Finite Mathematics, Finite Quantum Theory And A Conjecture On The Nature Of Time

TL;DR

This paper demonstrates that classical mathematics and standard quantum theory are special cases of finite mathematics and finite quantum theory respectively, which do not require infinities or classical space-time, offering a more fundamental framework.

Contribution

It introduces finite quantum theory as a more fundamental approach, proving classical mathematics and quantum theory emerge as limits, and explores implications for cosmology without classical space-time.

Findings

Classical mathematics is a limit case of finite mathematics as p→∞.

Standard quantum theory is a special case of finite quantum theory in the limit p→∞.

Classical equations of motion can emerge from quantum quantities without space and time.

Abstract

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 $p\to\infty$. 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.