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Journal of Lie Theory

Spectral Properties of Convex Bodies

H. Barnum, J. Hilgert

Published Jan 1, 2020• 0 citations
Publisher

Abstract

We use the Madden-Robertson classification of regular convex bodies to show that convex bodies are spectral and strongly symmetric if and only if they are affinely isomorphic to the normalized state spaces of simple euclidean Jordan algebras, or to simplices.Further, we discuss the relevance of this result for general probabilistic theories of quantum and classical physical systems, and its relation to other characterizations of various classes of euclidean Jordan algebra state spaces.

DOI
10.5802/jolt.1118

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•Jan 1, 2020•22 citations
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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.
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