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

TL;DR

This paper introduces a quantum theory-based representation of real and complex numbers using qukit string states, enabling the construction of 3D reference frame fields that incorporate different bases, gauge choices, and iterative stages, linking mathematical structures to physical theories.

Contribution

It presents a novel quantum representation of real and complex numbers via qukit strings, forming a basis for 3D reference frame fields that integrate mathematical and physical structures.

Findings

Quantum representations differ from classical by basis choice and mathematical structure.

Reference frames are built from these quantum number representations, supporting multiple bases and gauge choices.

Physical quantities are viewed as equivalence classes of sequences in parent frames.

Abstract

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations 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. These aspects enable the description of 3 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. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.