Solving the mysteries of quantum mechanics: why nature abhors a continuum

Abstract Feynman asserted that interference is the only mystery in quantum mechanics (QM). It is concluded that the reason for this mystery, and thereby the related mysteries of complementarity, non-commutativity of observables, the uncertainty principle and violation of Bell's equality, is that the Hilbert space continuum of QM is unphysical. We describe a theory of quantum physics—rational QM (RaQM)—in which Hilbert space is (gravitationally) discretized. The key to solving the mysteries of QM in RaQM is a ubiquitous number-theoretic property of the cosine function, concealed in QM when angles range over the continuum. This property implies that the laws of physics are profoundly holistic, but not non-local. In the former, correlations between remote parts of the universe exist, but an intervention at one place cannot instantaneously influence affairs at the other. In the latter, such interventions can spread non-causally. Examples of holistic theories include Mach's principle, Penrose spin networks or the fractal geometry of chaotic attractors. Indeed, RaQM solves the deepest mystery of all; why nature makes use of complex numbers. It is to permit realistic incommensurate discretizations of incompatible observables. Such a solution is wholly concealed in the continuum theory of QM.