Transition Probabilities of Positive Linear Functionals on $*$-Algebras

TL;DR

This paper investigates the transition probabilities between positive linear functionals on unital *-algebras using unbounded Hilbert space representations, with applications to density matrices, integrals, and Weyl algebra functionals.

Contribution

It establishes fundamental results on transition probabilities assuming essential self-adjointness of GNS representations, extending previous work to unbounded cases.

Findings

Derived transition probability formulas under self-adjointness assumptions

Applied results to density matrix and integral functionals

Analyzed vector functionals on the Weyl algebra

Abstract

Using unbounded Hilbert space representations basic results on the transition probability of positive linear functionals $f$ and $g$ on a unital *-algebra are obtained. The main assumption is the essential self-adjointness of GNS representations $\pi_f$ and $\pi_g$. Applications to functionals given by density matrices and by integrals and to vector functionals on the Weyl algebra are given.