Why quantum dynamics is linear

Abstract

A seed George planted 45 years ago is still producing fruit now. In 1961, George set out the fundamental proposition that quantum dynamics is described most generally by linear maps of density matrices. Since the first sprout from George's seed appeared in 1962, we have known that George's fundamental proposition can be used to derive the linear Schrodinger equation in cases where it can be expected to apply. Now we have a proof of George's proposition that density matrices are mapped linearly to density matrices, that there can be no nonlinear generalization of this. That completes the derivation of the linear Schrodinger equation. The proof of George's proposition replaces Wigner's theorem that a symmetry transformation is represented by a linear or antilinear operator. The assumption needed to prove George's proposition is just that the dynamics does not depend on anything outside the system but must allow the system to be described as part of a larger system. This replaces the physically less compelling assumption of Wigner's theorem that absolute values of inner products are preserved.The history of this question is reviewed. Nonlinear generalizations of quantum mechanics have been proposed. They predict small but clear nonlinear effects, which very accurate experiments have not seen. This begs the question. Is there a reason in principle why nonlinearity is not found? Is it impossible? Does quantum dynamics have to be linear? Attempts to prove this have not been decisive, because either their assumptions are not compelling or their arguments are not conclusive. The question has been left unsettled. The simple answer, based on a simple assumption, was found in two steps separated by 44 years.