MadSci Network: Physics

Re: Why is Bellīs theorem, and itīs practical results seen as so spectacular?

Date: Sun Jun 10 21:00:13 2001
Posted By: Benjamin Monreal, Grad student, Physics, MIT
Area of science: Physics
ID: 991945430.Ph

Hi Mikael,

The ideas behind Bell's Theorem, and the related Einstein-Rosen-Podolsky Paradox (or EPR paradox) are right at the foundations of quantum mechanics. I'm not sure that I myself fully appreciate the depth of Bell's Theorem, nonlocality, many-worlds, and the other weirdnesses of quantum theory. But I'll take a shot at addressing your question. Have a seat, this is sort of lengthy. :)

First let me try to describe quantum theory briefly, in my own words:

"In order to properly describe the behavior of a particle, you have to figure out all possible behaviors of that particle. A real particle will actually behave as if all of the allowable behaviors were present, some more likely than others, all interfering with one another, superimposed or "mixed" with one another, etc.. However, whenever the particle interacts with something else it must choose one and only one behavior. (If you already know a bit of quantum mechanics, I hope that this statement makes sense. I just want to make clear what terms I use, and what aspects of the theory I will talk about below.) Behavior can mean "position", "speed", "energy" ... for example, in the classic "Young's double-slit experiment", you observe that single electrons exit in multiple positions, that they interfere with one another, but when the time comes to measure them they "choose" a single position-state. Good examples of Bell's Theorem use "spin direction" as the behavior. The real question involved is: when I force a mixed state to "choose" one of its available options, how exactly does the choice occur? Especially if the mixture extends over a large region of space, and the choice-forcing measurement only touches a small neighborhood?

Actually, the usual example of the Bell experiment (dealing with spin-directions) can be sort of confusing. Let me raise some of the same issues in a simpler context. Suppose I put a very light particle in a big sealed box, in such a way that I have no idea where it is. So the "behavior" to look at is "position". In quantum mechanics, the true state of the particle within the box says that the particle is in a mixture of all possible values of "position". A measurement has a certain probability of causing a "choice" of any particular position. The mixed state of the particle extends over the entire box, and if we force the particle to "choose", it will no longer be mixed and will no longer extend over the whole box. So somehow the particle jumps from a large, extended state, into a small localized state, when we measure it.

The problem is that the box is very large. If I do a measurement and find the particle in (say) the upper northeast corner, then a simultaneous measurement is certain not to find the particle in the opposite corner. But, hang on, we said that the particle's true state was distributed all over the box - how does the "part of the particle" in the southwest corner know that a choice has been forced up in the northeast? If there really is some sort of "probability field" on one end of the box, how does that field know to go away when the particle decides to emerge on the other end? Even if the two ends are separated by meters, kilometers, or light-years? It seems as though the choice affects the whole collection of behaviors at once, no matter how far apart they are or what the speed of light is.

The natural objection: maybe the particle isn't some mysterious mixture extending over the whole box, but has actually existed in one state all along. If the particle actually is in the northeast, of course it can't be found in the southwest! But this is to deny that the particle was in a mixed state extending over a large space. This contradicts other aspects of quantum mechanics - indeed, the whole of quantum mechanics rests on the idea of mixed-states, and the idea that they choose particular states at particular times. There really is a difference between a mixed or superposition state and a "well-defined state that we haven't made note of". A box containing a particle in the northeast corner is different than a box containing a particle in a superposition of all possible locations.

In technical terms, the "mixed" state of a particle is described by something called a wavefunction, and the particular states that can be chosen are called "eigenfunctions".

The usual Bell's Theorem example, involving electron spins or photon polarizations, is important because it shows exactly how a mixed state is different than an unmixed state. It's subtle, but it works. This is the key to your question - you cannot say that "Of course we observe a single behavior everywhere at once, since it was behaving that way all along, whether we looked at it or not." It is demonstrably not true that the system was in a single state all along. It was in a genuinely mixed, undetermined state. (In the usual Bell example, it's true that the particles had one spin up and one spin down, but they really don't choose which one gets which spin, until the measurement! There are two available configurations (up-down or down-up), and the initial state mixes both of them!)

Understanding superposition-of-states, understanding the actual experiments with Bell's Theorem, and following the debate - which continues even today - about what quantum states really mean, is a big task. I hope this answer helps a little bit. You'll have to do lots of reading (and, ideally, actual problem-solving) to really get a grip on these things. In particular, understanding how a "mixed state" really behaves (and how it differs from a "well-defined state that nobody has measured yet") is a challenge, but a surmountable one!

On the truly practical side, I should point out that nonlocality can not be used to communicate faster than light. See Nick Herbert's book for details.

-Ben Monreal

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