Josef Janyška

Covariant Schrödinger operator

We analyse the Schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field. We approach the Schrodinger operator in three independent ways: in terms of covariant differentials induced by the quantum connection, via a quantum Lagrangian and directly by the only requirement of general covariance. In particular, in the flat case, our Schrodinger operator coincides with the standard one.

Hermitian vector fields and special phase functions

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

Higher-order utiyama invariant interaction*

In this paper we prove higher order version of the Utiyamas invariant interaction of a particle and a gauge field (a principal connection on a principal bundle). To describe the Utiyamas interaction in order

GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

r\ge 2

Covariant Quantum Mechanics and Quantum Symmetries

we have to use an auxiliary gravitational field (a linear symmetric connection on the base manifold). We prove that any natural (invariant) operator of order

Higher order Utiyama-like theorem

s

On the characterization of infinitesimal symmetries of the relativistic phase space

in the gravitational field, of order

Higher order valued reduction theorems for classical connections

r

Special phase functions and phase infinitesimal symmetries in classical general relativity

in the gauge field and of order

Natural principal connections on the principal gauge prolongation of a principal bundle

k