Special conformal transformations and locality

Question

In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal transformations, which are not linear. I feel that this non - linearity could cause a conflict with locality, in the following sense:

• Let us consider two wave packets 1 and 2, separated by a space-like interval, corresponding to impulsions P1 and P2. If we make a linear transformation, for example a Lorentz transformation, we obtain impulsions P'1 and P'2. Now let us consider the total system 1 + 2; it has an impulsion P, which is equal to P1 + P2 because both systems are not causally related. By this same Lorentz transformation, the total impulsion P becomes P', but because the Lorentz transformation is linear, we have P' = P'1 + P'2, and P' is linear in P'1 and P'2, and is linear in P1 and P2. This is coherent with the fact that there is still no interaction between the systems 1 and 2.

• On the other hand, imagine we had taken, for example, a pseudo-Lorentz transformation , which would be, for example, quadratic, then, in the expression of P', we would have had cross-terms of type P1.P2, what would mean an interaction between systems not causally connected, which is incoherent.

So, I need a detailed argumentation, to infirm or confirm my feeling : non-linear special conformal transformations could cause a conflict with locality.

Answer 1

One way of defining conformal transformations are by a (positive) local scaling of the metric, of the form $e^{2 \phi}$. Such a transformation always preserves the sign of spacetime distances. In particular, the light-cone remains unchanged since null distances map to null distances. What was inside the lightcone stays inside and things outside stay outside. So there's no conflict with causality.

Answer 2

By definition light-like events are mapped into light-like events under conformal transformations. But you can map two space-like events by a finite transformation into two time-like events, actually you can map every pair of non-isotropic events into every other.

Basically this implies that if you impose that Wightman fuctions should be invariant under finite conformal transformations, then the fields commute for space-like and time-like separations and this is sometimes called Huygens locality. See eg. http://arxiv.org/abs/hep-th/0009004 and the following articles by the authors for details.

Answer 3

I will try to repeat your argument more precisely, to make sure I understand: you are saying that if you transform coordinates linearly, you take linear trajectories to linear trajectories, but if you transform coordinates nonlinearly, you take linear trajectories to curved trajectories, and this makes you think that an incoming pair of particles coming in at a straight lines will be transformed into particles which move along curvilinear trajectories, and therefore feel an instant force at a distance.

This is not a problem at all, because the conformal invariance is not an invariance of particle trajectories. A particle trajectory is constructed as a state using an operator which is necessarily smeared over a large region of space, where the function is periodic along planes corresponding to wavefronts to the extent it is extended in time (you are free to create a particle with a localized pulse, but I am assuming you are doing the creation gradually in time), and when you do a conformal transformation, you transform the smearing function to produce a nonlocal creation process which produces a curved trajectory.

The invariance of the correlation functions under conformal transformation just doesn't imply an invariance of particle trajectories under conformal transformations.