# Radial quantization and infrared divergences

I am reading Ginsparg lectures "Applied CFT" https://arxiv.org/abs/hep-th/9108028 which is not my first material on the subject. He tries to motivates radial quantization on the reason that compactifying the spatial direction avoids potential IR divergences, but I previously learned this as motivated by the use of contour integrals and the powerful tools of complex analysis to relate straightforwardly the OPE to the Virasoro algebras. Are these motivations related? Do most CFT suffer from IR divergences? By definition they cure UV divergences but now that I think I bit more they may suffer from strong IR divergences having a spectrum composed only of massless particles (the same way a massless photons brings IR divergences to QED).

Yes, the motivations are related but not equivalent.

Ginsparg wants to say a simple point that if we quantize a theory on an infinite 1+1-dimensional spacetime, there will be modes of any field e.g. $\tilde j(k)$ for an arbitrarily low momentum $k$ and the very soft quanta with these arbitrarily low momenta may be produced in scattering processes etc. which implies that the amplitudes for these scattering processes will inevitably suffer from infrared divergences.

When we take the 1+1-dimensional spacetime to have the form $S^1\times R$, i.e. if the spatial direction is a circle, then $k$ is quantized, effectively becomes an integer $n$, and there are only "discretely/countably many" components of the fields in the momentum representation. The zero modes $n=0$ may be treated in isolation, giving us the center-of-mass degrees of freedom of a sort, and all other modes are massive starting from $n=\pm 1$: the IR problems disappear.

It's related to the motivation by the contour integrals simply because by a "contour", we mean a closed curve parameterized by an angle $\phi$ which belongs to a compact circle. The compactness in both justifications is the "virtue" of the radial quantization. This compactness has many basic consequences: the absence of the arbitrarily soft modes, something that removes the IR divergences on the world sheet, as well as the possibility to get a well-defined contour integral (equivalent to the residue inside the contour) which is what allows us to to link OPEs to group action, relate operators with states, and so on.

In your questions about curing IR and UV divergences, you seem to confuse the spacetime (obtained via string theory) and the world sheet. The CFT on the world sheet (not discussing its string theory application) doesn't distinguish IR and UV (scales on the world sheet) because it's conformal i.e. also scale-invariant: it's the same theory at all scales. If a consistent theory like that exists at all, it's automatically free of such problems. The compactification of the world sheet breaks this scale invariance, of course, but the symmetry still constrains the short-distance physics on the world sheet and other things.

Moreover, the IR problems don't signal any inconsistency of a theory; they only imply that we have asked a wrong question. For example, we should be asking what is the inclusive cross section that allows an arbitrary number of soft quanta below some $E_{\rm soft}$ to be created at the same moment. These soft quanta are produced in any real-world process and the infrared divergences obtained when calculating using the assumption that there are no soft quanta simply mean that the perturbative expansion is trying to tell us that the probability of this strict process without any soft quanta is zero and uninteresting. However, the inclusive cross section is still finite and its value is interesting.

We avoid these IR problems on the world sheet, which would be self-inflicted wound, by considering compact spaces (closed string, cylinder etc.) and compact Riemann surfaces. The UV problems are absent if the theory is really conformal, too. For example, all the beta-functions vanish so the theory isn't getting any stronger (and out of control) at high energies or short distances. Note that QCD in $d=4$ is OK in the UV as it's getting weaker (asymptotic freedom) and CFTs are marginally the same thing (they don't get stronger or weaker).

In the string theory interpretation, the UV divergences in the spacetime are absent because every degenerate shape of the Riemann surface (history of merging strings) that could yield divergences may be interpreted as the opposite IR-like limit (a very thin torus is a thick torus when interpreted in the way rotated by 90 degrees, for example). So all would-be spacetime UV divergences in the string theory-produced spacetime may be interpreted as spacetime IR divergences. The spacetime IR divergences may still be there – for example, they surely do exist in bosonic string theory because of the tachyon and dilaton that have pathological effects on long distance physics – but they may be cancelled e.g. in the 10D superstring as well because of some supersymmetric cancellations.