The conditions under which a stress tensor $T^{\mu\nu}$ exists I used to believe that the existence of the stress tensor in a QFT has to do with translation invariance: "If a theory is translation invariant, then one can construct a conserved $T^{\mu\nu}$ by Noether theorem". But it has been pointed to me that the existence of a conserved $T^{\mu\nu}$ is related to the locality of the theory. How can we show this? and is there some counterexamples?
 A: Translation invariance guarantees only the existence of conserved charges: $P_i$ and the Hamiltonian $H$ itself. It is an extra assumption that those charges can be written in terms of local currents: $P_\mu = \int T_{0\mu}(x) d^{d-1}x$. Theories given by an action which is written in terms of local Lagrangian densities will possess a conserved EMT (provided they are translation invariant).
A: Indeed, nonlocal CFT's like the scaling limit of long-range Ising models will typically lack a local stress tensor. In Euclidean signature, the kinetic term is $\phi (-\Delta)^{\alpha}\phi$ which is still unitary if $0<\alpha<1$.
Some references on this:

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*Paulos, M. F., Rychkov, S., van Rees, B. C., & Zan, B. (2016). Conformal invariance in the long-range Ising model. Nuclear Physics B, 902, 246-291.

*Behan, C., Rastelli, L., Rychkov, S., & Zan, B. (2017). Long-range critical exponents near the short-range crossover. Physical Review Letters, 118(24), 241601.

*Behan, C., Rastelli, L., Rychkov, S., & Zan, B. (2017). A scaling theory for the long-range to short-range crossover and an infrared duality. Journal of Physics A: Mathematical and Theoretical, 50(35), 354002.

*Behan, C. (2019). Bootstrapping the long-range Ising model in three dimensions. Journal of Physics A: Mathematical and Theoretical, 52(7), 075401.

