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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?

Prof. Legolasov
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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 $(E/E_P)^2$.