A Simple Proof That Finite Quantum Theory And Finite Mathematics Are More Fundamental Than Standard Quantum Theory And Classical Mathematics, Respectively

Abstract

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 mathematics itself is a special degenerate case of finite mathematics in the formal limit when the characteristic of the ring or field in the latter goes to infinity. In general, introducing infinity automatically implies transition to a degenerate theory because in that case all operations modulo a number are lost. So, {\it 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. Motivation and implications are discussed.