Continuous symmetry entails the Jordan algebra structure of quantum theory

• URL: http://arxiv.org/abs/2411.19672v2
• Date: Fri, 13 Dec 2024 12:46:14 GMT
• Title: Continuous symmetry entails the Jordan algebra structure of quantum theory
• Authors: Gerd Niestegge,
• Abstract summary: 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.
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• 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, 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.

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