Relativistic Quantum Mechanics in Schwartz distribution spaces defined on Minkowski space-time

The intent of the general analysis conducted in this keynote is to start from Laurent Schwartz distribution spaces based on the Minkowski space-time, following the spirit itself of distribution Theory, and give precise mathematical meanings and rigorous support to many dubious calculus methods of Quantum Mechanics, starting from the renowned Dirac Calculus. Our approach not only provides a rigorous justification for the use of many quantum mathematical tools substantially “as they usually show up in the physical practice”, but - by a “correct“ formulation of the calculus methods in terms of contemporary mathematics - it helps us to reach a deeper comprehension of the physical structures studied in Quantum Mechanics. In particular, in this direction, we consider a new definition of state spaces for quantum systems. These structures are, often erroneously, identified only with separable Hilbert spaces, but, on the contrary, because of the existence of so called “non-normalizable” states - which reveal fundamental in the development of the quantum mechanics - they should be something different. Some physicists call them “physical Hilbert spaces”, without providing a clear mathematical definition. We have already shown in the past that “physical Hilbert spaces” can smoothly be identified with distribution spaces on suitable Euclidean spaces, depending from the nature of the quantum system considered. Such distribution spaces should be endowed with some algebraic-topological structures such that continuous operations of superposition and the extended Dirac product. In particular, the extended Dirac product allows us to introduce new Scalar products upon some distinguished subspaces of distribution space, endowing those subspaces with nonseparable Hilbert structures, which clarify definitely the role of the so-called non-normalizable states; for example, the singular Dirac distributions and the celebrated De Broglie waves become elements of those new non separable Hilbert spaces and consequently they acquire the status of normalizable states, as it seems completely natural because of the physical usual probability interpretation of such states. The new operation of continuous-superposition revealed the right tool which allows us to build - in a mathematically rigorous way - the extended Linear Algebra of Maurice Dirac, in distribution spaces, using the Schwartz natural topological-linear structures of those spaces. More precisely, we saw that the natural algebraic-topological structure of those spaces allows to define an extension of the finite linear combination, when the sets indexing the families of vectors are continuous sets, even in the case in which the systems of coefficients show a continuous-infinity of terms different from zero. In particular, we solve the problem of quantizing the relativistic Hamiltonian of a free massive particle massive particle (rest mass different from 0). In distribution state spaces, we find a natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation. We, then, deduce the equivalent continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine the possible probability currents and flux velocity fields associated with the particle evolution.