Waves on noncommutative spacetimes

Waves on “commutative” spacetimes like ${\mathbb{R}}^d$ are elements of the commutative algebra $C^0({\mathbb{R}}^d)$ of functions on ${\mathbb{R}}^d$. When $C^0({\mathbb{R}}^d)$ is deformed to a noncommutative algebra ${\mathcal{A}}_{\theta }({\mathbb{R}}^d)$ with deformation parameter $\theta$ (${\mathcal{A}}_0({\mathbb{R}}^d)=C^0({\mathbb{R}}^d)$), waves being its elements, are no longer complex-valued functions on ${\mathbb{R}}^d$. 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 $d\le 4$ and with time-space noncommutativity. Novel phenomena are encountered. Thus when the time of observation $T$ is so brief that $T\le 2\theta w$, 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 $\frac{\theta w}{T}\to 0$. As an application, we discuss interference of starlight due to cosmic strings.

Figure 1

Variation of intensity $I$ as a function of $w{x}_1$, for fixed $\frac{\theta w}{T}<\frac{1}{2}$. The plots are for $r=\frac{\theta w}{T}=\frac{1}{3},\frac{1}{4},\frac{1}{16}$ and $\theta w^2=\frac{\pi }{2}$. Clearly, as the ratio $\frac{\theta w}{T}$ gets smaller, it converges to the commutative result.

Figure 2

Variation of intensity $I$ as a function of $y=\theta$. It is plotted at $x_1=0$, $\omega ={10}^{-7}{\mathrm{m}}^{-1}$ for the values of $T=2.5\times {10}^7$, $4\times {10}^7$, $6.5\times {10}^7\mathrm{m}$. Note that for $\theta =0$, $I=4$ and for any given $T$, $I$ takes the value $2$ and becomes independent of $\theta$ at $\theta =\frac{T}{2}\times {10}^7$ and thereon.

Figure 3

Double image of $Q$ is observed by $O\equiv O^{\prime }$.

Figure 4

The observer $\epsilon$ at $O\equiv O^{\prime }$ can barely observe a double image of the source $Q$. Keeping her distance from the string fixed, she can shift her position along the arc $A$ and can observe a double image of $Q$ from $\tilde{O}\equiv {\tilde{O}}^{\prime }$. At $\tilde{\tilde{O}}\equiv {\tilde{\tilde{O}}}^{\prime }$ she again can only barely observe a double image of $Q$.

Figure 5

The diagram provides the definition of angles $\psi ,{\psi }^{\prime }$ and $\beta$.