Doubly special relativity modifies not only the particle dispersion relation but also the form of the energy conservation equations. The situation is therefore very different from that in EFT. The first difference between DSR and EFT is that DSR evades all of the photon decay and vacuum Čerenkov constraints that give strong limits on EFT Lorentz violation. Since there is no EFT type description of particles and fields in a DSR framework, one has no dynamics and cannot calculate reaction rates. However, one still can use the DSR conservation laws to analyze the threshold kinematics. By using the pseudo-momentum and and energy one can show that if a reaction does not occur in ordinary Lorentz invariant physics, it does not occur in DSR . Physically, this is obvious. If the vacuum Čerenkov effect for, say, electrons began to occur at some energy , in a different reference frame the reaction would occur at some other energy , as the threshold energy is not an invariant. Therefore frames could be distinguished by labelling them according to the energy when the vacuum Čerenkov effect for electrons begins to occur. This violates the equivalence of all inertial frames that is postulated in DSR theories.
A signal of DSR in threshold reactions would be a shift of the threshold energies for reactions that do occur, such as the GZK reaction or -ray annihilation off the infrared background . However, the actual shift of threshold energies due to DSR is negligible at the level of sensitivity we have with astrophysical observations . Hence DSR cannot be ruled out or confirmed by any threshold type analysis we currently have. The observational signature of DSR would therefore be a possible energy dependence of the speed of light (see Section 6.2) without any appreciable change in particle thresholds .
Similar to DSR, the lack of dynamics in the non-systematic dispersion framework of Section 3.5 makes it more problematic to set bounds on the parameters . In [160, 12, 11, 24], the authors assume that the net effect of spacetime foam can be derived by considering energy conservation and non-systematic dispersions at a point. There is a difficulty with this, which we shall address, but for now let us assume that this approach is correct.
As an example of the consequences of non-systematic dispersion let us consider the analysis of the GZK reaction in . The authors consider nonsystematic dispersion relations with normally distributed coefficients that can take either sign and have a variance of . Looking solely at the kinematical threshold condition, they find that all cosmic ray protons would undergo photo-pion production at energies above . This is perhaps expected, as the energy scale at which an term becomes important is for of . There is a large region of the parameter space that is susceptible to the vacuum Čerenkov effect with pion emission  and hence a significant amount of the time the random coefficients will fall in this region of parameter space. If an ultra-high energy proton can emit a pion without scattering off of the CMBR, then certainly it can scatter as well, which implies that GZK reaction is also accessible. This same type of argument can be rapidly extended to dispersion, yielding a cutoff in the spectrum at . The cutoff could easily be pushed above GZK energies if the coefficients had a variance slightly less than . In short, since we see high energy cosmic rays at energies of , the results of [160, 12, 11, 24] imply that we could not have non-systematic dispersion unless the coefficients are , while for the coefficients would only have to be an order of magnitude or two below .
We now return to a possible problem with this type of analysis, which has been raised in . Performing threshold analyses on non-systematic dispersion assumes that energy-momentum conservation can be applied with a single fluctuation (i.e. the reaction effectively happens at a point). It further assumes that the matrix element is roughly unchanged. In GZK or Čerenkov reactions, however, one of the outgoing particles is much softer than the incoming particle. In this situation the interaction region is much larger than the de Broglie wavelength of the high energy incoming particle, which means that many dispersion fluctuations will occur during the interaction. The amplitude of low energy emission in regular quantum field theory changes dramatically in this situation (e.g., Bremsstrahlung with a rapidly wiggling source) as opposed to the case in which there is only one fluctuation (e.g., the Čerenkov effect). The above approach, modified conservation plus unchanged matrix element/rate when the reaction is allowed, is not correct when a low energy particle is involved. If the outgoing particle has an energy comparable to the incoming particle, then it may be possible to avoid this problem. However, in this case the reverse reaction is also kinematically possible with a different fluctuation of the same order of magnitude, so it is unclear what the net effect on the spectrum should be. Note, finally, that these arguments only concern the rate of decay - the conclusion that high energy particles would decay in this framework is unchanged.
© Max Planck Society and the author(s)