Before I say what it is, let be introduce a bit of the background. We are in the context of theories with gauge invariances which are as everybody knows redundancies in the degrees of freedom. One might think that in quantising the theory one should directly work with the gauge invariant observables but this is often not the case since the description with gauge invariances often has ma much simpler structure, e.g. the space of connections is affine, as discussed elsewhere.

The price one has to pay are is the gauge invariance one drags around and which one has to mod out after the quantization. This can turn out to be impossible and for chiral theories it generically is. Of course I have just described the fact that the theory is anomalous.

Let me discuss this in a concrete example, the bosonic string in which the Virasoro algebra plays the role of the gauge invariance. In this example, the modes of everything are labeled by integers rather than by continuous variables (as for example for the axial anomaly) so there are fewer pitfalls from integrals etc one has to avoid. Plus we have discussed this case in detail in our paper so I don't have to repeat myself too much (for this discussion ignore all the parts on polymer states and the LQG way of doing things, just focus on the mathematical description of what one usually does (Fock space, Gupta Bleuler etc)).

The task in quantization is to turn the classical algebra of observables (functions on phase space) into a quantum algebra with representations on Hilbert spaces. The problem is that the classical algebra is a Poisson algebra with two multiplicative structures, the usual (pointwise) product of functions and the Poisson bracket. Both are supposed to map into in single product in the quantum algebra such that the Poisson bracket becomes the commutator for that product. Already in quantum mechanics of a single degree of freedom you know that this does not work exactly but only "up to higher order h-bar terms".

What does this mean in practice? The usual procedure is to take a subset of observables (typically coordinates of the phase space, or x and p and 1, or the field and its canonical momentum) which have simple Poisson brackets and which generate the classical algebra in terms of the pointwise product. Then one 'promotes' them to operators such that the Poisson bracket goes to commutator rule holds exactly. For all the other observables, one fixes a way of wring them in terms of the simple ones (aka one fixes an operator ordering prescription) and uses this and the product in the operator algebra to define their quantum versions.

Now, what about the gauge symmetry? In the classical theory, Noether's theorem tells us, that all symmetries are inner, that is, for each symmetry transformation, there is a function of phase space which generates it via Poisson brackets. Furthermore, the group relations for the transformations map to Poisson brackets of the generators.

In the quantum theory, you now have to deal with the gauge symmetry. There are two slightly different ways of saying what goes on: The first is quite abstract and is the one we used in the LQG string paper: You take your quantum algebra as above and have your symmetry act on it by an automorphism. Now, in a representation on a Hilbert space, you demand that this automorphism is implemented by unitary operators U(S) (for a gauge transformation S). There is no direct way to obtain these, educated guessing is probably best. The property you demand is that when A is in the algebra and p is the representation you have .

This implies that the U(S) nearly implement the group law: where is a phase. Of course, the above constistency condition for U(S) does not change if you change U(S) by a phase. The question is if you can find a consistent assignment of phases for all U(S) such that all the go away. If this is impossible, you have an anomaly.

In the other approach you use your knowledge of the classical symmetry generators and quantise them as all the other functions on phase space. Often as in the case of the bosonic string, they are quadratic in the basic fields which you quantised directly. This implies that the ordering ambiguity is just a complex number (imaginary for anti-hermitean generators). Again, the difficult step is to find an assignment of these such that the group law holds in terms of commutators.

If you don't succeed you could subtract the left hand side from the right hand side of your expression of the commutator and have the physical state condition that this anomaly (a complex non-zero number) annihilates physical states. This condition of course immediately empties your physical Hilbert space and you are left with nothing.

So the upshot of all this is: In this canonical quantisation approach, the way the anomaly manifests itself is in the inability to get the quantum symmetry algebra to work.

Now we come to the Master Constraint Trick: Assume, we write all our symmetry generators as C_i for i in some index set (let's not worry for a second that this will be infinite in the examples and thus one should worry about existence of the sums). Then form for some positive a_i.

As you can see, M annihilates a state exactly iff all C_i annihilate the state. Thus this constraint contains all the other constraints! Even better, as we only have one constraint, the algebra is trivial and for obvious reasons it also holds in the quantised version.

One is of course not yet done as again the kernel of M could be empty and thus the spectrum of this positive operator could be bound away from zero. But Giesel and Thiemann instruct as what to do:

This can be cured by subtracting from the Master Constraint the minimum of the spectrum provided of course that it is finite and vanishes as so that the modified constrain still has the same classical limit as the original one. One then defines the physical Hilbert space as the (generalised) kernel of the Master Constraint,...

Great, now we finally know how to get rid of these stupid anomalies!

## 5 comments:

Robert wrote:

"[...] M annihilates a state exactly iff all C_i annihilate the state. [...]"

I am guessing you want to demand that M annihilates states not for _some_ a_i but for _all_ a_i. Otherwise the implication does not follow.

So there is not just one constraint M instead of n of them, but an n-parameter family of operators M(a_i).

No? Maybe I should go and read the paper...

Sorry, I take that back. I missed that the C_i enter as squares into M.

Reminds me a bit of a quip which I've heard attributed to Feynman. You could ask: is there "one" equation of the universe or "many"? Feynman remarked that since all equations can be written as E_i=0, then the equation of the universe could be expressed as \sum_i E_i^2=0, i.e as just one equation.

A completely trivial observation, which I made in math-ph/0603024, is this: the class of gauge transformation matters. The existence of anomalies depends on whether you consider polynomials or Laurent polynomials, or more generally on whether your algebra admits a one-sided or two-sided grading (or filtration).

If you have a two-sided grading with non-zero global charge generators sitting in the middle, neither of the nilpotent subalgebras can be represented trivially. If you consider a one-sided grading, OTOH, you can combine non-zero charge with local gauge transformations being trivial.

E.g., the charge corresponding to conformal symmetry is the anomalous dimension L_0. You can combine L_0 = h != 0 with all L_m with m > 0 acting trivially, but only if you restrict attention to m >= -1. If you consider the Laurent polynomial version of the Virasoro algebra, a non-zero charge L_0 is incompatible with conformal invariance.

The same is true for Yang-Mills theory: with Laurent polynomials (in r rather than z), non-zero charge implies that no gauge transformations act trivially and thus that there are anomalies.

So why are Laurent polynomials forbidden in Yang-Mills theory but allowed in string theory?

07 28 06

Good post. I am always suspicious when I see things like: Master Constraint etc. Nice blog. I just finished reading the seminal paper you and Policastro did a couple of years back on the Fock v. LQG quantization of the string. I really liked the way you explained the singular nature of the state he chose and why that led to a non separable Hilbert space, and non definable momenta etc. Pardon my lateness, I have just began to seriously read LQG and some string theory. Just figured I would stop by to say thanks:)

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