On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups
Benjamin Hinrichs, Fumio Hiroshima
TL;DR
This paper proves that certain quantum semigroups related to Nelson Hamiltonians are ergodic and positivity improving, simplifying previous proofs and extending results to semi-relativistic cases.
Contribution
It provides a simplified proof of ergodicity for ultraviolet-renormalized Nelson semigroups, including the semi-relativistic case, using functional integration.
Findings
Semigroups are positivity improving and ergodic for all total momenta.
The proof simplifies existing methods for establishing ergodicity.
The result extends to semi-relativistic Nelson Hamiltonians.
Abstract
We present a simple functional integration based proof that the semigroups generated by the ultraviolet-renormalized translation-invariant non- and semi-relativistic Nelson Hamiltonians are positivity improving (and hence ergodic) with respect to the Fr\"ohlich cone for arbitrary values of the total momentum. Our argument simplifies known proofs for ergodicity and the result is new in the semi-relativistic case.
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