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

TL;DR

The paper investigates how continuous symmetry constrains finite-dimensional generalized probabilistic theories. By combining spectral order unit spaces, a strongly order-determining state space, the gbit-property (equivalent to the covering property), and weak/continuous symmetry, the author proves that the observable structure must be a simple Euclidean Jordan algebra, specifically spin factors or Jordan matrix algebras over the real, complex, or quaternionic numbers (with a notable exception at m=3). The approach relies on orthomodular lattice representation and a Gleason-type theorem for Jordan matrix algebras, yielding a principled reconstruction of quantum-like finite-dimensional theories from symmetry and information-theoretic assumptions. The results clarify the role of the covering property and continuous symmetry in excluding classical and reducible models, offering a broad reconstruction framework for quantum theory while highlighting open cases (notably m=3) and potential infinite-dimensional extensions.

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.