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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It would have looked more natural to impose the stronger condition \(\deg _\infty \mathfrak {t}^{(k)} = \deg _\infty \mathfrak {t} - |k|\) as before. One may further think that in this case one would be able to extend Theorem 4.3 to all diagrams Γ, not just those in \(\mathcal {H}_+\). This is wrong in general, although we expect it to be true after performing a suitable form of positive renormalisation as in [2, 3]. This is not performed here, and as a consequence we are unable to take advantage of the additional large-scale cancellations that the stronger condition \(\deg _\infty \mathfrak {t}^{(k)} = \deg _\infty \mathfrak {t} - |k|\) would offer.
- 2.
I.e. T i is such that if u ∈ T i and v ≤ u, then v ∈ T i.
References
Bogoliubow, N.N., Parasiuk, O.S.: Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder. Acta Math. 97, 227–266 (1957). https://doi.org/10.1007/BF02392399
Bruned, Y., Hairer, M., Zambotti, L.: Algebraic renormalisation of regularity structures (2016). ArXiv e-prints. http://arxiv.org/abs/1610.08468
Chandra, A., Hairer, M.: An analytic BPHZ theorem for regularity structures (2016). ArXiv e-prints. http://arxiv.org/abs/1612.08138
Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199(1), 203–242 (1998). http://arxiv.org/abs/hep-th/9808042, https://doi.org/10.1007/s002200050499
Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210(1), 249–273 (2000). http://arxiv.org/abs/hep-th/9912092, https://doi.org/10.1007/s002200050779
Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216(1), 215–241 (2001). http://arxiv.org/abs/hep-th/0003188, https://doi.org/10.1007/PL00005547
de Calan, C., Rivasseau, V.: Local existence of the Borel transform in Euclidean \(\Phi ^{4}_{4}\). Commun. Math. Phys. 82(1), 69–100 (1981/1982)
Dyson, F.J.: The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. (2) 75, 486–502 (1949). https://doi.org/10.1103/PhysRev.75.486
Dyson, F.J.: The S matrix in quantum electrodynamics. Phys. Rev. (2) 75, 1736–1755 (1949). https://doi.org/10.1103/PhysRev.75.1736
Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 211–295 (1973/1974)
Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Bounds on renormalized Feynman graphs. Commun. Math. Phys. 100(1), 23–55 (1985). https://doi.org/10.1007/BF01212686
Gallavotti, G., Nicolò, F.: Renormalization theory in four-dimensional scalar fields. I. Commun. Math. Phys. 100(4), 545–590 (1985)
Gallavotti, G., Nicolò, F.: Renormalization theory in four-dimensional scalar fields. II. Commun. Math. Phys. 101(2), 247–282 (1985)
Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014). http://arxiv.org/abs/1303.5113, https://doi.org/10.1007/s00222-014-0505-4
Hairer, M.: The motion of a random string (2016). ArXiv e-prints. http://arxiv.org/abs/1605.02192
Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015). ArXiv e-prints. http://arxiv.org/abs/1512.07845
Hepp, K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys. 2(4), 301–326 (1966). https://doi.org/10.1007/BF01773358
Salam, A.: Divergent integrals in renormalizable field theories. Phys. Rev. (2) 84, 426–431 (1951). https://doi.org/10.1103/PhysRev.84.426
Salam, A.: Overlapping divergences and the S-matrix. Phys. Rev. (2) 82, 217–227 (1951). https://doi.org/10.1103/PhysRev.82.217
Weinberg, S.: High-energy behavior in quantum field-theory. Phys. Rev. (2) 118, 838–849 (1960). https://doi.org/10.1103/PhysRev.118.838
Zimmermann, W.: Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208–234 (1969). https://doi.org/10.1007/BF01645676
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Hairer, M. (2018). An Analyst’s Take on the BPHZ Theorem. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-01593-0_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01592-3
Online ISBN: 978-3-030-01593-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)