Noncommutative Geometry, Quantum Fields and Motives

1-dimensional 2-dimensional Q-lattices abelian adèle class space affine group scheme automorphisms bundle C*-algebra Chapter coefficients cohomology commensurability commutative compact consider construction corresponding counterterms cyclic cyclic cohomology defined DEFINITION denote dimension Dirac operator dual elements endomorphisms endomotives equation equisingular equivalence extremal KMS fact factor fermions Feynman field theory finite following result Fourier transform functor Galois group gauge given gives graph group scheme groupoid Haar measure Hilbert space homomorphism Hopf algebra idèle inner fluctuations integral isomorphism L-functions Lagrangian Lemma linear matrix modular module morphism motives multiplication neutrino noncommutative geometry noncommutative space notation obtained pair PROOF Proposition quark quotient relation renormalization representation restriction Riemann Shimura variety shows spectral action spectral triple Standard Model subalgebra subgroup symmetries Tannakian category tensor Theorem trace formula vector von Neumann algebra zero zeta function