A rough path perspective on renormalization. (English) Zbl 1436.60086
As physicist, renormalisation methods are known to me in two areas: in phase transitions of multiparticle systems like water at the critical point or a ferromagnet, and in perturbative quantum field theory. Both areas consider Kenneth G. Wilson as the “father” of these renormalisation methods. Therefore, it came to me as a surprise to find the concept of algebraic renormalisation in a quite different subject, namely for rough paths. Still, in reading the publication I became aware that this is not a surprise at all.
Rough paths, as they were introduced by T. J. Lyons [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)], are differential equations driven by irregular signals. A popular example for this is the Brownian motion of suspended particles in a fluid. Such kind of differential equations can be integrated by a shuffle of integrations, leading either to the convergence of the series (enhancement) or to a branched result. Ito, Lyons and other have worked on this integration procedure, and even if the reader (like me) is not able to penetrate the details of these calculations containing topological tensor products and a lot more, the authors were so kind to mark their central findings by boxes, understandable also for the not-so-involved reader.
I think that the central finding of the authors is that the renormalisation of the model is equivalent to the translation of rough paths. To my understanding, the effect which has to be renormalised is the branching of the integration procedure, like UV divergences from loop integrals have to be renormalised in quantum field theory. In the same way as we stress in quantum field theory that knowing the whole theory for instance for the strong interaction would remove UV divergences ar all, for rough paths we could say that branching is only a result of insufficient knowledge of the system. Though a further analogy is possible: the translation of rough paths can be compared with the gauge degree of freedom, unveiling another kind of singularity to be “renormalised” by the method of Faddeev and Popov.
To conclude: even though the subject is very difficult to understand, the authors were successful in provoking such kind of deep questions in readers like me who understand renormalisation in a different way.
Rough paths, as they were introduced by T. J. Lyons [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)], are differential equations driven by irregular signals. A popular example for this is the Brownian motion of suspended particles in a fluid. Such kind of differential equations can be integrated by a shuffle of integrations, leading either to the convergence of the series (enhancement) or to a branched result. Ito, Lyons and other have worked on this integration procedure, and even if the reader (like me) is not able to penetrate the details of these calculations containing topological tensor products and a lot more, the authors were so kind to mark their central findings by boxes, understandable also for the not-so-involved reader.
I think that the central finding of the authors is that the renormalisation of the model is equivalent to the translation of rough paths. To my understanding, the effect which has to be renormalised is the branching of the integration procedure, like UV divergences from loop integrals have to be renormalised in quantum field theory. In the same way as we stress in quantum field theory that knowing the whole theory for instance for the strong interaction would remove UV divergences ar all, for rough paths we could say that branching is only a result of insufficient knowledge of the system. Though a further analogy is possible: the translation of rough paths can be compared with the gauge degree of freedom, unveiling another kind of singularity to be “renormalised” by the method of Faddeev and Popov.
To conclude: even though the subject is very difficult to understand, the authors were successful in provoking such kind of deep questions in readers like me who understand renormalisation in a different way.
Reviewer: Stefan Groote (Tartu)
MSC:
| 60L20 | Rough paths |
| 60L30 | Regularity structures |
| 60L40 | Paracontrolled distributions and alternative approaches |
Citations:
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