Past and Future Blurring at Fundamental Length Scale

We obtain the $\kappa$-deformed versions of the retarded and advanced Green functions and show that their causality properties are blurred in a time interval of the order of a length parameter $q=1/(2\kappa )$. The functions also indicate a smearing of the light cone. These results favor the interpretation of $q$ as a fundamental length scale below which the concept of a point in space-time should be substituted by the concept of a fuzzy region of radius $q$, as proposed long ago by Heisenberg.

Figure 1

A plot of ${\delta }_q$ given in (8) with $q=0.001\mathrm{a}.\mathrm{u}.$.

Figure 2

A contour plot with contours at $-0.5$, 0 and 3.0 a.u. (darker at lower levels) of a light cone leg.