A quantum number theory

Abstract

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a Hilbert space that generate classical numbers ($c$-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$-number $\textbf{N}$, such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$-numbers $n \in \mathbb{N}$. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$-number $\textbf{Z}$, such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$, $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$, albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$-numbers as `$q$-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu$n$its. To interconnect QNSV of different dimensions, associated to the same $c$-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.