Minimal Scales from an Extended Hilbert Space

TL;DR

This paper extends the quantum Heisenberg algebra to include noncommutative coordinates and momenta, leading to minimal length and mass scales, and develops a quantum field theory with finite propagators in both ultraviolet and infrared regimes.

Contribution

It introduces a novel extension of the Heisenberg algebra with noncommutative relations, providing a framework for minimal scales and finite propagators in quantum field theory.

Findings

Propagators are finite in ultraviolet and infrared regimes.

Extended algebra includes nontrivial commutation relations for coordinates and momenta.

Developed a quantum field theory model in (2+1) and (3+1) dimensions.

Abstract

We consider an extension of the conventional quantum Heisenberg algebra, assuming that coordinates as well as momenta fulfil nontrivial commutation relations. As a consequence, a minimal length and a minimal mass scale are implemented. Our commutators do not depend on positions and momenta and we provide an extension of the coordinate coherent state approach to Noncommutative Geometry. We explore, as toy model, the corresponding quantum field theory in a (2+1)-dimensional spacetime. Then we investigate the more realistic case of a (3+1)-dimensional spacetime, foliated into noncommutative planes. As a result, we obtain propagators, which are finite in the ultraviolet as well as the infrared regime.