The mathematical foundations of quantum mechanics

Von Neumann’s 1932 book “The mathematical foundations of quantum mechanics” was a cornerstone for the development of quantum theory, yet his insights have been mostly ignored by physicists. Many of us know that the book exists and that it formalizes the mathematics of operators on a Hilbert space as the basic language of quantum theory, but few have bothered to read it; feeling that as long as the foundations are there, we don’t need to examine them. One reason the book has never been popular as a textbook is that von Neumann does not Dirac’s notation. He admits that Dirac’s representation of quantum mechanics is “scarcely to be surpassed in brevity and elegance” but then goes on to criticize its mathematical rigor, leaving us with a mathematically rigorous treatment but a notation that is difficult to follow and a book that nobody ever reads.

Today, von Neumann’s book is brought up often in connection with foundations research, which is perhaps not surprising given the author’s original intent as stated in the preface:

the principle emphasis shell be placed on the general and fundamental questions which have arisen in connection with this theory. In particular , the difficult problems of interpretation, many of which are even now not fully resolved, will be investigated in detail.

Two particular results are often cited

  1. Von Neumann’s (presumably incorrect) proof of impossibility for a hidden variable model of quantum theory.
  2. The von Neumann measurement scheme, which is the standard formalism for describing a quantum measurement.

The description of the measurement scheme is (or at least was to me) a little surprising. The usual textbook description of von Neumann’s work is to start with the Hamiltonian that couples the measurement device to the system under measurement, and to show that it produces the right dynamics (up to the necessity for state collapse). It turns out that von Neumann’s motivation was in some sense the opposite.

He begins by considering a classical measurement and showing that the observer must be external to the measurement, i.e there are three distinct objects, the system under observation, the measurement device, and the observer. Keeping this as a guide, von Neumann provides a dynamical process where the observer is not quantum mechanical while the measurement device is. The important point is that the precise cut between the (classical) observer and the (quantum) measurement device, is not relevant to the physics of the system, just as in classical physics.

Von Neumann does not suggest that he solves the measurement problem, but he does make it clear that the problem can be pushed as far back as we want, making it irrelevant for most practical purposes, and in some ways just as problematic as it would be in classical physics. Many of us know the mathematics, and could re-derive the result, but few appreciate von Neumann’s motivation: understanding the role of the observer.

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