Relativistic Free Schrödinger Equation for Massive Particles in Schwartz Distribution Spaces

Abstract

In this work, we pose and solve, in tempered distribution spaces, an open problem proposed by Schrödinger in 1925. In particular, on the Schwartz distribution spaces, we define the linear continuous quantum operators associated with relativistic Hamiltonians of massive particles—particles with rest mass different from 0 and evolving in the four-dimensional Minkowski vector space M4. In other words, upon the tempered distribution state-space S′(M4,C), we have found the most natural way to introduce the free-particle relativistic Hamiltonian operator and its corresponding Schrödinger equation (together with its conjugate equation, standing for antiparticles). We have found the entire solution space of our relativistic linear continuous evolution equation by completely solving a division problem in tempered distribution space. We define the Hamiltonian (Schwartz diagonalizable) operator as the principal square root of a strictly positive, Schwartz diagonalizable second-order differential operator (linked with the “Klein–Gordon operator” on the tempered distribution space S4′). The principal square root of a Schwartz nondefective operator is defined in a straightforward way—following the heuristic fashion of some classic and greatly efficient quantum theoretical approach—in the paper itself.