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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?

Abdelmalek Abdesselam
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Bastam Tajik
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    What does "nonlocality" mean here? – ACuriousMind Nov 14 '21 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 – Bastam Tajik Nov 14 '21 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 – Bastam Tajik Nov 14 '21 at 12:07
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    All QFT Lagrangians are local. I suspect you have a non-standard definition of what "non-local" means. – ACuriousMind Nov 14 '21 at 12:27
  • https://en.wikipedia.org/wiki/Nonlocal_Lagrangian @ACuriousMind – Bastam Tajik Nov 14 '21 at 13:32
  • Definitely violation of Unitarity is an option.(In the quantum mechanical case) – Bastam Tajik Nov 14 '21 at 13:34
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    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. – ACuriousMind Nov 14 '21 at 13:58
  • A conformal field theory is unitary if its space of states has a positive definite scalar product such that the dilation operator is self-adjoint. Then the scalar product endows the space of states with the structure of a Hilbert space.(https://en.wikipedia.org/wiki/Conformal_field_theory) @ACuriousMind – Bastam Tajik Nov 14 '21 at 14:42
  • I guess now you can understand what I really mean now. – Bastam Tajik Nov 14 '21 at 14:43
  • @ACuriousMind: It is not clear why you mean by "explain how QFT with such non-local Lagrangian is supposed to work". What exactly do you need to work? And what definition of QFT are you using when saying "All QFT Lagrangians are local". – Abdelmalek Abdesselam Nov 16 '21 at 19:06
  • @AbdelmalekAbdesselam It is not obvious to me that just writing down a non-local action functional defines a quantum theory - for starters, without a Hamiltonian, what even is time evolution? (you don't need to answer if it doesn't fit in a comment - my point is that the question should do a little more to make its setup clear) – ACuriousMind Nov 16 '21 at 20:46
  • As far as I know, to define a Quantum Theory one needs a classical one to start with. Perhaps you have some specific quantization method in mind, like canonical that doesn't work in the case of non-local actions. The statement "all QFTs are local" is quite incorrect as you can integrate out fields from path integral and get non-local and pretty much quantum mechanical effective actions at the end. @ACuriousMind – Bastam Tajik Nov 16 '21 at 22:32

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Everyone knows the difference between a local operator and a nonlocal operator. But the interesting question is what it means for a theory to be nonlocal. This has been given some poor definitions in the past, e.g. by people who say that everything quantum mechanical is nonlocal because of entanglement.

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 \begin{equation} S = \int d^dx \int d^dy \frac{\phi(x)\phi(y)}{|x - y|^{2(d - \Delta_\phi)}}. \quad (1) \end{equation} 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.

Connor Behan
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  • So there are non-local field theories that are truly(Quantum mechanically) scale-invariant. That's beautiful. Can you elaborate in case of gravity? – Bastam Tajik Nov 14 '21 at 14:46
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    The question of conserved currents there is moot because gravity doesn't have continuous global symmetries. But gravity could be called nonlocal in another sense because local operators are not gauge invariant unless you send them to infinity to define the S-matrix. – Connor Behan Nov 14 '21 at 15:15
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    But having said that, it is hard to find gravitational theories where there is some notion of scale invariance. Conformal supergravity and fixed points of "asymptotically safe" gravity proposals are the only examples I can think of. – Connor Behan Nov 14 '21 at 15:19
  • Sorry, but are there also any instances of interacting non-local CFTs? I'd be grateful if you could introduce some. – Bastam Tajik Nov 15 '21 at 13:44
  • Perhaps one can localize the scale invariance symmetry for the aforementioned non-local CFT and introduce interactions. – Bastam Tajik Nov 15 '21 at 15:48
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    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. – Connor Behan Nov 15 '21 at 16:18
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    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. – Abdelmalek Abdesselam Nov 16 '21 at 19:03