Information dynamics and new geometric foundations of quantum theory

TL;DR

This paper introduces a novel geometric framework for quantum theory that does not rely on traditional Hilbert spaces or measure spaces, instead using non-linear geometry and entropy maximization to unify different approaches.

Contribution

It presents a new mathematical foundation for quantum theory based on non-linear geometry and entropy, recovering standard Hilbert space and measure-theoretic methods as special cases.

Findings

Unified geometric framework for quantum theory.

Recovery of Hilbert space and measure-theoretic approaches.

New insights into quantum dynamics via entropy maximization.

Abstract

We discuss new approach to mathematical foundations of quantum theory, which is completely independent of Hilbert spaces and measure spaces. New kinematics is defined by non-linear geometry of spaces of integrals on abstract non-commutative algebras. New dynamics is defined by constrained maximisation of quantum relative entropy. We recover Hilbert space based approach (including unitary evolution and the von Neumann--L\"{u}ders rule) and measure theoretic approach to probability theory (including Bayes' rule) as special cases of our approach.