**Colloquium Publications**

Volume: 55;
2008;
785 pp;
Hardcover

MSC: Primary 58; 11; 81; 14; 34;

Print ISBN: 978-0-8218-4210-2

Product Code: COLL/55

List Price: $102.00

AMS Member Price: $81.60

MAA member Price: $91.80

**Electronic ISBN: 978-1-4704-3201-0
Product Code: COLL/55.E**

List Price: $102.00

AMS Member Price: $81.60

MAA member Price: $91.80

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#### Supplemental Materials

# Noncommutative Geometry, Quantum Fields and Motives

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*Alain Connes; Matilde Marcolli*

A co-publication of the AMS and Hindustan Book Agency

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.

#### Readership

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

#### Reviews & Endorsements

...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

#### Table of Contents

# Table of Contents

## Noncommutative Geometry, Quantum Fields and Motives

- Cover Cover11
- Title page i2
- Dedication iii4
- Contents v6
- Preface xiii14
- Quantum fields, noncommutative spaces, and motives 124
- The Riemann zeta function and noncommutative geometry 341364
- Quantum statistical mechanics and Galois symmetries 437460
- Endomotives, thermodynamics, and the Weil explicit formula 577600
- Appendix 733756
- Bibliography 749772
- Index 763786
- Back Cover Back Cover1809