Timeline for Non-perturbative Renormalization in the sense of Polchinski's equation. Do we have a mathematical formulation?
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| May 19, 2020 at 22:39 | comment | added | Abdelmalek Abdesselam | @AlexArvanitakis: one should also remark that "perturbative vs. nonperturbative" can mean different things for different people. Polchinski's equation does look nonperturbative in the sense that it is about the full connected Green's functions and not individual Feynman diagrams. The rigorous proofs of perturbative renormalizability based on it (work of Kopper et al. I mentioned) can be done without a single Feynman diagram in sight. | |
| May 19, 2020 at 22:21 | comment | added | AlexArvanitakis | Just seconding here @AbdelmalekAbdesselam's observation that the Polchinski RG equation doesn't look especially non-perturbative | |
| May 19, 2020 at 22:07 | comment | added | user158305 | Thanks a lot for the answer. Really appreciate it. | |
| May 19, 2020 at 22:00 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
fixed typos
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| May 19, 2020 at 21:57 | comment | added | Abdelmalek Abdesselam | Also, not only the couplings but the value of the UV cutoff must be space-dependent. So one is in fact dealing with a cutoff hypersurface in the AdS bulk approaching the boundary, possibly in a curved way. What we did so far, explained in my talk above, is the flat cutoff surface case, i.e., the traditional Fourier cutoff. | |
| May 19, 2020 at 21:52 | comment | added | Abdelmalek Abdesselam | Shore sciencedirect.com/science/article/pii/0550321387904457 Jack and Osborn sciencedirect.com/science/article/pii/055032139090584Z Osborn sciencedirect.com/science/article/pii/055032139180030P | |
| May 19, 2020 at 21:49 | comment | added | user158305 | Also, thank you very much for the very detailed answer. | |
| May 19, 2020 at 21:48 | comment | added | user158305 | I completely agree with you on the point that the RG to study must involve space-dependent coupling constants, but I'm not familiar with local RG you mentioned. Could you please point out the reference relating to the work by Shore, Jack, and Osborn which you mentioned in the response? | |
| May 19, 2020 at 20:18 | comment | added | Abdelmalek Abdesselam | ...on that a couple years ago, see birs.ca/events/2018/5-day-workshops/18w5015/videos/watch/… | |
| May 19, 2020 at 20:18 | comment | added | Abdelmalek Abdesselam | I believe, essential to this understanding of the extra direction as a scale and of holography as a geometrization of the RG is a notion of Wilsonian local RG. The local RG is what I talked about regarding RG for space-dependent coupling constants. The version of it in the physics literature developed by Shore, Jack and Osborn and many others, is a Gell-Mann Low RG rather than a Wisonian one. On a simplified toy model related to the work of the late Steven Gubser on p-adic AdS/CFT, I and collaborators developed a rigorous Wilsonian local (inhomogeneous or space-dependent) RG. I gave a talk... | |
| May 19, 2020 at 20:11 | comment | added | Abdelmalek Abdesselam | Very worthy goal and motivation. In fact it is one my main motivations too, related to what I said in bold face font in this other MO post mathoverflow.net/questions/268540/… | |
| May 19, 2020 at 19:11 | comment | added | user158305 | Thanks a lot for the reference. I will take a closer look at those. But first maybe I should say a few words about my motivation. The problem arise from holography - more specifically the idea that holography is a geometric version of renormalization group. Personally unsatisfied by the lack of mathematical rigor in various statements made in holographic duality research, I am motivated to see if I can work out a more rigorous way of implementing the idea that RG equation can produce the classical equation of motion of the dual bulk action. | |
| May 19, 2020 at 17:43 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
added 136 characters in body
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| May 19, 2020 at 17:38 | history | answered | Abdelmalek Abdesselam | CC BY-SA 4.0 |