A mathematical framework for Dirac's calculus
In the same spirit as in his previous works on relative set theory (RST), the author introduces a mathematical framework closer to the physical intuition, and specially to the Dirac calculus. This framework avoids the introduction of abstract spaces (spaces of Schwartz distributions, ultradistributions, etc.)\ or other kinds of spaces of nonlinear generalized functions to tackle these mathematical problems. In some sense, those spaces are replaced by levels of improperness, which give rise to levels of infinitesimals. Using these levels of improperness, two main tools are introduced. First, they are used here to introduce a notion of observed derivative for large classes of functions. For example, the Dirac function (which can be well defined in this framework through principal evaluations) admits (as one may expect) such an observed derivative. Further, apart from the usual equality, Dirac equalities are introduced. Roughly speaking, these equalities play the role of weak equalities in nonlinear theories of generalized functions. Namely, the similarity that can be pointed out is that the problem of multiplying distributions (such as \(\delta\)) is replaced by the problem of compatibility of the weak equalities with the product. With these tools, the author is able to develop a functional analysis which respects the basic rules of usual calculus, while remaining very close to the intuition. The article is enriched by many examples of practical calculus and very readable and only needs a very basic knowledge in RST. It should interest both mathematicians and theoretical physicists.
scientific article; zbMATH DE number 5178651