Softcover ISBN:  9781470450458 
Product Code:  COLL/55.S 
List Price:  $102.00 
MAA Member Price:  $91.80 
AMS Member Price:  $81.60 
Electronic ISBN:  9781470432010 
Product Code:  COLL/55.E 
List Price:  $102.00 
MAA Member Price:  $91.80 
AMS Member Price:  $81.60 

Book DetailsColloquium PublicationsVolume: 55; 2008; 785 ppMSC: 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: spacetime, 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 longstanding 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 RiemannHilbert 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 Qlattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of Lfunctions. The quantum statistical mechanics of the space of Qlattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the lowtemperature 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 Qlattices used in the study of the zeta function.ReadershipGraduate and research mathematicians interested in noncommutative geometry, quantum field theory and particle physics, number theory, and arithmetic algebraic geometry.

Table of Contents

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


Additional Material

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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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: spacetime, 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 longstanding 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 RiemannHilbert 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 Qlattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of Lfunctions. The quantum statistical mechanics of the space of Qlattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the lowtemperature 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 Qlattices 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