### 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:

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

iscentrally-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 subtleto 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 youcouldblindly copy over the classical symmetries to the quantum theory, you’d reach absurd conclusions.Thiemann: Umh … Look, a plane!

## 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?