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arXiv:2108.10145

A quantum number theory

Lucas Daiha, Roberto Rivelino

Published Aug 18, 2021• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2108.10145

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AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
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In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. The number of partitions of n is given by the partition function p(n). Inspired by quantum information processing, we extend the concept of partitions in number theory as follows: for an integer n, we treat each partition as a basis state of a quantum system representing that number n, so that the Hilbert‐space that corresponds to that integer n is of dimension p(n); the “classical integer” n can thus be generalized into a (pure) quantum state | |ψ(n) > which is a superposition of the partitions of n, in the same way that a quantum bit (qubit) is a generalization of a classical bit. More generally, ρ(n) is a density matrix in that same Hilbert‐space (a probability distribution over pure states). Inspired by the notion of quantum numbers in quantum theory (such as in Bohr’s model of the atom), we then try to go beyond the partitions, by defining (via recursion) the notion of “sub‐partitions” in number theory. Combining the two notions mentioned above, sub‐partitions and quantum integers, we finally provide an alternative definition of the quantum integers [the pure‐state |ψ′(n)> and the mixed‐state ρ′(n), ] this time using the sub‐partitions as the basis states instead of the partitions, for describing the quantum number that corresponds to the integer n.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
Quantum mechanics elevates non-commutativity from a mathematical curiosity to a physical axiom: observables are promoted to operators, and Poisson brackets are replaced by commutators. This ``quantization procedure'' has remained a postulate for a century, with no explanation of \textit{why} nature demands non-commuting observables. We argue that the answer has been hiding in plain sight since 1843: the quaternion algebra $\HH$, whose defining relation ij=−ji=k\mathbf{i}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k}ij=−ji=k makes it the unique non-commutative normed division algebra over R\RR.We develop this argument in three tiers of increasing rigor. First, we prove that the 44-vector Z3\mathbb{Z}_3Z3​-triality vacuum lattice---a zero-parameter discrete structure established in prior work---is \textit{exactly} the orbit closure of the quaternion basis under $\mathbb{Z}_3 \subset \SU(2)$. The three closure operations are unified as conjugations by a single $q_T \in \SU(2)$ (Proposition~3.1), and a group-theoretic proof via the A2A_2A2​ root lattice decomposition of R3=V0⊕V1\R^3 = V_0 \oplus V_1R3=V0​⊕V1​ yields ∣L44∣=44|L_{44}| = 44∣L44​∣=44 (Theorem~3.1). Second, we derive the canonical commutation relation [x^,p^]=iℏ[\hat{x},\hat{p}] = i\hbar[x^,p^​]=iℏ from quaternion algebra through the constructive chain $\HH \to \HH_\C \cong M(2,\C) \to \mathcal{F}_B(\C^2) \to$ Heisenberg algebra, with ℏ=3 S0\hbar = \sqrt{3}\,S_0ℏ=3​S0​ where 3\sqrt{3}3​ is the geometric quantum number of the Z3\mathbb{Z}_3Z3​ rotation angle and S0S_0S0​ the fundamental action scale of the lattice. Third, we construct the full Hilbert space framework via the GNS representation of $\HH$ as a C∗^*∗-algebra (Appendix~A).The Pauli algebra, Dirac algebra, and spin-statistics connection admit progressively weaker claims: the Pauli algebra is identically the quaternion algebra (theorem); the Dirac algebra inherits quaternion structure through $\mathrm{Cl}(3,0) \cong \HH \oplus \HH$ (structural); the spin-statistics connection is a geometric interpretation suggested by the framework. What was discovered as a mathematical structure in 1843, applied as a computational technique in 1925, and axiomatized as a postulate ever since, is revealed as the geometric foundation of quantum non-commutativity: \textit{quaternion multiplication is the mathematical origin of why observables fail to commute.