UV and IR modes in ground state

[ShayanJ]

I'm trying to understand chapter 19 of these lecture notes. But I have some difficulties with what the author explains:

1) In page 176, under equation 19.3 he says:

This is for the same reason that when we do renormalization, we are only allowed to add local counterterms to the Lagrangian; non-local terms come from IR physics.

This is weird. If we are considering a local QFT, then how can he say IR physics can cause non-locality? What is he talking about?

2) In page 178, under equation 19.7 he says:

In a highly excited random state, the IR modes that contribute to $\tilde S$ should all be highly entangled with the outside, and the number of such modes scales with volume.

He's contrasting the vacuum state with any random state. What he says implies that in the vacuum state, IR modes are either not entangled with outside, or are not excited. But the vacuum state is itself an energy eigenstate and so I don't understand how it can be consisted of different modes. What does he mean by "mode"? What is he talking about here?

Thanks

 

[king vitamin]

1) The IR physics of the ground state wave function (the ground state config of your quantum fields) is a non-local object: it exists throughout your whole space and there is entanglement between different points in space which are far apart. If your theory is relativistic, it is still causal (so there's no issue with Lorentz invariance), but entanglement in the quantum theory is still "non-loca" in the usual EPR way. And in CFTs, the universal term in the EE (an inherently non-local quantity) is independent of the UV regulator, so it is clearly a property of IR physics.

2) I agree that Hartman is being a little heuristic here, but I'll give a stab at it. Imagine you discretize your space into a lattice - you can then consider each lattice point to be a "mode." As he explains earlier in the notes, one expects that in a highly excited state, each lattice point in the region should have a large amount of entanglement with the outside, so tracing out all of these lattice points gives an entropy proportional to the number of lattice points, see pages 169-170.

In contrast, for gapped systems, the ground state does not have correlations along far distances and modes inside the entangling region do not all couple to the outside. Of course, CFTs are not gapless, but maybe you could argue that the ground state of the CFT cannot depend on any energy scale, so only the cutoff can enter into a volume term - but Hartman already argued that the leading-order divergent term is an area law. (But the area law is not 100% rigorous, so these are all just plausible arguments which can fail).