One of the topics raised by the recent Joe Polchinski's review that led to some continued discussions is the topic of the hypothetical violations of the Lorentz symmetry and various proposed modifications of special relativity such as the so-called doubly special relativity or deformed special relativity. See cosmicvariance.com and backreaction.blogspot.com.
Figure 1: Discovery space shuttle took off again tonight. With a good enough eyesight or digital camera, you could see it from Boston, Copley Plaza. Can you find it on the picture?
I view these speculations as another example of unmotivated, bad physics, and a waste of paper. Let me explain why.
Back to 1905
First of all, Einstein's discovery of special relativity in 1905 was the real beginning of modern theoretical physics. Einstein was able to discover the true kinematics of space and time that morally respected the previously believed symmetries of Nature and that preserved the degree of their symmetry but it extended their validity. Einstein strengthened some of the holy principles of the physics before him - in fact, he modestly described special relativity as a minor update of Newton's framework; what he abandoned were assumptions that were so obvious to everyone that they didn't even doubt about them in public.
The postulates of special relativity say that
- the laws of physics have the same form in all inertial reference frames
- the speed of light is constant in all reference frames
The first postulate was already valid in the Newtonian mechanics. This principle requires that the laws of physics are invariant under a ten-dimensional group, generated by 1 temporal and 3 spatial translations, 3 rotations, and 3 Galilean boosts. Special relativity has just replaced 3 Galilean boosts by 3 Lorentz boosts. The resulting Poincaré algebra is more non-Abelian than its non-relativistic limit. And it picks a preferred speed, the speed of light. But it doesn't really reduce the power of the old principles.
The second postulate of special relativity morally follows from the first one once you promote the value of the speed of light to a law of physics which is what Einstein did. In classical Newtonian mechanics, it was not a law of physics. The speed of light according to Newton depended on your speed and the speed of the source; something that was in tension with Maxwell's equations. According to Einstein, it must be a constant for all observers. Einstein preserved everything that was beautiful about the previous theory and reproduced all of its successful predictions; on the other hand, his new theory was compatible with the newer experiments by Morley and Michelson and it ignited the modern 20th century physics.
The constraining power of special relativity
In quantum field theory, the principles of special relativity play a crucial role. They're not only experimentally tested but they are also critical for us to be able to say what the theory actually is. The Standard Model is a renormalizable quantum field theory with a few dozens of parameters. The finiteness of the number of its parameters critically depends on Lorentz invariance. A generic theory without Lorentz invariance can be deformed by infinitely many perturbations. In a non-relativistic theory, the form factors for all particles and interactions can be chosen in an arbitrary way. The result would be a theory with infinitely many undetermined constants. I personally don't use the word "theory" for such structures.
If you only allow theories that are renormalizable perturbations of a relativistic theory, you obtain different results. Coleman and Glashow have shown that there are 46 CPT-even gauge-invariant renormalizable perturbations of the Standard Model that preserve the rotational symmetry in a preferred reference frame. Their paper is what I call an important physical result about the subject: it describes things and phenomena that could actually be observed. Every realistic theory has to reduce to an effective quantum field theory at low energies, and because the Lorentz invariance is now very accurately verified, every realistic theory must be at most a small perturbation of a Lorentz-invariant theory. If this violation is indeed small, it should be possible to construct the idealized limiting theory in which the symmetry is fully restored and to treat the Lorentz-violating theory as its small perturbation.
I am convinced that this conclusion is quite general and it applies to speculative theories such as doubly special relativity, too.
If you're puzzled why the number of parameters in a fully non-relativistic theory was infinite while the number of Lorentz-breaking deformations of the Standard Model is finite, namely 46, don't forget about the adjective "renormalizable". The number 46 only counts the deformations that are renormalizable - relevant or marginal - according to the relativistic counting of dimensions of the operators.
The dimensions of the operators are computed from a classical relativistic theory. An entirely non-relativistic theory has a different notion of dimensions - the energy goes like "p^2" as opposed to "p" at high momenta - and the space of conceivable non-relativistically renormalizable perturbations is different and, in fact, infinite-dimensional.
Exactness of Lorentz symmetry
So is the Lorentz invariance in the absence of gravity an exact symmetry? Well, we can't ever be quite certain and one can always speculate that any principle we hold dear will suddenly break down without any known good reason. And it has happened to some principles in the past. But I think that it is fair to say that according to everything we know, the symmetry is exact. It has been confirmed by all experiments. It is essential for quantum field theory to be predictive. Its validity at distance scales shorter than the curvature radius can be derived from closed string pertubative string theory. The hypothetical violations don't help to solve any outstanding problems. They are ugly. They are unjustified.
Doubly special relativity
But doubly special relativity is not a violation of special relativity, is it? I am convinced that the answer is either "it is a violation" or "it is equivalent to what we have".
In the past, we discussed the special case of doubly special relativity in three dimensions. In this low-dimensional spacetime, the equations of DSR are related to an interesting kind of a star-product. However, it is pretty clear that these results can't be extended to four dimensions (or higher) just like the cross product giving you a vector out of two vectors can't. In these realistic cases, it is more important that DSR is just a change of variables. Whatever can be expressed in the natural variables of DSR can also be interpreted in the usual variables. DSR doesn't give you any natural Lagrangian.
And the DSR speculators so far only have some bizarre dispersion relations, not an actual dynamical theory.
The only way how it could hypothetically give you some new interesting physics would materialize if there were some dynamics that is natural in the DSR variables even though you would never guess it in the normal variables. I find this possibility extremely unlikely. The transformations that you need for such a redefinition of momentum map a local theory into a non-local theory.
It is hard to believe that there are non-local theories that are much more acceptable than other non-local theories. Locality is another face of relativity. It is related to the additivity of energy. The additivity of energy or the action is a principle that allows two separated regions in spacetime to evolve more or less independently. Nothing guarantees that this principle is exact. But if you look at perturbative string theory, its amplitudes actually preserve all the analytic rules that you expect from a fully local theory. A priori, you could think that the locality rules will be heavily violated because the fundamental object of string theory is an extended object: a string. But it just turns out that the analytic rules hold even in perturbative string theory. String theory actually saturates some inequalities for the high-energy growth of the amplitudes that can be proven from locality even though no point-like quantum field theory ever came close to saturating this bound.
Don't mess up with the dispersion relations
Replacing the dispersion relation
- E2 = m2 + p2
by any non-homogeneous equation that is not invariant under the scaling of "(E,m,p)" is a bad idea that leads, among other difficulties, to the soccer ball problem. The problem is the following. The usual relativistic relation above holds for elementary particles as well as composite objects. It can hold for all of them because it is a uniform second-degree equation.
However, if you replace it by a more non-linear relation, it can't simultaneously hold for the elementary as well as composite particles. You must make a choice. This fact itself is a reason for a serious concern - or a reason to abandon the idea - because it is incompatible with the principles of the renormalization group that tell you that you should be able to work with the proton as with an elementary particle as long as you study long-distance phenomena only.
If the non-linear dispersion relations only hold for quarks - and they probably hold for quarks if they are fundamental - they won't hold for the proton. The assumption that these relations hold even for macroscopic objects leads to an even more obvious disaster: you can't play soccer. It is impossible for objects to exceed the Planck energy which soccer balls often do whenever the game is being played. The prediction that there can't be any soccer - or any other motion of macroscopic objects - is a pretty good reason to abandon a hypothesis, I think.
All these facts make it quite clear that the attempts to violate or deform the principles of special relativity lead us to a lose-lose situation. The result is an ugly theory that probably disagrees with the known experiments and doesn't have any advantages.
Gravity and special relativity
Many people outside the particle physics community are often confused about the relation between special relativity and general relativity. They imagine that general relativity has rejected special relativity. Quite on the contrary. General relativity is an extension of the principles of special relativity in which all coordinate systems, not just inertial reference frames, are equally good for our formulation of the physical laws. It is a theory of curved space where the laws of special relativity are locally satisfied in all freely falling reference frames. General relativity without the principles of special relativity inside it is no theory of relativity.
The proponents of deformed special relativity are often seen to justify their interest in this bizarre structure by a clearly flawed argument. They say that all observers should agree what is the shortest time interval or the highest possible energy which is why the normal Lorentz transformations are not allowed. This sentence may look like an argument but in reality, it is just another unsubstantiated rant against the theory of relativity.According to special relativity, which is the nice theory that has been confirmed by all experiments that were made by 2006, different observers will never agree about concrete time intervals, distances, energies, or momenta by which I mean components of various four-vectors. This is a universal law of Nature that applies to all phenomena including QCD and gravity, among others, and we have working full theories including QCD and string theory that show that this principle can indeed be satisfied even in theories of strong interactions or gravity. Different observers can only agree about invariant quantities such as the proper times and "p^2" for various four-vectors. Only these Lorentz-invariant quantities may become a subject of various fundamental inequalities.
The deformed relativists simply deny this insight and other principles of relativity but according to everything I know, they have no rational justification for their denial.
Spin networks, aether, and special relativity
The real reason why some people keep on promoting these pretty much falsified and unattractive theories - such as doubly special relativity - is to create a friendly atmosphere for other ideas that are often even worse. What I have in mind are various discrete "theories" of gravity among which the so-called loop quantum gravity is the most familiar example. There is no direct logical relationship between loop quantum gravity on one side and doubly special relativity on the other side. But both of them violate some basic and well-established principles of physics which is why both of these inexpensive theoretical structures are often sold together.
Any theory that assumes that the vacuum has a discrete, naively visualizable structure inevitably breaks the Lorentz symmetry because it is nothing else than a new form of the luminiferous aether. It is something that was bad physics already in 1905 if not earlier and be sure that in 2006, without having any new evidence or a more attractive reincarnation of these ideas, it is even worse physics than it was 101 years ago. Any kind of luminiferous aether is unmotivated. It has always been even though some great physicists had some irrational reasons to pursue these concepts.
150 years ago, the theories of aether could at least reproduce Maxwell's equations in the continuum limit; FitzGerald has even created a working prototype. Today, the counterparts of these theories that are speculated to be relevant for gravity can't even do that: they haven't done it and there are good reasons to think that they can't ever do it. It is obvious that any kind of aether, spin network, or anything else that is hypothesized to describe the structure of spacetime is a part of bad physics movement that has become even worse than it was 150 years ago. The progress in this line of reasoning was clearly negative during the last 150 years.
There could be some discrete structures behind spacetime but they will certainly not become an exciting part of theoretical physics until someone shows why they satisfy the rules of special relativity. Quantum foam and melting crystals are some of the allowed exceptions because they are relevant for topological theories only. Topological theories have no local degrees of freedom which is why the presence of an aether doesn't imply any lethal Lorentz violation.
String theory and principles of physics
String theory is a canonical example, and to a large extend unique example, of physics that goes beyond the effective relativistic quantum field theory. Various ingredients are modified and you might guess that such a theory will violate many of the previous principles such as the Lorentz invariance of the physical laws or the postulates of quantum mechanics.
A priori, you might believe that this is indeed the case. Also, you might think about various reasons why it could be a good thing to break these principles - for example to solve the information loss paradox. But when you study string theory carefully enough, you find out that it is actually not necessary to break any of the principles in order to solve any of these problems. More importantly, you find out that string theory doesn't really modify these principles at all. The postulates of quantum mechanics hold exactly and they don't contradict any known properties of the black holes. What string theory modifies are things that can be expressed as the old-fashioned modifications that were always thought to be possible - like the higher-derivative terms in general relativity. There can however exist other, equivalent ways to describe the same situation in string theory.
The special relativity also holds exactly in string theory: all of its violations can be interpreted as a spontaneous violation - as the presence of an object (or B-field) that could also be removed so that the Lorentz symmetry is restored.
Does string theory have a minimal length? It depends what you exactly mean. String theory certainly does imply that the usual laws of geometry break down at sufficiently short distances: in some respects it is already the string length, in other thought experiments it is the Planck length. But the details of this breaking are certainly very different from the first stupid idea that someone might invent - e.g. the idea that the areas are multiples of a minimum area "A0".
The breaking of these ideas is much more continuous and subtle in string theory and one needs much more sophisticated and delicate mathematical tools to quantitatively describe how the breakdown works than the first elephant-in-china-shop proposal that a random person on the street could write down.
The search for the new exciting ideas in physics is also the search for the perfect balance between conservatism and progressivism of the new ideas. Some topics may become too boring and conventional; some new ideas are too radical and they can often be shown to be wrong (and their proponents often don't care). Physics must continue to search for exciting ideas that are moreover true and these ideas can't emerge if one is overly conservative or overly progressive.
And that's the memo.