  Quantum dynamics is linear. How do we know? From theory or experiment? 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. 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. There is a simple answer, based on a simple
assumption. It was found in two steps separated by 44 years. They are steps
back to simpler and more compelling assumptions. A proof of the assumptions of
the Wigner-Bargmann proof has been known since 1962. It assumes that the maps
of density matrices in time are linear. For this step, it is also assumed that
density matrices are mapped one-to-one onto density matrices. An alternative is
to assume that pure states are mapped one-to-one onto pure states and that
entropy does not decrease. In a step taken in 2006, it is proved that the maps
of density matrices in time are linear. It is assumed, as in the earlier step,
that at each time the physical quantities and states are described by the usual
linear structures of quantum mechanics, so the question is only about how
things change in time. Beyond that, the proof assumes only 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.
