Excitation of an inertial Unruh detector in the Minkowski vacuum: a numerical calculation using spherical modes

We consider the excitation of a finite-length inertial Unruh detector in the Minkowski vacuum with an adiabatic switch on of the interaction in the infinite past and a sudden switch off at finite times, and obtain the excitation probability via a numerical calculation using the expansion of the quantum field in spherical modes. We evaluate first the excitation probabilities for the final states of the field with one particle per mode, and then we sum over the modes. An interesting feature is that, despite of the inertial trajectory and of the vacuum state of the field, the multipole components of the excitation probability are time-dependent quantities. We make clear how the multipole sum yields the time-independent probability characteristic to an inertial trajectory. In passing, we point out that the excitation probability for a sudden switch on of the interaction in the infinite past is precisely twice as large as that for an adiabatic switch on. The procedure can be easily extended to obtain the response of the detector along radial trajectories in spherically symmetric spacetimes.Comment: 29 pages, 10 figures; submitted to Proceedings of TIM 17 Physics Conferenc

An index theorem for families invariant with respect to a bundle of Lie groups

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle \GR of Lie groups. If the fibers of \GR \to B are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singer type formula. We also study traces on the corresponding algebras of pseudodifferential operators and obtain a local index formula for such families of invariant operators, using the Fedosov product. For topologically non-trivial bundles we have to use methods of non-commutative geometry. We discuss then as an application the construction of ``higher-eta invariants,'' which are morphisms K_n(\PsS {\infty}Y) \to \CC. The algebras of invariant pseudodifferential operators that we study, \Psm {\infty}Y and \PsS {\infty}Y, are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in \RR^q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators.Comment: AMS-Latex, 39 pages, references, corrections, and new results adde

Competitiveness of Romania’s South-East Region in the European Context

The European interventionist policy is much stronger on the regional level than on the level of national states. Each economic activity in Europe’s regions has now its own place on the European market economy. Creating a European market is important due to its size and economic potential. The Central and Eastern Europe is a potential area of new markets expansion and organization. Moreover expansion and trade are becoming important to the entire European economy as well as all its regions.region, regional development, competitiveness, European Union