Meaning
A Markov coarse-graining shadow is a finite-rank, gauge-compatible averaging operation that preserves positivity and constants.
GAT VIII constructs a normalized nonnegative central kernel kρ(g)=|χρ(g)|²/∫|χρ|² from a finite-dimensional unitary representation. Convolution has finite Peter–Weyl support. Tensoring such kernels over edges and applying Gauss averaging gives the finite-rank Markov shadow.
Example (illustrative)
As a finite probability example, (f₁,f₂)↦((f₁+f₂)/2,(f₁+f₂)/2) has rank one, preserves nonnegative functions, and fixes constants. It illustrates the averaging properties; the source construction additionally certifies gauge compatibility.
Scope and limits
This averaging shadow is distinct from the sharp physical detector. A general projection or finite-rank compression need not be a positive Markov operation.
Source
GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory. GAT VIII, Section 15.1.