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3.5 Non-systematic dispersion

As mentioned in Section 2.6, Lorentz violation is only one possibility for a signal of quantum gravity. Another common idea about quantum gravity is that spacetime should have a stochastic [99255] or “foamy” [231] structure at very small scales. Combining these two ideas has lead a number of authors to the idea of stochastic/non-systematic dispersion where the modifications to the dispersion relation fluctuate over time. Such dispersion modifications have been phenomenologically parameterized by three numbers, the usual coefficient (n) f and exponent n of Section 3.1, and a length scale L which determines the length over which the dispersion is roughly constant. After a particle has travelled a distance L a new coefficient f(n) is chosen based upon some model dependent probability distribution P that reflects the underlying stochasticity. Usually P is assumed to be a gaussian about 0, such that the average energy of the particle is given by its Lorentz invariant value. As well, L is in general taken to be the de Broglie wavelength of the particle in question. Note that in these models n is not required to be an integer, the most common choices are n = 5/2,n = 8/3, n = 3 [232].4 The only existing constraints on non-systematic dispersion come from threshold reactions (see Section 6.6.2) and the phase coherence of light (see Section 6.9).
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