Motives, Quantum Field Theory, and Pseudodifferential Operators (Clay Mathematics Proceedings) by Alan Carey (PDF)

Description

This volume contains articles related to the conference “Motives, Quantum Field Theory, and Pseudodifferntial Operators” held at Boston University in June 2008, with partial support from the Clay Mathematics Institute, Boston University, and the National Science Foundation. There are deep but only partially understood connections between the three conference fields, so this book is intended both to explain the known connections and to offer directions for further research. In keeping with the organization of the conference, this book contains introductory lectures on each of the conference themes and research articles on current topics in these fields. The introductory lectures are suitable for graduate students and new Ph.D.’s in both mathematics and theoretical physics, as well as for senior researchers, since few mathematicians are expert in any two of the conference areas. Among the topics discussed in the introductory lectures are the appearance of multiple zeta values both as periods of motives and in Feynman integral calculations in perturbative QFT, the use of Hopf algebra techniques for renormalization in QFT, and regularized traces of pseudodifferential operators. The motivic interpretation of multiple zeta values points to a fundamental link between motives and QFT, and there are strong parallels between regularized traces and Feynman integral techniques. The research articles cover a range of topics in areas related to the conference themes, including geometric, Hopf algebraic, analytic, motivic and computational aspects of quantum field theory and mirror symmetry. There is no unifying theory of the conference areas at present, so the research articles present the current state of the art pointing towards such a unification. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is a very good contribution to the mathematical physics of pseudodifferential operator theory. Most of us in mathematical physics discover and see the intuitive concept of a motive in the Atiyah-Singer theorem, anomalies, index theory and Chern-Weil theory. Quite obviously you sum differential forms of different degrees, and so we see sums of objects corresponding to different dimensions. Same thing with the Dyson expansion, there you see motives. Also in string theory it is natural to consider branes of different dimensions at the same time, so again motives arise.Now, pseudodifferential operator theory is among other things relevant to achieve and analyze asymptotics of operators in Quantum Field Theory, (QFT). It has a very obvious relation to QFT, and one of the very interesting things is that there are workhorse theorems that allow you to compute asymptotics via simple differentiation with respect to phase space(symplectic or tangent bundle) coordinates. This book introduces the reader to this without being too technical, and may be a first reading in this interesting subject.Now, inasfar as this book, I would like to recommend it (strongly).The people that were in that conference are among the foremost in mathematical physics. The only pitty is that there are so few books of this nature and relevance. Also, it helps seeing recent trends in mathematical physics.

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