On Nonlocality, Lattices and Internal Symmetries

TL;DR

This paper explores how certain quantum gravity-inspired modifications to the Heisenberg algebra imply a minimal length scale, leading to lattice structures and internal symmetries, and investigates related unitarily inequivalent representations.

Contribution

It introduces and analyzes two types of correction terms to the Heisenberg algebra, linking them to minimal length scales, lattice eigenvalues, and internal symmetries, including the first study of their Wick-rotated counterparts.

Findings

Lattices of eigenvalues form representations of unitary groups.

Correction terms imply a finite resolution limit consistent with quantum gravity.

Unitarily inequivalent representations are examined for the second correction type.

Abstract

We study functional analytic aspects of two types of correction terms to the Heisenberg algebra. One type is known to induce a finite lower bound $\Delta x_0$ to the resolution of distances, a short distance cutoff which is motivated from string theory and quantum gravity. It implies the existence of families of self-adjoint extensions of the position operators with lattices of eigenvalues. These lattices, which form representations of certain unitary groups cannot be resolved on the given geometry. This leads us to conjecture that, within this framework, degrees of freedom that correspond to structure smaller than the resolvable (Planck) scale turn into internal degrees of freedom with these unitary groups as symmetries. The second type of correction terms is related to the previous essentially by "Wick rotation", and its basics are here considered for the first time. In particular, we investigate unitarily inequivalent representations.