Abstract:
These lecture notes give a pedagogical introduction to the use of dispersion relations in loop calculations. We first derive dispersion relations which allow us to recover the real part of a physical amplitude from the knowledge of its absorptive part along the branch cut. In perturbative calculations, the latter may be constructed by means of Cutkosky's rule, which is briefly discussed. For illustration, we apply this procedure at one loop to the photon vacuum-polarization function induced by leptons as well as to the $\gamma f\bar f$ vertex form factor generated by the exchange of a massive vector boson between the two fermion legs. We also show how the hadronic contribution to the photon vacuum polarization may be extracted from the total cross section of hadron production in $e^+e^-$ annihilation measured as a function of energy. Finally, we outline the application of dispersive techniques at the two-loop level, considering as an example the bosonic decay width of a high-mass Higgs boson.

Abstract:
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory.

Abstract:
We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.

Abstract:
Recent results on two--loop calculations in the MSSM are reviewed. The computation of the QCD corrections to \Delta\rho, \Delta r and to the mass of the lightest Higgs boso in the MSSM are presented.

Abstract:
We study the 2-loop part of the rational Kontsevich integral of a knot in an integer homology sphere. We give a general formula which explains how the 2-loop part of the Kontsevich integral of a knot changes after surgery on a single clasper whose leaves are not linked to the knot. As an application, we relate this formula with a conjecture of L. Rozansky about integrality of the 2-loop polynomial of a knot.

Abstract:
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory as well as number theory and report on recent results in support of this connection.

Abstract:
Three programs are presented for automatically generating and calculating Feynman diagrams: the diagrams are generated with FeynArts, algebraically simplified with FormCalc, and finally evaluated numerically using the LoopTools package. The calculations are performed analytically as far as possible, with results given in a form well suited for numerical evaluation. The latter is then straightforward using the implementations of the one-loop integrals in LoopTools.

Abstract:
The question addressed by thls paper is, how close is the tunnel number of a knot to the minimum number of relators in a presentation of the knot group? A dubious, but useful conjecture, is that these two invariants are equal. (The analogous assertion applied to 3-manifolds is known to be false. [1]). It has been shown recently [2] that not all presentations of a knot group are “geometric ”. The main result in this paper asserts that the tunnel number is equal to the minimum number of relators among presentations satisfying a somewhat restrictive condition, that is, that such presentations are always geometric.

Abstract:
These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar's variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field.