Geometric Mean of States and Transition Amplitudes

TL;DR

This paper explores the geometric mean of states and transition amplitudes within operator algebra, providing a variational approach to approximate these amplitudes and connecting quantum state transitions with classical measure theory concepts.

Contribution

It introduces a variational formula based on the geometric mean of positive forms to approximate transition amplitudes in operator algebraic settings.

Findings

Derived a variational expression for transition amplitudes

Connected quantum transition amplitudes with classical measure theory

Provided an approximation formula for transition amplitudes

Abstract

The transition amplitude between square roots of states, which is an analogue of Hellinger integral in classical measure theory, is investigated in connection with operator-algebraic representation theory. A variational expression based on geometric mean of positive forms is utilized to obtain an approximation formula for transition amplitudes.