From Physics to Mathematics

Abstract

The foundations of quantum mechanics are concerned with Hilbert spaces, linear operators on Hilbert spaces, unitary operators, and self-adjoint operators and their spectra. The foundations of Bohmian mechanics contain nothing of that sort and nothing of that sort seems relevant. Of course, the Schrödinger equation is a partial differential equation and contains differential operators, but so does the Maxwell-Lorentz theory of electromagnetism, which one learns about without all those abstract notions. Why is quantum mechanics different? Why does it need to be based on such abstract mathematical notions? The quantity which determines the empirical import of Bohmian mechanics is the effective wave function, the "collapsed" wave packet which guides the particles.