Homer nodded once more. Von Neumann's misreading of the Compton-Simon experiment and its fallout

Abstract

In his book `Mathematical Foundations of Quantum Mechanics', von Neumann asserted the following: the Compton-Simon experiment showed that the state vector must collapse upon measurement of any self-adjoint operator. Comparing von Neumann's account with the Compton-Simon paper, we find that von Neumann had misinterpreted the experiment as consisting of two successive measurements (which gave identical results), whereas the experiment only measured two angles on the same photographic plate. Note, however, that the state vector must collapse upon measurement of an additively-conserved quantity; otherwise the conservation law could be violated. Next, it turns out that the mathematical problem of explaining collapse is not fully defined until one specifies the nature of the apparatus. If the apparatus does not have a `classical description', the problem is insoluble, even if the measurement is only approximate (Fine, Simony, Brown, Simony and Busch); but if it does, the problem is soluble within Schroedinger dynamics (with a time-dependent hamiltonian) for additively-conserved observables. The solution, a modification of Sewell's, shows that the state vector has collapsed, but it does not reveal the eigenvalue of the collapsed state. The collapse is irreversible, and results from the interplay of additive conservation laws with the quantum measurement postulate. Indeed, the quantum measurement problem - as expounded by Wigner - may be better understood as the problem of establishing that the two are compatible with each other; it has little relevance to actual measurements.