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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
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Noncommutative Geometry, Quantum Fields and Motives (Paperback)
Loot Price: R2,925
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Noncommutative Geometry, Quantum Fields and Motives (Paperback)
Series: Colloquium Publications
Expected to ship within 10 - 15 working days
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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: 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 adele 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.
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