# Is nonlocality consistent with scale invariance?

For sure I'm excluding gravity at first step, the question is that if nonlocality is compatible with scale invariance. At the classical and quantum levels for field theory in Minkowski spacetime.

Then what about the case of gravity?

• What does "nonlocality" mean here? Nov 14, 2021 at 11:12
• Nonlocal lagrangian. Suppose your lagrangian is nonlocal. Is it necessarily scale-dependent or not? Some instances in both scenarios would be desirable. @ACuriousMind Nov 14, 2021 at 12:04
• Another question would be: can one extract features like locality and scale-invariance from a more general perspective like the properties of the general S-Matrix rather than the lagrangian? @ACuriousMind Nov 14, 2021 at 12:07
• All QFT Lagrangians are local. I suspect you have a non-standard definition of what "non-local" means. Nov 14, 2021 at 12:27
• You have to explain how QFT with such non-local Lagrangians is supposed to work in the first place before asking specific questions about scale invariance or other symmetries. Already classical field theory is complicated for these cases, see e.g. this answer by Qmechanic for issues with the Hamiltonian formalism in such cases. Nov 14, 2021 at 13:58

A much better definition is that nonlocal field theories are the ones that fail to have a local conserved current for continuous symmetries. Any QFT with a nonlocal Lagrangian will be an example. One of the simplest is $$$$S = \int d^dx \int d^dy \frac{\phi(x)\phi(y)}{|x - y|^{2(d - \Delta_\phi)}}. \quad (1)$$$$ Since this is a well defined theory (it obeys Wick's theorem), it makes perfect sense to call the $$y$$ integral a nonlocal Lagrangian. It also clearly has no local stress tensor which means energy is conserved only globally... not through a continuity equation. The last thing to notice about (1) is that is is scale invariant (it is in fact the so called generalized free CFT). So the answer to the question is yes.
• Sure. You can just add $\lambda \phi^4$ to the above nonlocal free term to get something interacting. If you want nonlocality, interactions and scale invariance, you can tune $\lambda$ to the critical point to reach the so called long-range Ising model in much the same way that local $\phi^4$ theory flows to the (short-range) Ising model. Nov 15, 2021 at 16:18
• It might be good to also mention that for $\Delta_{\phi}\ge\frac{d-2}{2}$ this theory is unitary and does (rigorously) give rise to a Wightman QFT by OS reconstruction. Nov 16, 2021 at 19:03