- “Minimal Length in Quantum Gravity and the Fate of Lorentz Invariance”
Studies of History and Philosophy of Modern Physics 40(3): 259-267 (PDF)
Amit Hagar is an assistant professor at Indiana University's Department of History and Philosophy of Science, and he has taken an interest in the history of a minimal length and the current discussion about deformations of Lorentz invariance. And it is true indeed that the existence and implementation of a minimal length in quantum gravity is an intriguing open question. Hagar uses it as “a case study for highlighting the delicate balance between conservatism and innovation that characterizes the process of constructing new physics.”
I find his paper very refreshing, though in some aspects misleading and the argumentation incomplete.
To briefly summarize the basics,
there are many motivations, stemming from different approaches to quantum gravity and various thought experiments, that there is a fundamental limit to how well we can resolve structures (for a summary see eg “Quantum gravity and minimum length” by Luis J. Garay and “On Gravity and the Uncertainty Principle” by Adler and Santiago). This limit is generally thought to be at or close by the Planck scale. This is far off what we can experimentally test today, thus the lack of experimental data.
However, such a finite minimal distance scale causes a problem with Lorentz invariance since Special Relativity tells us a ruler in motion towards us appears shortened. A minimal length better shouldn't appear shorter than minimal. This reasoning thus creates the need to modify Special Relativity, which is very hard to do in a self-consistent and observer independent way. Attempts have become known under the name “Deformed Special Relativity.” Such modifications of Special Relativity can imply modified dispersion relations and an energy dependent speed of light, though the theoretical basis for these theories is presently incomplete and partially inconsistent. Note that modified dispersion relations are quite easily obtained also from preferred frame effects. The point of DSR has been that it does respect relativity of inertial frames.
I have argued in this paper that the alleged problem isn't since there is no observation and thus no contradiction without an interaction. The only thing necessary for self-consistency is then that no interaction can ever resolve structures below the Planck scale, but there is no need to modify the Lorentz boosts for freely propagating particles. This is, in a nutshell, the main difference between my model and the standard DSR approach.
DSR is generally thought to be not a fundamental theory on its own, but an approximate description applicable to incorporate effects of quantum gravity in the particle/quantum field context. People differ on what approximation it is supposed to describe, but the point is there might not be an obvious way to find such a modification in the fundamental theory since it could only be an effective description. Take as an example friction. There's no friction inside the atom, and there's no friction in planetary orbits either. Yet on intermediate scales Cosmopolitan avidly advocates lubricants.
The point of view I've been taking (which of course isn't shared by everybody) is that quantum field theory with a minimal length and DSR is a way to incorporate still little understood quantum gravitational effects that would be described by a fully consistent yet-to-be-found theory into the old-fashioned theories we already have by adding a generalized uncertainty principle, a modification of dispersion relations and a deformation of momentum space. I like this approach because it bridges the gap towards phenomenology. It is however unsatisfactory there is presently no derivation from fundamental principles.
But to come back to Hagar's paper,
he studies some arguments that have been raised against such deformations of Lorentz invariance and finds the criticism wanting. On the other hand, he also finds the theoretical motivation for having such a modification unconvincing, though the attempt to do so makes a nice object of study for the philosopher
“While so far there seems to be little physical motivation for deforming the standard energy-momentum dispersion relations (apart from the fact that there are good reasons to think that a fundamental QG theory will involve spatial discreteness), from the methodological perspective I am interested in here the attitude within the QG community towards DSR exemplifies nicely the aforementioned delicate balance between conservatism and innovation.”I can basically picture the string theorists among the readers grinding their teeth. I'll leave it up to you whether you think reasons for spatial discreteness are “good,” since it is actually a different question than whether there is a finite resolution, and the matter of discreteness is thus not relevant to the topic under discussion. One can have a fundamentally finite resolution of structures without spatial discreteness, and one can also have spatial discreteness without violations of Lorenz invariance. Unfortunately, these issues are quite often confused. Hagar does mention these differences later on, but this introduction of his paper is somewhat misleading.
Hagar discusses an argument by Schützhold and Unruh according to which a position space description of DSR either involves large scale non-locality inconsistent with our current theories and observations, or it necessitates a preferred frame. Hagar concludes the argument is unconvincing since it makes use of unwarranted assumptions about the Fourier transformations in such a framework. While I agree with Hagar's criticism, I did a similar analysis in this paper without making use of Fourier transformations and came to essentially the same conclusion: If one has an energy dependent speed of light, one either needs a preferred frame, or one needs an external parameter to label Lorentz transformations. This parameter is commonly chosen to be an energy (don't ask the energy of what), but besides the ambiguous interpretation this is a non-local modification that seems to me as unnatural as implausible .
Anyway, despite me finding the argumentation in Hagar's paper rather incomplete, I very much like the attempt to disentangle the discussion and approach it from a logical and objective basis. You see, I have stakes in the issue, so has everybody else who has worked on the topic. If you read a random paper on DSR it will tell you how natural such a modification is, how plenty the motivations, how great the prospects to experimentally test it - and be kinda brief on the “well-known” inconsistencies. Hagar's paper makes a nice contrast to this by telling the story as it really is.
I wrote an email to Amit Hagar,
and he kindly replied, letting me know he is “an avid reader of [my] blog and papers, and the truth is they have very much inspired [his] looking into this interesting debate.” I am very flattered. But what's even better is that he tells me he plans to write a book on the history and philosophy of the minimal length, starting from Heisenberg up to now. I think it is a great idea. The history of the topic is full with beautiful thought experiments and arguments about their implications, and the whole field would benefit from a clear summary.
- The Minimal Length Scale
- Dangerous Implications
- Deformed Special Relativity
- Constraining Modified Dispersion Relations with Gamma Ray Bursts
- That Photon from GRB090510
 Such modifications run under the keyword “energy dependent metric.” Note that we are talking here about an energy dependent metric in position space, not momentum space.