On Arguments for Linear Quantum Dynamics

Abstract

Two recent arguments for linear dynamics in quantum theory are critically re-examined. Neither argument is found to be satisfactory as it stands, although an improved version of one of the arguments can in fact be given. This improved version turns out to be still not completely unproblematic, but it is argued that it contains only a single actual loophole, which is identical to a loophole that remains in experimental proofs of nonlocality of Bell-type. It is concluded that - within the context of the standard quantum kinematical framework and in agreement with what has been concluded by earlier authors - a nonlinear dynamics of density operators is inconsistent with relativistic causality. However, it is also stressed that this conclusion in itself has little implication for the nature of dynamics at the Hilbert space level - in particular, it does not force dynamics to be linear at this level - nor does it continue to be valid in contexts that go beyond the standard quantum kinematical framework. Despite their seeming triviality, these last two points have not always been appreciated in the literature. Finally, it is also pointed out that the argument for complete positivity, as given in conjunction with one of the two recent arguments for linear dynamics, in fact only establishes a condition that is weaker than complete positivity.