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Noncommutative Geometry, Quantum Fields and Motives

Alain Connes Collège de France, Paris, France
Matilde Marcolli Max-Planck-Institut für Mathematik, Bonn, Germany
A co-publication of the AMS and Hindustan Book Agency
Available Formats:
Softcover ISBN: 978-1-4704-5045-8
Product Code: COLL/55.S
List Price: $102.00 MAA Member Price:$91.80
AMS Member Price: $81.60 Electronic ISBN: 978-1-4704-3201-0 Product Code: COLL/55.E List Price:$102.00
MAA Member Price: $91.80 AMS Member Price:$81.60
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List Price: $153.00 MAA Member Price:$137.70
AMS Member Price: $122.40 Click above image for expanded view Noncommutative Geometry, Quantum Fields and Motives Alain Connes Collège de France, Paris, France Matilde Marcolli Max-Planck-Institut für Mathematik, Bonn, Germany A co-publication of the AMS and Hindustan Book Agency Available Formats:  Softcover ISBN: 978-1-4704-5045-8 Product Code: COLL/55.S  List Price:$102.00 MAA Member Price: $91.80 AMS Member Price:$81.60
 Electronic ISBN: 978-1-4704-3201-0 Product Code: COLL/55.E
 List Price: $102.00 MAA Member Price:$91.80 AMS Member Price: $81.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$153.00 MAA Member Price: $137.70 AMS Member Price:$122.40
• Book Details

Colloquium Publications
Volume: 552008; 785 pp
MSC: Primary 58; 11; 81; 14; 34;

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.

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

• Chapters
• Chapter 1. Quantum fields, noncommutative spaces, and motives
• Chapter 2. The Riemann zeta function and noncommutative geometry
• Chapter 3. Quantum statistical mechanics and Galois symmetries
• Chapter 4. Endomotives, thermodynamics, and the Weil explicit formula
• Appendix

• 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
• Request Review Copy
Volume: 552008; 785 pp
MSC: Primary 58; 11; 81; 14; 34;

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.

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

• Chapters
• Chapter 1. Quantum fields, noncommutative spaces, and motives
• Chapter 2. The Riemann zeta function and noncommutative geometry
• Chapter 3. Quantum statistical mechanics and Galois symmetries
• Chapter 4. Endomotives, thermodynamics, and the Weil explicit formula
• Appendix
• ...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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