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Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics)
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Feynman Motives (Paperback)
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Feynman Motives (Paperback)
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This book presents recent and ongoing research work aimed at
understanding the mysterious relation between the computations of
Feynman integrals in perturbative quantum field theory and the
theory of motives of algebraic varieties and their periods. One of
the main questions in the field is understanding when the residues
of Feynman integrals in perturbative quantum field theory evaluate
to periods of mixed Tate motives. The question originates from the
occurrence of multiple zeta values in Feynman integrals
calculations observed by Broadhurst and Kreimer.Two different
approaches to the subject are described. The first, a "bottom-up"
approach, constructs explicit algebraic varieties and periods from
Feynman graphs and parametric Feynman integrals. This approach,
which grew out of work of Bloch-Esnault-Kreimer and was more
recently developed in joint work of Paolo Aluffi and the author,
leads to algebro-geometric and motivic versions of the Feynman
rules of quantum field theory and concentrates on explicit
constructions of motives and classes in the Grothendieck ring of
varieties associated to Feynman integrals. While the varieties
obtained in this way can be arbitrarily complicated as motives, the
part of the cohomology that is involved in the Feynman integral
computation might still be of the special mixed Tate kind. A
second, "top-down" approach to the problem, developed in the work
of Alain Connes and the author, consists of comparing a Tannakian
category constructed out of the data of renormalization of
perturbative scalar field theories, obtained in the form of a
Riemann-Hilbert correspondence, with Tannakian categories of mixed
Tate motives. The book draws connections between these two
approaches and gives an overview of other ongoing directions of
research in the field, outlining the many connections of
perturbative quantum field theory and renormalization to motives,
singularity theory, Hodge structures, arithmetic geometry,
supermanifolds, algebraic and non-commutative geometry.The text is
aimed at researchers in mathematical physics, high energy physics,
number theory and algebraic geometry. Partly based on lecture notes
for a graduate course given by the author at Caltech in the fall of
2008, it can also be used by graduate students interested in
working in this area.
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