## Monday, January 17, 2005

### Lee Smolin responds

Lee Smolin asked me to publish his reply to my criticism of his recent paper. Why not? :-)

Dear Lubos,

Thanks for giving me a chance to reply to your criticisms of hep-th/0501091.

First, your criticisms of LQG are off the mark, but I won't take the time to reply to them here as the main results of the paper employ only the very weak assumptions that the configuration space is a space of connections. As shown in the references cited, this is very general, applies to all known classical gravity theories, in all dimensions, with and without supersymmetry. A detailed example given in section VI is based on loop quantum gravity. But for the main results, all I need assume is that whatever quantum gravity is, it has an effective low energy description in terms of a theory of forms and connections, from which I can draw predictions to low order in hbar using standard semiclassical methods. This is the same assumption that your colleagues Gukov, Nietske and Vafa are making about topological string theory in their recent work.

The results of the paper are called "semiclassical" because they are based on the use of a wavefunctional which is an exponential of a Hamilton-Jacobi functional, S (eq. 4). This a common meaning of the word, because such wavefuctionals solve the quantum dynamical equations toleading order in hbar. As you say I do not do perturbation theory around a classical metric, instead I study the action of the operator for the full frame field, (eq. 3) acting on such a semiclassical state.

I then study how quantum field theory on that classical manifold emerges from the full quantum theory by the Born-Oppenheimer approximation. This, including eq. 3, is standard stuff, introduced into quantum gravity by Banks, Starobinsky, deWitt and others, and often used in semiclassical approaches to quantum cosmology. The only novelty is to work on the configuration space given by a connection rather than the spatial metric, but as this exists generically one can't object to it.

I find that there is a leading effect, to order square root of hbar, which hence dominates order hbar effects normally studied. This must be there, I argue, if a standard notion of time on the configuration space is to be related to a physically meaningful time coordinate on a spacetime, where that spacetime results by evolving a classical trajectory on configuration space, following gradients of S.

The result is that the metric becomes frequency dependent to order squareroot of hbar. This is a form of DSR, developed with Magueijo in gr-qc/0305055, Class.Quant.Grav. 21 (2004) 1725-1736. It is a standard notion that the parameters governing an effective quantum field theory become energy dependent. We are not the first to apply this to the spacetime metric, all we do is show that this can be described by a modification of Lorentz invariance. The predictions follow from this.

Regarding the issue of whether there could be a version of string theory with DSR symmetry, you guess no, and so do I. But can I suggest a challenge? We studied this question with Magueijo in hep-th/0401087. We studied only the question of whether a free string with deformed dispersion relations as in DSR could propagate consistently. We got an answer which surprised us, which is that we found evidence for the existence of such a theory. My guess is that it breaks down when one tries to include interactions or checks unitarity to one loop. I've been waiting to give this problem to a student, but I would guess its something you could settle in a few hours work.

As the relevant experiments may report within two years, it is good to get predictions on the table. This seems to be the only chance string theory has to make an up or down falsifiable prediction that can be tested in the near future.

A brief reply to some of the things you say about LQG can be made by noting that LQG does start with the assumption that the configuration space is a space of connections. Therefore any semiclassical approximation to LQG is going to satisfy the assumptions made here. Tofully connect these results with LQG, as rigorously defined by Ashtekar,Baez, Lewandowski, Thiemann and others, one would have to show that the semiclassical state (4) approximates an exact, physical state. WKB states can fail to be normalizable, one normally resolves this by constructing wavepackets. As we argue in hep-th/0309045 with Alexander and Malecki, this may work here, but it has yet to be shown in full generality and rigor.

Some criticisms of LQG, including yours, have the form that if everything is not done, and understood rigorously, nothing can be trusted. Critics of string theory use the same logic, indeed it can be used to attack cheaply any research program in progress. My stance has always been that science would progress faster if we forget what we are for and against, rise above ideology, and try to take what can be learned from the partial results of each program, and see if they give us new insights and new predictions concerning real experiments.

Thanks,
Lee

1. Dear Lee,

the proposal that the gravitational configuration space is a space of gauge connections is not a "weak assumption". It is a far-fetched speculation that, as of today, has not been supported by a single consistency check (and certainly not by a single observation or an experiment). On the contrary, there is growing evidence that this assumption is not correct, and the article below this one describes a paper that lists a lot of serious problems with this picture.

Vafa, Neitzke, Gukov, and Dijkgraaf (please look how Neitzke should be spelled - it may become an important name) don't claim that these theories based on gauge fields and form fields describe physical gravity in 4D. They don't say that the form fields (or even gauge fields) describe *any* theory of gravity, which is what you explicitly say. On the contrary, they list a very small set of topological gravity theories that are the only known solutions to the condition that they can be expressed as form fields, and unify them under the "topological M-theory".

Concerning your comments about the Born-Oppenheimer approximation. One can use various approximations in various calculations that are not under control. But it's not enough to "use" semiclassical approximation somewhere in the paper to argue that the paper is a "semiclassical prediction". A prediction can only be semiclassical if it is based *only* on the semiclassical approximation of physics, while your particular predictions are based on very nontrivial (and very unlikely) assumptions about the new physics that regularizes gravity at the Planck scale - which is certainly not a semiclassical regime. Once you express the metric using the gauge theoretical degrees of freedom, you implicitly make identifications on the configuration space which don't follow from quantum gravity itself. These identifications emerge as the "area quantization" which is clearly not a consequence of semiclassical gravity.

You also say that "It is a standard notion that the parameters governing an effective quantum field theory become energy dependent" when you defend the deformed dispersion relations. The only way how I am able to interpret this sentence is that you confuse the renormalization group and the Lorentz symmetry. It is certainly not a standard notion that the rest mass is energy-dependent.

Concerning the free string with modified dispersion relations, I think that one does not need "several hours" to show that a particular model like that is inconsistent. I will happily explain you why these models are inconsistent. Yes, it has to do with unitarity and one-loop physics, but such things can already be discussed in the language of "tree level physics".

Concerning the "predictions on the table", I am not sure whether you really believe your predictions. In my opinion, they're extremely far-fetched and unjustified, and the probability that they will be confirmed is 10^{-n} where n is a rather large number. A particular model is unlikely, but it can have particular predictions. If it fails, OK - there are other models. But as long as one does not know what kind of physics there is above the electroweak scale, it would be foolish to claim that we have "consensus predictions". There are many predictions what will happen, and most of them will be incorrect.

I don't know for sure where supersymmetry is broken and so on, and I am not going to pretend that I know these things. In my opinion, it is a very bad policy to pretend that we know something that we do not know. A scientist must always be ready to admit his or her ignorance, despite those - kind of annoying - calls for "falsifíable predictions".

My guess still is that you realize that your prediction can't be confirmed. Will you then admit that your theory that the gravity configuration space is the space of connections has been falsified? Unfortunately, there have already been too many wrong predictions of LQG, and their falsification never meant anything, so I suppose that this will be the case with the Lorentz violation, too.

Sincerely
Luboš

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4. Dear Lee,

I looked at hep-th/0401087. It's obvious that one-loop physics will fail. But one needs much less than five minutes to see it.

It's just the whole concept of that paper that is flawed. The paper wants to redefine the worldsheet action (incidentally, a very unusual form of it) by inserting two functions f,g at random places in the Lagrangian.

Which functions? The functions f,g that are functions of the "total energy P_0", which is an integral of p_0, a variable that is treated as an independent field, over the string.

Nothing like that can ever lead to a consistent string theory. This construction creates a non-local theory (dynamics at every point depends on all points that contribute to the total energy) that breaks not only the Weyl symmetry. It even breaks the diffeomorphism and the local Lorentz symmetry on the worldsheet: when one defines P_0 in that theory, it must be an integral over a particular, priviliged curve on the worldsheet. There are no priviliged curves on a Lorentz invariant worldsheet - different observers slice the worldsheet in different ways.

For very special choices of f,g you may obtain an equivalent description of a single usual string. But once you allow the interactions, the local physics can't be defined in terms of some integrals over the string - because there is nothing such as the "total P_0" on an interacting worldsheet. This is the whole reason why perturbative string theory is UV soft. The point where the strings first interact is not well-defined in string theory - which is why the UV problems are "smeared out". This is covered beautifully even in Brian's book and the corresponding TV show which I wholehartedly recommend to those who are interested in the heuristic explanation why string theory works in the UV.

If local physics depends on the center-of-mass degrees of freedom of the string, then one is back to particle physics and no extra consistent features of string theory work. String theory is only string theory because various observables such as the momentum are distributed over the string, and physics is local on the worldsheet. In string theory, a point on the worldsheet can't decide whether the interaction has already occur. In your theory, it would be possible.

I don't know what it means to verify your string theory at one-loop level. As far as I can tell, you don't seem to have any string theory. It can't be quantized, it does not have any of the necessary symmetries, it's not local, it's not diff invariant, and the conformal anomaly is a function of the zero modes, too - well, it's not a conformal theory so one should not be talking about the Virasoro algebra at all because it's not a symmetry.

As far as I see, what you have is a deformation of something that could otherwise look like the classical description of worldsheet dynamics of a string theory, a deformation by extra functions inserted at random places. I apologize, Lee, but this is really not how string theory works. String theory is an extremely constrained, unique structure. It's not a structure where "anything goes", and my feeling is - please accept my apologies once again - that you will have to study it more carefully to see why.

All the best
Luboš

P.S. Incidentally, I erased an off-topic posting that said nothing factual about the questions discussed in this article - but that was, on the contrary, focusing on personal things, and I will keep on deleting such comments here.

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6. Incidentally, it is extremely unlikely that "Quantoken" will be able to create a comment that will have the power to survive here. Quantoken, feel free to send your e-mails to Lee - I wish your mails will make him happy. And maybe he will even agree with you! :-) Nevertheless, your texts don't pass the tests to be kept on my blog.

7. LM said: In my opinion, it is a very bad policy to pretend that we know something that we do not know. A scientist must always be ready to admit his or her ignorance,