Space- and time-dependent scaling of numbers in mathematical structures: effects on physical and geometric quantities

The relationship between the foundations of mathematics and physics is a topic of of much interest. This paper continues this exploration by examination of the effect of space- and time- dependent number scaling on theoretical descriptions of some physical and geometric quantities. Fiber bundles provide a good framework to introduce a space- and time- or space–time-dependent number scaling field. The effect of the scaling field on a few nonlocal physical and geometric quantities is described. The effect on gauge theories is to introduce a new complex scalar field into the derivatives appearing in Lagrangians. U(1) invariance of Lagrangian terms does not affect the real part of the scaling field. For this field, any mass is possible. The scaling field is also shown to affect quantum wave packets and path lengths, and geodesic equations even on flat space. Scalar fields described so far in physics are possible candidates for the scaling field. The lack of direct evidence for the field in physics restricts the scaling field in that the gradient of the field must be close to zero in a local region of cosmological space and time. There are no restrictions outside the region. It is also seen that the scaling field does not affect comparisons of computation or measurements outputs with one another. However, it does affect the assignment of numerical values to the outputs of computations or measurements. These are needed because theory predictions are in terms of numerical values.