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Crisis in Quantum Theory and Its Possible Resolution

Felix M. Lev

Published May 1, 2014• 0 citations
OpenAlex

Abstract

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.

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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.
Sabine Hossenfelder argues in her blog [1] that the present situation in foundation of physics should be called not crisis but stagnation. I argue that the main reason of the stagnation is that quantum theory inherited from classical one several notions which should not be present in quantum theory. In particular , quantum theory should not involve the notion of space-time background and, since nature is discrete and even finite, quantum theory should not be based on classical mathematics involving the notions of infinitely small/large and continuity. I discuss uncertainty relations, paradox with observation of stars, symmetry on quantum level, cosmological acceleration, gravity and particle theory. My main conclusion is that the most general quantum theory should be based on finite mathematics and, as a consequence: Mathematics describing nature at the most fundamental level involves only a finite number of numbers while the notions of limit and infinitely small/large and the notions constructed from them (e.g. continuity, derivative and integral) are needed only in calculations describing nature approximately.
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the situation is the opposite: classical mathematics is only a degenerate special case of finite one and finite mathematics is more pertinent for describing nature than standard one. Then we describe results of a quantum theory based on finite mathematics. Implications for foundation of mathematics are discussed.
We first give a rigorous mathematical proof that classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a special degenerate case of finite one in the formal limit when the characteristic p of the field or ring in finite mathematics goes to infinity. We consider a finite quantum theory (FQT) based on finite mathematics and prove that standard continuous quantum theory is a special case of FQT in the formal limit p→∞.p \to \infty .p→∞. The description of states in standard quantum theory contains a big redundancy of elements: the theory is based on real numbers while with any desired accuracy the states can be described by using only integers, i.e. rational and real numbers play only auxiliary role. Therefore, in FQT infinities cannot exist in principle, FQT is based on a more fundamental mathematics than standard quantum theory and the description of states in FQT is much more thrifty than in standard quantum theory. Space and time are purely classical notions and are not present in FQT at all. In the present paper we discuss how classical equations of motions arise as a consequence of the fact that p changes, i.e. p is the evolution parameter. It is shown that there exist scenarios when classical equations of motion for cosmological acceleration and gravity can be formulated exclusively in terms of quantum quantities without using space, time and standard semiclassical approximation.
We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers.