Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta

TL;DR

This paper develops a noncommutative geometric extension of quantum field theory incorporating minimal uncertainties in positions and momenta, demonstrating potential regularization of divergences possibly linked to gravity effects.

Contribution

It introduces a generalized framework with nonzero minimal uncertainties in positions and momenta, using algebraic quantum group techniques, and shows regularization of ultraviolet divergences in a specific quantum field theory example.

Findings

Nonzero minimal uncertainties can regularize ultraviolet divergences.

Algebraic techniques from quantum groups are effective in this framework.

Motivated by gravity, these uncertainties may have fundamental physical implications.

Abstract

A noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations. There appear nonzero minimal uncertainties in positions and in momenta. As the main result it is shown with the example of a quadratically ultraviolet divergent graph in $\phi^4$ theory that nonzero minimal uncertainties in positions do have the power to regularise. These studies are motivated with the ansatz that nonzero minimal uncertainties in positions and in momenta arise from gravity. Algebraic techniques are used that have been developed in the field of quantum groups.