We calculate the integrated trace anomaly for a real spin-0 scalar field in six dimensions in a torsionless curved space without a boundary. We use a path integral approach for a corresponding supersymmetric quantum mechanical model. Weyl ordering the corresponding Hamiltonian in phase space, an extra two-loop counterterm 1 8 (

Canonical transformations are studied in the bosonic Fock space using coherent and ultracoherent vectors. The connection between the Weyl operator and the homogeneous canonical transformations (Bogoliubov transformations) is presented. The unitary ray representation of the symplectic group is defined in the bosonic Fock space by the action of the group on the ultracoherent vectors.

We compute the Weyl anomaly for an abelian Wilson surface by using a regularization that respects the gauge invariance. We then study the loop space on which lives a one-form connection. We restrict ourselves to the subsector consisting of only concentrical circular loops, and derive a Maxwell type action on this restricted loop space.

We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.

5 1. Prologue 5 2. INTRODUCTION 8 2.1. The general concerns 8 2.2. The geometry of finite dimensional Cartan-Lie groups and abstract Lie groups 9 2.3. The information geometry of complex systems 9 2.4. The gauge group 9 2.5. The overview of the main results: Solutions to some problems EX(S) 10 2.6. Geometry of Lie groups 13 2.7. A comment on Figues 17 3. BASIC NOTIONS. 18 3.1. Cartan-Lie groups...

Vortex configurations in the two-dimensional torus are considered in noncommu-tative space. We analyze the BPS equations of the Abelian Higgs model. Numerical solutions are constructed for the self-dual and anti-self dual cases by extending an algorithm originally developed for ordinary commutative space. We work within the Fock space approach to noncommutative theories and the Moyal-Weyl conne...