Quantum Mechanics and Its Evolving Formulations

In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.

Generalized Maps

Spaces of generalised sections (also called section-distributions) are introduced, and their fundamental properties are described. Several special cases are considered, with particular attention to the case of semi-densities; when a Hermitian structure on the underlying classical bundle is given, these determine a rigged Hilbert space, which can be regarded as a basic notion in quantum geometry. The essentials of tensor products in distributional spaces, kernels and Fourier transforms are exposed.

Dynamics of algebras in quantum unstable systems

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is nonunitary. This allows to describe observables that are initially noncommutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.