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.