Theorem. Finite gauge averaging [fdip-006P]
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  • November 1, 2026

For the finite Wilson law, define \[(\mathcal AF)(U)=\frac 1{|\mathcal G|} \sum _{g\in \mathcal G}F(gU),\qquad \mathcal G=G^V.\] Then \(\mathcal A\) projects onto gauge-invariant observables and \[\langle F\rangle _W=\langle \mathcal AF\rangle _W.\] In particular \(\langle F\rangle _W=0\) whenever \(\mathcal AF=0\). A general noninvariant observable may have a nonzero invariant part; only \(F-\mathcal AF\) is always forced to have zero expectation.

Proof. 

  • November 1, 2026

For each \(g\), the map \(U\mapsto gU\) is a bijection of \(G^E\) with inverse \(U\mapsto g^{-1}U\). Gauge invariance of the action and change of variable in a finite sum give \[\sum _U F(gU)e^{-S_W(U)}=\sum _U F(U)e^{-S_W(U)}.\] Divide by \(Z_W\) and average over \(g\). Reindexing \(g\mapsto gh\) gives \((\mathcal AF)(hU)=(\mathcal AF)(U)\). If \(F\) is already invariant then \(\mathcal AF=F\), proving \(\mathcal A^2=\mathcal A\) and the asserted range.

Let \(\sigma \) be a nontrivial irreducible complex representation of \(G\), and let \(e\) have distinct endpoints. Then \[\langle \sigma (U_e)\rangle _W=0.\] To see this, set \(R=|G|^{-1}\sum _{a\in G}\sigma (a)\). Reindexing shows that \(\sigma (b)R=R\) for every \(b\), so \(R\) maps into the fixed-vector space. It is the identity on that space. A nonzero fixed vector would span a trivial subrepresentation; irreducibility and nontriviality therefore make the fixed-vector space zero and \(R=0\). Averaging the gauge transformation at \(h(e)\) while fixing the other vertices now replaces \(\sigma (U_e)\) by \(R\sigma (U_e)=0\). The expectation is unchanged by this averaging, by the same finite change-of-variable proof. Distinct endpoints are essential here.

For a self-loop, the transformation is only conjugation. The one-loop example shows that its expectation need not vanish.

This is a direct finite averaging proof, related to Maschke averaging. Mathlib's Representation.isProj_averageMap proves projection onto invariants for a finite group representation over a commutative ring in which the group order is invertible. Over \(\mathbb C\), apply it to the inverse-pullback representation \((R_gF)(U)=F(g^{-1}U)\), or to \(\sigma \) above. This supports the linear averaging step, not the Wilson probability or torus formulas.