Revisiting the Algebraic and Analytic Descriptions of Quantum Mechanics

TL;DR

This paper explores Heisenberg's matrix mechanics using an algebraic pre-Hilbert approach, revealing how boundary effects and algebraic structures influence quantum spectra, uncertainties, and the nature of quantum randomness.

Contribution

It introduces an algebraic pre-Hilbert framework for finite-dimensional quantum mechanics, connecting discrete kernels with continuous spectra and analyzing the origins of quantum randomness.

Findings

Reproduces standard spectra and uncertainty relations in the algebraic framework.

Identifies boundary contributions as sources of unbounded operators.

Discusses the algebraic origins of quantum randomness.

Abstract

We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character originates from boundary contributions and whose structure induces a discrete analogue of the Cauchy-Hilbert kernel. Compared with the separable Hilbert-space completion, the algebraic framework reproduces the standard spectra, canonical commutation relations, and Heisenberg uncertainty relation for finite-energy states, while the discrete kernel is absorbed into its continuous integral counterpart under completion. The comparison shows that both formulations require restrictions on admissible states for effective calculations -- analytic domain restrictions in Hilbert space and finite-energy restrictions in the pre-Hilbert framework. Finally, we discuss to what extent quantum randomness arises from the algebraic structure of the pre-Hilbert framework.