I am not talking about conventional anomalies proportional to the third Casimir, which indeed are inconsistent – the anomalous algebra does not possess any unitary lowest-weight reps. If you introduce the observer’s trajectory and quantize it together with the fields, there are also new anomalies proportional to the second Casimir. This is necessary to do canonical quantization in a manifestly covariant way, see hep-th/0501043.

It should be possible to describe these new anomalies also in the conventional, non-covariant Hamiltonian formalism, although I have not thought so much about it. The YM gauge algebra can be cast in the form

[J^a(m_0,m_i),J^b(n_0,n_i)]
= f^abc J^c(m_0+n_0,m_i+n_i)
+ k \delta^ab m_0 \delta(m_0+n_0)\delta(m_i+n_i),

where four-momentum m = (m_0,m_i) has been split into temporal and spatial components. The extension can be expressed covariantly, but this form is suitable to make my point. Note that this is a 4D generalization of the affine algebra, and it is easy to show that it is indeed a Lie algebra.

The spatial subalgebra, generated by J^a(0,m_i), is anomaly free. This means that you can construct the Hilbert space as usual, and mod out spatial gauges. However, a rarely observed fact is that the temporal gauges are implemented as time-dependent canonical transformations. If the extension is non-zero, you will run into serious trouble with this.

This does not happen for the free Maxwell field, because the adjoint rep of U(1) is trivial. But it does happen in interacting theories, where the second Casimir k != 0. I do not understand this in detail, because I have only quantized interacting theories in a formal sense, but it must be related to renormalization.

Anyway, the second-Casimir extension is simply there, and it always arises when you build lowest-energy reps of the gauge algebra, see math-ph/9810003.

Now I think about, they missed the opportunity to stab the muon as a susy partner of the pion.

I did not say nor imply that anomalous local gauge theories would
be consistent, rather that you’d be warmly invited to try to prove this

In fact, it is standard knowledge that it is not possible; unitarity
will be lost (and renormalizabilty: the anomalous graphs scale not
in the way the non-anomalous ones do, and there is no renormalization
scheme where you could cancel the divergences of both types of graphs
simultaneously).

If nature is any guidance: the spectrum of the standard model is
supposed to be anomaly free, and this had in the past led to the
prediction that given the tau lepton there should be a bottom quark
(and similarly a top quark given the tau neutrino). This prediction
has been experimentally verified later, and one may view this as a
spectacular triumph where theory made a prediction based on
consistency, ahead of experimental data.

Summa summarum, I don’t see why one would insist on abandonding
gauge invariance and giving up consistency for no good reasons.

As for gravity, things may be more subtle and it looks indeed that
background (in-)dependence may have something deep to do with
anomalies. In topological strings there is a beautiful story relating
certain anomalies to an apparent background dependence, and this
could be a prototype for something more general; there are some
recent, extremely interesting papers on that. However, the involved
“holomorphic” anomalies are not crucial for the consistency of the
theory, so there is no parallel to local gauge symmetry; rather, it is
more the other way around, namely the holomorphic anomalies arise
_because_ one insists on a consistent geometric interpretation of
the theory.

I thought we agreed that gauge anomalies cannot in general be dismissed on the grounds of unitarity violation – the subcritical free string. The chiral Schwinger model in 2D is another example, at least according to Roman Jackiw. As for 4D gravity, here is what I have and have not done.

I don’t see how to use path integrals, so I prefer a version of canonical quantization which is more directly connected to representation theory. But any correct quantization method should do.

Moreover, I don’t know how to prove unitarity, because I don’t know what the invariant inner product is. But I do know that every non-trivial, unitary rep of the diffeomorphism algebra must be anomalous (in 1D, c=0 implies h=0), and I know how construct anomalous reps. So at least I satisfy a necessary condition.

The problem, both for path integrals and unitarity, is this: In order to avoid infinities, I must first expand all fields in a Taylor series around the trajectory q(t), and truncate at some finite order p. This gives me a classical, non-linear realization on finitely many fields of a single variable, which is exactly when the normal-ordering prescription works. Without this step, normal ordering gives infinities and diffeomorphisms do not act in a meaningful way.

Thus everything is expressed in terms of Taylor data instead of field data. Classically, this is nothing, but I don’t see how to make sense of a path integral over Taylor data. And although one can readily write down inner products, I haven’t found an invariant one. Another problem is that truncation to order p is a regularization, which must be removed at the end. Infinities resurface in this limit, and can only be cancelled with a clever choice of field content.

I don’t claim to have quantized gravity. However, I have quantized (on a linear space rather than a Hilbert space) a regularized form of gravity, while maintaining manifest diffeomorphism covariance, constructed the relevant anomalies, and derived conditions when the regularization can be removed. I think that that is a rather significant achievement, in particular since there is no abundance of good new ideas around.

I contacted with Eotvos group and are waiting for reply.

I cannot “rebate” your arguments now, since that i discover this posible weakness of Newtonian potential today.

However, let me say that like in some well-known LED models (e.g. mm-scale string extradimensions), perhaps the reduction of dimensionality could be undetected with usual non-gravitational thecniques being real.

the issue of background independence in QG is a subtle one and I
prefer not to get drawn into this. But as far as local gauge anomalies
are concerned, I wouldn’t see how an anomalous theory would make
sense from a path integral point of view. And I would expect important
basic properties like unitary getting violated – so in order to be
convincing, why don’t you cook up a proof that anomalous gauge
theories are unitary, that would help your case !

Not to defend the string M theory, but it should be clear that the “recent experimental verification” of weaker gravity force at shorter distance is very weak in credibility, and does not say anything either way. Further, it has nothing to do with reduction of dimentionality, even if the reduction of force is credible and verifiable. Clearly the distance scale at which the allerged force reduction happens is well above atomic scale, and we have plenty of solid evidence that at atomic scale everything is just as 4-D as the macroscopic scale. Should dimentions start to reduce, which could be possible, it should start at a much much smaller scale.

Quantoken

Basically the idea is as follow if recent sugestion of experimental verification of weakenes of Newton force to short scales is correct. This would be a final knock to String M theory.

Ignoring possible dependence on relative velocity, one obtains strong effective gravitational interaction to shorter distances, I take like good the rule 1/r^(2+d) for d extra dimensions (some recent RS brane model introduces Yukawa like exponential correction from extra 5th dimension), we can observe that smooth behavior is obtained formally with

d 0 imply formally elimination of divergencies on (1/r^2) force strengh since (1/r^2) —-> (1/r^0) at short scales without appeal to an arbitrary (by hand) add cut-off.

Are not these exciting news?

Although people know that subcritical strings are fine, they still seem to believe that gauge anomalies are inconsistent, and that gauge symmetries are redundancies of the description. Lubos has repeated that phrase for five years. Moreover, in a discussion a long time ago, Jacques Distler implied that there is a fundamental difference between conformal gauge symmetries, relevant in string theory, and conformal global symmetries, relevant to 2D critical phenomena. Why would he do that if he realized that gauge symmetries can become global upon quantization?

So one thing I want to do is to eliminate the widespread myth that all gauge anomalies are inconsistent and must be cancelled.

This issue comes up in the context of diffeomorphism symmetry in QG. It was always obvious to me that the diffeomorphism group will acquire anomalies in 4D QG, which is pretty obvious since already 2D QG has gauge anomalies. (The anomaly can be traded between the Weyl and diff sectors, but it can not be removed.) With this is mind, I generalized the Virasoro algebra to higher dimensions (in particular 4D), and worked out its Fock representations. Fortunately, the same problem was simultaneously addressed by mathematicians Rao and Moody (of Kac-Moody fame), which helped me overcome the crucial obstacles.

In case you think that there are no pure gravitational anomalies in 4D, it is only true if you quantize the fields alone. A crucial insight is that one must also explicitly specify where observation takes place, and quantize the observer’s trajectory together with the fields. This is mandatory because the relevant Virasoro-like cocycles are functionals of this trajectory.

One can view the controversy between ST and LQG in the light of this result. The key lesson of GR is background independence, and the key lesson of QM is that it is QM, in the Fock sense. However, Lee Smolin has informed me that a rigorous theorem rules out anomaly-free Fock quantization of background-independent theories, and I see no reason to doubt that assertion, partly because I have proven similar (but very non-rigorous) theorems myself. Locally, this leaves three possibilities:

1. QG is not background independent. A lot of people would dislike this possibility, because it would violate the spirit of GR, but it is a logical possibility.

2. One should not quantize in the Fock sense, but only in the weaker LQG sense. A lot of people would certainly dislike this, especially the part of the unbounded harmonic oscillator spectrum.

3. The diffeomorphism symmetry is anomalous. This neither violates the spirit of QM nor GR, it is known to happen in 2D, and much of the math is now here. However, the anti-gauge-anomaly myth prevents this idea from being taken seriously.

that strings with c less than 26 are fine and make sense is known since a long time, there are hundreds of papers (I guessimate) on this issue called non-critical strings. This includes the well-investigated c=1 model, etc. What is what you want to convey – that professional physicists wouldn’t know about this ?

As for desinformation, it is one of the unfortunate virtues the internet has brought to us, namely that laymen can just go ahead and spread nonsense, and other laymen are sadly influenced by this as they have no way to distinguish crap from serious science. This is what I meant when I was referring Q’s statements, in relation to someone expressing appreciation for them.