Wednesday, August 30, 2006

Thomas Thiemann and the Master Constraint

The last hep-th paper tonight was written by Thomas Thiemann and it is called

The title is meant to be a parody of the modest title of the most cited 2005 paper on loop quantum gravity called

that was written by Nicolai, Peeters, and Zamaklar. We have discussed the paper on this blog. Thiemann repeats his bizarre opinion that Nicolai et al. are "outsiders" while he is an "insider" about five times in his paper. Needless to say, I think that his opinion is unrealistic because Nicolai et al. know what they're talking about, unlike Thiemann himself.

Let me try to explain. We will start with some details.

Spin foam fad

In the middle of page 21, Thiemann criticizes Nicolai et al. because they didn't spend enough time with the spin foam models. That's quite a cute criticism because Thiemann dedicates approximately one page, namely page 38, to these models. The section about them contains one displayed equation, namely an ill-defined path integral with the delta-functions (4.50). For comparison: Nicolai and Peeters dedicate 1/3 of their newer paper to the spin foam models.

Harmonic oscillators

One of Thiemann's many amusing constructions appears on page 45. Thiemann confirms that he, apparently with the rest of the loop quantum gravity community and perhaps also the algebraic quantum field theory community, disagrees with the rest of the physicists how to quantize the harmonic oscillator. Helling and Policastro have shown that if the loop quantum gravity methods are applied to the harmonic oscillator, we don't obtain the usual spectrum "(n+1/2) hbar.omega" that has been derived in a dozen of formalisms, tested experimentally, and that has become one of the main textbook results of quantum mechanics. Robert Helling adds some comments and links about this work in the slow comments.




Thiemann says that Helling and Policastro's construction is a technically correct loop quantum gravity calculation but their physical conclusion is "completely wrong". In order to show what this bizarre self-contradictory statement means, he offers a brand new quantization of the harmonic oscillator :-) with the strange sines of "epsilon.x" and sines of "epsilon.p" we've seen in Helling & Policastro but also with roughly three new ad hoc unjustifiable steps that go beyond Helling and Policastro.

Thiemann's result? What you have learned at school about the quantum harmonic oscillator is essentially correct but only up to a level "E_0" and measurement error "delta", while the ground state energy is zero instead of "hbar.omega/2". No kidding. ;-)

Thiemann's landscape confusion

On page 4, he "rationally" addresses the problems of loop quantum gravity by pointing out that the "infinite number of vacua, recently found by Douglas and Denef, show that string theory is not mathematically unique". Well, this statement is completely off-topic which shows how difficult it is for Thiemann to rationally focus on a well-defined question. Second of all, the number of classical solutions of a theory has nothing whatsoever to do with the mathematical uniqueness of the theory. Third, paradoxically, it was exactly Douglas who recently argued, in his paper with Acharya, that the number of the realistic vacua is finite.

The main point: how to cure all anomalies

There are hundreds of other absurdities about loop quantum gravity and its relations to other ideas in physics mentioned in the paper. Some of them have been discussed on this blog.

But an even more incredible part of the paper is what Thiemann considers to be an answer to one of the main criticisms by Nicolai et al., namely the fact that the constraint algebra of loop quantum gravity isn't closed off-shell which makes it impossible to impose the constraints at the quantum level. On page 35, eighth line from the bottom or so, Thiemann agrees that at the quantum level - as far as the corrections that are of higher order in hbar go - it is indeed true because there are anomalies in the algebra.

But he argues that it doesn't matter. Let me spell it again: gauge anomalies don't matter because of his Master Constraint program. Wow. So a reader returns back to page 9 to see how is the mysterious M(aster) Constraint program supposed to cure anomalies if the anomalies are really there. ;-)

Feynman's joke = Thiemann's work

You will have to use the following trick due to Feynman. Feynman once said - and wrote in Feynman's Lectures on Physics - that the theory of everything can be written in the following form:

  • M = 0.

Originally, Feynman used either "U" (for "Universality" or "Unworldliness" as Frank Wilczek cleverly calls it) or "A" instead of "M" but we will use "M" to make the letter closer to M-theory as well as to the notation of Thiemann's Master Constraint program. ;-) The only detail that Feynman needed in order to clarify his theory of everything was to explain how to define "M". He defined it like this:

  • M = (F - ma)^2 + integral (div D - rho)^2 + ...

You see, it is a quadratic, positively definite form involving all equations of motion we need. As you can see, "M=0" really means "F=ma", "div D = rho", and so forth. How smart. ;-)

Thiemann, apparently convinced that he is being original, is using the same "M" to impose not all the equations of motion but just the constraints. The constraints are normally a subset of those equations of motion that don't involve any time derivatives; however, in loop quantum gravity, the word stands for all equations of motion - in fact, especially those deduced from the Hamiltonian that generate the time evolution - and Thiemann's biggest discovery is thus exactly equivalent to Feynman's joke. Instead of the constraints

  • C1=0, C2=0, C3=0 ... ,

he simply requires that

  • M = 0

where

  • M = sum_{IJ} C_I K_{IJ} C_J

is constructed from a positively definite matrix "K_{IJ}" on the space of constraints that are visualized as functions on the classical phase space. Using "M" instead of "C_I", he thinks that he can perhaps solve all the problems of loop quantum gravity. What a great idea.

Note that the more general form of "M" can always be rewritten as a sum of squares using the procedure called the diagonalization of "M" - something that the pigs will never learn but the freshmen should. The content of the diagonal "M" and the general "M" is identical.

The difference between Feynman and Thiemann is that Feynman offered his universality function as a joke and he has also explained very well why it can never solve or simplify anything: the very point of his joke was to teach students to avoid vacuous concepts and dumb, unnecessary superconstructions. Thiemann, on the other hand, is dead serious when he says that it is an important step in physics that solves many of his problems. He believes that his idea makes all anomalies harmless.

He wants "M" to annihilate the physical states. The only problem that can happen, he believes, is that "0" can be absent from the spectrum of "M". In that case, he just postulates that the physical states must be eigenstates of "M" with the minimum eigenvalue, see page 11 (point 3). He won't tell you why it's not the second minimum eigenvalue - that can actually be zero - but he thinks it's a good idea. I mentioned the second minimum eigenvalue because it is the eigenvalue of "m^2" that actually gives us massless states in bosonic string theory that admit "p=0" modes. If Thiemann had ever tried to compute any one-loop effect, he would have known that the properly regulated ordering constants are often negative, in sharp contradiction with Thiemann's assumption in the middle of page 10 about the positive definiteness of the unshifted operator "M".

In order to see how incredibly childish Thiemann's trick is, it is useful to look at a particular theory with an anomaly in the algebra of constraints that is analogous to the anomaly pointed out by Nicolai et al. It doesn't really matter whether you take the Virasoro anomaly or a gauge anomaly in gauge theory or anything else. All of these anomalies are less severe than the gravitational ones - so Thiemann's trick should work even better in those familiar contexts - but the basic structure is universal.

Take the Virasoro algebra with a central extension. Instead of postulating that the generators "L_I" annihilate the physical states, you follow Thiemann's recommendation and construct something like

  • M = sum L_{-N} L_{N}

where the sum goes over integers. For free fields describing strings in the flat space, such an "M" is quartic in the oscillators. You can easily see that if you rewrite "M" in the continuum worldsheet variables, it will be an integral whose integrand has severe UV problems because the expression contains things like the integrated square of the delta-function. Thiemann is aware of this problem, as you can see on page 9, so he proposes a "better" kernel which, in my variables, has the form

  • M = sum L_{M} K_{MN} L_{N}

where the matrix "K_{MN}" is chosen in such a way that the UV problems are regulated away. Imagine that you deal with an anomalous Virasoro algebra in string theory. You define "M" by the formula above and require that the physical states are either annihilated by "M", or, if the vanishing eigenvalue is impossible, that are the eigenstates corresponding to the lowest eigenvalue.

Can you find the physical spectrum? For a non-anomalous algebra or algebra that is related to a non-anomalous one, you might actually be able to reproduce the usual physical spectrum in the old covariant quantization if you take "M" with the original, diagonal kernel regulated properly. It's because there exists a natural non-anomalous extension (with the b,c ghosts) of the whole system and the minimum eigenvalue of "M", if you do it right, will impose the original Virasoro constraints or at least one half of them as explained by Gupta and Bleuler.

However, it is not hard to convince yourself that if such an extension - an underlying theory with the exact symmetry - didn't exist, you couldn't have ever obtained any meaninful result. One of the problems would be that the precise structure of your physical Hilbert space would depend on the kinematical Hilbert space you started with. More importantly and more quantitatively, it would depend on the matrix "K_{IJ}".

That's of course a disaster because the role of "K_{IJ}", as Thiemann admits on page 9, is to act as a UV regulator or a cutoff. Because the nature of the physical Hilbert space is clearly going to depend on this kernel (how could eigenstates of an operator randomly constructed from a kernel be independent of this kernel? At least the quantum corrections to physics will surely be affected), we can say - using the normal language - that the resulting physical observables will be cutoff-dependent. We haven't constructed any meaningful theory that is independent of the regularization. The kernel "K_{IJ}" contains infinitely many undetermined parameters and each of them influences the resulting physics. Be sure that an "algebra" that doesn't close is not a "symmetry", and without a symmetry reason, how could the physical properties of "M" be independent on some kernel that explicitly enters the definition?

The situation is completely equivalent to a randomly picked prescription how to cut off or otherwise regulate UV divergent integrals, including the linearly divergent integrals that are responsible for anomalies. Of course that if you insert some cutoffs and functions that smooth the expressions out, you may get finite results. But these finite results will be, from a physics viewpoint, complete garbage because they will depend on the regulator which, moreover, will break the general consistency criteria such as unitarity and, of course, also Lorentz symmetry (that was broken in loop quantum gravity from the beginning). You haven't solved any problems whatsoever.

The dependence on "K_{IJ}" is one of several examples showing how unjustifiable are Thiemann's statements on the top of page 5 and elsewhere that he has eliminated the infinite ambiguities that were also pointed out by Nicolai et al., among others including Jacques Distler and myself.

If someone had proposed this obviously silly idea of Feynman's universality function in the world of high-energy physics, he or she would be instantly explained why the idea is silly and he or she would be encouraged to learn or re-learn some basic things about UV cutoffs and anomalies. In the world of loop quantum gravity in particular and discrete approaches to physics in general, however, there is no one who could explain Thiemann why he is being silly because, as far as I know, there is no one who understands the very basic things about the UV divergences and anomalies in quantum field theory.

There is no one who understands that if you impose a cutoff on a UV divergent, non-renormalizable, anomalous, or otherwise inconsistent theory, you remain very far from finding a finite physical theory. It is still an inconsistent theory with a cutoff. They keep on celebrating these worthless constructions with infinitely many undetermined parameters - constructions in which all real problems are just swept under the rug - as solutions to the key problems of physics.

More concretely, Thiemann, being an important representative of a "serious competitor" to string theory, as he modestly calls all this nonsense on the top of page 3, can sell this obviously unhelpful and silly idea as an important advance in physics. That's an example what happens if the society implicitly supports affirmative action for the proponents of bad ideas who were skipping their courses of quantum field theory and, as it became clear from Thiemann's reply to Helling and Policastro, also introductory lectures of quantum mechanics.

And that's the memo.

P.S.: I would like to believe that similar blog articles as well as preprints by Nicolai, Peeters, Zamaklar, Helling, Policastro, and others will ultimately lead complaining Lee Smolin to avoid similar untrue statements like his recent false assertions in the Wired magazine that the string theorists don't read the papers of others. Incidentally, William Dembski predicts that next time, Lee Smolin will start to create room for Intelligent Design. So far, the most important supporter of Lee's book is Evelyn Fox Keller, a professional feminist science-hater.

1 comment:

  1. I have a slightly more technical comment re the harmonic oscillator in my blog.

    An earlier post on the master constraint is here.

    Besides ads, let me mention why in addition to going to the smalles eigenvalues (which should go to 0 as hbar goes to 0 which is why it is selected) doing away with squared constraints is not such a bright idea:

    The nice thing about the constraints is that they generate gauge transformations via Poisson brackets or commutators. However C^2 acts on an operator A as [C^2,A]= C[C,A]+[C,A]C and thus this action vanishes on the contraint surface C=C^2=0! Thus the master constraint does not generate any transformations on the contraint surface.

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