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Consider the following quote from Dirac's book, The Principles of Quantum Mechanics (4th ed., p. 36):

When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second. Hence, after the first measurement has been made, the system is in an eigenstate of the dynamical variable ξ, the eigenvalue it belongs to being equal to the result of the first measurement. This conclusion must still hold if the second measurement is not actually made. In this way we see that a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement.

This is usually presented as a postulate in a system of axioms for QM; here Dirac is arguing for them, and not simply asserting them.

What is curious here, given the association of positivism with science, and particularly in physics, is that Dirac's argument, though going through by instrumentalism, relies on 'measurements ... not made'; and in fact it appears that the underlying principle that he relies on is 'physical continuity'; continuity is of course something that Aristotle goes on at great length in his Physics - there is a whole chapter devoted to it.

On the positivistic interpretation of physics, how does one argue for the principle of physical continuity? And is it correct to say, that once this has been shown, then positivistic sense can be made of 'measurements ... not made?'; or is there some further subtlety I've missed?

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First off, I'm not sure that Dirac was wedded to logical positivism, so his writing need not be consistent with this view of physics.

A good example framework for getting your head around this is by considering iterated Stern-Gerlach experiments. A Stern-Gerlach experiment uses spatially varying magnetic fields to separate beams of particles on the basis of their (discrete) magnetic moment. I.e. one beam comes in, and the apparatus splits the beam into two (spin-1/2 e.g. silver/sodium atoms) or three (spin-1 particles as in the linked discussion) beam, each with proportionately lower intensity.

The first point is that it is an empirical fact that in this type of experiment taking one of the split beams, e.g. the +S beam, and running it through an identically oriented apparatus doesn't further split the beam; this is an empirical example of physical continuity. In principle (I'm not sure how far experiments have actually gone down this route) you can chain various combinations of apparatuses in differing orientations and see what comes out to probe what happens when you perturb the beams. All of this is measurable (in principle).

The second point is that now we need to construct an appropriate language to describe these results. So we start labeling the filtered beams, `+S,0S,-S' etc., and define a mathematical framework that connects them together with the observable outcomes of various experiments. (which ends up showing that these labels are identical to eigenvalues and the representation of the states themselves correspond to the eigenvectors) and so on. At the end of this, you end up with a mathematical formalism that can be used to calculate the statistical outcome of any arrangement of iterated Stern-Gerlach apparatuses, i.e. to describe describe what would happen if a measurement were made even if it is not.

To address each bolded statement in turn:

  1. Dirac just asserts this, but in fact this is an empirical observation
  2. Dirac just asserts this, but in fact it is an empirical observation
  3. This is LP if it is interpreted as a statement about the formal mathematical theory, and not a statement about "reality".

Items 1 and 2 are somewhat wishy-washy in that the principle of continuity applies when we don't perturb the system (basically by definition); and when we don't don't perturb the system, the physical observables satisfy the principle of continuity (basically by definition). One of the nice things about the theory of QM is that it formalizes this notion: if the operator representing the observable commutes with the QM propagator, then the expected value of that operator doesn't change. And since, at least in a Logical Positivist's ideal world, QM is a formal theory built up on the basis of empirical observations, this idea of continuity is empirically based.

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