Space-time uncertainty relation and Lorentz invariance

TL;DR

This paper proposes a Lorentz covariant space-time uncertainty relation that aligns with existing models at larger scales and introduces a minimal area concept at smaller scales, with implications for non-commutative geometry.

Contribution

It introduces a Lorentz covariant uncertainty relation, explores its implications for minimal area in four dimensions, and develops a non-commutative space-time field theory consistent with this relation.

Findings

At large time scales, the relation matches existing models.

At small scales, it implies a minimal area rather than a minimal length.

The algebraic structure may violate the Jacobi identity in four dimensions.

Abstract

We discuss a Lorentz covariant space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period delta t is larger than the Planck time lp. At delta t < lp, this uncertainty relation takes roughly the form delta t delta x > lp^2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the representation space. In four-dimensional space-time, the Jacobi identity should be violated in the algebraic representation of the uncertainty relation.