Iterated integrals in quantum field theory

doi:10.1017/CBO9781139208642.006

5 - Iterated integrals in quantum field theory

Published online by Cambridge University Press: 05 May 2013

Edited by

Iván Contreras and

Francis Brown: Affiliation:
CNRS – Institut de Mathématiques de Jussieu, Paris, France
Alexander Cardona: Affiliation:
Universidad de los Andes, Colombia
Iván Contreras: Affiliation:
Universität Zürich
Andrés F. Reyes-Lega: Affiliation:
Universidad de los Andes, Colombia

Book contents

Get access

Summary

Abstract

These notes are based on a series of lectures given to a mixed audience of mathematics and physics students at Villa de Leyva in Colombia in 2009. The first half is an introduction to iterated integrals and polylogarithms, with emphasis on the case ℙ1\{0, 1, ∞}. The second half gives an overviewof some recent results connecting them with Feynman diagrams in perturbative quantum field theory.

Introduction

The theory of iterated integralswas first invented by K. T. Chen in order to construct functions on the (infinite-dimensional) space of paths on a manifold, and has since become an important tool in various branches of algebraic geometry, topology and number theory. It turns out that this theory makes contact with physics in (at least) the following ways:

the theory of Dyson series,
conformal field theory and the KZ equation,
the Feynman path integral and calculus of variations,
Feynman diagram computations in perturbative quantum field theory (QFT).

The relation between Dyson series and Chen's iterated integrals is more or less tautological. The relationship with conformal field theory is well-documented, and we discuss a special case of the KZ equation in these notes. The relationship with the Feynman path integral is perhaps the deepest and most mysterious, and we say nothing about it here. Our belief is that a complete understanding of the path integral will only be possible via the perturbative approach, and by first understanding the relationship with (2) and (4). Thus the first goal of these notes is to try to explain why iterated integrals should occur in perturbative quantum field theory.

Type: Chapter
Information: Geometric and Topological Methods for Quantum Field Theory
Proceedings of the 2009 Villa de Leyva Summer School
, pp. 188 - 240

DOI: https://doi.org/10.1017/CBO9781139208642.006 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

[1] P., Belkale, P., Brosnan: Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116: 1 (2003), 147–188.Google Scholar

[2] S., Bloch, H., Esnault, D., Kreimer: On motives associated to graph polynomials, Comm. Math. Phys. 267: 1(2006), 181–225.Google Scholar

[3] D., Broadhurst, D., Kreimer: Knots and numbers in ϕ4 theory to 7 loops and beyond, Int. J. Mod. Phys. C 6 (1995), 519.Google Scholar

[4] F. C. S., Brown: Multiple zeta values and periods of moduli spaces M0, n, Ann. Scient. Éc. Norm. Sup. 4e série, t. 42, 373–491, math.AG/0606419.

[5] F. C. S., Brown: On the periods of some Feynman integrals, arXiv:0910.0114v1 (2009), 1–69.

[6] F. C. S., Brown, K., Yeats: Spanning forest polynomials and the transcendental weight of Feynman graphs, arXiv:0910.5429 (2009), 1–30.

[7] F. C. S., Brown, O., Schnetz: A K3 in ϕ4, Duke Math. J. 161(2012), 1817–1862.Google Scholar

[8] P., Cartier: Jacobiennes généralisées, monodromie unipotente et intégrales itérées, Séminaire Bourbaki, 1987–1988, exp. no. 687, pp. 31–52.

[9] P., Cartier: Polylogarithmes, polyzetas et groupes pro-unipotents, Séminaire Bourbaki, 2000–2001; Asterisque 282 (2002) 137–173.Google Scholar

[10] K. T., Chen: Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831–879.Google Scholar

[11] K. T., Chen: Integration of paths-a faithfulr epresentation of paths by non-commutative formal powers eries, Trans. Amer. Math. Soc. 89 (1958), 395–407.Google Scholar

[12] P., Deligne, A., Goncharov: Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. (4) 38: 1(2005), 1–56.Google Scholar

[13] A. B., Goncharov: Multiple polylogarithms and mixed Tate motives, preprint (2001), arXiv:math.AG/0103059v4.

[14] R. M., Hain: The geometry of the mixed Hodge structure on t he fundamental group, Proc. Sympos. Pure Math. 46 (1987), 247–282.Google Scholar

[15] R. M., Hain: Classical polylogarithms, Proc. Sympos. Pure Math. 55 (1994), Part 2.Google Scholar

[16] C., Itzykson, J. B., Zuber: Quantum field theory, International Series in Pure and Applied Physics. New York: McGraw-Hill, 1980.

[17] J., Oesterlé: Polylogarithmes, Séminaire Bourbaki 1992–1993, exp. no. 762, Astérisque 216 (1993), 49–67.Google Scholar

[18] O., Schnetz: Quantum periods: A census of ϕ4 transcendentals, arXiv:0801.2856 (2008).

[19] V. A., Smirnov: Evaluating Feynman integrals, Springer Tracts in Modern Physics 211. Berlin: Springer, 2004.

[20] J., Stembridge: Counting points on varieties over finite fields related to a conjecture of Kontsevich, Ann. Combin. 2 (1998), 365–385.Google Scholar

[21] M., Waldschmidt: Valeurs zêtas multiples: une introduction, J. Théorie Nomb. Bordeaux 12 (2002), 581–595.Google Scholar

[22] S., Weinzierl: The art of computing loop integrals, arXiv:hep-ph/0604068v1 (2006).

[23] W., Zudilin: Algebraic relations for multiple zeta values, Russ. Math. Surveys 58: 1(2003), 1–29.Google Scholar