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In the context of quantum gravity, but also in other places, I have heard the question "does this effective field theory admit an UV completion?". However, I really don't understand this question, in the following sense.

It is my understanding that an effective field theory is a field theory that is valid at low energies up to some scale, and that can be obtained (but not necessarily) by integrating "high-energy degrees of freedom", by renormalisation for example. From this point of view, how could an effective field theory have no UV completion? By definition an effective field theory describes approximatively well some system at low energies, but if there is no UV completion, would it mean that the system doesn't exist at high energies?

Take the example of general relativity, which is an EFT. What would it mean for it to not have an UV completion? What happens at higher energies? There is just no theory?

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One way to understand the concept of a UV completion is in terms of fixed points in the space theories generated by the renormalization group. The RG flow, interpreted as a course-graining procedure, generates a vector field that is determined by the criterion that as we move along the vector field flow, we obtain theories that better and better describe the IR physics of the initial point on the curve. These initial points of these flows are called sources of the RG flow, where the language comes from ODEs; the end points of these curves are called sinks. Expressed in using this language, a UV completion is nothing but a source of an RG flow. The IR physics is then described by the sinks of the RG flow.

Generically, it is possible that in a space of theories not all curves generated by the RG flow will have sources or sinks. One may think that complexifying the space of theories will help (or even compactifying), with the hope that the additional theories may reveal a hidden fixed point (either sources or sinks), but this may not be true. For instance, one may have closed curve solutions (see for instance figure one of this paper).

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  • $\begingroup$ Ok, now I see that mathematically how it could be that there isn't any UV-completion. However, I still don't understand what it means physically. Our EFT supposedly describes some "coarse grained theory", but as you resolve the details (reverse RG-flow) you don't get any UV theory ? Is it that physically, the UV-fixed point is at "infinity" in the RG-space, so that it is a theory with "infinite" degrees of freedom, or something like that ? $\endgroup$
    – Frotaur
    Commented Jul 6, 2020 at 10:20
  • $\begingroup$ I like to think of the running energy scale as a resolution of the system. In that case, we can assume that the system already has infinitely many degrees of freedom (say in position space), however the running energy scale tells how well we can resolve intricate features or correlations. In particular, the fixed point at infinity (if it exists) should be interpreted as the limit of these theories. (In my mind, I imagine some analogy to that of fractals). $\endgroup$
    – linebyline
    Commented Jul 8, 2020 at 2:39
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All theories are incomplete, but it does not mean they all need renormalizations. See, for example, this explanation here.

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  • $\begingroup$ What do you mean by incomplete ? If I have a theory at and UV fixed point, is it not UV-complete by design ? Edit: I just saw you defined "incomplete" in your paper, but the question still stands $\endgroup$
    – Frotaur
    Commented Jul 6, 2020 at 10:17
  • $\begingroup$ @Frotaur: Nobody knows high energy excitations, so any theory will disagree with experiment in this region. However, any incomplete theory has its own high energy behavior. $\endgroup$
    – Vladimir Kalitvianski
    Commented Jul 6, 2020 at 10:49

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