Why is radial ordering necessary?

Question

Suppose I have some conserved charge in a 2 dimensional CFT $$Q(|z|)=\int_{w=|z|}\text{d}w\,T(w).\tag{1}$$ The infinitesimal transformation induced on a field $\phi$ at $z$ is then $$[Q(|z|),\phi(z)]=\int_{w=|z|}[T(w),\phi(z)].\tag{2}$$ Books in CFT claim this is not a well defined quantity. On the one hand this seems reasonable to me since at a point in the integral we are taking $[T(z),\phi(z)]$. If $T$ and $\phi$ are distributional one should expect this to run into trouble. On the other hand, in the usual canonical quantization of the scalar field we do not run into such trouble. Indeed, taking $$H(t)=\int\frac{\text{d}^3\vec{p}}{(2\pi)^32E_\vec{p}}\,E_\vec{p}a_\vec{p}^\dagger a_\vec{p},\quad\phi(x)=\int\frac{\text{d}^3\vec{p}}{(2\pi)^32E_\vec{p}}\left(e^{-ipx}a_\vec{p}+e^{ipx}a_\vec{p}^\dagger\right),\tag{3}$$ or $$H(t)=\int\text{d}^3\,\vec{x}\frac{1}{2}\left(\Pi(t,\vec{x})^2+\vec{\nabla}\phi(t,\vec{x})^2+m^2\phi(t,\vec{x})^2\right)\tag{4}$$ one can easily compute $[H(t),\phi(t,\vec{x})]$. Why don't we see singularities in this case?

Another way of phrasing this would be: in the usual canonical quantization of the scalar field in Minkowki spacetime there is a compatibility between the commutators being taken at equal times and the Hamiltonian being constant in time. In CFT one seems to loose this compatibility at some point in the Euclidean field theory. Namely, while the commutators are taken at equal radius, the conservation equation guarantees that the density is holomorphic.

Answer 1

1. Correlation functions in 2D CFT are radial ordered for a similar reason that correlation functions in QFT are time-ordered. In fact the radial worldsheet coordinate is often identified with time.

2. Presumably OP's eq. (2) refers to eq. (6.15) in Ref. 1, cf. e.g. this related Phys.SE post. That commutator definition only applies to the holomorphic sector of a 2D CFT.

3. It is in principle possible to transcribe the holomorphic/anti-holomorpic variables and OPEs of 2D CFT into real variables. However, the holomorphic/anti-holomorpic formalism is more powerful, as we can rely on complex function theory.

4. In contrast, OP's scalar example uses real fields in 3+1D. We do see singularities in equal-time commutators, such as e.g. $[\phi(\vec{x},t),\pi(\vec{y},t)]=i\hbar \delta^3(\vec{x}\!-\!\vec{y})$ in form of a Dirac delta distribution. For non-equal times, the singularities can be more complicated.

References:

1. P. Di Francesco, P. Mathieu and D. Senechal, CFT, 1997; subsection 6.1.2.