By Alain Connes, Matilde Marcolli
The unifying subject of this ebook is the interaction between noncommutative geometry, physics, and quantity idea. the 2 major items of research are areas the place either the noncommutative and the motivic features come to play a task: space-time, the place the guideline is the matter of constructing a quantum thought of gravity, and the distance of primes, the place it is easy to regard the Riemann speculation as a long-standing challenge motivating the advance of latest geometric instruments. The ebook stresses the relevance of noncommutative geometry in facing those areas. the 1st a part of the booklet bargains with quantum box conception and the geometric constitution of renormalization as a Riemann-Hilbert correspondence. It additionally provides a version of undemanding particle physics in line with noncommutative geometry. the most result's an entire derivation of the total usual version Lagrangian from a very easy mathematical enter. different issues lined within the first a part of the e-book are a noncommutative geometry version of dimensional regularization and its position in anomaly computations, and a short creation to explanations and their conjectural relation to quantum box concept. the second one a part of the e-book supplies an interpretation of the Weil particular formulation as a hint formulation and a spectral cognizance of the zeros of the Riemann zeta functionality. this is often in response to the noncommutative geometry of the adele classification area, that's additionally defined because the area of commensurability periods of Q-lattices, and is twin to a noncommutative intent (endomotive) whose cyclic homology offers a normal surroundings for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the distance of Q-lattices, in a single and dimensions, indicates spontaneous symmetry breaking. within the low-temperature regime, the equilibrium states of the corresponding platforms are with regards to issues of classical moduli areas and the symmetries to the category box thought of the sector of rational numbers and of imaginary quadratic fields, in addition to to the automorphisms of the sphere of modular services. The booklet ends with a suite of analogies among the noncommutative geometries underlying the mathematical formula of the normal version minimally coupled to gravity and the moduli areas of Q-lattices utilized in the research of the zeta functionality.
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