Properties of Frame Fields based on Quantum Theory Representations of Real and Complex Numbers

Here quantum theory representations of real (R) and complex (C) numbers are described as equivalence classes of Cauchy sequences of states of single, finite strings of qukits where the qukit string states represent rational numbers. This work extends earlier work with qubit string states to qukit string states for any base k ≥ 2. Quantum theory representations differ from the usual classical representations as states of kit strings in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. In particular, states of qukit strings are elements of Hilbert spaces, which are vector spaces over the complex field. These aspects enable the description of three dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. One result of note is that the R and C values of physical quantities, which are viewed in a frame as elementary points, are seen in a parent frame as equivalence classes of Cauchy sequences of states of qukit