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Quantum mechanics and quantum field theory on one hand and Gravity as a theory of curved space-time on the other are the two great conc- tual schemes of modern theoretical physics. For many decades they have lived peacefully together for a simple reason: it was a coexistence wi- out much interaction. There has been the family of relativists and the other family of elementary particle physicists and both sides have been convinced that their problems have not very much to do with the problems of the respective other side. This was a situation which could not last forever, because the two theoretical schemes have a particular structural trait in common: their claim for totality and universality. Namely on one hand all physical theories have to be formulated in a quantum mechanical manner, and on the other hand gravity as curved space-time influences all processes and vice versa. It was therefore only a question of time that physically relevant domains of application would attract a general int- est, which demand a combined application of both theoretical schemes. But it is immediately obvious that such an application of both schemes is - possible if the schemes are taken as they are. Something new is needed which reconciles gravity and quantum mechanics. During the last two de- des we are now doing the first steps towards this more general theory and we are confronted with fundamental difficulties.
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.
This book provides an up-to-date understanding of the progress and current problems of the interplay of nonlocality in the classical theories of gravitation and quantum theory. These problems lie on the border between general relativity and quantum physics, including quantum gravity.
Mathematics provides a language in which to formulate the laws that govern nature. It is a language proven to be both powerful and effective. In the quest for a deeper understanding of the fundamental laws of physics, one is led to theories that are increasingly difficult to put to the test. In recent years, many novel questions have emerged in mathematical physics, particularly in quantum field theory. Indeed, several areas of mathematics have lately become increasingly influentialin physics and, in turn, have become influenced by developments in physics. Over the last two decades, interactions between mathematicians and physicists have increased enormously and have resulted in a fruitful cross-fertilization of the two communities. This volume contains the plenary talks fromthe international symposium on Noncommutative Geometry and Representation Theory in Mathematical Physics held at Karlstad University (Sweden) as a satellite conference to the Fourth European Congress of Mathematics. The scope of the volume is large and its content is relevant to various scientific communities interested in noncommutative geometry and representation theory. It offers a comprehensive view of the state of affairs for these two branches of mathematical physics. The book is suitablefor graduate students and researchers interested in mathematical physics.
This book provides an up-to-date understanding of the progress and current problems of the interplay of nonlocality in the classical theories of gravitation and quantum theory. These problems lie on the border between general relativity and quantum physics, including quantum gravity.
First published in 1997, this title is a sequel to Dr Noel Curran's first book The Logical Universe: The Real Universe (published by Ashgate under the Avebury imprint, 1994). The philosophy of mathematics in this book is based on ideas of Sir William Rowan Hamilton on the ordinal character of numbers, the real numbers, the measure numbers, scalar numbers and the extension to vectors. The final extension is to Hamilton’s quaternions. This algebra is interpreted as the mathematics of spin. This led to a a new theory of time and space which is Euclidian. The motion of spin is absolute, no frame of reference is required. If time is assumed to have a beginning it would be asymmetric with an arrow. This concept is applied to the laws of nature, which are symmetrical. This is another Copernican Revolution in three aspects: absolute time is restored, time has an arrow - is asymmetric, and thirdly the theory is based on the motion of spin which is absolute and more fundamental than the motion of translation. This opens the way to the final unification of physics.
In modern mathematical physics, classical together with quantum, geometrical and functional analytic methods are used simultaneously. Non-commutative geometry in particular is becoming a useful tool in quantum field theories. This book, aimed at advanced students and researchers, provides an introduction to these ideas. Researchers will benefit particularly from the extensive survey articles on models relating to quantum gravity, string theory, and non-commutative geometry, as well as Connes' approach to the standard model.