By Alain Connes

The unifying subject matter of this e-book is the interaction between noncommutative geometry, physics, and quantity idea. the 2 major gadgets of research are areas the place either the noncommutative and the motivic facets come to play a job: space-time, the place the tenet is the matter of constructing a quantum thought of gravity, and the gap of primes, the place you will regard the Riemann speculation as a long-standing challenge motivating the advance of latest geometric instruments. The e-book stresses the relevance of noncommutative geometry in facing those areas. the 1st a part of the publication offers with quantum box conception and the geometric constitution of renormalization as a Riemann-Hilbert correspondence. It additionally provides a version of hassle-free particle physics in line with noncommutative geometry. the most result's a whole derivation of the whole common version Lagrangian from an easy mathematical enter. different themes lined within the first a part of the ebook are a noncommutative geometry version of dimensional regularization and its position in anomaly computations, and a quick creation to explanations and their conjectural relation to quantum box concept. the second one a part of the ebook offers an interpretation of the Weil particular formulation as a hint formulation and a spectral consciousness of the zeros of the Riemann zeta functionality. this can be in accordance with the noncommutative geometry of the adele type house, that is additionally defined because the house of commensurability sessions of Q-lattices, and is twin to a noncommutative purpose (endomotive) whose cyclic homology offers a basic environment 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 structures are relating to issues of classical moduli areas and the symmetries to the category box idea of the sphere of rational numbers and of imaginary quadratic fields, in addition to to the automorphisms of the sphere of modular capabilities. The publication 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 examine of the zeta functionality.

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XN ) dx1 · · · dxN , and is similar to the free energy of a statistical mechanical system in the presence of the source JE . 89) W = Γ connected V (Γ) . σ(Γ) 3. FEYNMAN DIAGRAMS 37 Figure 12. 91) V (Γ1 ∪ Γ2 ) = V (Γ1 ) V (Γ2 ) , with σ(Γ) = σ(Γj )nj nj ! where the Γj are the distinct graphs that appear as components of Γ and the nj are the multiplicities of their occurrences. 93) V (Γ)(J) = 1 N! ˆ 1 ) · · · J(p ˆ N ) V (Γ(p1 , . . , pN )) J(p j dpj . 5. The eﬀective action and one-particle irreducible graphs.

Around 1947, motivated by the experimental ﬁndings of spectroscopy about the ﬁne structure of spectra, physicists were able to exploit the distinction between these two notions of mass (bare and observed), and similar distinctions for the charge and ﬁeld strength, in order to eliminate the unwanted inﬁnities associated to the pointwise nature of the electron. This marked the beginning of renormalization techniques in quantum ﬁeld theory. We refer the interested reader to [125] for a detailed account of the historical developments of the subject.

Double tadpole ˆ ˆ In the other terms, the factors of φ(p) and φ(−p) get paired with the ˆ ˆ ˆ ˆ ˆ ˆ terms of the monomial φ(k1 ) φ(k2 ) φ(k3 ) φ(q1 ) φ(q2 ) φ(q3 ). e. with a k and a q). These cases are represented pictorially by the diagrams of Figure 5 for the ﬁrst case, and Figure 6 and Figure 7 for the second case. The latter, in fact, splits into two subcases according to the pairings of the remaining terms in the two groups. e. momentum conservation) shows that in all the tadpole graphs the momentum k ﬂowing through the single 26 1.