Waves on Noncommutative Spacetimes

TL;DR

This paper explores how wave phenomena like interference and diffraction are altered on noncommutative spacetimes, revealing novel effects such as interference suppression at short observation times and pattern deformation depending on noncommutativity parameters.

Contribution

It develops rules for interpreting waves on noncommutative spacetimes and applies them to interference and diffraction, uncovering new phenomena related to noncommutativity effects.

Findings

Interference is suppressed when observation time T ≤ 2θw.

Interference patterns deform depending on θw/T ratio.

Patterns approach classical behavior as θw/T approaches zero.

Abstract

Waves on ``commutative'' spacetimes like R^d are elements of the commutative algebra C^0(R^d) of functions on R^d. When C^0(R^d) is deformed to a noncommutative algebra {\cal A}_\theta (R^d) with deformation parameter \theta ({\cal A}_0 (R^d) = C^0(R^d)), waves being its elements, are no longer complex-valued functions on R^d. Rules for their interpretation, such as measurement of their intensity, and energy, thus need to be stated. We address this task here. We then apply the rules to interference and diffraction for d \leq 4 and with time-space noncommutativity. Novel phenomena are encountered. Thus when the time of observation T is so brief that T \leq 2 \theta w, where w is the frequency of incident waves, no interference can be observed. For larger times, the interference pattern is deformed and depends on \frac{\theta w}{T}. It approaches the commutative pattern only when \frac{\theta w}{T} goes to 0. As an application, we discuss interference of star light due to cosmic strings.