Comment on “Noncommutativity as a Possible Origin of the Ultrahigh-energy Cosmic Ray and the TeV Photon Paradoxes”

A Lorentz-noninvariant modification of the kinematic dispersion law was proposed in Ref [1], claimed to be derivable from from q-deformed noncommutative theory, and argued to evade ultrahigh energy threshold anomalies (trans-GKZ-cutoff cosmic rays and TeV-photons) by raising the respective thresholds. It is pointed out that such dispersion laws do not follow from deformed oscillator systems, and the proposed dispersion law is invalidated by tachyonic propagation, as well as photon instability, in addition to the process considered. Reported anomalous cosmic threshold events serve to constrain hypothesized violations of Lorentz invariance [2, 3, 4, 5]. E.g., observations of roughly isotropic cosmic rays with p ≥ 4 × 10eV, which are above the Greisen-Zatsepin-Kuz’min (GZK) threshold, or else, detection of up to 20 TeV photons from the active galaxy Markarian 501. The GZK threshold for p + γ → p + π, ensures absorption, within about 50 Mpc, of cosmic rays such as protons with energies above pGKZ = ((mp + mπ) 2 − m2p)/(4k), where k is an active energy of available cosmic microwave background photons, ∼ 10 − 10eV . Similarly, reported 20 TeV photons from Markarian 501 (at a distance of 157 Mpc, much longer than the mean free path for photons cut off by collisions with infrared photons to pair-produce, γ + γ → e + e) should have been absorbed: the cutoff can effectively set in at about p = m2e/k, where k is an active energy of universal IR background photons, ∼ 10eV , thought to result out of star formation in the early universe [4]. These thresholds might be evaded by small deformations of Lorentz kinematics, for instance by raising them to higher energies which allow for the reportedly observed events [2, 3, 4, 5]. One such kinematic energy-momentum dispersion law was proposed in ref [1], namely, E = ω 2   [√ m + p ω − 1 2 ] q + [√ m + p ω + 1 2 ] q   = ω 2 [√ m + p ω/2 ]