Events in a Non-Commutative Space-Time

TL;DR

This paper explores quantum events in a noncommutative space-time framework, extending POVM-based measurement theory to Snyder's model, revealing a minimal length compatible with Lorentz covariance and discussing the implications for space-time concepts.

Contribution

It generalizes POVM measurement theory to noncommutative space-time, specifically analyzing Snyder's model and its implications for minimal length and Lorentz covariance.

Findings

Derived POVMs for single coordinate measurement in Snyder's model

Established lower bounds for coordinate dispersions

Showed minimal length is compatible with Lorentz covariance

Abstract

We treat the events determined by a quantum physical state in a noncommutative space-time, generalizing the analogous treatment in the usual Minkowski space-time based on positive-operator-valued measures (POVMs). We consider in detail the model proposed by Snyder in 1947 and calculate the POVMs defined on the real line that describe the measurement of a single coordinate. The approximate joint measurement of all the four space-time coordinates is described in terms of a generalized Wigner function (GWF). We derive lower bounds for the dispersion of the coordinate observables and discuss the covariance of the model under the Poincare' group. The unusual transformation law of the coordinates under space-time translations is interpreted as a failure of the absolute character of the concept of space-time coincidence. The model shows that a minimal length is compatible with Lorents covariance.