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Effect Algebras Are Not Adequate Models for Quantum Mechanics

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Abstract

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states.

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  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    Stan Gudder

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  1. Stan Gudder
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Correspondence to Stan Gudder.

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Gudder, S. Effect Algebras Are Not Adequate Models for Quantum Mechanics. Found Phys 40, 1566–1577 (2010). https://doi.org/10.1007/s10701-009-9369-7

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  • DOI: https://doi.org/10.1007/s10701-009-9369-7

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