Evolution integrals

TL;DR

This paper introduces a novel framework called evolution integrals, inspired by quantum path integrals, for abstract dynamical systems within a W*-algebraic setting, leading to a new perspective on quantum-like dynamics.

Contribution

It develops a new mathematical framework for evolution integrals in W*-algebras, extending path integral concepts to abstract dynamical systems.

Findings

Defines spaces of evolutions as analogues of classical paths.

Constructs projection-valued measures for evolution integrals.

Demonstrates how these integrals induce dynamics in a Hilbert space.

Abstract

A framework analogous to path integrals in quantum physics is set up for abstract dynamical systems in a W*-algebraic setting. We consider spaces of evolutions, defined in a specific way, of a W*-algebra A as an analogue of spaces of classical paths, and show how integrals over such spaces, which we call ``evolution integrals'', lead to dynamics in a Hilbert space on a ``higher level'' which is viewed as an analogue of quantum dynamics obtained from path integrals. The measures with respect to which these integrals are performed are projection valued.