Dirac Calculus in distribution spaces for Quantum Mechanics and Quantum Field Theory

In his famous treatise “Principles of Quantum Mechanics” (1930), the great mathematician and physicist Paul Dirac introduced several “manipulation rules” for vectors and operators of linear spaces, which together constitute the so-called “Dirac Calculus”. This Calculus is nothing more than a wide set of formal extensions of the basic properties of the finite-dimensional Linear Algebra to the case of infinite-dimensional vector spaces. The discourse is elegant and surprisingly efficient, but it is far from being a rigorous mathematical treatment. As mathematicians well know, the passage from the finite to the infinite dimensional case does not amount to a mere substitution of finite linear combinations with formal integrals! The goal of the research introduced in this talk is to give a precise mathematical meaning and rigorous support to many analytic methods of Quantum Mechanics, starting from the fundamental Dirac Calculus, using the Weak Duality Theory of L. Schwartz and J. A. Dieudonné and the L. Schwartz Theory of Distributions. This approach will give a rigorous justification for the use of Dirac's tools, leaving them substantially “as they are” in Quantum Mechanics practice. Moreover, by providing a correct interpretation of these heuristic methods in terms of new solid and efficient mathematical concepts, we will be helped in reaching a deeper understanding of the physical structures studied in Quantum Mechanics. The new operations of continuous-superposition and that of the Dirac product allow us to build - in a mathematically rigorous way - the "Dirac extended Linear Algebra" in the spaces of tempered distributions, via their natural topological linear structures. More precisely, we shall see that the algebraic-topological structure of tempered distribution spaces allows us to define - naturally - the linear combinations of a continuous family of vectors and operators and a scalar product (of a vector by such continuous families of vectors) which are necessary for the modern theoretical development of Quantum Mechanics. The application of the rigorous Dirac Calculus to QFT is yet an open and stimulating problem.