Local Availability of mathematics and number scaling: Effects on quantum physics

TL;DR

This paper introduces a framework where mathematics and number systems are locally available at each spacetime point, with number scaling affecting quantum theory, and shows that scaling effects are negligible at small scales but may be significant cosmologically.

Contribution

It proposes a novel approach assigning separate mathematical universes to each spacetime point and extends gauge freedom to number systems, impacting quantum theory.

Findings

Scaling has external and internal components affecting quantum expressions.

Scaling effects are negligible within a few light years, including the solar system.

Outside this region, at cosmological scales, scaling could have significant effects.

Abstract

Local availability of mathematics and number scaling provide an approach to a coherent theory of physics and mathematics. Local availability of mathematics assigns separate mathematical universes, U_{x}, to each space time point, x. The mathematics available to an observer, O_{x}, at x is contained in U_{x}. Number scaling is based on extending the choice freedom of vector space bases in gauge theories to choice freedom of underlying number systems. Scaling arises in the description, in U_{x}, of mathematical systems in U_{y}. If a_{y} or \psi_{y} is a number or a quantum state in U_{y}, then the corresponding number or state in U_{x} is r_{y,x}a_{x} or r_{y,x}\psi_{x}. Here a_{x} and \psi_{x} are the same number and state in U_{x} as a_{y} and \psi_{y} are in U_{y}. If y=x+\hat{\mu}dx is a neighbor point of x, then the scaling factor is r_{y,x}=\exp(\vec{A}(x)\cdot\hat{\mu}dx) where \vec{A} is a vector field, assumed here to be the gradient of a scalar field. The effects of scaling and local availability of mathematics on quantum theory show that scaling has two components, external and internal. External scaling is shown above for a_{y} and \psi_{y}. Internal scaling occurs in expressions with integrals or derivatives over space or space time. An example is the replacement of the position expectation value, \int\psi^{*}(y)y\psi(y)dy, by \int_{x}r_{y,x}\psi^{*}_{x}(y_{x})y_{x}\psi_{x}(y_{x})dy_{x}. This is an integral in U_{x}. The good agreement between quantum theory and experiment shows that scaling is negligible in a space region, L, in which experiments and calculations can be done, and results compared. L includes the solar system, but the speed of light limits the size of L to a few light years. Outside of $L$, at cosmological distances, the limits on scaling are not present.