I simply can't resist the temptation to comment on the following paper that just appeared as the #1 on hep-th tonight:
This reminds me of a layman who was writing about loop quantum gravity on a newsgroup and used the term "one-loop quantum gravity" instead of "loop quantum gravity" because the multiplicative factor "one" does not change anything. ;-) Is it the same way how the author of that paper determined that "loop gravity" may be renamed to "one-loop, i.e. semiclassical gravity"? :-)
There is a lot of difference between semiclassical gravity and loop gravity, for example these theories have a very different meanings:
- loop gravity is a combinatorial toy model that has nothing to do with gravity and probably nothing to do with any kind of physics in nearly smooth spacetimes; see the article about problems with loop quantum gravity
- on the other hand, semiclassical gravity is a perturbative treatment of quantized general relativity that takes classical physics and the first correction proportional to hbar - this first correction is called semiclassical physics - into account; semiclassical gravity is useful to explain the existence of gravitons, their leading scattering, and effects like black hole evaporation
- in the commercials, loop quantum gravity is described as a "non-perturbative" description of quantum gravity that has a simple form if one wants to describes the effects that take place on the Planck scale which is a very short distance - for example the geometry of Planckian surfaces
- on the other hand, semiclassical gravity is an expansion of gravity valid at very long distances at which all quantum corrections - non-perturbative as well as higher-order perturbative ones - can be neglected
First of all, it was realized by many people - including the proponents of loop quantum gravity - that this approach couldn't agree with the usual Lorentz invariance of local physics. It's very simple to see why - the spin networks are essentially a new kind of luminiferous aether that picks a priviliged reference frame, even in the hyper-optimistic case that they can conspire in such a way that smooth spacetime appears at long distances.
Note that General Relativity is based on a basic postulate - the principle of equivalence - that says , in one possible interpretation, that the freely falling observer sees local physics that is indistinguishable from physics without gravitational fields - which means physics of special relativity. It's the whole point of General Relativity that it is a theory of gravity that is compatible with special relativity and reduces to special relativity if the spacetime curvature can be neglected. This is why Einstein was looking for General Relativity in the first place.
A theory of gravity that does not respect special relativity in these limits is equally unacceptable for post-1905 physics as Newton's theory of gravity.
OK, let's forget about all these obvious facts. Loop quantum gravity offers us a "prediction" of the quantized areas. Is it really a prediction? No way. It's a basic assumption of loop quantum gravity - namely that the metric tensor can be expressed as a dual variable to a gauge field. Because the monodromies of the gauge field live in a compact space, the canonically dual variable to the gauge field is quantized (that's seen as the area quantization). Obviously, the area quantization does not follow from quantum gravity - and the metric tensor degrees of freedom themselves; it follows from the field redefinition that relates the metric tensor to the gauge field. This proves that this field redefinition, although it may be valid locally on the configuration space, is invalid globally.
Spin foam breaks Lorentz symmetry
This field redefinition - and the unphysical quantization of the areas - is also the primary source of the Lorentz violation in loop quantum gravity and the majority of other physical problems of loop quantum gravity. There cannot really be a sharp bound for a minimum positive area (or distance) in a Lorentz invariant theory because the Lorentz contraction can always lower this area (or distance), roughly speaking.
I used to think that this point had be completely obvious to anyone who has heard about the "luminiferous aether" and the story that Einstein had to abandon aether in order to preserve the principle of relativity: any discrete non-singular structure inserted to space picks a preferred frame which is not allowed by special relativity. But judging the number of questions I am getting all the time, this question is not so clear.
OK, let me formulate the proof in different words. Consider a spin foam - the time evolution of a spin network (a spin foam a structure made of 2-dimensional surfaces embedded in 3+1 dimensional space, and their junctions) - a spin foam that you believe to be typical configuration contributing to the path integral of what you want to call the "Minkowski vacuum". Assume that at long distances, the Lorentz symmetry is approximately preserved. Choose a large piece of a nearly null (but slightly space-like) 2-plane in the Minkowski space (a rectangle) and try to calculate its proper area. According to loop quantum gravity, the area is proportional to the number of intersections of this rectangle with your spin foam (weighted by coefficients of order one in Planck units, separated from zero by the assumption of area quantization). By Lorentz symmetry, the area must go to zero if the rectangle is becoming null because a boost can map it onto a very small, localized spacelike rectangle whose area should go to zero. This means that there can't really be any intersections of the spin foam with the null rectangle, and because it must be true for any null 2-plane, all faces of the spin foam must be parallel to all null 2-planes, which is easily seen to be impossible because different null 2-planes are not parallel to each other. In other words, every 2-plane in the Minkowski space intersects some null 2-planes, and therefore these null 2-planes will be assigned significant positive proper area by LQG, which is incorrect. If you have a spin foam, it either breaks Lorentz symmetry completely, or it will have to be infinitely dense. But you definitely can't get a non-singular Lorentz-invariant theory. Note that the violation of the Lorentz symmetry derived above is not "small" in any sense. The Lorentz contraction factor simply stays finite (of order one) if you approach the speed of light. There's nothing you can do about it: aether (or a spin foam) is a clearly wrong model of the Minkowski space.
Once we know that the usual Lorentz invariance can't hold in a discrete model of gravity, one can try to propose weaker conjectures. For example, loop quantum gravity could agree with "doubly special relativity" (DSR), also known as "deformed special relativity". That's a nonlinear deformation of the Poincare algebra whose Lorentz subgroup is not modified at all, but the action on the momenta is altered. It has a parameter kappa that can be identified with the Planck scale. It's not such a big deal that such a deformation exists because we're really deforming the translational part of the Poincare symmetry only - and because the momenta commuted with each other anyway, it's not hard to preserve the Jacobi identity. Technically, we may also obtain this kappa-Poincare algebra as a limit of a q-deformed de Sitter algebra SO_q(d,1). The quantum deformation is claimed to be the reformulation of the cosmological constant term to the language of the LQG gauge fields - a potentially interesting relation that is, however, made irrelevant in LQG by the non-existence of any Lorentz or de Sitter symmetry in the first place. Nevertheless, even in string theory people (David Lowe etc.) have studied the possibility that the de Sitter space should carry a quantum deformation of the corresponding isometry group.
Doubly special relativity is a highly problematic picture as it stands - for example, the appearance of non-linear functions of momenta prove that the action of the Poincare transformations is non-local in spacetime and a Lorentz transformation can actually map a function (lion) in the interior of a region (cage) into a function (lion) outside the region, which is pretty dangerous. Moreover, the proposed modifications of the dispersion relations will give you different values of the speed of light for a composite particle than if you consider the particle to be elementary (imagine the proton as an example), which is pretty bad. Incidentally, on every normal Hilbert space - i.e. an infinite-dimensional separable Hilbert space - we can find a collection of 10 operators that form the four-dimensional Poincare algebra: even for an ordinary harmonic oscillator. It's simply because all Hilbert spaces are isomorphic to each other, and therefore the harmonic oscillator Hilbert space may be mapped to the Hilbert space of the Standard Model if we don't care that such a map is physically ludicrous. But that does not mean that all theories are Lorentz-invariant; we only accept the Poincare generators as "real symmetries" if they have simple commutators with the usual notions of time etc. (for example, we require that the rotation generators commute with the Hamiltonian).
But even if we forget about all these problems of doubly special relativity, I think that it is pretty clear that the conjectured relation of loop quantum gravity to doubly special relativity is just another speculation. Of course, the smooth space at long distance probably can't appear in LQG anyway (there is definitely no proof that it does appear, to say the least), and therefore LQG can't agree with DSR that does reduce to flat space at long distances.
If someone will tell you that semiclassical gravity predicts a violation of the relativistic dispersion relations for photons, for example, don't believe her. It's a conclusion based on shaky, speculative connections between various unrelated inconsistent ideas, each of which can easily be shown to disagree with reality.