>> And the general expectation of some space time foam model is, that a,b should be roughly 1.
> OK. Any idea how generic this expectation really is?
I can't comment on this case specifically, but the entire notion of the "Planck energy" is based on that idea: that once you've found the combination of fundamental constants with the right units, the numerical coefficient that multiplies it will wind up being within one or two orders of magnitude of 1. By some miracle, the vast majority of systems in physics seem to match such expectations.
No no, I get the basic idea of making estimates based on dimensional analysis. My question is: how generically do we expect the dispersion relation to be of this form? I could imagine, for example, that differences in speed are exponentially suppressed by the ratio E_gamma/M_p.
It is just a Taylor expansion around c_0. And since we do not observe any strange effects we know that it should kind of work. (That the speed of light should be well behaved.) In addition the planck mass is the only constant, that we know of, which fits there.
> OK. Any idea how generic this expectation really is?
I can't comment on this case specifically, but the entire notion of the "Planck energy" is based on that idea: that once you've found the combination of fundamental constants with the right units, the numerical coefficient that multiplies it will wind up being within one or two orders of magnitude of 1. By some miracle, the vast majority of systems in physics seem to match such expectations.