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