Theorem. Integrating and conditioning the matter field [fdip-0077]
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  • November 1, 2026

For the joint law, let \(W(U)=\frac 12\log \det _{\mathbb R}Q_U\) and set \[Z_U=\int _{G^E}e^{-S_W(U)-W(U)}\,d\mu (U).\] Then \[\mathfrak Z=(2\pi )^{N/2}Z_U,\qquad d\mathbb P_U(U)=Z_U^{-1}e^{-S_W(U)-W(U)}\,d\mu (U).\] A conditional matter kernel is \[K_U(d\phi )=(2\pi )^{-N/2}(\det Q_U)^{1/2} e^{-\frac 12\langle \phi ,Q_U\phi \rangle } \,d\operatorname {vol}(\phi ),\] so, conditionally on \(U\), the field has law \(N(0,Q_U^{-1})\). In particular the induced action adds to the bare gauge action exactly, with no large-mass approximation.

Proof. 

  • November 1, 2026

The real Gaussian integral, converted from its normalized reference volume to ordinary volume, gives \[\int _{\mathcal H}e^{-\frac 12\langle \phi ,Q_U\phi \rangle } \,d\operatorname {vol}(\phi ) =(2\pi )^{N/2}e^{-W(U)}.\] Tonelli's theorem applies to the nonnegative joint density. Integrating first over \(\phi \) proves the normalization and marginal formulas. Dividing the fixed-\(U\) matter density by this Gaussian integral yields \(K_U\). Multiplying the two displayed densities recovers the joint density, proving the disintegration identity \[\mathbb P(dU,d\phi )=\mathbb P_U(dU)K_U(d\phi ).\] This explicitly chosen kernel supplies a conditional version at every connection.

Mathlib's MeasureTheory.lintegral_prod states Tonelli's identity for an almost-everywhere measurable nonnegative extended-real function, with an s-finite inner measure. The gauge probability and finite-dimensional Lebesgue measure satisfy these measure conditions; the density here is measurable. This is partial support for the product-integration step, not for the Gaussian evaluation or the marginal and conditional formulas derived above.

The selected Gaussian marginal and conditional propagator formulas therefore apply separately at each fixed \(U\). For example, in orthonormal retained and eliminated coordinates, write \[Q_U=\begin {pmatrix}A_U&B_U\\B_U^T&C_U\end {pmatrix},\qquad C_U>0.\] Then the retained precision at that connection is \(A_U-B_UC_U^{-1}B_U^T\), and \[\phi _{\mathrm {elim}}\mid (\phi _{\mathrm {ret}}=x,U) \sim N(-C_U^{-1}B_U^Tx,C_U^{-1}).\] These statements keep the connection in the conditioning data.

After averaging over \(U\), the matter law is generally a mixture of Gaussian laws, rather than one Gaussian law. Its covariance is \[\mathbb E_{\mathbb P_U}[Q_U^{-1}],\] by the conditional covariance identity and the zero conditional mean. Observing retained matter coordinates also reweights the distribution of \(U\) by their conditional likelihood. One cannot obtain the joint conditional law by keeping the original connection probabilities fixed and applying a single Gaussian formula.