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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Hidden models and logical structures in quantum theory

F. Thieffine

Published Mar 1, 1978• 0 citations
Publisher
DOI
10.1007/bf02730339

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Quantum Theory and Logic

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In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.
Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes occurring in quantum mechanics, which Kochen and Specker in [5] showed were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker [5]), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker [5], and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.
Abstract We present a notion of an un-measurable state, which we call an Anti-state and derive a simple and intuitive logic for Anti-states. Anti-states are then used to generate structures, which we call Anti-Structures. Theorems 3.25 and 4.2 yield that Anti-Structures are able to model the context changes occurring in quantum mechanics, which Kochen and Specker in 5 showed were un-modellable with local hidden variables. In our Examples 3.28, 4.3, and 4.7 , we show that, respectively, the orthohelium experiment (outlined in Kochen, Specker 5), wave functions, and entanglement all have Anti-Structure models. Our results give a good contrast to the results of Kochen and Specker 5, and establish a limitation on self reference for partial Boolean algebras. This limitation is not held by Anti-Structures in general, and also does not seem to be a limitation that should be put on our physics. The theorems and examples of this paper lead to a series of questions regarding the expressiveness and a measure of contrivance for models of quantum mechanics.