Quantum Mechanics — How Weird for Bayesians?

Abstract

Quantum mechanics is spectacularly successful on the technical level. The meaning of its rules appears, however, shrouded in mystery even today, more than sixty years after its inception. Quantum-mechanical probabilities are often said to be “operator-valued” and therefore fundamentally different from “classical” probabilities, in disregard of the work of Cox (1946) — and of Schrodinger (1947) — on the foundations of probability theory. One central question concerns the superposition principle, i. e. the need to work with interfering wave functions, the absolute squares of which are the probabilities. Other questions concern the collapse of the wave function when new data become available. These questions are reconsidered from the Bayesian viewpoint. The superposition principle is found to be a consequence of an apparently little-known theorem for non-negative Fourier polynomials published by Fejer (1915). Combined with the classical Hamiltonian equations for point particles, it yields all basic features of the quantum-mechanical formalism. It is further shown that the correlations in the spin pair version of the Einstein-Podolsky-Rosen experiment can easily be calculated classically, in contrast to EPR lore. All this demystifies the quantum-mechanical formalism to quite some extent. Questions about the origin and the empirical value of Planck’s quantum of action remain; finite particle size may be part of the answer.