Hi selfAdjoint -
you wrote:
It seems to me that Thiemann is saying "Ignore everything in sections 1 through 5, ignore group averaging and all of that. Here in section 6.1 is what I am really doing." And indeed if we look at 6.1, it does seem to be independent of what has gone before.
It kind of looks this way, yes. The most recent discussion at the Coffee Table shows, though, that Thomas might, after all, have made the same mistake that I did in the beginning, namely assuming that there is a quantization of the Virasoro algebra without an anomaly.
What he does is take the Borel intervals on the circle (which he did remark in your discussion are orthogonal if they differ anywhere - as you pointed out to me earlier!). He smears them in a particular special way with functions fk and asserts that the "handed" smeared functions Yk close to a Poisson *-algebra.
Yes, that's totally uncontoversial. It is, after all, nothing but an exotic reformulation of the fact that the usual worldsheet oscillators form a Poisson algebra.
Then he introduces the Weyl elements W = exp(iYk), and invokes the Baker-Campbell-Hausdorff formula to get a value for their product and concludes from this that the W's for right handed and left handed Y's commute.
Here a certain problem is beginning to show, which, at least for me, is a general one in this paper: It is not clear what, at this point, is assumption, definition and derivation.
The problem is that the Ys themselves are not represented as operators on Thomas Thiemann's Hilbert space. So how can we apply BCH to them, if they are not even operators? Of course we know that the Ys could be easily represented on some Hilbert space and we could compute their commutator there and it is the one that Thomas is using in the exponent of the BCH formula. But that's no real help either, because on Hilbert spaces where the Ys are represented (such as the usual Fock Hilbert space) their exponentiations are not unambiguously defined, unless we specify some rule of normal ordering. This gives, in the usual treatment, rise to the peculiar conformal dimension of such exponentiated operators, that you can see for instance in equation (2.4.17) of Polchinski. Therefore, whichever way I try to look at Thomas' equation (6.7) as something derived from previous input it makes me feel uneasy. I can accept (6.7) as a definition of the algebra of the Ws, though. But, just as with the definition of the Us by fiat, this is, while mathematically consistent, not manifestly related to physics-as-we-know-it, I think.
He then deduces from the general intersection geometry of intervals on the circle that "a general element of A (that is, a Weyl element W) can be written as a finite, complex linear combination of elements of the form [...]
Ok, given the algebra of the Ws, somehow, this follows without doubt.
He now defines the gauge group to by two copies of the diffeomorphism group of the circle plus the Poincare group
This is the point that we have been discussing in some detail with Thomas over at the Coffee Table. This way of
defining the quantum gauge group means to simply copy the classical gauge group. That's mathematically possible, but not related to any standard quantization procedures. Jacques Distler has today given a
further example for why this procedure is usually unphysical.
is there any anomaly visible to you in this work? Is there any reson why the GNS will not work?
No, the anomaly is indeed not there in this approach. But the reason is that by definition Thiemann is using a rep of the classical symmetry group on his Hilbert space. This is not the usual quantization procedure. There is no standard quantum anomaly because there is also no standard quantization.
The GNS theorem will work fine for the algebra of the Ws. The problem is that it is not clear what this algebra has to do with the standard quantization of the system at hand.
I can see that you are trying hard to escape the conclusion that is beginning to force itself upon us. I very much appreciate it. In a way I am delighted that the LQG-string is doing exactly what Nicolai has intended it to do: To show in terms of a simple example what is really going on in LQG. As long as we are dealing with 3+1d nonperturbative quantum gravity nobodoy knows what to expect and hence criticism of new proposals is very difficult. But now we are dealing with a case where we know much better what to expect and it has been possible to spot a very crucial difference of the LQG quantization approach to the standard procdedure:
LQG does not attempt to canonically quantize all the first-class constraints.
Actually, this is hardly a suprprise because, as Jacques has kindly reminded me, the ADM constraints of gravity simply cannot, even in priciple, be canonically quantized. LQG apparently circumvents this by not representing the constraints themselves on some Hilbert space but instead representing the symmetry group generated classically by them (at least for the spatial diffeos).
But this means breaking with a fundamental principle of quantum mechanics and can, at best, be addressed as an
alternative quantization procedure. There are many people who are proposing alternatives to standard quantization, for various reasons. I am open-minded and willing to consider all alternatives to standard physics as potentially interesting. But one should be fully aware of what is standard physics and what is a radically new and speculative proposal.
In fact, I am currently thinking about asking Ashtekar, or someone similar, if it is really technically correct to say that LQG is about
canonical quantization.
and plan to take a look at it, but in the meantime I was wondering if there are in fact experimental results that REQUIRE anomalies for their explanation and what those experiments are? Basically, I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.
