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I'm considering the $\kappa$-Minkowski space – a certain model from non-commutative geometry which reduces to the usual Minkowski space in the limit $\kappa \rightarrow \infty$. The parameter $\kappa$ is a new fundamental constant of order $\text{length}^{-1}$, which is conventionally taken to be $l_P^{-1}$ where $l_P$ is the Planck's length.

It was proven in hep-th/9405107 that the lowest order polynomial in the non-commutative coordinates invariant under the full Poincare quantum group is

$$ c^2 t^2 - \vec{x}^2 + \frac{3 c t}{\kappa}, $$

which becomes the ordinary Lorentzian invariant interval in the $\kappa \rightarrow \infty$ limit.

According to a straightforward calculation, this invariant polynomial leads to the following dispersion relations for light in vacuum:

$$ p^{2}(E)=\frac{E^{2}}{c^{2}}\left(1-\frac{9}{4}\left(\frac{E}{E_{p}}\right)^{2}+\frac{27}{4}\left(\frac{E}{E_{p}}\right)^{3}+\dots\right), $$

where $E_p$ is the Planck energy.

Note the absense of the 1-st order term $E/E_p$ in the expression above. This reminds me of this PSE question of mine, where an excellent answer by kleingordon@ referenced a paper with a specific experimental bound on the dimensionless coefficient before the 1-st order term. Because the 1-st order term obtained from non-commutative geometry vanishes, that bound is satisfied.

Question: does this mean that $\kappa$-Minkowski has not been experimentally falsified by the gamma ray burst measurements? Or are there any other bounds on subsequent terms that also have to be taken into account?

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Nemiroff et al. published an analysis of this sort of thing in 2012. Although there may be more recent work, I assume it hasn't improved the bounds by many orders of magnitude. For photons with energies of up to about 10 GeV, they found a limit on dispersion of $\Delta v/v \lesssim 10^{-20}$. The value of $E/E_P$ is about $10^{-18}$, so it looks like they only rule out a first-order effect, not one proportional to $(E/E_P)^2$.

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