Definition. The joint gauge and real-matter measure [fdip-0076]

✍️sourceAGENTDRAFTED

Fix the graph, positive edge weights and invariant fiber metric of a connection background. When using a Wilson action, also supply square faces and their attaching walks. Let \(G\) be a finite group, or a compact Hausdorff group supplied with a normalized bi-invariant Borel probability measure \(\mu _G\). In the finite case use uniform probability measure. Let \(\rho :G\to O(F)\) be a continuous orthogonal representation on a finite-dimensional real inner-product space; continuity is automatic for a finite group. The compact-group measure is assumed as part of the data here.

Put \(\mathcal H=F^V\), \(N=\dim _{\mathbb R}\mathcal H\) and \(d\mu (U)=\prod _{e\in E}d\mu _G(U_e)\). Choose \(m>0\) and write \(Q_U=L_U+m^2I\). Use a continuous real gauge-invariant action \(S_W(U)\), such as the Wilson action with its declared representation and coupling. With ordinary Lebesgue volume \(d\operatorname {vol}(\phi )\) in the fixed metric, define \[\begin {aligned} \mathfrak Z &=\int _{G^E}\int _{\mathcal H} e^{-S_W(U)-\frac 12\langle \phi ,Q_U\phi \rangle } \,d\operatorname {vol}(\phi )\,d\mu (U),\\ d\mathbb P(U,\phi ) &=\mathfrak Z^{-1} e^{-S_W(U)-\frac 12\langle \phi ,Q_U\phi \rangle } \,d\mu (U)\,d\operatorname {vol}(\phi ). \end {aligned}\] This is a law on both the connection and the real matter field.

The normalization is finite and positive. Indeed, \(Q_U\ge m^2I\), and the continuous action has a finite lower bound on compact \(G^E\). Thus the integrand is bounded by a constant times \(e^{-m^2\|\phi \|^2/2}\), which is integrable. It is everywhere positive. A gauge transformation preserves the edge measure by left and right invariance, the matter volume by orthogonality, and the quadratic form by gauge covariance. Hence the joint law is gauge invariant.

The edge measure differs from the unnormalized counting convention in Definition [fdip-006O]: for finite \(G\), replacing \(\mu \) by counting measure multiplies \(\mathfrak Z\) by \(|G|^{|E|}\) and leaves normalized expectations unchanged. The matter volume is also explicit: it has not been divided by \((2\pi )^{N/2}\).