An analyst's take on the BPHZ theorem

TL;DR

This paper offers a clear, self-contained mathematical formulation of the BPHZ theorem, providing a systematic way to handle divergent integrals in Euclidean space without relying on quantum field theory background.

Contribution

It presents a new, self-contained formulation of the BPHZ theorem that clarifies the renormalization process for divergent integrals in a purely analytical setting.

Findings

Provides a systematic renormalization procedure for divergent integrals

Clarifies the role of arbitrariness in renormalization

Appeals to an analytically minded audience without quantum field theory background

Abstract

We provide a self-contained formulation of the BPHZ theorem in the Euclidean context, which yields a systematic procedure to "renormalise" otherwise divergent integrals appearing in generalised convolutions of functions with a singularity of prescribed order at their origin. We hope that the formulation given in this article will appeal to an analytically minded audience and that it will help to clarify to what extent such renormalisations are arbitrary (or not). In particular, we do not assume any background whatsoever in quantum field theory and we stay away from any discussion of the physical context in which such problems typically arise.

Equations

$$({\mathfrak{t}}^{(k)}{)}^{(\ell )}={\mathfrak{t}}^{(k+\ell )},\mathrm{deg}{\mathfrak{t}}^{(k)}=\mathrm{deg}\mathfrak{t}-|k|.(1):propLab=e01$$

Full text

1. 1 Introduction
2. 2 An analytical form of the BPHZ theorem

1 Introduction

The BPHZ renormalisation procedure named after Bogoliubov, Parasiuk, Hepp and Zimmerman [BP, hepp1966, Zimmermann] (but see also the foundational results by Dyson and Salam [Dyson, Dyson2, Salam, Salam2]) provides a consistent way to renormalise probability amplitudes associated to Feynman diagrams in perturbative quantum field theory (pQFT). The main aim of this article is to provide an analytical result, Theorem LABEL:theo:BPHZ below, which is a general form of the “BPHZ theorem” in the Euclidean context. To a large extent, this theorem has been part of the folklore of mathematical physics since the publication of the abovementioned works (see for example the article [FMRS85] which gives rather sharp analytical bounds and is close in formulation to our statement, as well as the series of articles [CK, CK1, CK2] which elucidate some of the algebraic aspects of the theory, but focus on dimensional regularisation which is not available in the general context considered here), but it seems difficult to find precise analytical statements in the literature that go beyond the specific context of pQFT. One reason seems to be that, in the context of the perturbative expansions arising in pQFT, there are three related problems. The first is to control the small-scale behaviour of the integrands appearing in Feynman diagrams (the “ultraviolet behaviour”), the second is to control their large scale (“infrared”) behaviour, and the final problem is to show that the renormalisation required to deal with the first problem can be implemented by modifying (in a scale-dependent way) the finitely many coupling constants appearing in the Lagrangian of the theory at hand, so that one still has a physical theory.

The approach we take in the present article is purely analytic and completely unrelated to any physical theory, so we do not worry about the potential physical interpretation of the renormalisation procedure. We do however show in Section LABEL:sec:props that it has a number of very nice mathematical properties so that the renormalised integrals inherit many natural properties from their unrenormalised counterparts. We also completely discard the infrared problem by assuming that all the kernels (“propagators”) under consideration are compactly supported. For the reader who might worry that this could render our main result all but useless, we give a simple separate argument showing how kernels with algebraic decay at infinity can be dealt with as well. Note also, that contrary to much of the related theoretical and mathematical physics literature, all of our arguments take place in configuration space, rather than in Fourier space. In particular, the analysis presented in this article shares similarities with a number of previous works, see for example [MR0342091, CalRiv, FMRS85] and references therein.

The approach taken here is informed by some results recently obtained in the context of the analysis of rough stochastic PDEs in [Proceedings, BHZalg, BPHZana]. Indeed, the algebraic structure described in Sections LABEL:sec:Hopf and LABEL:sec:twisted below is very similar to the one described in [Proceedings, BHZalg], with the exception that there is no “positive renormalisation” in the present context. In this sense, this article can be seen as a perhaps gentler introduction to these results, with the content of Section 2 roughly parallel to [BHZalg], while the content of Section LABEL:sec:analytical is rather close to that of [BPHZana]. In particular, Section 2 is rather algebraic in nature and allows to conceptualise the structure of the counterterms appearing in the renormalisation procedure, while Section LABEL:sec:analytical is rather analytical in nature and contains the multiscale analysis underpinning our main continuity result, Theorem LABEL:theo:BPHZ. Finally, in Section LABEL:sec:largeScale, we deal with kernels exhibiting only algebraic decay at infinity. While the conditions given in this section are sharp in the absence of any cancellations in the large-scale behaviour, we do not introduce an analogue of the “positive renormalisation” of [BPHZana], so that the argument remains relatively concise.

Acknowledgements

The author would like to thank Ajay Chandra and Philipp Schönbauer for several useful discussions during the preparation of this article. Financial support through ERC consolidator grant 615897 and a Leverhulme leadership award is gratefully acknowledged.

2 An analytical form of the BPHZ theorem

Fix a countable set $\mathfrak{L}$ of labels, a map $\mathop{\mathrm{deg}}\nolimits\colon\mathfrak{L}\to{{\mathchoice{\hbox{\bf R}}{\hbox{\bf R}}{\hbox{\scriptsize\bf R}}{\hbox{\tiny\bf R}}}}$, and an integer dimension $d>0$. We assume that the set of labels has a distinguished element which we denote by $\delta\in\mathfrak{L}$ satisfying $\mathop{\mathrm{deg}}\nolimits\delta=-d$ and that, for every multiindex $k$, there is an injective map $\mathfrak{t}\mapsto\mathfrak{t}^{(k)}$ on $\mathfrak{L}$ with $\mathfrak{t}^{(0)}=\mathfrak{t}$ and such that

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