Von Neumann, (re-)constructions and hidden variables

Abstract

I claim that the idea of "reconstructing" quantum mechanics along the lines of quantum probability was there from the beginning of the theory as we know it, in von Neumann's paper "Probabilistic construction of quantum mechanics". In this paper, von Neumann derives (and often introduces for the first time) a slew of now-familiar concepts in quantum mechanics, such as both pure and mixed states (respectively as wavefunctions and—generally unnormalized—density operators), the general form of the Born rule for expectation values, a rather general form of transition probabilities, and a recognisable version of the collapse postulate for the general case of maximal measurements and in a special case of non-maximal measurements. He does so axiomatically, starting from two probabilistic axioms that are neutral between classical and quantum probability and two axioms characterising quantum mechanical quantities in terms of a non-commutative algebra (that of self-adjoint operators in Hilbert space). The presentation of this material in von Neumann's 1932 book is usually understood as a "no-hidden-variables" theorem. However, the sense in which a "completion" of the theory is ruled out is merely that a theory with dispersion-free states would require a different algebraic structure. Grete Hermann pointed this out already in the 1930s (she later retracted the additional claim of "circularity"), and it is arguably also how von Neumann himself saw his result. The more fundamental disagreement between von Neumann and Hermann (and arguably between von Neumann and Bohr) turns out to be about the nature of the quantum state and the status of collapse.