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February 7, 2004

Three Card Monte

I’ve been pondering why I find the discussion of Thomas Thiemann’s recent paper over at the String Coffee Table so disturbing.

Finally, Thiemann’s latest comment made it all fall into place for me (emphasis added):

In any case, whether or not you should have an exact or projective rep. of a symmetry depends on the physical system under study and hence must be decided ultimately by experiment. In case of the string everything is allowed until we have quantum gravity experiments.

For the love of God, no! In the absence of direct experimental tests, we have to be all the more careful to ensure that what we do is mathematically self-consistent. Thiemann’s “anything goes” attitude is exactly the sort of thing that string theory sceptics warned us against.

In case you haven’t been following, here’s the executive summary:

Thiemann: I can quantize the bosonic string with no Virasoro central extension (and hence no restriction on the critical dimension) and no tachyon.

Us: No you can’t. The Virasoro central extension follows directly from the canonical commutation relation and the necessity of regularizing the Virasoro generators.

Thiemann: Your proof doesn’t apply to my quantization because it makes unwarranted assumptions about the Hilbert space, and the Hermiticity properties of the operators.

Us: No such assumptions were made. The proof relies only on the canonical commutation relations and the necessity of regularizing the Virasoro generators.

Thiemann: OK, so maybe the algebra of constraints is centrally-extended. But I wish to work with the exponentiated constraints (with the group, rather than the Lie algebra). I can just copy those over verbatim from the classical theory.

Us: The group elements are even more subtle to regularize, You are only shifting the problem to where we, hopefully, won’t be able to see it. Besides, if you take group elements infinitesimally close to the identity, you will just reproduce the Lie algebra computation. Moreover, if you could blindly copy over the classical symmetries to the quantum theory, you’d reach absurd conclusions.

Thiemann: Umh … Look, a plane!

Posted by distler at February 7, 2004 10:05 PM

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Re: Three Card Monte

Well, this is all highly amusing, and shows that teaching and learning string theory properly is a rather nontrivial activity.

Unfortunately (despite bearing the name of string phenomenologist) I haven’t had the chance to do much of it myself.

In fact no-one even got round to teaching me the operator product expansion. What’s a good textbook for OPE?

Someone in that thread said something like “If a quantum theory Q is supposed to be the quantization of a classical theory C then they should have the same symmetries”. Isn’t this exactly what the argument is about?

Posted by: Thomas Dent on February 11, 2004 7:49 AM | Permalink | Reply to this


In fact no-one even got round to teaching me the operator product expansion. What’s a good textbook for OPE?

You mean the OPE in Quantum Field Theory? I would suggest the Les Houches 1975 lectures (especially, the ones by Gross).

The OPE in 2D Conformal Field Theory is just a special case. You might want to look at di Francesco et al’s book, or Ginsparg’s “Applied Conformal Field Theory” Les Houches lectures.

“If a quantum theory Q is supposed to be the quantization of a classical theory C then they should have the same symmetries”. Isn’t this exactly what the argument is about?

Isn’t the first thing you learn about Quantum Field Theory that that is not true? I gave the example of broken scale-invariance in Yang-Mills theory but you can, surely, come up with many, many more examples.

Posted by: Jacques Distler on February 11, 2004 8:20 AM | Permalink | Reply to this

Re: Education

Correct me if I am wrong.
But, isn’t preserving classical symmetry the point of critical dimension in string theory?
I never understood the moral behind the argument. People like Emil Mottola argues that anomalies of QFT should be treated as a real physical thing, but string theorist want them to go away.


Posted by: Demian Cho on February 12, 2004 4:55 PM | Permalink | Reply to this

Old Covariant Quantization

OK, let me explain the way the “Old Covariant Quantization” of the bosonic string works.

The algebra of contraints receives a central extension, but this proves not to be an obstruction to imposing the constraints weakly:

(1)L n|phys = 0,n>0 phys|L n = 0,n<0 (L 0a)|phys = phys|(L 0a)=0\array{\arrayopts{\colalign{right center left} \equalcols{false}} L_n\, | \text{phys}\rangle&=& 0, \quad n\gt 0\\ \langle \text{phys} |\, L_n&=& 0, \quad n\lt 0\\ (L_0-a)\, | \text{phys}\rangle&=&\langle \text{phys} |\, (L_0-a) = 0}

That procedure clearly works for any dimension. So where does d=26d=26 come from? Well, one finds that for d>26d\gt 26, the physical Hilbert space (the subspace on which the above equations hold) still contains negative norm states. So we reject d>26d\gt 26 as leading to a nonsensical theory.

For d26d\leq 26, all the negative norm states are projected out. However, there are still states of zero norm (“null states”). We would like to quotient out by the null states, and have the induced inner product on phys/ null\mathcal{H}_{\text{phys}}/\mathcal{H}_{\text{null}} be positive-definite. This fails for d<26d\lt 26. Miraculously, for d=26d=26 (and a=1a=1), an extra set of null states appear (think of these as states which are positive norm for d<26d\lt 26 and negative-norm for d>26d\gt 26). Modding out by this larger space of null states, we find a positive-definite inner product on phys/ null\mathcal{H}_{\text{phys}}/\mathcal{H}_{\text{null}}.

The “Modern Covariant Quantization” (which you can find, say, in Polchinski) takes a different approach. It enlarges the Hilbert space to include the ghosts associated to the conformal gauge-fixing. On this larger space, one constructs a BRST operator, which is nilpotent only for d=26d=26. The space of states annihilated by QQ (at ghost number 1) is isomorphic to phys\mathcal{H}_{\text{phys}} found above, and the space of states (at this ghost number) which are QQ-exact are isomorphic to the null states found above.

The BRST cohomology at this ghost number is thus isomorphic to the desired positive-definite Hilbert space. Moreover, one proves that there is no cohomology at other ghost numbers.

This “Modern” approach is the one you want to think about in terms of “canceling” the Virasoro anomaly. The nilpotency of QQ is basically a consequence of the fact the c ghost=26c_{\text{ghost}}=-26, while c matter=dc_{\text{matter}}=d, so c total=0c_{\text{total}}=0 for d=26d=26.

Posted by: Jacques Distler on February 12, 2004 5:27 PM | Permalink | Reply to this

Re: Education

I think that the strange aspect of Thomas Thiemann’s quantization can be seen in an even simpler example than the bosonic string, where not even an anomaly appears and his quantization still yields something strange:

Just consider the Nambu-Goto action not in 1+1 but in 1+0 dimensions, i.e. the relativistic particle. This theory has just a single constraint, which, when quantized, is the Klein-Gordon operator. Exponentiating this guy, classically or quantumly, gives us a symmetry group isomorphic to the reals.

The standard Dirac quantization of this theory yields the familiar Klein-Gordon equation.

According to the ‘LQG-string’ quantization prescription, however, quantization consists of finding any representation of the classical symmetry group on some Hilbert space. As Thomas Thiemann has confirmed upon request, this implies a large ambiguity in choosing such representations. For instance this prescription would allow me to quantize the free relativistic particle by choosing the Hilbert space L 2(R)L^2(R) and use the translation operator U(a)=exp(ia x)U(a) = \exp({-i a\partial_x}) as the symmetry generator. Or should I rather use U(a)=exp(iaX)U(a) = \exp(i a X)? Or any other 1-parameter family of unitary operators? All of this gives ‘quantizations’ of the KG particle which have nothing to do with the real thing.

Posted by: Urs Schreiber on February 14, 2004 7:45 AM | Permalink | Reply to this

Pet Tricks

But, at least in this (quantum mechanics) example, the correct quantization exists among the multitude of alternatives allowed by Thiemann’s procedure.

Not so in (any?) quantum field theoretic examples, and certainly not for the Nambu-Goto string.

Posted by: Jacques Distler on February 14, 2004 8:24 AM | Permalink | Reply to this

Re: Pet Tricks

In fact, Ashtekar, Fairhurst,Willis is curretly studying Maxwell field. I’ve just talked to Stephen Fairhurst. He said that they almost finished the work and believe that they can recover all the “physical” predictions of usual QED, from their unconventional quantization. So, it seems to me that jury is still out there.

Posted by: Demian Cho on February 16, 2004 5:00 PM | Permalink | Reply to this


“Maxwell Theory,” or QED?

If the latter, then there’s a laundry list of things I’d be surprised if they got right.

Posted by: Jacques Distler on February 16, 2004 9:17 PM | Permalink | Reply to this

Re: QED?

I used “Maxwell’s Theory” to distinguish it from usual formulation of QED.
I personally don’t know neither of the theory well enough to have own judgement, but I think that Thiemann’s paper is interesting because it brings out the fundamental difference in two approachs.
In fact I am anxious to see whether LQG will survive this.

Posted by: Demian Cho on February 17, 2004 8:34 AM | Permalink | Reply to this


In the usual parlance, “Maxwell Theory” is the the theory of the quantized electromagnetic field (coupled to external charged sources). That’s a free field theory and if they couldn’t get free field theory right, I’d be really worried …

“QED” is the theory of quantized electrons coupled to the quantized electromagnetic field. That’s a much more nontrivial theory.

(“Nontrivial” in the colloquial, rather than the technical sense. Most experts believe it is “trivial” in the technical sense.)

Posted by: Jacques Distler on February 17, 2004 8:46 AM | Permalink | Reply to this

Re: Nomenclature

Jacques Distler wrote:

That’s a free field theory and if they couldn’t get free field theory right, I’d be really worried …

Apparently they don’t get the usual results for free field theory, at least not exactly. This is analyzed for instance in

M. Varadarajan, Photons from quantized electric flux representations, 2001

(We have once discussed this a little on s.p.r.)

That’s no surprise if the LQG-like quantization even of the nonrelativistic particle in 1d is different from the usual one.

Posted by: Urs Schreiber on February 17, 2004 9:23 AM | Permalink | Reply to this

Little steps

Life is short.

If — as appears to be the case — they can’t get free field theory right, why should one pay the slightest attention to what they have to say about interacting theories (especially the notoriously difficult interacting theory that is quantum gravity)?

Posted by: Jacques Distler on February 17, 2004 9:57 AM | Permalink | Reply to this

Re: Pet Tricks

I think it is no surprise that the essential aspects of the usual theory can be reproduced when proceeding as in

A. Ashtekar, S. Fairhurst, J. Willis, Quantum gravity, shadow states and quantum mechanics.

When one looks at the equation right above equation (IV.5) of that paper one sees that care is taken that the ordinary quantum algebra is reproduced. Imagine that this equation had instead been modeled after the classical algebra. Then the factor α 2/2-\alpha^2/2 that comes from the Baker-Campbell-Hausdorff rule would be absent and (IV.5) would not reproduce the standard result (IV.7).

If in the LQG-like quantization of the string one would similarly take care that the quantum algebra of the exponentiated Virasoro constraints is preserved (even in the absence of a representation of these constraints themselves), the anomaly would be present and everything would be as usual (except for the non-separability of the Hilbert space).

But the crucial point in Thomas Thiemann’s ‘quantization’ is that, in contrast to what is done on p. 14 of the above paper, the classical exponentiated algebra is represented by operators directly, not the usual quantum algebra. The same is true for the spatial diffeomorphism constraints in the LQG quantization of gravity, I think.

If you are in contact with Stephen Fairhurst and Joshua Willis I would very much enjoy hearing their comment on this assessment.

Posted by: Urs Schreiber on February 17, 2004 5:51 AM | Permalink | Reply to this

Re: Three Card Monte

I mean, I know what OPE is, but never got anywhere near a lecture course on the subject. It’s the 7-year programme they have in the UK - 4 years undergraduate and 3 years doctorate. OPE was a ‘special subject’ confined to places that had strong QCD or formal string interests. I never came across a summer school that paid any attention to it.

I hadn’t heard of Di Francesco et al. before. And it does help to have the advice of someone who knows which of the myriad sets of lecture notes are actually worth reading.

Posted by: Thomas Dent on February 12, 2004 6:44 AM | Permalink | Reply to this

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