Geometry and symmetry of quantum and classical-quantum variational principles

This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincar\'e reduction theory is applied to the Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This construction leads to new variational principles for the description of mixed quantum states. The corresponding momentum map properties are presented as they arise from the underlying unitary symmetries. Finally, certain semidirect-product group structures are shown to produce new variational principles for Dirac's interaction picture and the equations of hybrid classical-quantum dynamics.Comment: First version. 23 pages. Comments welcom

Wheeler-DeWitt equation and Lie symmetries in Bianchi scalar-field cosmology

Lie symmetries are discussed for the Wheeler-De Witt equation in Bianchi Class A cosmologies. In particular, we consider General Relativity, minimally coupled scalar field gravity and Hybrid Gravity as paradigmatic examples of the approach. Several invariant solutions are determined and classified according to the form of the scalar field potential. The approach gives rise to a suitable method to select classical solutions and it is based on the first principle of the existence of symmetries.Comment: 17 page

A matrix-based approach to properness and inversion problems for rational surfaces

We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.Comment: 12 pages, latex, revised version accepted for publication in the Journal AAEC