Thursday, March 24, 2005

False anomalies?

One of the main topic of the discussions at Harvard today is the new paper
by Mitsuo Abe and Noboru Nakanishi. Although they're Japanese, this article may be a good opportunity to thank Samsung Electronics for the 256 MB USB Flash Drive that I've finally obtained. ;-)

What is this paper about? The gentlemen claim that
  • Alvarez-Gaume and Witten were sloppy in their derivation of gravitational anomalies in 1984
  • the textbook of Superstring Theory by Green, Schwarz, Witten is also sloppy
  • the sloppiness is based on confusion between two types of a time-ordered product
  • these two time-orderings are "T" and "T*"; the former is the literal time-ordering in the Hamiltonian/operator formalism while the latter is the type of time-ordering that one naturally obtains from the covariant path-integral approach; "T" commutes with the time-derivatives while "T*" does not
  • as an example, the Japanese physicists argue that what Alvarez-Gaume and Witten considered to be a gravitational anomaly in two dimensions, is actually just the difference between the expressions containing the "T" and "T*" products
There are obvious reasons to be skeptical about the Japanese claims:
  • sociological ones: Alvarez-Gaume and Witten are careful (and not just careful) physicists

  • the path-integral evaluation of the anomalies is what has been used and tested for a long time, and may be skeptical about the conclusions based purely
  • the Feynmanian covariant approach is closely connected with important phenomena of particle physics understood in the past decades while the alternative is closer to axiomatic quantum field theory that is known to have led to many incorrect physical conclusions (the discussion below "Sidneyfest" where someone argued, using AQFT, that the implications of the Weinberg-Witten theorem don't exist is an example of the failures of AQFT)
There may exist reasons why one may tend to believe Abe and Nakanishi:
  • sociological ones: these physicists may be careful and picky, and their paper is newer
  • the operator formalism is something that many people still find more well-defined than the path integral approach
  • more precisely, AQFT is still considered to be "the" rigorous approach to quantum field theory, although its tools have not been refined well to deal easily with the novel phenomena in gauge theory, renormalization, and dualities, among other things
Abe and Nakanishi prefer the operator approach and the "T" product, and they criticize others for the fact that their formulae are really based on the "T*" product which is more natural in the path-integral context.

I have personally no experience whatsoever with the Lorentz-non-covariant, operator treatment of the anomalies which is why I prefer to trust Alvarez-Gaume before I look at the details. The anomalies result from the inability to define the integration measure of the path-integral in such a way that all the classical symmetries are preserved. In order to define the measure properly, one needs to gauge-fix and introduce the Faddeev-Popov ghosts etc., and it is mostly unknown to me how their loop effects realize the same tasks in the operator formalism (without Feynman's path integrals).

On the other hand, I find it plausible that there may exist a wrong attempt to define the measure that leads one to believe that there is an anomaly, although there exists a better way in which the anomaly is absent. There are also terminological issues: I would only use the word "anomaly" for a violation of a classical symmetry that can't be fixed by adding a local counterterm. One should also be careful about the local anomalies and global anomalies - anomalies in small or large transformations, respectively.

Comments welcome.


  1. The cited paper tells us that the result of the "equal time anomalous commutator" (5-7) is what would arise from what is claimed to be the correct computation (5-12), which is not anomalous (italics in the original).

    If you will forgive what is likely to be a dumb question, why if the two computations give the same result, why should it matter if it is "anomalous" or "not anomalous"? Shouldn't the physical conclusions that follow be the same?

  2. Hi Arun,

    I think Lubos answered your question in his article already. The Japanese authors claim that what Alvarez-Gaume and Witten compute is really the difference between two results, obtained using different time-ordering prescriptions, rather than an anomaly. So they show that they obtain the same result as Alvarez-Gaume, but that it does not permit the interpretation of being an anomaly.

    I do not wish to imply that I believe that the Japanese paper is correct. I do not yet have an opinion on that.

    Best wishes,

  3. Let me rephrase - are they saying the Virasoro anomaly doesn't show up or are they saying it shows up, but this particular computation (5-1 to 5-7) is wrong?

    It seems to me that Green, Schwarz, Witten (3.2.49) or this paper, 5.12, derived by hook or by crook, is all that is needed to evaluate the anomaly coefficient a in GSW 3.2.34
    [L_m,L_n]=(m-n)L_{m+n} + ( a m^3 + b m ) delta_{m+n}

  4. I read the paper and found that it is quite readable but that it contained only odd pages! I would have liked to see the lacking page about conformal anomaly.

    Matti Pitkanen

  5. Matti said: "I read the paper and found that it is quite readable but that it contained only odd pages! I would have liked to see the lacking page about conformal anomaly."

    That's because the paper has exactly 11 pages, and you live on a brane damaged world where all the even number of dimensions compactify so you only see the odd pages. And because of an unusual cosmetic string induced by the blackhole created at brokhaven, all the dimensions decompactify and therefore the rest of us see all 11 pages :-)


  6. That is an interesting choice of words that they made; totally confused... I wonder if there is a simmering conflict going on...

  7. Oops, their actual choice of words was "totally careless", but my comment still applies.