Continuous symmetry entails the Jordan algebra structure of finite-dimensional quantum theory

TL;DR

This paper demonstrates that continuous symmetries, combined with specific conditions, uniquely determine the mathematical structure of finite-dimensional quantum theories as Euclidean Jordan algebras.

Contribution

It shows that continuous symmetry, along with spectrality, a strong state space, and the gbit property, leads to the Jordan algebra structure in quantum theory reconstruction.

Findings

Continuous symmetry and additional conditions imply Euclidean Jordan algebra structure.

Representation theory and Gleason's theorem are key tools in the proof.

The approach clarifies the foundational role of symmetry in quantum theory.

Abstract

Symmetry postulates play a crucial role in various approaches to reconstruct quantum theory from a few basic principles. Discrete and continuous symmetries are under consideration. The continuous case better matches the physical needs for mathematical models of dynamical processes and is studied here. Applying the representation theory of the orthomodular lattices and a generalized version of Gleason's theorem for Jordan matrix algebras, we show that the continuous symmetry, together with three further requirements, entails that the underlying mathematical structure of a finite-dimensional generalized probabilistic theory becomes a simple Euclidean Jordan algebra. The further requirements are: spectrality, a strong state space and a condition called gbit property.