By Alain Connes

The unifying topic of this booklet is the interaction between noncommutative geometry, physics, and quantity thought. the 2 major gadgets of research are areas the place either the noncommutative and the motivic features come to play a job: space-time, the place the tenet is the matter of constructing a quantum conception of gravity, and the gap of primes, the place you'll 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 thought and the geometric constitution of renormalization as a Riemann-Hilbert correspondence. It additionally offers a version of straightforward particle physics in keeping with noncommutative geometry. the most result's a whole derivation of the whole general version Lagrangian from a very easy mathematical enter. different subject matters coated within the first a part of the publication are a noncommutative geometry version of dimensional regularization and its function in anomaly computations, and a quick advent to factors and their conjectural relation to quantum box thought. the second one a part of the publication supplies an interpretation of the Weil specific formulation as a hint formulation and a spectral consciousness of the zeros of the Riemann zeta functionality. this is often in line with the noncommutative geometry of the adele classification house, that is additionally defined because the house of commensurability periods of Q-lattices, and is twin to a noncommutative reason (endomotive) whose cyclic homology presents a basic atmosphere for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the gap 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 concerning issues of classical moduli areas and the symmetries to the category box concept of the sphere of rational numbers and of imaginary quadratic fields, in addition to to the automorphisms of the sector of modular features. The publication ends with a suite of analogies among the noncommutative geometries underlying the mathematical formula of the traditional version minimally coupled to gravity and the moduli areas of Q-lattices utilized in the learn of the zeta functionality.

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**Example text**

There are 9 such pairings. An example of the second case (which yields 6 pairings) is given by k1 ←→ q1 , k2 ←→ q2 , k3 ←→ q3 . These two possibilities are represented pictorially by the vacuum bubble diagrams of Figures 3, 4. 3. FEYNMAN DIAGRAMS 25 Figure 4. Vacuum bubble Figure 5. Tadpole Figure 6. 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.

Its knowledge makes it possible to calculate Green’s functions in a direct manner. 99) P (φ) e−S(φ) D[φ] = N N P (φ) e−Seﬀ (φ) D[φ] . tree−level To compute the right-hand side one isolates the quadratic part of the functional Seﬀ (φ) and then applies perturbation theory, but one drops all non-tree graphs. The basic facts that show how this works are the following. a) Let Γ be a connected graph with two external legs. Then one can write Γ uniquely as a tree graph with interaction vertices given by 1PI graphs Π with two external legs.

Where g is the coupling constant. 53) where, by construction, we have Sint (φE ) = g 3! g ˆ 1 )φ(k ˆ 2 )φ(k ˆ 3 )δ(k1 + k2 + k3 ) φ(k dD kj . The simple mnemonic, in order to keep track of the powers of 2π, is to attach a factor of (2π)−D to each momentum integration dD k and a factor of (2π)D to each δ-function. Using φE (xj ) = (2π)−D ˆ eipxj dD p φ(p) the problem is then reduced, up to an overall factor and to the product (2π)2D δ(k1 + k2 + k3 )δ(q1 + q2 + q3 ), 24 1. QUANTUM FIELDS, NC SPACES, MOTIVES Figure 3.

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