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Nakanishi proves 2-d string theories inconsistent(?)

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Jan30-08, 08:49 AM
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From Peter Woit's "Not Even Wrong" blog
some guy named Nakanishi claims to have proven that 2-d string theories are inconsistent:

Eric Says:

The main reason for string theory being considered the ‘leading’ (really, the only) contender for a theory of everything is that it is presently the only known way to consistently combine gravity with quantum mechanics. String theory only works because of number of nearly miraculous anomaly cancellations. It is because of this that it is studied, despite the fact that it has not yet been possible to make definitive experimental predictions.
# N. Nakanishi Says:
January 29th, 2008 at 10:59 pm

I believe the anomaly cancellation in superstring is a meaningful condition only if the corresponding QFT has gravitational anomaly. The existence of gravitational anomaly in QFT was claimed by Alvarez-Gaume and Witten (Nucl. Phys. B 234 (1984) 269), but their reasoning contained a serious mistake: They were not aware of the fundamental difference between T-product quantities and T*-product ones. Both coincide for chiral current but not for energy-momentum tensor, because the expression for the latter contains time differentiation. The genuine anomaly must be considered for T-product quantities, but what they considered are T*-product ones,
because only T*-product quantities can be calculated by Feynman integrals and path integrals. I have explicitly shown in the 2dimensional case that what they called gravitational anomaly arises from the difference between T-product and T*-product. Thus,
at least in the 2-dimensional case, the gravitational anomaly in the genuine sense is non-existent in QFT. It is quite likely that the same is true in the 10-dimensional case.
B. Schroer completely agreed with me.
For details, see Abe and Nakanishi, Prog. Theor. Phys. 115 (2006) 1151 or arXiv hep-th/0503172 v2.
Back to the drawing board?
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Jan30-08, 10:18 AM
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Jan30-08, 11:11 AM
P: 3
Quote Quote by sluser View Post
Back to the drawing board?
Lubos Motl replied:

Dear metamars, I don't know what all string theorists would say. I only know what sane string theorists say about it. They will say that what is written above is 100% rubbish.

First of all, a UV-complete QFT by definition doesn't contain gravity. Second of all, the existence of gravitational and gauge anomalies in anomalous effective field theories is an established fact well beyond any sensible doubts and the comments above don't exceed confused comments of a generic anti-talented student who should be failed in her QFT II or III course because this is where the folks should learn how to compute anomalies. It can be done in many a priori inequivalent ways and all of them agree.

The paper by Abe and Nakanishi has all the 1-loop contributions to everything wrong - because it uses an incorrect Hamiltonian formalism where the required gauge invariance of the physical states is largely neglected - so it is not surprising that it can "derive" wrong anomalies (it's just like neglecting FP ghosts or divergent 1-loop diagrams in general). If you can't follow what I am saying, note that after 2 years, the paper still has 0 citations. I wrote above that paper, very politely, when it appeared here: anomalies.html

See also my analysis of a day in hep-th where stupid papers happened to be contributed both by Nakanishi and Schroer: friday.html

Did you copy it from a crackpot forum? I guess so. On the other hand, it doesn't surprise me that Mr Schroer doesn't understand why anomalies exist in QFT either - much like hundreds of other basic things about quantum theory.

I don't know why you think that there's something wrong with 2D string theories and how is this surprising statement related to the silly quotations that you added.

Jan30-08, 04:20 PM
P: 587
Nakanishi proves 2-d string theories inconsistent(?)

But if this were true, it wouldn't demonstrate 2D string theory "inconsistent", would it? It would only invalidate one proof of anomoly-free-ness.

Very interesting either way, I would love to see some explanation of what this means and some people weighing in on whether Nakanishi's argument is valid. For example, why were Witten et al "not aware of the fundamental difference between T-product quantities and T*-product ones" as Nakanishi claims? Was this divergence between the two products something which was discovered only more recently? Heck, what is a T-product?

Unfortunately I do have to say I have a lot of trouble taking anything Lubos Motl says seriously at face value.

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