Ground-State Markovization

Source result: GAT VIII, Theorem 15.1.
Status: Source-bound theorem in the public preprint.

Catalog summary

A certified Dirichlet realization with a normalized strictly positive rank-one ground admits a ground-state transform whose semigroup preserves positivity and constants and is symmetric in the ground-weighted measure.

What the theorem says

Subtracting the ground energy from a Hamiltonian does not, by itself, turn it into a probability-preserving evolution. The ground-state transform also changes the representation and the measure, using the positive ground function. In that representation the ground becomes the constant function one, and the transformed evolution preserves constants.

Hypotheses and scope

At a selected endpoint, assume an independently certified Dirichlet realization on with self-adjoint operator . The physical application uses the certified bridge from the configured physical form to this realization. Assume a normalized strictly positive ground function with

The rank-one and positivity requirements are specific to this result. They are not asserted for every complete-ground sector in the program.

Precise conclusion

Set and . Multiplication by is unitary from to . The nonnegative operator

has domain . Its semigroup is conservative, symmetric, and Markov on . Its form is

“Conservative” means . “Markov” includes preservation of positivity and of the interval between zero and one. The positive operator is written in the convention ; the infinitesimal semigroup generator is .

Why it works

The original Dirichlet semigroup preserves positivity. Conjugation by the positive ground function retains positivity, while the ground-state equation makes the transformed semigroup fix one. Unitary transport gives symmetry in the new measure.

Example

On two points with counting measure, take . Its normalized positive ground is , with ground energy one. The ground-weighted measure assigns probability to each point, and the transformed operator is . Thus is the symmetric two-state Markov evolution and fixes the constant vector.

This is an elementary model of the transform, not a physical Yang–Mills endpoint.

Claim limits

The result does not make a general sharp detector compression positivity preserving, establish a continuum stochastic process, or derive a measurement probability rule. The certified Dirichlet realization and its positive ground must actually be supplied.

Source

GAT VIII, §15, equations (15.1)–(15.5), Theorem 15.1. The ground-state/Doob transform is standard; the physical bridge is an additional certified input here.