Confusion about two definitions of anomalies

Question

As I am currently studying for an exam about quantum field theory and string theory, I got confused about the notion of "anomalies" and how they are actually defined. Similar questions have already been asked here and here, but the given answers did not really satisfy me.

So let's suppose we have some classical symmetry, i.e. a symmetry of the action. The two definitions I keep seeing are those:

1. (usually in texts on string theory) A symmetry is anomalous if it can not be represented unitarily on the Hilbert space (for any regularisation). Equivalently(?), if we can't find an ordering prescription such that the symmetry generators obey the correct algebra after quantisation and are still unitary and positive-energy.
   Note that if the symmetry is still a ray symmetry on the Hilbert space (is that always the case?), we can still find a projective representation. If the representation is genuinely projective, there are two options (as described e.g. in Weinberg): Either, if the symmetry group is not simply connected, we may have to use a covering group instead. Or, if the algebra is not semisimple, there might be central charges that can't be made vanishing.
   Here it is always assumed that the measure is still invariant and the Ward identities hold. Actually, in string theory the Ward identities are often used to derive certain OPEs and with them to derive the quantum Virasoro algebra.

   Examples for this are:

   • The Witt algebra in string theory, which gets a central extension when quantised and becomes the Virasoro algebra.
   • SO(3) acting on a non-integer spin. SO(3) is not simply connected and we have to consider representations of the double cover SU(2).
   • The algebra of the Galilei group acting on non-relativistic 1-particle states has a central charge in the relation $[K_i, P_j] = -i \delta_{ij} m$.
2. (usually in QFT texts) The symmetry is anomalous if we can't find a regularisation procedure for the path integral measure $\mathcal D \Phi$ such that $\mathcal D \Phi = \mathcal D \Phi'$.
   Then it is clear that the Ward-Takahashi identities will be violated: The equation $\left\langle \nabla_\mu j^\mu \right\rangle = 0$ (away from insertions) picks up new terms on the right hand side. The algebra of the symmetry generators apparently doesn't necessarily change.
   An example for this is the anomaly of the (global) chiral symmetry in QED. By the way, here we only have one symmetry generator, so we can not get a non-trivial central extension of the algebra anyway.

Actually, right now it seems to me that those two are just completely different things. In version 1 we do have the invariant measure and Ward identities, which are absent in version 2. On the other hand, in 2 we do not have central charges or other algebra modifications (as opposed to version 1).

Still it feels like there should be some kind of connection between the two. But maybe this is just because they are called the same. So my question is: Are those notions related? Maybe even the same thing from different viewpoints? (For example, I could imagine that from the viewpoint of canonical quantisation, we get the anomaly in the operator algebra, whereas from the viewpoint of path integral quantisation, we get the anomaly in the PI measure.)

Answer 1

The two notions are indeed related. Take for example the Weyl anomaly of bosonic string theory: the classical (Polyakov) action $S$ is invariant under Weyl rescalings of the worldsheet metric $\gamma_{ab}$, i.e. $$S[\gamma_{ab}(\tau,\sigma)]=S[\exp(2\omega(\tau,\sigma))\gamma_{ab}(\tau,\sigma)]=S[\gamma'_{ab}(\tau,\sigma)].$$ Since this is a conformal symmetry, the trace of the energy-momentum tensor has to vanish: $T_a^a=0$. Quantizing the theory without specifying the number of spacetime dimensions will result in an anomaly; Weyl invariance is no longer a symmetry of the quantum string theory. The stress energy tensor will acquire a trace proportional to the Ricci scalar $R$ corresponding to the world sheet geometry, i.e. $T_a^a=-\frac{c}{12}R$, where $c$ is the central charge. The latter is equal to zero only in the case of $D=26$, so the anomaly does not appear at the so-called critical dimension. (Note that it also vanishes when the world sheet geometry is flat.) The existence of the central charge induces a Virasoro algebra for the modes of the stress-energy tensor, which is consistent with what you write.

The arguments above can be made without reference to the path integral and its measure; this is the viewpoint of the symmetry algebra. One can, however, adopt a different one and ask what happens to the path integral. It turns out that under a Weyl transformation, the variation of the measure and the rest of the integral consists of insertions of the trace of the energy-momentum tensor, and will only vanish if there is no conformal anomaly. Hence, the two pictures are equivalent.