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My question is about mathematical treatment of exact renormalization group in the sense of Polchinski's flow equation. In a heuristic form, Polchinski's equation looks like: $\partial_t S[\phi] = \frac{\delta}{\delta \phi} \cdot \Delta \cdot \frac{\delta}{\delta \phi} S[\phi] - \frac{\delta}{\delta \phi}S[\phi]\cdot \Delta \cdot \frac{\delta}{\delta \phi}S[\phi]$, where $S[\phi]$ is the action, $\phi$ is the field, $\Delta$ contains a cut-off version of parametrix associated with the classical equation of motion of the field (for example Klein Gordon equation for scalar bosonic fields). In addition, t is the renormalization "time" which is the log of energy scale, and $\Delta$ depends on t since it involves high-energy cut-off.

Note that Polchinski's equation is meant to be a non-perturbative field-theoretic formulation of the Wilsonian renormalization.

By now, I think there are several expositions of mathematical aspects of renormalization. For example, Costello's formulation of perturbative renormalization in BV formalism. We also have Kreimer-Connes formulation of BPHZ renormalization emphasizing on Hopf algebra structure and non-commutative aspects. However neither seems to work with non-perturbative aspects of renormalization as in the sense of Polchinski.

Therefore, my question is whether there is an attempt to study Polchinski equation in a rigorous mathematical setting. If we write $S[\phi]$ in terms of formal power series in $\phi$, and study the corresponding equations term by term, an important point is that this set of infinitely many equations hold "within path integral" which means they are to be understood as equations of operators. Therefore preferably as a first step, I want to ask if there are formulations of differential equations in operator algebras.

To be more concrete, as a toy model, if we consider finite dimensional self-adjoint matrices $A$ and perturb it by self-adjoint matrices $A_s = A + sK$, where $K$ is self adjoint, and we consider $f \in B^1_{\infty,1}(\mathbb{R})$ (Besov space), it is known that we have a formulation of differentiation in terms of double operator integral. Then what can we say about solution to the ODE: $\frac{d}{ds}f(A_s) = B$ where B is a fixed self adjoint matrix.

Furthermore, suppose we keep the above setting and take $f\in B^2_{\infty,1}(\mathbb{R})$, we can now consider second order derivative in terms of a multi-operator integral. Then following the usual Laplacian on $\mathbb{R}^n$ and let $\{ E_{ij} \}$ be a basis of n-dim Hermitian matrices, consider the analog of Laplacian $\Sigma_{ij}\frac{d^2}{ds^2}f(A^{ij}_s)$ where $A^{ij}_s = A + sE_{ij}$. Can we formulate and prove a maximum principle for such an analog of Laplacian?

Anyhow, this is a super long question. Any thoughts are helpful. The comment on operator differential equation is just my very pre-mature thoughts. Any direction regarding the Polchinski's equation (not necessarily related to operator algebra) is greatly appreciated. Thanks a lot for the help.

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Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, you should at least read my previous answer

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability/375571#375571

If you have time, also look at

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

There is nothing sacrosanct about Polchinski's approach. It is just one among many ways of implementing the RG method. It is characterized by the use of a continuous flow, or ODE on a very infinite-dimensional space of effective actions/potentials. It looks nonperturbative but it isn't really because no one was able to find norms on space of functionals (functions of functions) where one can prove a local in time well posedness result for this ODE. As far as I know, there is only one rigorous nonperturbative result with a continuous flow RG: the article "Continuous Constructive Fermionic Renormalization" by Disertori and Rivasseau in Annales Henri Poincaré 2000. They didn't use Polchinski's equation but the older Callan-Symanzik equation, and this only works for Fermions which are easier than Bosons because perturbation series converge in the case of Fermion (with cutoffs).

As to what has been done rigorously with Polchinski's equations, it concerns rigorous proofs of perturbative renormalizability, i.e., in the sense of formal power series. A good introduction to this is the book "Renormalization: An Introduction" by Manfred Salmhofer. For recent results with this approach see the works of Christoph Kopper and Stefan Hollands.

For nonperturbative rigorous results for Bosons, at least for what is known so far, one has to abandon the idea of continuous flow as in Polchinski's equation and work with a discrete RG transformation. Again there are several approaches. If you want something that is closest to the Polchinski philosophy, then you might want to look at the approach by David Brydges and collaborators developed over many years and culminating with a series of five articles in J. Stat. Phys., with Gordon Slade and Roland Bauerschmidt. A pedagogical introduction is now available with the book "Introduction to a Renormalisation Group Method" by these three authors.

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    $\begingroup$ 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. $\endgroup$ – user158305 May 19 at 19:11
  • $\begingroup$ 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/… $\endgroup$ – Abdelmalek Abdesselam May 19 at 20:11
  • $\begingroup$ 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... $\endgroup$ – Abdelmalek Abdesselam May 19 at 20:18
  • $\begingroup$ ...on that a couple years ago, see birs.ca/events/2018/5-day-workshops/18w5015/videos/watch/… $\endgroup$ – Abdelmalek Abdesselam May 19 at 20:18
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    $\begingroup$ @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. $\endgroup$ – Abdelmalek Abdesselam May 19 at 22:39

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