From Schwartz Distribution Theory to Relativistic Quantum Mechanics (and Quantum Field Theory) via Schrödinger Equation

The problem of the relativistic description of fermions (half-spin particles) was historically solved by Dirac in the twenties of the last century by the formulation of his celebrated equation. The Dirac equation can be rightfully considered the relativistic counterpart of the Pauli equation, although it strangely acts on the state space S4, instead of the Pauli state space S2, where S is the Schrödinger equation state space. In the scientific literature, the problem of finding the relativistic counterpart of the classic Schrödinger’s equation was still not solved, nor does it seem fully understood, as it has been simplified to the problem of a relativistic equation for spin-1 particles (and subsequent ones). In this research, we found an extremely natural way to formulate (essentially) one (Lorentz-invariant) relativistic equation for all the above cases (spin = 0,1/2,1,3/2,2,...), starting from the original idea of Schrödinger himself and from some fruitful developments of Laurent Schwartz distribution theory. Indeed, the classic Dirac equation deals with half-spin particles; on the contrary, the first relativistic Schrödinger equation we propose here deals with zero-spin relativistic particles. When applying the same equation to bi-waves (bi-tempered distributions) or tri-waves (tri-tempered distributions), we simply obtain the free evolution equation for “positive” fermions (Dirac particles with positive energy) and “positive” spin-1 particles. The equation we propose and solve, in its Lorentz-invariant form, is |g|(P)u = (m0 c)u, where: • g represents the Minkowski metric; • |g| is the positive square root of the Minkowski quadratic form |g|2; • P is the 4-momentum operator; • m0 is the rest mass of the single quantum of the particle-field u; • u is the tempered distribution (or n-tempered distribution) representing the particle-field under exam; • c is the speed of light in a vacuum.