Quantum Harmonic Oscillator as a Zariski Geometry

TL;DR

This paper models the Heisenberg algebra as a Zariski geometry, revealing its non-classical nature and connecting it to quantum harmonic oscillators through geometric and model-theoretic analysis.

Contribution

It introduces a novel geometric structure associated with the Heisenberg algebra, demonstrating its non-classical Zariski geometry and linking it to quantum operators.

Findings

Heisenberg algebra corresponds to a Zariski geometry

The geometry is non-classical and not interpretable in algebraically closed fields

A discrete substructure emerges under self-adjointness assumptions

Abstract

We carry out a model-theoretic analysis of the Heisenberg algebra. To this end, a geometric structure is associated to the Heisenberg algebra and is shown to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be non-classical, in the sense that it is not interpretable in an algebraically closed field. On assuming self-adjointness of the position and momentum operators, one obtains a discrete substructure of which the original Zariski geometry is seen as the complexification.