Sunday, October 17, 2004 ... //

Objections to loop quantum gravity

It seems that one can easily repost one of my articles from Wikipedia, so that even the links are preserved. Let's try.

In theoretical physics, is one speculative approach to quantum gravity, sometimes cited as a competitor theory to string theory. The loop quantum gravity page summarises the theory as it appears to those working in the field.

As a physical theory, loop quantum gravity has been subject to some heavy criticisms. Some objections to the ideas of loop quantum gravity are given here.

Too many assumptions

Loop quantum gravity makes too many assumptions about the behavior of geometry at very short distances. It assumes that the metric tensor is a good variable at all distance scales, and it is the only relevant variable. It even assumes that Einstein's equations are more or less exact in the Planckian regime.

The spacetime dimensionality (four) is another assumption that cannot be questioned, much like the field content. Each of these assumptions is challenged in a general enough theory of quantum gravity, for example all the models that emerge from string theory. These assumptions have neither theoretical nor experimental justification. Particular examples will be listed in a separate entry.

The most basic, underlying assumption is that the existence of a meaningful classical theory, of general relativity, implies that there must exist a "quantization" of this theory. This is commonly challenged. Many reasons are known why some classical theories do not have a quantum counterpart. Gauge anomalies are a prominent example. General relativity is usually taken to be another example, because its quantum version is not renormalizable. It is known, therefore, that a classical theory is not always a good starting point for a quantum theory. Theorists of loop quantum gravity work with the assumption that "quantization" can be done, and continue to study it even if their picture seems inconsistent.

Commentary from the renormalization group aspect

According to the logic of the renormalization group, the Einstein-Hilbert action is just an effective description at long distances; and it is guaranteed that it receives corrections at shorter distances. String theory even allows us to calculate these corrections in many cases.

There can be additional spatial dimensions; they have emerged in string theory and they are also naturally used in many other modern models of particle physics such as the Randall-Sundrum models. An infinite amount of new fields and variables associated with various objects (strings and branes) can appear, and indeed does appear according to string theory. Geometry underlying physics may become noncommutative, fuzzy, non-local, and so on. Loop quantum gravity ignores all these 20th and 21st century possibilities, and it insists on a 19th century image of the world which has become naive after the 20th century breakthroughs.

On one hand, loop quantum gravity prohibits many types of new physics that have become natural and likely in the recent era. On the other hand, it brings no order to the structure and values of the higher-derivative terms. Indeed, the infinitely many parameters that govern the effective action and make quantized general relativity non-renormalizable and unpredictive are simply replaced by an infinite number of parameters of the Hamiltonian constraint and/or the spin foam Feynman rules.

As a predictive theory

Loop quantum gravity is therefore not a predictive theory. Moreover, it does not offer any possibility to predict new particles, forces and phenomena at shorter distances: all these objects must be added to the theory by hand. Loop quantum gravity therefore also makes it impossible to explain any relations between the known physical objects and laws.

Loop quantum gravity is not a unifying theory. This is not just an aesthetic imperfection: it is impossible to find a regime in real physics of this Universe in which non-gravitational forces can be completely neglected, except for classical physics of neutral stars and galaxies that also ignores quantum mechanics. For example, the electromagnetic and strong force are rather strong even at the Planck scale, and the character of the black hole evaporation would change dramatically had the Nature omitted the other forces and particles. Also, the loop quantum gravity advocates often claim that the framework of loop quantum gravity regularizes all possible UV divergences of gravity as well as other fields coupled to it. That would be a real catastrophy because any quantum field theory - including all non-renormalizable theories with any fields and any interactions - could be coupled to loop quantum gravity and the results of the calculations could be equal to anything in the world. The predictive power would be exactly equal to zero, much like in the case of a generic non-renormalizable theory. There is absolutely no uniqueness found in the realistic models based on loop quantum gravity. The only universal predictions - such as the Lorentz symmetry breaking discussed below - seem to be more or less ruled out on experimental grounds.

Self-consistency

Unlike string theory, loop quantum gravity has not offered any non-trivial self-consistency checks of its statements and it has had no impact on the world of mathematics. While string theory smells by God, loop quantum gravity smells by Man. It seems that the people are constructing it, instead of discovering it. There are no nice surprises in loop quantum gravity - the amount of consistency in the results never exceeds the amount of assumptions and input. For example, no answer has ever been calculated in two different ways so that the results would match. Whenever a really interesting question is asked - even if it is apparently a universal question, for example: "Can topology of space change?" - one can propose two versions of loop quantum gravity which lead to different answers.

There are many reasons to think that loop quantum gravity is internally inconsistent, or at least that it is inconsistent with the desired long-distance limit (which should be smooth space). Too many physical wisdoms seem to be violated. Unfortunately the loop quantum gravity advocates usually choose to ignore the problems. For example, the spin foam (path-integral) version of loop quantum gravity is believed to break unitarity. The usual reaction of the loop quantum gravity practitioners is the statement that unitarity follows from time-translation symmetry, and because this symmetry is broken (by a generic background) in GR, we do not have to require unitarity anymore. But this is a serious misunderstanding of the meaning and origin of unitarity. Unitarity is the requirement that the total probability of all alternatives (the squared length of a vector in the Hilbert space) must be conserved (well, it must always be 100%), and this requirement - or an equally strong generalization of it - must hold under any circumstances, in any physically meaningful theory, including the case of the curved, time-dependent spacetime. Incidentally, the time-translation symmetry is related, via Noether's theorem, to a time-independent, conserved Hamiltonian, which is a completely different thing than unitarity.

A similar type of "anything goes" approach seems to be applied to other no-go theorems in physics.

Gap to high-energy physics

Loop quantum gravity is isolated from particle physics. While extra fields must be added by hand, even this ad hoc procedure seems to be impossible in some cases. Scalar fields can't really work well within loop quantum gravity, and therefore this theory potentially contradicts the observed electroweak symmetry breaking; the violation of the CP symmetry, and other well-known and tested properties of particle physics.

Loop quantum gravity also may deny the importance of many methods and tools of particle physics - e.g. the perturbative techniques; the S-matrix, and so on. Loop quantum gravity therefore potentially disagrees with 99% of physics as we know it. Unfortunately, the isolation from particle physics follows from the basic opinions of loop quantum gravity practitioners and it seems very hard to imagine that a deeper theory can be created if the successful older theories, insights, and methods (and exciting newer ones) in the same or closely related fields are ignored.

Moreover, it has been recently argued that in every consistent theory of quantum gravity, gravitational force must be the weakest one (much like in the real world). More precisely, there must always exist objects whose repulsive force of any kind exceeds the attractive force of gravity. Loop quantum gravity, much like all other attempts to start with "pure quantum gravity", violates this insight maximally because it is an expansion around the point in which the other interactions are turned off. This point in the parameter space is inconsistent.

Smooth space as limiting case

Loop quantum gravity does not guarantee that smooth space as we know it will emerge as the correct approximation of the theory at long distances; there are in fact many reasons to be almost certain that the smooth space cannot emerge, and these problems of loop quantum gravity are analogous to other attempts to discretize gravity (e.g. putting gravity on lattice).

While string theory confirms general relativity or its extensions at long distances - where GR is tested - and modifies it at the shorter ones, loop quantum gravity does just the opposite. It claims that GR is formally exact at the Planck scale, but implies nothing about the correct behavior at long distances. It is reasonable to assume that the usual ultraviolet problems in quantum gravity are simply transmuted into infrared problems, except that the UV problems seem to be present in loop quantum gravity, too.

Clash with special relativity

Loop quantum gravity violates the rules of special relativity that must be valid for all local physical observations. Spin networks represent a new reincarnation of the 19th century idea of the luminiferous aether - environment whose entropy density is probably Planckian and that picks a priviliged reference frame. In other words, the very concept of a minimal distance (or area) is not compatible with the Lorentz contractions. The Lorentz invariance was the only real reason why Einstein had to find a new theory of gravity - Newton's gravitational laws were not compatible with his special relativity.

Despite claims about the background independence, loop quantum gravity does not respect even the special 1905 rules of Einstein; it is a non-relativistic theory. It conceptually belongs to the pre-1905 era and even if we imagine that loop quantum gravity has a realistic long-distance limit, loop quantum gravity has even less symmetries and nice properties than Newton's gravitational laws (which have an extra Galilean symmetry, and can also be written in a "background independent" way - and moreover, they allow us to calculate most of the observed gravitational effects well, unlike loop quantum gravity). It is a well-known fact that general relativity is called "general" because it has the same form for all observers including those undergoing a general accelerated motion - it is symmetric under all coordinate transformations - while "special" relativity is only symmetric under a subset of special (Lorentz and Poincare) transformations that interchange inertial observers. The symmetry under any coordinate transformation is only broken spontaneously in general relativity, by the vacuum expectation value of the metric tensor, not explicitly (by the physical laws), and the local physics of all backgrounds is invariant under the Lorentz transformations.

Loop quantum gravity proponents often and explicitly state that they think that general relativity does not have to respect the Lorentz symmetry in any way - which displays a misunderstanding of the symmetry structure of special and general relativity (the symmetries in general relativity extend those in special relativity), as well as of the overwhelming experimental support for the postulates of special relativity. Loop quantum gravity also depends on the background in a lot of other ways - for example, the Hamiltonian version of loop quantum gravity requires us to choose a pre-determined spacetime topology which cannot change.

One can imagine that the Lorentz invariance is restored by fine-tuning of an infinite number of parameters, but nothing is known about the question whether it is possible, how such a fine-tuning should be done, and what it would mean. Also, it has been speculated that special relativity in loop quantum gravity may be superseded by the so-called doubly special relativity, but doubly special relativity is even more problematic than loop quantum gravity itself. For example, its new Lorentz transformations are non-local (two observers will not agree whether the lion is caught inside the cage) and their action on an object depends on whether the object is described as elementary or composite.

Global justification of variables

The discrete area spectrum is not a consequence, but a questionable assumption of loop quantum gravity. The redefinition of the variables - the formulae to express the metric in terms of the Ashtekar variables (a gauge field) - is legitimate locally on the configuration space, but it is not justified globally because it imposes new periodicities and quantization laws that do not follow from the metric itself. The area quantization does not represent physics of quantum gravity but rather specific properties of this not-quite-legitimate field redefinition. One can construct infinitely many similar field redefinitions (sibblings of loop quantum gravity) that would lead to other quantization rules for other quantities. It is probably not consistent to require any of these new quantization rules - for instance, one can see that these choices inevitably break the Lorentz invariance which is clearly a bad thing.

Testability of the discrete area spectrum

The discrete area spectrum is not testable, not even in principle. Loop quantum gravity does not provide us with any "sticks" that could measure distances and areas with a sub-Planckian precision, and therefore a prediction about the exact sub-Planckian pattern of the spectrum is not verifiable. One would have to convert this spectrum into a statement about the scattering amplitudes.

The S-matrix

Loop quantum gravity provides us with no tools to calculate the S-matrix, scattering cross sections, or any other truly physical observable. It is not surprising; if loop quantum gravity cannot predict the existence of space itself, it is even more difficult to decide whether it predicts the existence of gravitons and their interactions. The S-matrix is believed to be essentially the only gauge-invariant observable in quantum gravity, and any meaningful theory of quantum gravity should allow us to calculate it, at least in principle.

Ultraviolet divergences

Loop quantum gravity does not really solve any UV problems. Quantized eigenvalues of geometry are not enough, and one can see UV singular and ambiguous terms in the volume operators and most other operators, especially the Hamiltonian constraint. Because the Hamiltonian defines all of dynamics, which contains most of the information about a physical theory, it is a serious object. The whole dynamics of loop quantum gravity is therefore at least as singular as it is in the usual perturbative treatment based on semiclassical physics.

We simply do have enough evidence that a pure theory of gravity, without any new degrees of freedom or new physics at the Planck scale, cannot be consistent at the quantum level, and loop quantum gravity advocates need to believe that the mathematical calculations leading to the infinite and inconsistent results (for example, the two-loop non-renormalizable terms in the effective action) must be incorrect, but they cannot say what is technically incorrect about them and how exactly is loop quantum gravity supposed to fix them. Moreover, the loop quantum gravity proponents seem to believe that the naive notion of "atoms of space" is the only way to fix the UV problems. String theory, which allows us to make real quantitative computations, proves that it is not the case and there are more natural ways to "smear out" the UV problems. In fact, a legitimate viewpoint implies that the discrete, sharp character of the metric tensor and other fields at very short distances makes the UV behavior worse, not better.

Moreover, as explained above, the "universal solution of the UV problems by discreteness of space" implies at least as serious loss of predictive power as in a generic non-renormalizable theory. Even if loop quantum gravity solved all the UV problems, it would mean that infinitely many coupling constants are undetermined - a situation analogous to a non-renormalizable theory.

Black hole entropy

Despite various claims, loop quantum gravity is not able to calculate the black hole entropy, unlike string theory. The fact that the entropy is proportional to the area does not follow from loop quantum gravity. It is rather an assumption of the calculation. The calculation assumes that the black hole interior can be neglected and the entropy comes from a new kind of dynamics attached to the surface area - there is no justification of this assumption. Not surprisingly, one is led to an area/entropy proportionality law. The only non-trivial check could be the coefficient, but it comes out incorrectly (see the Immirzi discrepancy).

The Immirzi discrepancy was believed to be proportional to the logarithm of two or three, and a speculative explanation in terms of quasinormal modes was proposed. However it only worked for one type of the black hole - a clear example of a numerical coincidence - and moreover it was realized in July 2004 that the original calculation of the Immirzi parameter was incorrect, and the correct value (described by Meissner) is not proportional to the logarithm of an integer. The value of the Immirzi parameter - even according to the optimists - remains unexplained. Another description of the situation goes as follows: Because the Immirzi parameter represents the renormalization of Newton's constant and there is no renormalization in a finite theory - and loop quantum gravity claims to be one - the Immirzi parameter should be equal to one which leads to a wrong value of the black hole entropy.

Nonseparable Hilbert space

While all useful quantum theories in physics are based on a separable Hilbert space, i.e. a Hilbert space with a countable basis, loop quantum gravity naturally leads to a non-separable Hilbert space, even after the states related by diffeomorphisms are identified. This space can be interpreted as a very large, uncountable set of superselection sectorsthat do not talk to each other and prevent physical observables from being changed continuously. All known procedures to derive a different, separable Hilbert space are physically unjustified. Equivalently, loop quantum gravity does not allow one to derive the conventional notion of continuity of space.

Foundational lacks

Loop quantum gravity has no tools and no solid foundations to answer other important questions of quantum gravity - the details of Hawking radiation; the information loss paradox; the existence of naked singularities in the full theory; the origin of holography and the AdS/CFT correspondence; mechanisms of appearance and disappearance of spacetime dimensions; the topology changing transitions (which are most likely forbidden in loop quantum gravity); the behavior of scattering at the Planck energy; physics of spacetime singularities; quantum corrections to geometry and Einstein's equations; the effect of the fluctuating metric tensor on locality, causality, CPT-symmetry, and the arrow of time; interpretation of quantum mechanics in non-geometric contexts including questions from quantum cosmology; the replacement for the S-matrix in de Sitter space and other causally subtle backgrounds; the interplay of gravity and other forces; the issues about T-duality and mirror symmetry.

Loop quantum gravity is criticised as a philosophical framework that wants us to believe that these questions should not be asked. As if general relativity is virtually a complete theory of everything (even though it apparently can't be) and all ideas in physics after 1915 can be ignored.

Prejudices claimed

The criticisms of loop quantum gravity regarding other fields of physics are misguided. They often dislike perturbative expansions. While it is a great advantage to look for a framework that allows us to calculate more than the perturbative expansions, it should never be less powerful. In other words, any meaningful theory should be able to allow us to perform (at least) approximative, perturbative calculations (e.g. around a well-defined classical solution, such as flat space). Loop quantum gravity cannot do this, definitely a huge disadvantage, not an advantage as some have claimed. A good quantum theory of gravity should also allow us to calculate the S-matrix.

Background independence

Loop quantum gravity's calls for "background independence" are misled. A first constraint for a correct physical theory is that it allows the (nearly) smooth space(time) - or the background - which we know to be necessary for all known physical phenomena in this Universe. If a theory does not admit such a smooth space, it can be called "background independent" or "background free", but it may be a useless theory and a physically incorrect theory.

It is a very different question whether a theory treats all possible shapes of spacetime on completely equal footing or whether all these solutions follow from a more fundamental starting point. However, it is not a priori clear on physical grounds whether it must be so (it can be just an aesthetic feature of a particular formulation of a theory, not the theory itself), and moreover, for a theory that does not predict many well-behaved backgrounds the question is meaningless altogether. Physics of string theory certainly does respect the basic rules of general relativity exactly - general covariance is seen as the decoupling of unphysical (pure gauge) modes of the graviton. This exact decoupling can be proved in string theory quite easily. It can also be seen in perturbative string theory that a condensation of gravitons is equivalent to a change of the background; therefore physics is independent of the background we start with, even if it is hard to see for the loop quantum gravity advocates.

Claims on non-principled approach

Loop quantum gravity is not science because every time a new calculation shows that some quantitative conjectures were incorrect, the loop quantum gravity advocates invent a non-quantitative, ad hoc explanation why it does not matter. Some borrow concepts from unrelated and fields, including noiseless information theory and philosophy, and some explanations why previous incorrect results should be kept are not easily credible.

snail feedback (16) :

Being a fan of the string camp, I can say that there is at least one weak point in your critique of LQG: the issue of a minimum length. Minimum length *is* compatible with special relativity (in fact, also string theory implies a minimum length, so I am puzzled by the argument) as many have shown. Minimum length leads to deviations form special relativity only at very high energies, near the Planck energy; for usual energies, there is no measureable effect.
The people studying "doubly special relativity" show this in detail.

In fact, the idea of minimum length originated in string theory (Veneziano and the like), so that it is dangerous to use it as a criticism.

A string tension (i.e. a maximum tension) plus quantum mechanics (no observation below hbar/2) *implies* a minimum length.
The simplest way to see it is the following. A tension T=< Tmax also implies, dividing by the speed of light c, a maximum mass change per time.

Now length squared is, dimensionally, an angular momentum divided by a mass change.
d^2 = L/(dm/dt)
With a minimum angular momentum and a maximum mass change, you thus get a minimum length.

Any theory that has a maximum tension and a minimum angular momentum (like quantum theory) must have a minimum length. That is true both for string theory and LQG.

I don't think that anything you said is correct.

The issue about the "minimum length" in loop quantum gravity is
especially about the "minimum nonzero length", which means the discrete spectrum of length: the minimal nonzero eigenvalue of length (or area) is positive, and this is incompatible with special relativity. You can easily see it if you boost a "spin network" or "spin form", constructed from these "atoms of space". If you want to get a Lorentz invariant theory with local physical excitations, such a spin foam must be averaged over the whole Lorentz group, and a typical element of the noncompact Lorentz group will stretch the "minimal length intervals" by an infinite aomunt, which will make the singular configurations dominate the path integral.

I believe that this is explained in the article.

This does not occur - and cannot occur - in (physical) string theory because the latter has an exact Lorentz symmetry. If you think it does occur, I would like to know the articles which, according to you, support the idea of discrete values of geometry in physical string theory.

String theory certainly modifies the UV behavior of all theories, but it does so but making them "smeared", more fuzzy - in some sense *more continuous* than at low energies, not in the other way around i.e. by making the eigenvalues of geometry discrete.

What I don't understand at all is why you think that *perturbative* string theory implies quantization of length (or even Veneziano amplitude?). I see nothing that would even incorrectly lead to this conclusion. Perturbative strings smooth the geometry at the string scale, and perturbatively everything becomes "very smooth" at this scale, but there is a lot of physics that occurs at much shorter distances nonperturbatively - namely dynamics of D-branes and quantum gravity. They don't have any discreteness in geometry either.

Your dimensional analysis "length is angular momentum over mass change" is totally confused and cannot imply anything about discreteness of either. Mass change is not bounded from above; why do you say it is? A black hole can be as heavy as you want, and it can still join other black holes. A black hole is a geometrically huge metastable resonance, from the particle physics viewpoint.

Moreover, it seems that you are thinking about the *mass spectrum* of excited strings. Yes, the spectrum at tree level id discrete. But that's a totally different issue than quantization of length. Quantum field theories have point-like particles with a completely discrete mass spectrum - well, there is one value of mass for each particle - but nevertheless they don't imply any length quantization; on the contrary, they are based on infinitely smooth space which is their background.

Your statements look so obviously wrong that it's a bit unclear why you did not show yourselves that they are wrong before you posted the comment.

And doubly special relativity is physically a nonsense, too, and it *is* mentioned in the article. Moreover, if something is just "doubly special invariant", you should not say that it is relativistic, strictly speaking.

The argument seems wrong; but it is not. I did not say nor mean to say that the length spectrum is discrete. I meant to say that it is bounded from below. Or better: there is a minimal *measurable* length. This statement is also made by many string papers. I believe Veneziano was one of the first; but Susskind any many others repeated it.

About the maximum mass change in nature (c^3/4G) the argument is the following. (1) There is no measured change of mass that is greater. Just check. (2) A higher value cannot even be imagined: You can try yourself: take any closed physical surface you want, and measure how mass-energy you can get out or in. It cannot exceed the value. Just try. If you get too high, a horizon will appear and stop you. (3) One can deduce the field equations of general relativity from the statement dm/dt =< c^3/4G. I'm happy to explain the last point, if you want. It is only a few lines. But the argument is not so well known. (Or see the 9 pages here.)

The steps are the same as when maximum speed was established. (1) No observation above c. (2) No imaginable way to beat it. (3) Deduce the Lorentz equations from v=< c.

Given that mass change is bounded from above, and given that angular momentum (or action, if you prefer) is bounded from below, one gets that length is bounded from below. That was the only reason to use of d^2=L/(dm/dt). One indeed gets no statement on the spectrum; I would tend to agree that everything is smeared out; but a measurement with a result below the minimum length is not possible. The result of a minimum length appears wehenever one combines general relativiy and quantum theory.

Part of our disagreement is a misunderstanding. I am not questioning that the usual notions of geometry break down at the Planck scale (or earlier).

But the reason in string theory is that it does not make sense to talk about shorter distances because the physics at "shorter" distances is not just normal geometry plus something else, but a stringy generalized fuzzy blah blah structure.

Loop quantum gravity, on the other hand, says that geometry is a good variable at all distance scales, and the areas etc. have discrete spectrum, which contradicts Lorentz invariance in any theory with local excitations.

c^4/4G is the maximum force in Nature (in some sense), I agree (the force between two black holes that are the horizon distance from each other). This conclusion only works like that in 4 dimensions though.

I will try to find time to look at your argument that the inequality about the force (mass change, as you say) implies Einstein's equations, it certainly sounds appealing - or, equivalently, most likely incorrect.

I see... Yes, I know Ted Jacobson's argument, and it is interesting.

On the other hand, it's not quite clear to me how you "connected" to Jacobson's argument using the inequality only. You also need "equalities" to get Jacobson's argument.

I wish I had more time to study your paper in more detail right now. Could you please post it to gr-qc at www.arxiv.org?

Also, once again, the force between two adjacent black holes of masses M,M in D dimensions is roughly

M^2 / (G.R^{D-2}}

The radius of the black hole satisfies R^{D-3}=G.M. Note that it is only in D=4 when the force becomes M-independent. Your statements don't seem to generalize to higher-dimensional gravity, which is for us, string theorists, a problem - as you can imagine. ;-)

The real derivation of general relativity is from the equality (dm/dt)_max = c^3/4G, indeed as you say. The inequality just describes general systems. It is like special relativity: the equality v_max=c is used to deduce the Lorentz equations, the inequality v =< c is valid for general systems.

I have to think about the higher-dimensional aspects. Interesting point!

Lubos,

My main beef about loop quantum gravity is that it doesn't seem to reproduce anything which looks remotely like gravity coupled to the Standard Model (Yang-Mills theory) in some low energy limit. With exception of string theory, many other "quantum gravity" models over the years don't seem to be very interested about Yang-Mills theory.

If any "quantum gravity" theory can't reproduce gravity coupled to Yang-Mills theory in some low energy limit, then I wonder what the theory is attempting to explain in the first place and why it should be taken seriously.

I agree with you, of course, that a more complete theory should always be trying to reproduce GR plus gauge theory at low energies.

On the other hand, the article "objections" is taken from its context. If you first read the article "Loop quantum gravity" on www.wikipedia.org, you will be told 50 times that the goal of LQG is not to find a TOE, but rather to just show that (pure) GR may be quantized canonically.

It seems that both of us disagree with that. ;-)

Might I interrrupt for one second.

In regards to Gerard Hooft's Panel lectureI have been trying to gain perspective on how LQG might be limited here as a good discriptor of quantum mechanical perspectives. Where strings, might have found limitations in computational avenues?

Have I approached this properly?

In previous post I gave link for consideration, that should have included, the instruction to immediate scroll down with the bar on the right hand side of that page linked.

Hope these instructions prevents deletion.

I am trying to follow this debate on minimum length, but am getting a little lost. I have heard that LQG breaks Lorentz invariance -- which makes sense to me, since they make spacetime discrete, and so there is a preferred rest frame in which the "lattice" is not squished along any axis. Is this a correct interpretation?

For someone in LQG, there is something very different about a world boosted so it is, relative to me, at the Planck scale, but this violates "physics is the same in every reference frame."

But I am definitely not following how string theory can break Lorentz invariance in general. Certainly there are various length scales introduced by string theory, but how are they different from the length scales introduced by, e.g., the electroweak theory?

So, I am curious -- does string theory break Lorentz invariance? (Of course, if you compactify -- it has to! But not necessarily in our 3+1 dimensional slice.)

Right. The idea about the preferred frame picked by the lattice - which breaks Lorentz invariance - is morally correct, even though the spin networks in loop quantum gravity don't look like regular uniform cubic lattices, for example. Nevertheless, they qualitatively look like that, and the conclusion that it picks a preferred frame and breaks Lorentz invariance is correct.

I think that you are also completely right that the Planck scale is just "another scale", much like the electroweak scale. It's just the highest energy scale where something new happens - above it, you start to produce (in high-energetic collissions) larger and larger black holes.

You can break the Lorentz symmetry spontaneously - e.g. if you have a nonzero gauge field or B-field - in string theory, but it is always spontaneous breaking. Normally, string theory preserves exact Lorentz symmetry. This is definitely a good thing, not a bad one! Did you want to say that you WANT to break Lorentz symmetry? Why do you want it? Experiments have shown no Lorentz symmetry violation in the physical laws, and it is a beautiful and constraining principle of physics. It's definitely a big problem for any theory if it can't explain that the Lorentz symmetry holds exactly - or very nearly exactly.

Hee hee, no I don't want string theory to break Lorentz invariance (although I am very interested in the possibility of spontaneous Lorentz violation, so if you do know of any review papers that cover these stories, I would love to hear of them!)

Now, one quibble -- if the vacuum has non-zero energy density, isn't LI broken? Then everything has to be done in [anti] de Sitter space, which *does* have a characteristic length scale (the horizon.) I am way out of my depth here, but given this, it seems like there *would* be a preferred reference frame -- since the horizon has a temperature, there would be a particular frame in which the temperature was uniform over the whole sky.

In other words, the horizon would define a CMB frame. Now, OK, maybe there is a way to see this as a spontaneous breaking of Lorentz invarience. Also, of course, on small scales, LI is preserved -- and it doesn't make sense to talk of Lorentz invariance except on a tiny patch of space time. On small scales all you see is the energy density, and that's a constant scalar and invariant under boosts, rotation, etc.

Anyway, perhaps you can see some easy way out of this confusion. If I want to talk about LI, should I only stick to a tiny flat patch of space?

Yes, the cosmological constant does break the global Lorentz invariance.

However, the Lorentz invariance still works at very short distance - shorter than the horizon radius - and moreover, even globally, the Lorentz invariance is replaced by de Sitter or anti de Sitter symmetry which is as big as the Lorentz invariance times translations (this product is called the Poincare symmetry).

So the amount of symmetry of empty space is not really changed by the cosmological constant. This is why the flat space, de Sitter space, anti de Sitter space are called "maximally symmetric spaces".

CMB is the cosmic microwave background, and it does pick a really priviliged frame, even locally. But this is a spontaneous symmetry breaking - breaking by the actual state of the Universe; a type of breaking that can be "unbroken" if you remove the breaking object.