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Noncommutative Geometry, Quantum Fields and Motives
Alain Connes, Collège de France, Paris, France, and Matilde Marcolli, Max-Planck-Institut für Mathematik, Bonn, Germany
A co-publication of the AMS and Hindustan Book Agency.
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Colloquium Publications
2008; 785 pp; hardcover
Volume: 55
ISBN-10: 0-8218-4210-2
ISBN-13: 978-0-8218-4210-2
List Price: US$102
Member Price: US$81.60
Order Code: COLL/55
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See also:

In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes - Michel L Lapidus

Quanta of Maths - Etienne Blanchard, David Ellwood, Masoud Khalkhali, Matilde Marcolli, Henri Moscovici and Sorin Popa

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces.

The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory.

The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions.

The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

This book is co-published with Hindustan Book Agency (New Delhi) and is distributed worldwide, except in India, Bangladesh, Bhutan, Nepal, Sri-Lanka, and the Maldives by the American Mathematical Society.

Readership

Graduate and research mathematicians interested in noncommutative geometry, quantum field theory and particle physics, number theory, and arithmetic algebraic geometry.

Reviews

"...the authors manage very well in filtering and presenting the central ideas whilst including a rich and precise list of references to the literature. ...will undoubtedly serve as an inspiration to the formidable mathematical question on the structure of the following two spaces: spacetime and the space of primes."

-- Mathematical Reviews

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