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?

Comments

# on 20 November 2004, 10:02

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.

# on 20 November 2004, 12:51

Many thanks! I didn't realize that my first suggestion implies discontinuous bouncing. But you're right that if there are no further laws governing the dynamics that's presumably to be expected.

I guess my problem with what you regard as the best view is that I don't understand how something can have a determinable property without having a corresponding determinate property: how something can be located in spacetime without occupying some point or region of spacetime, how something can have a colour without having any specific colour. Then if something like the GRW theory is true and there are no complete collapses (without 'tails'), your view implies that nothing at all is located in spacetime, nor does anything ever have a colour. Which I really find hard to believe.

# on 20 November 2004, 21:51

Well, this is an instance of the "tails" problem for GRW, which some take as a reason to reject GRW (either for alternative collapse dynamics or for non-collapse dynamics). But a GRW proponent can say that we just have to give a slightly revisionary analysis to locutions such as "has such-and-such location". Instead of these being understood as attributing microphysical spatial properties (the fundamental properties in the physics that generate the Hilbert space), they instead attribute certain properties that supervene on the wavefunction. E.g. "an object is located in such-and-such region" will be true iff enough of the relevant squared amplitude in the wavefunction is located in that region. This may have occasional odd consequences, but that's quantum mechanics. One might even justify this analysis quasi-two-dimensionally on the grounds that the relevant supervening properties are the normal causes of our spatial experiences.

# on 21 November 2004, 20:02

Ah ok, that might work, though I don't really see through it right now. It presumably involves saying that our spatiotemporal vocabulary ("is located in region R" etc.) is indeterminate between different ranges of fundmental 'position' properties. Strange!

It seems to me that other interpretations also require such an account. The GRW 'tails' are only especially long (infinite), but to my knowledge other collapse theories also accept not-quite-complete collapses. Moreover, since the collapse of one observable goes hand in hand with a superposition of others, I think any QM theory will have to claim that, say, the position or velocity of the cup in front is slightly 'indeterminate'. But that shouldn't entail that the cup is either not located in spacetime at all or that it completely lacks a velocity.

On no-collapse interpretations, the situation seems much worse, as on this view, things are virtually never even close to an eigenstate of position. So I don't think your explanation works here. Perhaps that's why in no-collapse accounts, indeterminacy is usually described as multiplicity or branching, corresponding to my last two proposals.

# on 23 November 2004, 18:09

I am sorry for my philosopher bashing yesterday evening, but I still hold to it. ;-)

First of all for the first comment by djc: there is nothing wrong with the electron bouncing around discontinousely. In the Feynman path integral picture of quantum mechanics, the electron takes every path leading from A to B weighted with a certain probability factor, this as far as I know, includes discontinous ones.

Second, on what I told you, was my picture of QM, which lead to the first view you present:
Namely, that quantum mechanical particles have a position and a spin etc. but there is just no way whatsoever to determine it, unless you do a measurement. (That's the view for short) I _really_didn't_ mean this statement as an epistemological one in any sense. I was always very uneasy about your statement that God then knew it all, because he is omniscient. Since this leads to a hidden variable theory, although the hidden variables are only apparent to God.
It's just that even he can't know, there is nothing for anybody to know. This is the way, I think, in which quantum mechanics is a indeterministic theory. If QM is right, then, say, the position of aparticle is indeterminate as long as it didn't hit a screen. There is no way to determine how a specific particle on a specific run propagated from A to B.

Saying that it nevertheless has a position means only: Given a certain set up,i.e. the potential in which it moves, you can determine the spectrum of the position operator, so you know all possible positions of the particle and since it is, according to the standard model, a point particle it has no extension and therefore is not smeared over all places. Or as another example, the electron has spin 1/2, then you know, that it must have spin up or spin down. Again this results in knowing _all_possible_values_ of a spin operator.
In other words, I only meant by this, somethink like: Everytime you measure a certain property, you will get a definit value.

All this amounts very much to something which djc calls your "best view", I think. And I really think, quantum mechanics just tells us something like this. Accepting it as the correct theory, leads just to accept this indeterminacy. Of course you can reject QM but then you are very much in the duty for proposing something better. And of course you are free to translate the quantum mechanical superposition state to a language you understand more easily as long as it doesn't conflict with any part of the theory.


# on 25 November 2004, 03:58

Hi Magdalena, thanks for your patience with my old-fashioned world view. Just a minor note on your last paragraph: my problem, I think, isn't really that I don't accept the indeterminacy. It is rather that I don't know what it amounts to. (I can neither accept nor reject a position if I don't understand it.) Here is the problem, once again, in a nutshell:

1) Something x is located in spacetime if and only if there is a spacetime point p such that some part of x is located at p.

This seems to me to be a conceptual truth: that's what I mean by "located in spacetime". Now let x be some determinate electron. You say that

2) for no spacetime point p is "x is located at p" true.

From (1) and (2) it follows that

3) the electron x is not located in spacetime.

However, (3) seems peculiar. My problem therefore is whether I must regard (2) as false or whether, given that "located in spacetime" means what (1) says, I must accept (3). The QM formalism, though it precisely defines superposition in Hilbert spaces, remains silent on this question.

Hidden variable theories and multiplicity/branching interpretations of the formalism reject (2). But they have other strange consequences. (They are not mere "translations into a language I understand more easily" of the QM formalism.)

You and Dave Chalmers accept (3) on the conception of "located in spacetime" expressed by (1). But you both suggest that "being located in spacetime" can be understood differently, either as "the position operator has a determinate value" or as "everytime the position is measured, one gets a determinate value". Dave Chalmers even made a good point that if the standard view of QM is true, something like this may be what, surprisingly, we ordinary people always meant by "being located in spacetime".

Oh well. Do you want the new Jolly and the Flytrap CD?

# on 26 November 2004, 18:25

Yes, I think that's quite correct. Of course the whole problem arises only when you believe that QM really tells you something about how the physical world really is.
There isn't probabely any mean to convince somebody of this if he doesn't want to believe it.
But then you could believe in any theory you like, as long as it provides experimentalists with the right data. (Hidden variables, gods playing quantum mechanics...)

But of course, as a physicist I pretty much do think that we are hunting down the real world.

And of course I also want that CD.

Add a comment

Please leave these fields blank (spam trap):

No HTML please.
You can edit this comment until 30 minutes after posting.