# How does locality decouple the UV and IR behaviour of a QFT?

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I came a comment in this paper: Scattering Amplitudes and the positive grassmannian in the last paragraph of page 104 which says: "One of the most fundamental consequences of space-time locality is that the ultraviolet and infrared singularities are completely independent."

How do I understand this?

EDIT: Since I asked this question, I've learned about non-commutative geometry and fields theories on such geometries -- where UV and IR aspects influence each other. Non-commutative field theories are non-local in the sense that you cannot specify an arbitrarily small volume in such a geometry. So I guess that's an indirect answer to the question. However, it's still not transparent to me how locality guarantees UV/IR decoupling.

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user Siva

edited Apr 16, 2014

The standard example of a theory where IR and UV divergences coincide is string theory. Here when you find a high energy divergence, it can always be reinterpreted as a long-distance infrared divergence, this is the UV-IR duality.

Interesting. But isn't string theory "local" in the sense that spacelike separate events don't talk to each other? So we're talking about some particulat notion of non/locality. Useful ref: http://physicsoverflow.org/216/is-string-theory-local

String theory just isn't local in this pointwise sense, you can see that it can't be local like that from the UV/IR duality, but it's obvious even in the formulation of the theory, because the interactions aren't constructed from local fields. String diagrams don't come from local splitting and joining of strings at space-time points when strings meet (there is string field theory, which does do this, but it does it on a light cone, it's not really local in the direction outward from the light-front, it's only just as local as possible, this was formulated by Mandelstam's students, Kaku, and Kikkawa).

The string theory series is built up from a hypothesis on the spectrum and then by a global diagrams which are built up by a bootstrap, that is using unitarity to reconstruct the series. They don't come from local fields describing local events at localized space-time points, otherwise it would be field theory, and it's not field theory.

There is a different version of locality in string theory, the proper version, which is formulated strictly on an asymptotic boundaries. I call it "Mandelstam locality", this principle doesn't have an accepted name as far as I know. This is saying that you can't outrun the speed of light asymptotically.

Asymptotically far away from scattering events, in a mostly-flat background where the space-time reduces to flat at the boundary, it surely makes sense to ask whether you can get from one boundary to the other while outrunning a ray of light. Mandelstam figured out the conditions which forbid this, by abstracting out the locality condition of field theory in the 1960s to talk only about the S-matrix. Unfortunately, it really is an involved analytic condition on the scattering amplitudes, it has to be, because it is a condition on the superposition integral of scattering amplitudes that prevents responses before causes, and this condition is mathematically saying that singularities only occur in certain places on a complex continuation of the integration path.

This weird analytic version of locality, the Mandelstam locality, is only a holographic locality, the thing that guarantees that the boundary theory ends up local in a holographic description. It isn't bulk-local, and that's ok, because within string theory, the interior bulk space-time is not even completely well defined in the full theory, outside of perturbation theory. You always need some scattering experiment to figure out where things are in the middle of space, to define the concept. The well-defined thing in string theory is S-matrix, this only tells you what the results of scattering experiments is.

So within string theory, you can't even properly formulate the question about spacelike separated events in the bulk. You must say "if I define the boundary of asymptotic communication by the instant when light first reaches the other side boundary from this side, can I exceed this limit by doing scattering in the middle?" String theory makes sure that the answer is no, because the singularities of the scattering amplitude are on proper branches of analytic continuation, so that when you superpose plane waves to make asymptotic packets, you never get an asymptotic response before the first moment when the asymptotic light can reach the other side.

It is annoying that you get such a mathematically involved notion of locality and causality, this type of thing is what made S-matrix theory so unpopular by the 1970s, local fields are so much easier, but this is really what you need to do when you don't have naive locality, no local fields.

I believe that the correct analog of this Mandelstam restriction in modern string theory, which isn't just formulated in flat space, but in AdS spaces, is the fact that boundary theories in AdS/CFT are local. I believe this, but it isn't shown in the literature, and I never actually did it, it's a hunch (but I'm pretty confident it's true--- if you have a violation of microcausality on the boundary, there are intuitive constructions which you can imagine which convert this into a asymptotic-to-asymptotic faster than light signal). Even though the boundary field theories reconstruct the bulk completely nonlocally, so you wouldn't expect them to be local theories, they themselves still are local field theories, because their basic states define asymptotic states of the full theory, and the asymptotic states need to obey normal field theory locality.

None of this is ever said in the modern (meaning past 1975) string theory literature, just because Mandelstam is being actively ignored. That's not right, he is a major founder of string theory, arguably the most significant one. He explained how S-matrix stuff is supposed to work abstractly, before Veneziano and others realized it explicitly.

I'll put a properly fleshed out version of this in the question you linked, it's really the right answer, and it's too old fasioned and taboo to talk about today. But I think one should work out the proper detailed relationship between boundary field theory locality, and Mandelstam analytic locality, to make a really proper answer, but it's like this.

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The connection to space-time locality is quite indirect, as it is just used to establish the following two facts:

UV singularities disappear if one discretizes spacetime by going to a lattice.

IR singularities disappear if one compactifies (Euclidean) spacetime by going to a torus.

Discretization and compactification can be done independently of each other, hence their origin (arbitrarily small distances, resp. unbounded distances) is independent.

answered Apr 16, 2014 by (13,969 points)

It's not clear to me exactly what role locality plays, in your reasoning.

Essentially none. But of course, the UV and IR problems appear in the first place due to locality, so one cannot argue without using locality that they go away upon performing the above approximations.

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On thing to keep in mind is that IR and UV divergences appear in different kinematical regimes: UV divergences are basically due to the fact that in loop integrals there are not sufficient propagators to make the integral fall off at infinity. E.g for a bubble integral

$\int d^4l \frac{1}{l^2(l-p)^2}$ will be logarithmically divergent. Do for instance a Taylor expansion of this expression for the loop momentum becoming large then this becomes obvious.

IR divergences however live in a completely different regime: they appear either when two particles becoming collinear $p_1\sim p_2$ or because some particles become soft $p_i\sim0$.

Or put a little more condensed:

UV: loop momentum becomes large

IR: external momenta become collinear/soft.

This is one way to see why these two kinds of divergence are not connected. Nima and company propably meant just this but in fancier terms.

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user A friendly helper
answered Feb 19, 2013 by (320 points)
Isn't your first integral (UV divergent example) diverging logarithmically and not linearly.

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user Learning is a mess
You're right. Corrected :) Cheers.

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user A friendly helper
I think the description is right, but I'm still not sure I understand the connection to the locality of the theory. Is there an easy way to see that, if I have a non local theory, this separation doesn't occur?

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user twistor59
Sorry for not commenting earlier, but I've been waiting to se eif there are any other responses. I have 2 questions with regards to this answer: 1. As @twistor59 says, I don't understand what role locality plays

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user Siva
Secondly, say we have a function $f(x)=\frac{1}{x^k}$. Then, in $n$ dimensions, the asymptotic behaviour ("divergence") of the integral $\int f(x) d^n x$ in the IR and UV is related (and complementary). So why should the divergences not be related?

This post imported from StackExchange Physics at 2014-04-15 16:45 (UCT), posted by SE-user Siva

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