标题: My Impression on Doubly Special Relativity
作者: sage
There has been a lot of effort in studying the Lorentz violating effect. A lot of those approaches necessarily involve deforming the dispersion relation of the Special Relativity
E^2=p^2+m^2. (p is spatial momentum, of course)
Notice that candidate of quantum gravity (unfortunately we have only one which barely qualifies this title: string theory) naturally modifies dispersion relation into
E^2 = p^2 + m^2 + f( k^2 / M^2)
where k is the four momentum and M is some fundamental scale (presumably the string scale). This is not Lorentz violation, since the UV theory (string theory in this case), is manifestly Lorentz invariant. We expect those things to be there, from an effective field theory point of view.
A true Lorentz violating deformation will have the form
E^2 = p^2 + m^2 + g(E,p)
where g is some generic (but not Lorentz inv. function of E and p). Since the Lorentz violation has not been observed, such effects are constrained to be small. Depending on the specific model, the deformation part above is either small because it has a small number in front of it, or suppressed by some large scale, such as the Planck scale.
There are many ways to introduce Lorentz violation. One systematical way is through the so called spurion analysis motivated by spontaneous symmetry breaking. It envisions that the fundamental theory is Lorentz invariant, and this symmetry is spontaneously broken. In field theory, we do this by introducing spurions. We first pretend that spurions are fields which transform in some representation of the symmetry we are interested in. We proceed, under this assumption, to write down the most general invariant Lagrangian. Then, we give spurion a vev and the symmetry is spontaneously broken.
For example, we introduce a_mu (a 4-vector) as a spurion. We could then write, with a scalar field
a_mu \phi^* \partial^mu \phi
which is Lorentz invariant if a_mu transforms as a Lorentz vector. Now we give a_mu a vev, i.e., making it a constant vector. Then, this is a Lorentz symmetry breaking term in the Lagrangian.
We could, in this way, introduce many terms of Lorentz violating terms this way. Many of them are renormalizable, which means their coefficient must be small. The others are non-renormalizable. Therefore, they should at least be suppressed by a high scale.
Now, we are breaking a global symmetry spontaneously. You should be asking what happens to the Goldstone. There is very nice story there which has just been worked out a couple of years ago. This is called ghost condensation. however, I will not go into details about that here.
This, in my point of view, is the only healthy and well-motivated way to introduce Lorentz violation, so far.
However, there are also other ways of motivating and organizing the introduction of Lorentz violation. All of them involving saying something special about energy (and/or momentum in a lorentz violating way, of course).
Doubly special relativity is one.
It says that Planck Energy must be invariant in nature. Therefore, we must extend Lorentz transformation incorporating a new constant, the Planck constant.
Apart from philosophical reasons, I don't see a motivation for doing that. The supporters of DSR like to say that this is just like going from Galilei transformation to Lorentz transformation by introducing a new constant (speed of light). However, notice that we have a very good motivation for including the speed of light. After all, there is Maxwell theory telling us speed of light is a constant. There is no indication why Plank energy must be a constant.
However, let's admit for now that it is nontheless possible and see its consequences.
There are many attempts trying to just write various forms of dispersion relations and transformation laws such that Planck Energy is invariant. It is possible to do so.
However, such transformation laws are generically problematic, due to an argument (still has not been fully addressed by the supporters) by Unruh et al (gr-qc/0308049). Here is the gist of that argument.
We know that linear Lorentz transformation (the one we all know), denoted as L here, has two fix points in energy, 0 and infinity. We write DSR transformation as L*F (where F is the modification). Since E_planck is invariant under L*F, F must map E_planck to infinity. A polynomial F cannot do it. Therefore, we must have non-polynomial behavior for F. Therefore, we also expect the dispersion relation, which is invariant under DSR, also have non-polynomial behavior in E. Therefore, the theory is not a finite expansion in powers of derivatives. Therefore, it is non-local. Moreover, since the non-locality will be integrated over the world-line, it is not small (could be much larger than the Planck length). This is problematic.
There are other criticisms of the DSR. Some original proposed DSR transformations turn out to be just a change of base of the normal Lorentz transformation (this has a very fancy mathematical name which I will not mention). Therefore, it is not really that revolutionary.
Since we don't have a clear definition of energy here (E can be transformed into E^2 in some frame), one could wonder whether an unambiguous choice is possible. The answer is not clear, but maybe possible with some work.
So far, I have been only talking about particles worldlines and DSR transformation on it. One might want to make a field theoretical Lagrangian which indeed has such dispersion relations. We are looking for a derivative (probably infinite) expansion with a scale. This fits of characteristic of the effective Largrangian from non-commutative geometry. Therefore, a lot of people jumped on and wrote papers on this subject. So far, it seems to me that not much has been acheived (in spite of a lot of highly mathematical papers). Even this is done, it is not clear how much more theoretical foundation it delivers.
Despite some claims, DSR has no root in quantum gravity yet. (Well, expect loop quantum gravity claims they can do it. However, LQG is not even a theory yet.)
All in all, with all these mess, I did not see a clean prediction from DSR (even assume it is somehow healthy). All I see is just some modification of dispersion relation. If it is not crazy, it will probably just be some non-renormalizable operators suppressed by Planck scale which are already there in many earlier consideration anyway.