The field of real numbers in axiomatic quantum mechanics

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Abstract
Consider a system of states and observables based on the logic L(V, D) of all φ‐closed subspaces of a vector space V over a division ring D with a definite Hermitian form φ. Gudder and Piron have shown that if the system admits a special kind of observable called the smooth maximal observable, then the division ring D must be an extension of the real field. In this paper it is shown that the same conclusion can be obtained from weaker and more physical assumptions which are in principle experiment verifiable.

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