,W_{\mu\nu}(x)\,\Omega^\dagger(x), \] so \(\operatorname{tr} W_{\mu\nu}(x)\) is gauge invariant.
5.5 Proof of Property (a)
The reversed Wilson line from \(x+ae_\mu\) to \(x\) is obtained by reversing the path and relabelling \(s\to a-s\): \[ W(x+ae_\mu\to x) = \mathcal{P}\exp\!\left\{-i\int_0^a ds\,A_\mu(x+se_\mu)\right\} = \left[\mathcal{P}\exp\!\left\{i\int_0^a ds\,A_\mu(x+se_\mu)\right\}\right]^\dagger = U_\mu^\dagger(x). \]
The minus sign arises because reversing the orientation of the path negates the exponent, and Hermitian conjugation is consistent with this since \(A_\mu^\dagger = A_\mu\).
5.6 Proof of Property (b)
We construct \(B_\mu(x)\) as the solution to a Cauchy problem. Define \[ U(a) \stackrel{\mathrm{def}}{=} \mathcal{P}\exp\!\left\{i\int_0^a ds\,A_\mu(x+se_\mu)\right\}. \] This satisfies the differential equation \[ \frac{d}{da}U(a) = iA_\mu(x+ae_\mu)\,U(a), \qquad U(0) = \mathbf{1}. \] Writing \(U(a) = e^{iB(a)}\) and expanding \(B(a) = aB_1 + a^2 B_2 + O(a^3)\), one finds from the equation of motion at each order in \(a\): \[ B_1 = A_\mu(x), \qquad B_2 = \frac{1}{2}\partial_\mu A_\mu(x). \] Therefore \[ B_\mu(x) \equiv B(a) = aA_\mu(x) + \frac{a^2}{2}\partial_\mu A_\mu(x) + O(a^3). \]
Verification. One checks that \(\frac{d}{da}e^{iB(a)}\big|_{a=a_0} = iA_\mu(x+a_0 e_\mu)\,e^{iB(a_0)}\) holds to \(O(a^2)\) by using \(\frac{d}{da}e^{iB} = i\dot{B}e^{iB} + O([B,\dot{B}])\) and the BCH identity for the commutator term.
5.7 ④ The Haar Measure
Theorem. On a compact Lie group \(G\) there exists a unique measure \(dU\) — the Haar measure — that is invariant under both left and right multiplication by any fixed \(V\in G\): \[ \int dU\, f(VU) = \int dU\, f(UV) = \int dU\, f(U), \] and normalised by \(\int dU = 1\).
Construction. Let \(\zeta = (\zeta^1,\ldots,\zeta^{\dim G})\) be local coordinates on \(G\), with \(U = U(\zeta)\). Define the induced metric \[ g_{\alpha\beta}(\zeta) \stackrel{\mathrm{def}}{=} \operatorname{tr}\!\left[\frac{\partial U^\dagger}{\partial\zeta^\alpha}\,\frac{\partial U}{\partial\zeta^\beta}\right]. \] The Haar measure is then \[ \int dU\, f(U) = \int_{\mathcal{D}} \Bigl(\prod_\alpha d\zeta^\alpha\Bigr)\,\sqrt{\det g(\zeta)}\; f(U(\zeta)), \] where the integration domain \(\mathcal{D}\) covers \(G\) exactly once. The factor \(\sqrt{\det g}\) is the Riemannian volume element of the group manifold.
Exercise (SU(2)). Show that \(SU(2)\) is diffeomorphic to the 3-sphere \(S^3\), and write the Haar measure in spherical coordinates. (See the appendix for the SU(2) parametrisation.)
5.8 Summary
The non-Abelian lattice theory shares the same structure as scalar QED on the lattice:
- The action \(S = S_g + S_M\) is gauge invariant by construction.
- The path-integral measure \(\prod_{x,\mu}dU_\mu(x)\prod_x d\varphi(x)\) is gauge invariant because the Haar measure satisfies \(d(VU) = dU\).
- The naive continuum limits are:
- \(U(1)\) lattice theory \(\to\) Scalar QED in the continuum.
- \(SU(N)\) lattice theory \(\to\) Yang–Mills theory coupled to a scalar in the \(\mathbf{N}\) representation.
- On a finite lattice, the partition function \(Z\) is finite without any gauge fixing, because the gauge-group volume \(\left(\int dU\right)^{N_{\mathrm{links}}} = 1\) is finite. This is a major practical advantage of the lattice formulation.
5.9 Appendix: SU(2) Parametrisation
The Pauli matrices are \[ \sigma_1 = \begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \sigma_2 = \begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_3 = \begin{pmatrix}1&0\\0&-1\end{pmatrix}. \] They satisfy \(\{\sigma_i,\sigma_j\} = 2\delta_{ij}\) and \([\sigma_i,\sigma_j] = 2i\varepsilon_{ijk}\sigma_k\).
A general \(SU(2)\) matrix can be written as \[ U = a_0\mathbf{1} + i\,\mathbf{a}\cdot\boldsymbol{\sigma}, \qquad a_0,\mathbf{a}\in\mathbb{R}, \] and \(U\in SU(2)\) if and only if \(\det U = a_0^2 + |\mathbf{a}|^2 = 1\). Writing \(a_0 = \cos\theta\) and \(\mathbf{a} = \hat{\mathbf{n}}\sin\theta\) gives \[ U = \cos\theta\,\mathbf{1} + i(\hat{\mathbf{n}}\cdot\boldsymbol{\sigma})\sin\theta, \] identifying \(SU(2)\) with the unit 3-sphere \(S^3 \subset \mathbb{R}^4\).