An introduction to transposable Schwartz families

Abstract

In this paper we start to construct some fundamental features of Dirac Calculus, specifically, we go inside the theory of Heisenberg continuous matrices, which, in our Schwartz Linear Algebra, are represented by Schwartz families. We distinguish the important subclass of transposable continuous matrices and give some basic and very important examples in Quantum Mechanics. So we define transposable Schwartz families and their transpose families, we prove the transposability of Dirac families and Fourier families. We find the transpose of regular-distribution families in a much general case. We define symmetric families, the analogous of symmetric ma- trices in the continuous case. We prove the symmetry of Dirac families and of Fourier families. We define Hermitian families, the analogous of Hermitian matrices in the continuous case. We prove the Hermitianity of Dirac families and of Fourier families. We define unitary families, the analogous of unitary matrices in the continuous case. We prove the unitarity of Dirac families and of the fundamental normalized de Broglie family. Then, we use the transpose of a family to find the components of the superpositions of transposable families, we give a general result and we apply this result to the Dirac families and the eigenfamilies of the vector-wave operator. We shall use the transposable families in next chapters to define the Dirac product in distribution spaces, basic product for the entire foundation of Dirac Calculus and Quantum Mechanics formalism.