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  • Print publication year: 2013
  • Online publication date: May 2013

5 - Iterated integrals in quantum field theory



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.


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.

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