Gauge-Invariant Markov Ancestor

Source result: GAT VIII, Theorem 14.1.
Status: Source-bound theorem in the public preprint.

Catalog summary

Before sharp detector compression, the finite electric–magnetic form is a closed symmetric Dirichlet form on the compact configuration space. Restricting it to gauge-invariant functions preserves that structure.

What the theorem says

The electric–magnetic form has a positivity-preserving analytic realization before the sharp detector is applied. Its energy decreases under normal contractions of real functions. A normal contraction fixes zero and changes distances by at most their original size; clipping a real function to the interval is one example.

This contraction property, together with a closed densely defined symmetric nonnegative form, is the Dirichlet-form structure. “Dirichlet” here names that analytic property, not an imposed spatial boundary condition.

Hypotheses and scope

Use the finite compact-group configuration space with product Haar probability measure and the electric–magnetic coefficients and Sobolev realization of GAT VIII §14. Electric terms are nonnegative weighted squared gradients. Magnetic terms use finite-dimensional unitary representations and the bounded nonnegative potentials

The closed form is the Sobolev electric form plus the stated bounded nonnegative magnetic potential, before sharp Peter–Weyl detector compression.

Precise conclusion

The form

is a closed symmetric Dirichlet form with smooth form core . Its restriction to the gauge-invariant subspace is also Dirichlet, with gauge-invariant smooth core .

Why it works

The Sobolev chain rule makes the gradient term decrease under normal contraction. The magnetic term decreases because the potential is nonnegative and the contraction does not increase absolute value. Compact-group Sobolev closedness and boundedness of the potential give closedness. Gauge invariance and gauge averaging preserve the invariant subspace and its smooth core.

Example

Clip a real configuration function to . The clipped function has no larger gradient magnitude almost everywhere and no larger absolute value. Neither the electric nor magnetic contribution increases. If is gauge invariant, so is .

Claim limits

The magnetic potential can produce killing, so this Dirichlet semigroup need not preserve constants. The conservative ground-state transform requires the additional hypotheses of Theorem 15.1. Nor is a sharp finite detector projection automatically positivity preserving: the ancestor’s contraction property does not automatically survive that compression.

Source

GAT VIII, §14, equations (14.1)–(14.4), Theorem 14.1, and the qualification after Proposition 14.2. The Sobolev and Dirichlet-form mechanisms are standard; the term “ancestor” identifies their position before detector compression.