Effects on quantum physics of the local availability of mathematics and space time dependent scaling factors for number systems

TL;DR

This paper explores how local mathematical structures and space-time dependent scaling factors influence quantum physics, revealing effects on wave packets, momenta, and gauge theories within a framework of locally available mathematics.

Contribution

It introduces a novel framework where local mathematical universes and space-time dependent scaling factors impact quantum physics and gauge theories.

Findings

Scaling factors affect wave packets and momenta in quantum theory.

Number scaling influences mathematical expressions with integrals and derivatives.

Properties of the gauge field $oldsymbol{A}$ are linked to quantum and gauge theory phenomena.

Abstract

The work is based on two premises: local availability of mathematics to an observer at any space time location, and the observation that number systems, as structures satisfying axioms for the number type being considered, can be scaled by arbitrary, positive real numbers. Local availability leads to the assignment of mathematical universes, $V_{x},$ to each point, $x,$ of space time. $V_{x}$ contains all the mathematics that an observer, $O_{x},$ at $x,$ can know. Each $V_{x}$ contains many types of mathematical systems. These include the different types of numbers (natural numbers, integers, rationals, and real and complex numbers), Hilbert spaces, algebras, and many other types of systems. Space time dependent scaling of number systems is used to define representations, in $V_{x}$, of real and complex number systems in $V_{y}$. The representations are scaled by a factor $r_{y,x}$ relative to the systems in $V_{x}.$ For $y$ a neighbor point of $x,$ $r_{y,x}$ is the exponential of the scalar product of a gauge field, $\vec{A}(x),$ and the vector from $x$ to $y.$ For $y$ distant from $x,$ $r_{y,x}$ is a path integral from $x$ to $y.$ Some consequences of the two premises will be examined. Number scaling has no effect on general comparisons of numbers obtained as computations or as experimental outputs. The effect is limited to mathematical expressions that include space or space time integrals or derivatives. The effect of $\vec{A}$ on wave packets and canonical momenta in quantum theory, and some properties of $\vec{A}$ in gauge theories, is described.