Causal perturbation theory

Short description: Mathematically rigorous approach to renormalization theory

Renormalization and regularization

Renormalization

Renormalization group
On-shell scheme
Minimal subtraction scheme

Regularization

Dimensional regularization
Pauli–Villars regularization
Lattice regularization
Zeta function regularization
Causal perturbation theory
Hadamard regularization
Point-splitting regularization

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Causal perturbation theory is a mathematically rigorous approach to renormalization theory,[1] which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis. It goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser.[2]

## Overview

When developing quantum electrodynamics in the 1940s, Shin'ichiro Tomonaga, Julian Schwinger, Richard Feynman, and Freeman Dyson discovered that, in perturbative calculations, problems with divergent integrals abounded. The divergences appeared in calculations involving Feynman diagrams with closed loops of virtual particles.[3] It is an important observation that in perturbative quantum field theory, time-ordered products of distributions arise in a natural way and may lead to ultraviolet divergences in the corresponding calculations. From the generalized functions point of view, the problem of divergences is rooted in the fact that the theory of distributions is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as was proved by Laurent Schwartz in the 1950s.[4]

Epstein and Glaser solved this problem for a special class of distributions that fulfill a causality condition, which itself is a basic requirement in axiomatic quantum field theory.[2] In their original work, Epstein and Glaser studied only theories involving scalar (spinless) particles. Since then, the causal approach has been applied also to a wide range of gauge theories, which represent the most important quantum field theories in modern physics.[5]

## References

1. ↑ Prange, Dirk (1999). "Epstein-Glaser renormalization and differential renormalization". Journal of Physics A: Mathematical and General 32 (11): 2225–2238. doi:10.1088/0305-4470/32/11/015. ISSN 0305-4470. Bibcode: 1999JPhA...32.2225P.
2. ↑ 2.0 2.1 Epstein, H.; Glaser, V. (1973). "The role of locality in perturbation theory". Annales de l'Institut Henri Poincaré A 29 (3): 211–295. http://www.numdam.org/item/AIHPA_1973__19_3_211_0/.
3. ↑ Nastase, Horatiu (2019). Introduction to Quantum Field Theory. Cambridge University Press. pp. 488. ISBN 9781108493994.
4. ↑ L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", Comptes Rendus de L'Académie des Sciences 239, pp. 847–848 [1]
5. ↑ Scharf, G. (2016). Gauge field theories : spin one and spin two : 100 years after general relativity (Revised ed.). Mineola, New York. ISBN 978-0-486-80524-5. OCLC 929591612. https://www.worldcat.org/oclc/929591612.

## Additional reading

* Scharf, G (1995). Finite Quantum Electrodynamics : The Causal Approach (2nd ed.). Berlin New York: Springer. ISBN 978-3-540-60142-5. OCLC 32890905.
* Scharf, G (2001). Quantum gauge theories : a true ghost-story (1st ed.). New York: John Wiley & Sons. ISBN 978-0-471-41480-3. OCLC 45394191.
* Dütsch, Michael; Scharf, Günter (1999). "Perturbative gauge invariance: the electroweak theory" (in de). Annalen der Physik (Wiley) 8 (5): 359–387. doi:10.1002/(sici)1521-3889(199905)8:5<359::aid-andp359>3.0.co;2-m. ISSN 0003-3804. Bibcode: 1999AnP...511..359D.

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