Time Discretization From Noncommutativity

TL;DR

This paper demonstrates that in a specific noncommutative geometry, the spectrum of time becomes discrete, leading to quantized time intervals despite continuous possible measurement values.

Contribution

It introduces the concept that noncommutative geometries like angular or ρ-Minkowski enforce a discrete spectrum for the time variable.

Findings

Time spectrum is discrete and equally spaced by noncommutativity scale.

Self-adjoint extensions allow real-valued measurements of time.

Time intervals are quantized even if measurements can be any real number.

Abstract

We show that a particular noncommutative geometry, sometimes called angular or $\rho$-Minkowski, requires that the spectrum of time be discrete. In this noncommutative space the time variable is not commuting with the angular variable in cylindrical coordinates. The possible values that the variable can take go from minus infinity to plus infinity, equally spaced by the scale of noncommmutativity. Possible self-adjoint extensions of the "time operator" are discussed. They give that a measurement of time can be any real value, but time intervals are still quantized.