In May, I gave a seminar at UCSB about my recent work on the topic, which I have briefly summarized in the post The Minimal Length Scale. Since I recall that Joe Polchinski was present that day, I sadly conclude that the seminar wasn't very illuminating, so I'll try to clarify some things. In the beginning though, I should add a note of caution since my work on DSR is not in complete agreement with what the standard approach is. You find further details in my papers
- Interpretation of Quantum Field Theories with a Minimal Length Scale
- Running Coupling with Minimal Length
- Signatures in the Planck Regime
To set the context, in the review Joe Polchinski writes:
Smolin addresses the problem of the Planck length (“It is a lie,” he says). Indeed, Planck’s calculation applies to a worst-case scenario. String theorists have identified at least half a dozen ways that new physics might arise at accessible scales , and Smolin points to another in the theories that he favors , but each of these is a long shot. [...]
With reference to the footnotes:
 The ones that came to mind were modifications of the gravitational force law on laboratory scales, strings, black holes, and extra dimensions at particle accelerators, cosmic superstrings, and trans-Planckian corrections to the CMB. One might also count more specific cosmic scenarios like DBI inflation, pre-Big-Bang cosmology, the ekpyrotic universe, and brane gas cosmologies.
 I have a question about violation of Lorentz invariance, perhaps this is the place to ask it. In the case of the four-Fermi theory of the weak interaction, one could have solved the UV problem in many ways by violating Lorentz invariance, but preservation of Lorentz invariance led almost uniquely to spontaneously broken Yang-Mills theory. Why weren’t Lorentz-breaking cutoffs tried? Because they would have spoiled the success of Lorentz invariance at low energies, through virtual effects. Now, the Standard Model has of order 25 renormalizable parameters, but it would have roughly as many more if Lorentz invariance were not imposed; most of the new LV parameters are known to be zero to high accuracy. So, if your UV theory of gravity violates Lorentz invariance, this should feed down into these low energy LV parameters through virtual effects. Does there exist a framework to calculate this effect? Has it been done?
I wanted to answer the question the question posed in .
In short, this is a significant problem for any theory that predicts Lorentz violation. [...]
The most explicit calculation of this that has been published is, I believe, in Collins, et al. Phys. Rev. Lett. 93, 191301 (2004). They take a Lorentz-violating cutoff and show how it affects one low-energy function. [...]
Which refers to this paper:
Lorentz invariance and quantum gravity: an additional fine-tuning problem?
Authors: John Collins, Alejandro Perez, Daniel Sudarsky, Luis Urrutia, Héctor Vucetich
IIRC the way Lorentz violation is supposed to show up in loopy physics is that the dispersion relation is violated and the speed of light depends on energy (showing up in early or late arrival of ultra high energy gamma ray burst photons compared to ones of lower energy). The idea is that even if the relative effect is quite small the absolute size could be measurable as these photons have traveled across half the universe. Does anybody have an understanding of how this effect arises? [...] Which calculation this referes to? What do I have to compute to get this energy dependent speed of light?
Then, in comment #43, Joe Polchinski partly answers his own question:
Brett #20,22: Thanks for the reference, this is certainly what I would expect. I understand that there is the hope for a `deformed algebra’ rather than a simple violation, but to an outsider it seems that what is being done in LQG is to return to pre-covariant methods of QFT, cut things off in that form, and hope for the best. It would be good to see some calculations.
Now let me add my comments:
The idea of deforming special relativity is to allow two invariant parameters of the transformations between reference frames. The one invariant is the speed of light, the other one is the regulator in the ultra violet, alias a maximal energy scale. This energy scale is usually identified with the Planck energy ~ 1019GeV. If one believes that the Planck energy acts as a maximal energy scale, then all observers should agree on this scale to be maximal. Since usual Lorentz transformations do not allow this (one can always boost an energy to arbitrarily high values), one needs a new type of transformations. These turn out to be non-linear in the momentum variables, which is the reason why they usually do not show up in standard derivations of Lorentz transformations, where one assumes linearity.
The construction of such transformations that respect the upper bound on the energy scale is possible, and they can be explicitly written down. The approach has been pushed forward notably by Giovanni Amelino-Camelia, who has written an enormous amount of papers on the topic. Unfortunately, I find his papers generally very hard to read and confusing. A very readable and clear introduction that I can recommend is e.g.
- Generalized Lorentz invariance with an invariant energy scale
Authors: Joao Magueijo, Lee Smolin
These theories do not break Lorentz invariance in the sense that they do not single out a preferred restframe. Instead, the Lorentz transformations (as functions of the boost parameter), are modified at high values of the boost parameter. This allows the maximal energy to be an invariant quantity. You can find an explicit example for such transformations e.g. in gr-qc/0303067, Eq. (19). What is deformed in this approach, as far as I understand it, is not the algebra itself, but the action of the generators on the elements of the space.
A deformation of Lorentz invariance consequently leads to a new invariant scalar product in momentum space, which means one has a modified dispersion relation. Under quantization, the approach is also known to imply a generalized uncertainty principle, which stems from the modified commutation relations. Theories of this type can but need not necessarily have an energy dependent speed of light (for details about these relations see e.g. hep-th/0510245).
In contrast to this, the paper mentioned by Brett in comment #20 by Collins et al explicitly examines a scenario with violation of Lorentz invariance. As they state already in the abstract "Here, we explain that combining known elementary particle interactions with a Planck-scale preferred frame gives rise to Lorentz violation at the percent level, some 20 orders of magnitude higher than earlier estimates[...]" I am reasonably sure this was not the scenario Lee Smolin is referring to in his book. I vaguely recall he actually writes something about Giovanni -- it implied a knife being put on somebodies throat or so. Unfortunately, I lent the book to my office mate, so I can't look it up.
If one introduces a hard cut-off in a momentum integration without making use of a modified Lorentz-symmetry one runs of course intro problems. With the use of deformed transformations however, this problem can be circumvented. A good way to think about it is in my opinion to picture momentum space not as being a flat, but a curved space. In this case, the integration over the volume in one or more directions can be finite. The non-flatness of the space shows up in the volume element via the square root of the determinant of the metric tensor, which can improve the convergence of loop integrals. By construction, the integration is invariant under the appropriate transformations in that space. In this approach, it is exactly the additional factor (square root of g) in the volume element that makes the integration invariant.
Another way to think about it is to consider a non-linear relation between wave-vector and momentum, in which case the role of the convergence-improving factor is played by the Jacobian determinant of the functional relation between both, see e.g. hep-ph/0405127.
A quantum field theory with DSR can be formulated as a theory with higher derivatives in the Lagrangian (see e.g. hep-th/0603032, or gr-qc/0603073). In fact, as I like to point out, in a power series expansion one needs arbitrarily high derivatives, since a finite polynomial could never reproduce an asymptotic limit. If one writes down a series expansion to get an effective theory, one has corrections in higher order interactions suppressed with powers of the Planck mass as one would expect. Each of these terms is Lorentz invariant, provided the quantities are transformed appropriately. However, in my opinion, such an expansion is not so very helpful, since the important thing is the convergence of the full series. These higher order terms come with the usual constraints on the interactions. I also don't see a point in examining them in great detail, since we don't know anyhow what other funny things might happen to the particle content at GUT or Planck scale energies.
The status of a full quantum field theory with DSR is presently unfortunately still very unsatisfactory. It is a topic I am working on myself, and I am very optimistic that there will be some progress soon. It is however possible to make some general predictions, using kinematic arguments, or just by applying the modified transformations. As mentioned by Robert above, the time of flight being energy dependent (in the case of DSR with an energy dependent speed of light) is an example for such a prediction. Some details about this can be found in
- Introduction to Quantum-Gravity Phenomenology
Authors: Giovanni Amelino-Camelia
- On the Problem of Detecting Quantum-Gravity Based Photon Dispersion in Gamma-Ray Bursts
Authors: Jeffrey D. Scargle, Jay P. Norris, Jerry T. Bonnell
My interest in DSR arises from the fact that it is based on a very general expectation that we have about quantum gravity, which is that the Planck energy acts as a regulator in the ultra violet. In my works, I have mainly examined in how far it is possible to include this property into standard quantum field theories as an effective description of what should actually be described by a full theory of quantum gravity. I am not an expert as to how DSR is related to LQG, and how strictly this connection can be established.
Update: See also what the expert says.
TAGS: PHYSICS , QUANTUM GRAVITY, SPECIAL RELATIVITY