Analogy Between Special Relativity and Finite Mathematics

viXra, 2019

Standard quantum theory is based on classical mathematics involving such notions as infinitely small/large and continuity. Those notions were proposed by Newton and Leibniz more than 300 years ago when people believed that every object can be divided by an arbitrarily large number of arbitrarily small parts. However, now it is obvious that when we reach the level of atoms and elementary particles then standard division loses its meaning and in nature there are no infinitely small objects and no continuity. In our previous publications we proposed a version of finite quantum theory (FQT) based on a finite ring or field with characteristic $p$. In the present paper we first define the notion when theory A is more general than theory B and theory B is a special degenerate case of theory A. Then we prove that standard quantum theory is a special degenerate case of FQT in the formal limit $p\to\infty$. Since quantum theory is the most general physics theory, this implies that classical m...

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 $p$ 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 $p\to\infty$: 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 $p$ is more general than standard quantum theory because the latter is a degenerate case of the former in the formal limit $p\to\infty$.

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 .

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.

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...

viXra, 2017

We discuss finite quantum theory (FQT) developed in our previous publications and give a simple explanation that standard quantum theory is a special case of FQT in the formal limit $p\to\infty$ where $p$ is the characteristic of the ring or field used in FQT. Then we argue that FQT is a more natural basis for quantum computing than standard quantum theory.

viXra, 2019

In our previous publications we have proved that quantum theory based on finite mathematics is more fundamental than standard quantum theory, and, as a consequence, finite mathematics is more fundamental than classical one. The goal of the present paper is to explain without formulas why those conclusions are natural.

2018

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. In our previous publications we argue that the situation is the opposite: classical mathematics is only a special degenerate case of finite one in the formal limit when the characteristic of the ring or field in finite mathematics goes to infinity. In the present paper we give a simple and rigorous proof of this fundamental fact. In general, introducing infinity automatically implies transition to a degenerate theory because in that case all operations modulo a number are lost. So, even from the pure mathematical point of view, the very notion of infinity cannot be fundamental, and theories involving infinities can be only approximations to more general theories. We also prove that standard quantum theory based on classical mathematics is a special degenerate case of quant...

Entropy 26, 989., 2024

The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.

Conference lecture (slides), Quantum Information and Probability: from Foundations to Engineering (QIP24), Linnaeus University, Växjö, Sweden11-14 June , 2024

Building on Martin Heidegger’s insight that, beginning with René Descartes and Galileo Galilei, “modern [physics] is experimental because of its mathematical project” [Heidegger 1967, p. 93], I argue that the advancement of modern physics has been defined by the invention of new mathematical schemes (possibly borrowing them from mathematics itself). Among the greatest such inventions, all using differential equations, are: *Classical physics, based on calculus; *Maxwell’s electromagnetic theory, based on the ideal of (classical) field and its mathematization, as represented by Maxwell’s equations; *Relativity, SR and especially GR, based on Riemannian geometry; *Quantum mechanics (QM) and quantum field theory (QFT), based on the mathematics of Hilbert spaces over C, and the operator algebras. I further argue that QM and QFT (to either of which the term quantum theory will refer hereafter) gave this thesis a radically new meaning: Quantum phenomena are defined physically, as essentially different from all previous physics, as is manifested in paradigmatic experiments such as the double-slit experiment or those dealing with quantum correlations. On the other hand, quantum theory, at least QM or QFT, is defined as different from classical physics on the basis of purely mathematical postulates, which connect it to quantum phenomena strictly in terms of probabilities, without, at least in the interpretation adopted here, representing or otherwise relating to how these phenomena come about.