Proposition. The classical small-spacing Wilson action [fdip-007F]
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  • November 1, 2026

Fix a flat \(d\)-dimensional torus with \(d\ge 2\), a compact Lie group \(G\), and a unitary representation \(\rho :G\to U(r)\), \(r>0\). Choose smooth periodic Hermitian fields \(A_j(x)\) such that \(iA_j(x)\) lies in the image of the derived Lie-algebra representation. All derivative bounds are fixed as the mesh spacing \(a\to 0\); take meshes whose periods fit the torus. Define head-directed midpoint links \[U_j(x)=\exp \left (ia A_j(x+\tfrac a2 e_j)\right ).\] With the rightmost-first transport convention, the \(j\)-then-\(k\) plaquette is \[U_{jk}(x)=U_k(x)^*U_j(x+ae_k)^*U_k(x+ae_j)U_j(x).\] Its curvature convention is \[F_{jk}=\partial _jA_k-\partial _kA_j-i[A_j,A_k].\] The matrices \(F_{jk}\) are Hermitian.

For the Wilson action \[S_W(a)=\beta \sum _x\sum _{j<k} \left (1-\frac {\operatorname {Re}\operatorname {tr}U_{jk}(x)}r\right ),\] one has \[S_W(a)=\frac {\beta a^{4-d}}{2r} \int \sum _{j<k}\operatorname {tr}(F_{jk}(x)^2)\,dx +O(\beta a^{5-d}).\] This is a statement about evaluating a discrete action on a fixed smooth field. The constant in the remainder depends on that field, the fixed torus and the representation.

Proof. 

  • November 1, 2026

Taylor-expand the 4 midpoint values at \(x\) and use \(e^{iaA}=I+iaA-\tfrac 12a^2A^2+O(a^3)\). The terms of order \(a\) cancel. The variation of the two transverse links contributes \(ia^2(\partial _jA_k-\partial _kA_j)\); the products of first-order terms contribute \(a^2[A_j,A_k]\). The midpoint derivatives along each link cancel between opposite traversals. Hence, uniformly in \(x\), \[U_{jk}(x)=I+ia^2F_{jk}(x)+O(a^3).\] To obtain the real trace to the needed order, use the exact identity for a unitary \(r\times r\) matrix: \[1-\frac {\operatorname {Re}\operatorname {tr}U}{r} =\frac 1{2r}\operatorname {tr}\bigl ((U-I)^*(U-I)\bigr ).\] Substitution yields \[1-\frac {\operatorname {Re}\operatorname {tr}U_{jk}(x)}r =\frac {a^4}{2r}\operatorname {tr}(F_{jk}(x)^2)+O(a^5).\] There are \(O(a^{-d})\) plaquettes. After multiplication by \(a^d\), the Riemann sum for \(\sum _{j<k}\operatorname {tr}(F_{jk}^2)\) differs from its integral by \(O(a)\). Summing the uniform error and this Riemann-sum error proves the formula.

To fix a component convention, suppose the Hermitian generators in the chosen representation satisfy \[\operatorname {tr}(T_\alpha T_\gamma ) =T(\rho )\delta _{\alpha \gamma },\qquad T(\rho )>0.\] Write \(F_{jk}=\sum _\alpha F_{jk}^\alpha T_\alpha \) and fix \(g>0\). Choosing \[\beta =\frac {r a^{d-4}}{T(\rho )g^2}\] gives the classical limit \[\begin {aligned} S_W(a)&=\frac 1{2g^2}\int \sum _{j<k,\alpha }(F_{jk}^\alpha )^2\,dx+O(a)\\ &=\frac 1{4g^2}\int \sum _{j,k,\alpha }(F_{jk}^\alpha )^2\,dx+O(a). \end {aligned}\] The second sum counts both orders, with \(F_{kj}=-F_{jk}\) and \(F_{jj}=0\). The trace normalization is an explicit assumption; it must not be imposed on directions annihilated by a nonfaithful representation. For the fundamental representation of \(SU(r)\) with \(T(\rho )=1/2\), this is \(\beta =2r a^{d-4}/g^2\). For unit-charge \(U(1)\), \(r=T(\rho )=1\), giving \(\beta =a^{d-4}/g^2\).

The matrices \(A_j\) are those appearing in the link exponential; any coupling absorbed into them must be retained when comparing component conventions. Wingate's equation (6.20) and Exercise 2 [wingate2025lattice, p. 8] give the Wilson-action context and the four-dimensional fundamental normalization. The ordered expansion and error estimate above are derived with the convention stated here. They do not prove convergence of lattice probability measures, renormalization or the existence of a quantum continuum theory.