Von Neumann’s Completeness Proof

Abstract

Unlike Bohr and Heisenberg, who attempted to ground the completeness of quantum mechanics on a thesis concerning the peculiarities of measurement at the microlevel — an argument depending on considerations extraneous to the theory — von Neumann saw the problem as that of proving the non-existence of hidden variables on the basis of certain structural features of quantum mechanics. The difference is important. For suppose quantum mechanics can be reformulated as a classical statistical theory — none of the arguments proposed by Bohr or Heisenberg proves the contrary. Von Neumann could argue that, insofar as quantum mechanics or its completion is an adequate theory, the Copenhagen interpretation is simply false as a thesis concerning micro-elements. But adherents to this interpretation would have to drop quantum mechanics, regretfully perhaps, and search for a new theory incorporating the ‘feature of wholeness’ or complementarity. The Copenhagen interpretation might conceivably explain why the description of the microlevel requires an ‘irreducibly statistical’ theory, but it cannot guarantee that the mathematical theory of quantum mechanics does in fact have this character.