abstract

In this series of three lectures, baryon number violation at high temperatures in the Weinberg–Salam model is discussed. The first lecture presents a discussion of anomalies, and how this is related to level crossing of energy levels in the Dirac equation for fermions in an external field. The second lecture discusses topological aspects of the Weinberg–Salam theory, and some related two dimensional models. The sphaleron solution of these theories is constructed. In the final lecture, the sphaleron is related to transition rates at finite temperature. In a simple quantum mechanics model, it is shown that sphalerons, not instantons, are responsible for transitions at high temperature. The sphaleron induced rate is then discussed in a solvable \(1 + 1\) dimensional model, which has many similarities to the Weinberg–Salam model. Finally, the result for the Weinberg–Salam model is derived, and is shown to be large for temperatures \(T\geq 1\) TeV.

abstract

A brief review is presented of recent developments in the understanding of the cancellation of gauge and gravitational anomalies in string theory, in relation to modular invariance.

abstract

In these lecture notes we discuss an analytic calculational scheme for SU(\(N\)) gauge theories in a finite volume. We will mainly concentrate on pure SU(2) gauge theory. The method relies on using an effective Hamiltonian for the zero momentum modes. The notorious problem of Gribov horizons is evaded by encoding the topological nontrivial nature of configuration space into boundary conditions for the zero momentum modes. This system then allows us to compute the low-lying energy spectrum in volumes up to about five times the size of the scalar glueball. These continuum results agree in general well with the lattice Monte Carlo results. We discuss in some detail the resolution of a discrepancy with Monte Carlo results for the \(T^+_2\) glueball.

abstract

The theory of strings should be the theory of random surfaces. I discuss the present understanding of this in view of the new results of Knizhnik, Polyakov and Zamolodchikov.

abstract

After a brief historical and contextual introduction to random networks of automata we review recent numerical and analytical results. Open questions and unsolved problems are pointed out and discussed. One such question is also answered; it is shown that the size of the stable core can be used as order parameter for a transition between phases of frozen and chaotic network behavior. A mean-field-like but exact self-consistency equation for the size of the stable core is given. A new derivation of critical parameter values follows from it.