# Special conformal transformations and locality

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.

• You were already answered at theoreticalphysics.stackexchange.com/questions/556/… and the question was closed. This question contains misconceptions that can be solved with a course on foundations on quantum field theory.
– Jon
Commented Nov 24, 2011 at 13:42
• @Jon, you definitively missed the point. There was no serious answer to my question, in theoreticalphysics.stackexchange.com. Look at the Ron's answer. I think it is very useful. Commented Nov 28, 2011 at 13:03

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.

• +1 for your interesting remark, but it applies only to the scale transformation part of the conformal group. I have always this problem with the non-linearity of special conformal transformations Commented May 20, 2013 at 10:29
• @Trimok, I'm confused as to what you mean. I think all conformal transformations (not just scale transformations) affect the metric by a local scale factor $e^{2 \phi(x)}$.
– Siva
Commented May 20, 2013 at 10:51
• If you think about the metric, you are perfectly right. So I am maybe a little confused. The problem is that I am thinking about the transformations on coordinates , so, in some sense, it could appear differently (so the damned non-linearity of the special conformal transformations). So I am going to think a little bit to your argument, and if I have no serious counter-argument, I will give you the credit of the answer. Commented May 20, 2013 at 11:23
• You might want to have a look at Section 1.1 of Ginsparg's notes. arxiv.org/abs/hep-th/9108028
– Siva
Commented May 20, 2013 at 11:32
• I knew the demonstration... Thanks however for the reference. In fact, my problem is maybe that I am a little hypnotized by some particular representation of the conformal algebra (and there are different representations), and so it is surely a very reductive point of view. So, in some sense, my question may be ill-posed... However, your answer is simple, serious, and elegant. so I give you credit for your answer. Commented May 20, 2013 at 11:45

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.

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.

• Your interpretation : linear trajectories transforming as curved trajectories by special conformal transformations, is extremely interesting. In my hypothesis, I used impulsions and not coordinates. In fact, I was considering a system made of two sub-systems (space-like separated wave packets). If you make a special conformal transformation, you'll find that the new total implusion may contain quadratic terms in the original impulsions of the two-sub-systems. But the idea is the same, it seems that there could be non-local interactions, and this is still extremely intriguing for me. Commented Nov 28, 2011 at 12:59