Space of Quantum Theory Representations of Natural Numbers, Integers, and Rational Numbers

TL;DR

This paper develops a mathematical framework for representing natural, integer, and rational numbers using quantum theory, parameterized by a space of representations and transformations, highlighting the role of prime bases.

Contribution

It introduces a parameterized space of quantum representations for numbers and explores transformations, invariance, and the significance of prime bases in these representations.

Findings

Representation space is parameterized by 4-tuples (k,m,h,g).

Transformations include base change and gauge adjustments.

Invariance of axioms under transformations depends on prime factorization.

Abstract

This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' and k. This suggests that one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the base k q's as composites of the prime number q's.