Based on the Hilbert $C^*$ -module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.Based on the Hilbert $C^*$ -module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.

Bulletin of the Korean Mathematical SocietyJan 31, 20125 citations

HTSD operators and local measurement theory in locally covariant quantum field theory

Christoph Hartmann

This paper is part of a broader project on the Hartmann Operator Geometry (HTSD) developed in the monograph “The Hartmann Operator Geometry – Volume I”. This preprint investigates the reduced Heisenberg evolution induced by local system–probe measurements in the Fewster–Verch framework of locally covariant quantum field theory. Starting from locally covariant system and probe functors and a local coupling in a compact region, the work considers the induced completely positive unital maps on the system algebra and assumes a weak–coupling / short–time regime that yields a norm–continuous Markovian semigroup. Under this local Markovianity assumption, the generator of the reduced dynamics is shown to have a purely diffusion–type Lindblad form built from finitely many selfadjoint “pointer” observables. Each pointer observable defines a local inner derivation, and the generator can be written as a Laplace–type operator of HTSD form, i.e. as a quadratic combination of these derivations. In this way, HTSD operator geometry appears as a natural differential structure underlying local measurement dynamics in locally covariant AQFT, with an associated Dirichlet form and curvature tensor encoding energy and non-integrability properties of the measurement–induced evolution.This paper is part of a broader project on the Hartmann Operator Geometry (HTSD) developed in the monograph “The Hartmann Operator Geometry – Volume I”. This preprint investigates the reduced Heisenberg evolution induced by local system–probe measurements in the Fewster–Verch framework of locally covariant quantum field theory. Starting from locally covariant system and probe functors and a local coupling in a compact region, the work considers the induced completely positive unital maps on the system algebra and assumes a weak–coupling / short–time regime that yields a norm–continuous Markovian semigroup. Under this local Markovianity assumption, the generator of the reduced dynamics is shown to have a purely diffusion–type Lindblad form built from finitely many selfadjoint “pointer” observables. Each pointer observable defines a local inner derivation, and the generator can be written as a Laplace–type operator of HTSD form, i.e. as a quadratic combination of these derivations. In this way, HTSD operator geometry appears as a natural differential structure underlying local measurement dynamics in locally covariant AQFT, with an associated Dirichlet form and curvature tensor encoding energy and non-integrability properties of the measurement–induced evolution.

Nov 24, 20250 citations

Based on the Hilbert

C^*

-module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.