tisdag 27 september 2016

What's Missing in Quantum Gravity IV: Objections

Today I sum up some of the more obvious objections to the arguments presented in the previous posts, and explain why these objections are not relevant.

1. The diffeomorphism algebra in $d$ dimensions has no central extension at all when $d>1$

This is correct, and would appear to be fatal blow to the existence of a multidimensional Virasoro algebra since the the ordinary Virasoro algebra is precisely the central extension of the diffeomorphism algebra on the circle. However, we saw in the previous post that the Virasoro extension in higher dimensions is not central, i.e. it does not commute with all diffeomorphisms. The Virasoro algebra in $d$ dimensions is essentially an extension of the diffeomorphism algebra by its module of closed $(d-1)$-forms. When $d=1$, a closed zero-form is a constant function, and the extension is a constant. In higher dimensions, a closed $(d-1)$-form does not commute with diffeomorphisms, but we still have a well-defined Lie algebra extension, not just a central one.

2. In QFT, there are no diff anomalies at all in four dimensions

Again this statement is correct and apparently fatal, because an extension of the diffeomorphism algebra is a diff anomaly by definition. However, the caveat is the phrase "in QFT". Virasoro-like extensions of the diffeomorphism algebra in four dimensions certainly exist, but they cannot arise within the framework of QFT. The reason is simple: as we saw in the previous post, the extension is a functional of the observer's trajectory $q^\mu(t)$, and since the observer does not explicitly appear in QFT, such a functional can not be written down within that framework. To formulate the relevant anomalies, a more general framework which explicitly involves the observer is needed.

3. Diff anomalies are gauge anomalies which are always inconsistent

In contrast to the first two objections, this statement is blatantly wrong. Counterexample: the free subcritical string, which according to the no-ghost theorem can be consistently quantized despite its conformal gauge anomaly. Of course, this does not mean that every kind of gauge anomaly can be rendered consistent; the free supercritical string and the interacting subcritical string are examples where the conformal anomaly leads to negative-norm states in the physical spectrum and hence to inconsistency. But the crucial condition is unitarity rather than triviality.

A necessary condition for a gauge anomaly to be consistent is clearly that the algebra of anomalous gauge transformation possesses non-trivial unitary representations. The Mickelsson-Faddeev (MF) algebra, which describes gauge anomalies in Yang-Mills theory, fails this criterion. It was shown by Pickrell a long time ago that the MF algebra has no "nice" non-trivial unitary representations at all. Therefore, Nature must abhor this kind of gauge anomaly, which of course is in agreement with experiments; gauge anomalies in the Standard Model do cancel. But this argument has no bearing on the situation where the algebra of gauge transformations does possess "nice" unitary representations.

4. Gauge symmetries are redundancies of the description rather than proper symmetries

This is only true classically, and quantum-mechanically in the absense of a gauge anomaly. A gauge anomaly converts a classical gauge symmetry into a ordinary quantum symmetry, which acts on the Hilbert space rather than reducing it. As an example we consider again the free string in $d$ dimensions. Classically, the physical dofs are the $d-2$ transverse modes, and this remains true in the critical case. In the subcritical case, however, there are $d-1$ physical dofs, because in addition to the transverse modes the longitudinal mode has become physical; time-like vibrations remain unphysical.

5. There are no local observables in Quantum Gravity

This is essentially the same objection as the previous one, and it assumes that there are no diff anomalies. There can be no local observables in a theory with proper diffeomorphism symmetry because the centerless Virasoro algebra has no nontrivial unitary representations. If the diffeomorphism algebra acquires a nontrivial extension upon quantization, there is no reason why local observables should not exist.

Note that the same argument applies to Conformal Field Theory (CFT). There are no local observables in a theory with infinite conformal symmetry, but that is not a problem in CFT because the relevant symmetry is not infinite conformal but Virasoro; the central charge makes a difference.

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