Mathematical surprises and Dirac's formalism in quantum mechanics

By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space formulation, we devote the second part of our paper to the latter approach: we discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided.Comment: Largely extended and reorganized version, with new title and abstract and with 2 figures added (published version), 54 page

Deep inelastic reactions with heavy ions. A probe for nuclear macrophysics studies

The purpose of this survey is to describe a new frontier in nuclear physics and to present prospects of this topic. Beams are now available for heavy ions from Li to Uranium, and they present themselves as a wide range of probes for studying nuclear matter. In the first chapter the main aspects of these probes as well as the general trends of the phenomena are sketched : deflection function, nucleus-nucleus potential, emphasize on collective aspects. Chapter II deals with complete fusion between two complex nuclei, which is considered as the extreme limit of inelastic collision. An analysis is made of the criteria for the existence of a compound nucleus, a well-defined system in which nucleons are in statistical equilibrium. The various limitations to complete fusion are indicated and briefly discussed: angular momentum, repulsive part of the potential due to nuclear matter incompressibility, shape deformation... The de-excitation of this compound nucleus which is highly excited and often shares large angular momenta, proceeds through particle emission, gamma radiation and eventually fission. After describing these decay-modes, a general picture is given of the various aspects of nuclear matter in different shapes and at different temperatures. Chapters III and IV deal of the discovery of a new class of reactions between complex nuclei, the so-called « Deep Inelastic Reactions » which take a great importance. The general description is given in chapter III and the theoretical analysis in chapter IV. They correspond to collisions which do not end up complete fusion, although a very strong change occurs in both partners, and a large kinetic energy dissipation. The nuclear exchange during the interaction might be either small or big, but there is a memory of the projectile and of the target concerning the mass asymmetry. The main feature of these phenomena is now rather well described. After a « deep » or a « hard » contact an intermediate species is formed, made of two nuclei connected by a neck. This short life system begins to rotate under the effect of the orbital angular momentum. The kinetic energy from the entrance channel is dissipated into individual and collective excitations. Then, the rupture into two fragments occurs from a very deformed object, at a time which depends strongly on the charge asymmetry, on the energy and on the angular momentum. The kinetic energy in the exit channel is attributed to Coulomb repulsion plus centrifugal effects after disruption. The statistical equilibrium has not been reached in the « composite » system, but nucleons of both partners have interacted in a potential well very different of the potentials of separated nuclei. Many level-crossings have occurred. The various conceptual ideas of non-equilibrium statistical mechanics, the treatment of transport phenomena and of fluid dynamics have been obviously introduced. This kind of theoretical approach is quite new in nuclear physics where single particle models issued from the atomic orbital quantal description, have been so successful. This novel description of the « nuclear fluid » consists in a « macrophysics approach » and is complementary of the « microscopic description ». This is obviously because we deal with different probes so the heavy ion accelerators bring another set of tools for exploring atomic nuclei

A Report On Gauge Invariant Forms And Variational Problems On The Bundle Of Connections Of A Principal U(1)-Bundle And On Associated Vector Bundles

Basic results on gauge invariance of differential forms on the bundle of connections of an arbitrary principal U(1)-bundle and its associated bundles, are reviewed in terms of the underlying geometry to such bundles, within the framework of classical electromagnetism