Finite Mathematics, Finite Quantum Theory and a Conjecture on the Nature of Time (original) (raw)
Related papers
Finite Mathematics, Finite Quantum Theory and Applications to Gravity and Particle Theory
HAL (Le Centre pour la Communication Scientifique Directe), 2014
We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular, from Gödel's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as kinematical manifestations of quantum de Sitter symmetry, i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed. In the second part we first prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic p of the field or ring in the latter goes to infinity. This implies that 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. In a quantum theory based on finite mathematics, the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application to particle theory gives that the notion of a particle and its antiparticle is only approximate and, as a consequence: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved; b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We argue that only Dirac singletons can be true elementary particles and discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e. p and not t is the true evolution parameter.
Finite Quantum Theory and Applications to Gravity and Particle Theory
arXiv: General Physics, 2011
We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from G\"{o}del's incompleteness theorems, any mathematics involving the set of all natural numbers has its own foundational problems which cannot be resolved. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as {\it kinematical} manifestations of de Sitter symmetry on quantum level ({\it i.e. for describing those phenomena the notions of dar...
Physics of Particles and Nuclei Letters, 2017
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.
A Conjecture on the Nature of Time
Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory, 2020
In our previous publications we argue that finite mathematics is fundamental, classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a degenerate special case of finite one, and ultimate quantum theory will be based on finite mathematics. We consider a finite quantum theory (FQT) based on a finite field or ring with a large characteristic p and show that standard continuous quantum theory is a special case of FQT in the formal limit p → ∞. 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.
Discussion of foundation of mathematics and quantum theory
Open Mathematics, 2022
Following the results of our recently published book [F. Lev, Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Applications to Gravity and Particle Theory, Springer, 2020, ISBN 978-3-030-61101-9], we discuss different aspects of classical and finite mathematics and explain why finite mathematics based on a finite ring of characteristic p p is more general (fundamental) than classical mathematics: the former does not have foundational problems, and the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty . In particular, quantum theory based on a finite ring of characteristic p p is more general than standard quantum theory because the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty .
Main problems in constructing quantum theory based on finite mathematics
Mathematics, vol. 12(23), 2024
As shown in our publications, quantum theory based on a finite ring of characteristic p (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit p → ∞. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second-antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at p → ∞. Consequently, most fundamental quantum theory will not contain the concepts of particle-antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value p is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.
Quantum Theory over a Galois Field and Applications to Gravity and Particle Theory
2011
We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular, from Gödel's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as kinematical manifestations of quantum de Sitter symmetry, i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed. In the second part we first prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic p of the field or ring in the latter goes to infinity. This implies that 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. In a quantum theory based on finite mathematics, the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application to particle theory gives that the notion of a particle and its antiparticle is only approximate and, as a consequence: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved; b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We argue that only Dirac singletons can be true elementary particles and discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e. p and not t is the true evolution parameter.
Crisis in Quantum Theory and Its Possible Resolution
2014
It is argued that the main reason of crisis in quantum theory is that nature, 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 involving 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.
Finite mathematics as the most general (fundamental) mathematics
Symmetry, 2024
The purpose of this paper is to explain at the simplest possible level why finite mathematics based on a finite ring of characteristic ppp is more general (fundamental) than standard mathematics. The belief of most mathematicians and physicists that standard mathematics is the most fundamental arose for historical reasons. However, simple {\it mathematical} arguments show that standard mathematics (involving the concept of infinities) is a degenerate case of finite mathematics in the formal limit ptoinftyp\to\inftyptoinfty: standard mathematics arises from finite mathematics in the degenerate case when operations modulo a number are discarded. Quantum theory based on a finite ring of characteristic ppp is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit ptoinftyp\to\inftyptoinfty.
Quantum Mechanics, Formalization and the Cosmological Constant Problem
Foundations of Science, 2020
Based on formal arguments from Zermelo-Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo-Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of ℝ 4. The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth ℝ 4 from which one determines the realistic, agreeing with observation, small value of the vacuum energy density.