Reference frame fields based on quantum theory representations of real and complex numbers

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 ≥ 2. Arithmetic and transformation properties of these states are given, both for basis states representing rational numbers and linear superpositions of these states. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of q k string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical representations as kit string states 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 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. Some approaches to integrating this work 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 states of qukit strings.

A Possible Approach to Inclusion of Space and Time in Frame Fields of Quantum Representations of Real and Complex Numbers

“…In this paper a possible approach to a coherent theory of physics and mathematics is described. The approach is based on the field of reference frames that follows from the properties of quantum mechanical representations of real and complex numbers [12,13].…”

“…All this and more is discussed in the rest of the paper. The next section is a brief summary of quantum theory representations and the resulting frame fields [12,13]. Section 3 describes a possible approach to a coherent theory of physics and mathematics as the inclusion of space and time lattices in each frame of the frame field.…”

“…This is especially the case when one considers that natural numbers, integers, rational numbers, and probably real and complex numbers, are even more fundamental to physics than is geometry. In earlier work [12,13] quantum theory representations of natural numbers (N ), integers (I), and rational numbers (Ra), were described by states of finite length qukit strings that include one qubit. To keep the description as simple as possible, the strings are considered to be finite sets of qukits and one qubit with the qukits and qubit parameterized by integer labels.…”

“…Gauge transformations correspond to a basis change (g to g ′ ) and base transformations correspond to a base change (k to k ′ ). Details are given in [13].…”