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

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 ψ_{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}ψ_{x}. Here a_{x} and ψ_{x} are the same number and state in U_{x} as a_{y} and ψ_{y} are in U_{y}. If y=x+\hatμdx is a neighbor point of x, then the scaling factor is r_{y,x}=\exp(\vec{A}(x)\cdot\hatμdx) 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 ψ_{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, \intψ^{*}(y)yψ(y)dy, by \int_{x}r_{y,x}ψ^{*}_{x}(y_{x})y_{x}ψ_{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.