Mathematical Foundations of Quantum Mechanics

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The book Mathematical Foundations of Quantum Mechanics (1932) by John von Neumann is an important early work in the development of quantum theory.

Quotes about Mathematical Foundations of Quantum Mechanics[edit]

  • As stated repeatedly in this book, John von Neumann's Mathematical Foundations of Quantum Mechanics was an extraordinarily influential work. It is important to recall that the language most commonly used to describe and discuss the measurement process in terms of a collapse or projection of the wave function essentially originates with this classic work. It was von Neumann who so clearly distinguished (in the mathematical sense) between the continuous time-symmetric quantum mechanical equations of motion and the discontinuous, time-asymmetric measurement process. Although much of his contribution to the development of the theory was made broadly within the boundaries of the Copenhagen view, he stepped beyond those boundaries in his interpretation of quantum measurement.
    • Jim Baggott, Beyond Measure (2004), Ch. 13 : I think, therefore ...
  • Thus the formal proof of von Neumann does not justify his informal conclusion: 'It is therefore not, as is often assumed, a question of reinterpretation of quantum mechanics - the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.' It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.
    • John Stewart Bell, "On the problem of hidden variables in quantum mechanics". Reviews of Modern Physics (1966)
  • The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!
  • There are (at least) two possible ways to formulate precisely (i.e. mathematically) elementary QM. The eldest one, historically speaking, is due to von Neumann in essence, and is formulated using the language of Hilbert spaces and the spectral theory of unbounded operators. A more recent and mature formulation was developed by several authors in the attempt to solve quantum field theory problems in mathematical physics. … The newer formulation can be considered an extension of the former one, in a very precise sense that we shall not go into here, also by virtue of the novel physical context it introduces and by the possibility of treating physical systems with infinitely many degrees of freedom, i.e. quantum fields. In particular, this second formulation makes precise sense of the demand for locality and covariance of relativistic quantum field theories, and allows to extend quantum field theories to a curved spacetime.
    • Valter Moretti, Spectral Theory and Quantum Mechanics, 2nd ed. (2017), p. 6

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