Musings

OK, let me explain the way the “Old Covariant Quantization” of the bosonic string works.

The algebra of contraints receives a central extension, but this proves not to be an obstruction to imposing the constraints weakly:

That procedure clearly works for any dimension. So where does $d=26$ come from? Well, one finds that for $d\gt 26$, the physical Hilbert space (the subspace on which the above equations hold) still contains negative norm states. So we reject $d\gt 26$ as leading to a nonsensical theory.

For $d\leq 26$, all the negative norm states are projected out. However, there are still states of zero norm (“null states”). We would like to quotient out by the null states, and have the induced inner product on $\mathcal{H}_{\text{phys}}/\mathcal{H}_{\text{null}}$ be positive-definite. This fails for $d\lt 26$. Miraculously, for $d=26$ (and $a=1$), an extra set of null states appear (think of these as states which are positive norm for $d\lt 26$ and negative-norm for $d\gt 26$). Modding out by this larger space of null states, we find a positive-definite inner product on $\mathcal{H}_{\text{phys}}/\mathcal{H}_{\text{null}}$.

The “Modern Covariant Quantization” (which you can find, say, in Polchinski) takes a different approach. It enlarges the Hilbert space to include the ghosts associated to the conformal gauge-fixing. On this larger space, one constructs a BRST operator, which is nilpotent only for $d=26$. The space of states annihilated by $Q$ (at ghost number 1) is isomorphic to $\mathcal{H}_{\text{phys}}$ found above, and the space of states (at this ghost number) which are $Q$-exact are isomorphic to the null states found above.

The BRST cohomology at this ghost number is thus isomorphic to the desired positive-definite Hilbert space. Moreover, one proves that there is no cohomology at other ghost numbers.

This “Modern” approach is the one you want to think about in terms of “canceling” the Virasoro anomaly. The nilpotency of $Q$ is basically a consequence of the fact the $c_{\text{ghost}}=-26$, while $c_{\text{matter}}=d$, so $c_{\text{total}}=0$ for $d=26$.

I think that the strange aspect of Thomas Thiemann’s quantization can be seen in an even simpler example than the bosonic string, where not even an anomaly appears and his quantization still yields something strange:

Just consider the Nambu-Goto action not in 1+1 but in 1+0 dimensions, i.e. the relativistic particle. This theory has just a single constraint, which, when quantized, is the Klein-Gordon operator. Exponentiating this guy, classically or quantumly, gives us a symmetry group isomorphic to the reals.

The standard Dirac quantization of this theory yields the familiar Klein-Gordon equation.

According to the ‘LQG-string’ quantization prescription, however, quantization consists of finding any representation of the classical symmetry group on some Hilbert space. As Thomas Thiemann has confirmed upon request, this implies a large ambiguity in choosing such representations. For instance this prescription would allow me to quantize the free relativistic particle by choosing the Hilbert space $L^2(R)$ and use the translation operator $U(a) = \exp({-i a\partial_x})$ as the symmetry generator. Or should I rather use $U(a) = \exp(i a X)$? Or any other 1-parameter family of unitary operators? All of this gives ‘quantizations’ of the KG particle which have nothing to do with the real thing.