What is Quantum Indeterminacy?
What can we say about physical systems when they are not in an eigenstate of a certain property? For instance, what can we say about an electron's x-spin when it is in a superposition of 'up' and 'down'?
We can say that a measurement of the property will (or rather, would) deliver such and such results with such and such probability. Most physicists apparently think that this is more or less all we can say. In particular, they argue that we should not interpret the superposition state as something like "the probability that the electron now actually has x-spin up is 0.5": having x-spin up (or down) requires being in an eigenstate of x-spin, but the electron is in no such eigenstate; thus the electron definitely has neither x-spin up nor x-spin down; it is in a superposition state, and that's all there is.
There are two things I don't quite understand about this answer: its content and its justification.
The justification usually offered is that rejecting this view -- i.e. assuming that the electron always has a determinate x-spin, even though we are somwhow unable to know it -- amounts to a hidden-variable theory, and hidden-variable theories either don't work or lead to something like Bohm's theory. But why can't I simply add the assumption to classical quantum mechanics without trying to develop an extra theory that directly captures the dynamics of the 'hidden variables'? Couldn't I say that something like the Schrödinger equation captures all that can be captured about the general behaviour of particles? The rest, the determinate values of the properties, is then a matter of irreducible chance, and not explainable in terms of further laws.
Of course, this interpretation would have very strange consequences about external events that influence the properties of a system: what (determinate) path an electron moving through the upper slit of the two-slit device will take depends on whether the lower slit is open or closed; the (determinate) x-spin of an electron sometimes changes when something happens to a specific different electron somewhere else; and so on. So while I don't quite understand the official justification for why this interpretation is bad, I see that it has serious drawbacks.
But what is the alternative, according to which electrons often are in a superposition of x-spin up and x-spin down, in which case their x-spin is "objectively indeterminate"? What does this mean? I know what it looks like in the electron's state space -- the electron's mathematical representation --, and I know what it means for the probability of certain measurements results. But what does it say about the electron itself, actually and now? I see several possibilities.
One is that all we can say about the electron's x-spin now is that it has a probabilistic disposition to cause such-and-such measurement results. This disposition has no categorical basis. Or perhaps its categorical basis is simply the electron's y-spin (or whatever spin direction its current state is an eigenstate of). The electron's x-spin is "indeterminate" only in the sense in which the name of my daughter is indeterminate. (I don't have a daughter, but one can still guess how a daughter I might have had would be called.) Which is not a reasonable sense of "indeterminate" at all.
Another possibility is to treat superpositions as genuine property values. The electron determinately has some property with value SQRT(1/2)|x-spin up> + SQRT(1/2)|x-spin down>. That's obviously not an ordinary x-spin property, but some altogether different kind of property, one that constantly changes by the dynamics of the wave function. So again, it isn't really "indeterminate" what x-spin the electron has. Like the 'hidden variable' spin mentioned above, this new kind of property exhibits a strange dependence on external events. Though entangled systems will presumably not be regarded as genuinely distinct and interacting systems on this view, but rather as one big, distributed entity, with lots of strange new properties.
These proposals both sound much more plausible for spin than for, say, position. It is very hard to believe that electrons usually don't have a spatiotemporal location at all -- somewhat like numbers --, but only a certain disposition to pop up at certain places from time to time, or a certain different property which is at best related to having a location.
When people introduce the 'measurement problem', they sometimes say that according to the ordinary wave dynamics, we should expect to find the pointers of a measuring device pointing at different directions at once. This suggests an interpretation of superposition on which systems in a superposition state have all the corresponding (superimposed) eigenvalues at once. As for location, the system would then be like a universal: multiply located. (Indeed, on some definitions, such a system would be a universal.) Unfortunately, it is hard to see how this view can distinguish between different superpositions of the same base states. One would have to say that the system has the various locations "to different degrees". But unless this can be satisfactorily explained, it amounts exactly to the previous (unacceptable) interpretation I believe. Perhaps one could say that the system is multiple located not only at different places, but also at the same place, so that it could be located more times at one place than at another.
The last possibility is similar except that the systems having the determinate eigenstates are not identified with each other: When our electron is in its x-spin superposition, there are really two electrons, one with x-spin up and another with x-spin down. Or rather, there are infinitely many electrons, as the spin space contains infinitely many observables with different eigenvectors. Again, the expansion coefficients might correspond to the number of electrons having the respective eigenvector. Ultimately, there are always (at least) continuum many electrons, continuum many of which are appearing and disappearing at any moment, where we say there is only one. It isn't clear whether these swarms of things even overlap to a substantial degree, like Geach's 1001 cats.
So all these proposals look rather bad to me. Have I missed an obvious solution?
I take it that your first suggestion is that the Schrodinger equation governs the wavefunction and that a particle's position (say) at a time is the appropriate probabilistic function of the wavefunction state at the time, and that's that. If there's no further constraint governing particle dynamics, then the proposal is compatible with the possibility that the particle bounces around discontinuously from moment to moment (a continuum number of bounces per finite interval!), and indeed such behavior would be expected according to the proposal. To avoid this bizarre behavior, one needs to add further constraints governing particle dynamics. By far the most natural way to do this, allowing continuous particle behavior, is the Bohmian dynamics.
On your main question: I think that given the standard dynamics, the best view is the view you find "hard to believe": the particle has a certain wavefunction property (e.g. a certain position in Hilbert space), and that wavefunction property is probabilistically connected to certain spatial (and other properties) on measurement, and that's that. That is, one can't really ask what the "position" of the particle is before measurement, except in some loose sense. It has the wavefunction property (which isn't really a position or even a set of positions), and that's all.