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M. Peskin and R. Schroeder, An Introduction to Quantum Field Theory. L. Ryder,
Quantum Field Theory. S. Weinberg, Quantum Theory of Fields, Vol. 1 & 2.
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Quantum Field Theory Demystifi ed David McMahon New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
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[8] C. Bender, F. Cooper, R. Kenway, and L. M. Simmons, Phys. ... [19] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, (Clarendon Press, 1996).
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ITERATED INTEGRALS IN QUANTUM FIELD THEORY
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 overview of some recent results connecting them with Feynman diagrams in perturbative quantum field theory.
1. Introduction The theory of iterated integrals was 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: (1) The theory of Dyson series (2) Conformal field theory and the KZ equation (3) The Feynman path integral and calculus of variations (4) Feynman diagram computations in perturbative 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. Our main example is the thrice punctured Riemann sphere M = P1 \{0, 1, ∞}. The iterated integrals on M (§2.2) can be written in terms of multiple polylogarithms, which go back to Poincar´e and Lappo-Danilevskyy and defined for integers n1 , . . . , nr ∈ N by X z kr (1.1) Lin1 ,...,nr (z) = . n1 k1 . . . krnr 0
