Uncertainty Principle, Hilbert Bases and Algebras of Operators

Say Song Goh, Charles A. Micchelli

Grigoriy Dedenko, Sergey Klykov, Alexey Orlovsky

We propose a unified framework where algebraic structures, rather than geometry, serve as the foundation of quantum theory. We derive the Hilbert space, canonical operators, and the uncertainty principle from an underlying algebra. The Number System Operator (NSO) model emerges as a concrete realization, linking precision (k), numerical base (B), and measurable quantities. We prove the non-commutativity of NSO projectors, introducing a “digital noise” as a new, testable form of uncertainty. We analyze its scale relative to Planck units and demonstrate Lorentz covariance of the NSO procedure, ensuring compatibility with special relativity. A roadmap for experimental verification is provided, suggesting high-precision spectroscopy, atomic clocks, and gravitational-wave interferometers as potential tests. This framework transforms conceptual ideas into a computable and physically relevant theory, bridging foundational physics, computation, and measurable predictions.

Bholanath Padhy, Bk Majhi, K. Navya +1 more

The Heisenberg uncertainty principle is foundational to quantum mechanics, yet its standard formulation is limited to Hilbert space operator commutators. Recent advances in noncommutative geometry (NCG) allow a reformulation of quantum observables and spacetime itself using operator algebras, providing a deeper framework for uncertainty relations. In this paper, we develop a generalized uncertainty relation using spectral triples, extending the Robertson– Schrödinger inequality into the noncommutative regime. Explicit derivations are given for operator-valued distances, modified commutators, and position–momentum operators in a noncommutative configuration space. Our results reveal the emergence of a minimal measurable length scale, consistent with predictions from quantum gravity, and demonstrate that uncertainty is fundamentally geometric in origin.

Leon W. Cohen

Thomas A. Brody

Zen Revista, 10 PHYSICS

Quantum mechanics' formal core is a handful of objects---the wavefunction that encodes a state, the operators that act on it, the commutators that measure their incompatibility, and the uncertainty principle that the commutators imply---and the theory's history is the story of how those objects were given. This article presents a narrative review of the primary literature that gave them, from Heisenberg's 1925 umdeutung---the reinterpretation that replaced orbits with arrays of transition amplitudes---and Schrödinger's 1926 quantization as an eigenvalue problem, which wrote the wave equation and proved the equivalence with matrix mechanics, through Born's 1926 probability interpretation of the wavefunction, Heisenberg's 1927 uncertainty paper, Robertson's 1929 general commutator inequality, Pauli's 1926 solution of the hydrogen spectrum by the new mechanics, Bohr's 1928 quantum postulate and complementarity, Weyl's 1928 group theory and quantum mechanics, Dirac's 1930 Principles, which organized the formalism's transformations, and von Neumann's 1932 mathematical foundations, which gave the theory its Hilbert space, to the Einstein-Podolsky-Rosen 1935 argument that tested what the formalism claims about reality. The synthesis is organized around three themes: the state's representations, in which matrix and wave mechanics were shown to be one theory; the uncertainty structure, in which commutators became the measure of incompatibility and the uncertainty principle its theorem; and the formalism's completion, in which Hilbert space, transformations, and the measurement problem gave the objects their rigorous home. It is concluded that the formal core's construction is physics' most consequential algebra---the theory's content located not in its pictures but in its operators' relations.