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 N, such that N 2 ≡ N 2 1 +N 2 2 , satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural c-numbers n ∈ N. A probabilistic interpretation of QNT is then inferred from this representation.Furthermore, we define a 3-component integer q-number Z, such thatand obeys a Lie algebra structure. The eigenvalues of each Z component generate a set, albeit all components do not generate 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 qunits. 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 ⊆ 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.