Finite Mathematics, Finite Quantum Theory and Applications to Gravity and Particle Theory

TL;DR

This paper proposes that the fundamental nature of reality is finite and discrete, reformulating quantum theory within finite mathematics, which resolves some inconsistencies and offers new perspectives on gravity and cosmology.

Contribution

It introduces a finite mathematical framework for quantum theory, showing how classical mathematics is a limit case and explaining gravity as a consequence of finiteness.

Findings

Standard quantum predictions conflict with astrophysical observations.

Cosmological acceleration can be explained by de Sitter symmetry without dark energy.

Gravity emerges from the finiteness of nature, vanishing as the characteristic p approaches infinity.

Abstract

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 G\"{o}del'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 {\it kinematical} manifestations of quantum de Sitter symmetry, {\it 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 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. {\bf 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 $p\to\infty$, i.e. gravity is a consequence of finiteness of nature.