On the equivalence of complex and quaternionic quantum mechanics

Abstract

Due to the existence of incompatible observables, the propositional calculus of a quantum system does not form a Boolean algebra, but an orthomodular lattice. Such lattice can be realised as a lattice of subspaces on a real, complex or quaternionic Hilbert space, which motivated the formulation of real and quaternionic quantum mechanics in addition to the usual complex formulation. It was argued that any real quantum system admits a complex structure that turns it into a complex quantum system and hence real quantum mechanics was soon discarded. Several authors, however, developed a quaternionic version of quantum mechanics and this version did not seem to be equivalent to the standard formulation on a complex Hilbert space. Motivated by some recently developed techniques from quaternionic operator theory, we conjecture in this article that this not correct and that showing the equivalence of real and complex quantum mechanics and the equivalence of complex and quaternionic quantum mechanics are dual problems that can be solved with the same techniques. We furthermore suggest that any quaternionic quantum system is actually simply the quaternionification of a complex quantum system and we show that this conjecture holds true for quaternionic relativistic elementary systems by applying some recent arguments that were used to show the equivalence of real and complex relativistic elementary systems. Finally, we conclude by discussing how the misconception that complex and quaternionic quantum mechanics are inequivalent arose from assuming the existence of a left multiplication on the Hilbert space, which is physically not justified.