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Quantum Systems based upon Galois Fields – from Sub-quantum to Super-quantum Correlations –

Lay Nam Chang, Zachary Lewis, Djordje Minić, Tatsu Takeuchi

Published Aug 15, 2016• 3 citations
OpenAlex

Abstract

In this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

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In this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.
We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of Galois fields (which is extremely simple and elegant) is discussed in detail and we also discuss the conditions when our description gives the same predictions as the conventional one. In quantum theory based on a Galois field, all operators are well defined and divergencies cannot not exist in principle. A particle and its antiparticle are described by the same modular irreducible representation of the symmetry algebra. This automatically explains the existence of antiparticles and shows that a particle and its antiparticle are the different states of the same object. As a consequence, a new symmetry arises, and the structure of the theory is considerably simplified. In particular, one can work with only creation operators or only annihilation ones since they are no more independent. In our approach the problem arises whether the existence of neutral elementary particles (e.g. the photon) is compatible with the usual relation between spin and statistics, or in other words whether neutral particles can be elementary or only composite.
It is argued that the main reason of crisis in quantum theory is that na-ture, which is fundamentally discrete, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from Gödel’s incompleteness theorems, any mathematics in-volving the set of all natural numbers has its own foundational problems which cannot be resolved. In the first part of the paper inconsistencies in standard approach to quantum theory are discussed and the theory is reformulated such that it can be naturally generalized to a formulation based on discrete and finite mathematics. Then the cosmological acceleration and gravity can be treated simply as kinematical manifestations of de Sitter symmetry on quantum level (i.e. for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed). In the second part of the paper motivation, ideas and main results of a quantum theory over a Galois field (GFQT) are described. In contrast to standard quantum theory, GFQT is based on a solid mathematics and therefore can be treated as a candidate for ultimate quantum theory. The presentation is non-technical and should be understandable by a wide audience of physicists and mathematicians.