Thursday, March 23, 2017

QM for dummies

Sometimes I think my "QM for dummies" requires too much understanding of tensor products on Hilbert spaces.

In light of that, here's a non-mathy intro taken from my journal. More to come later. Some background: Einstein, Podolsky, and Rosen published a paper claiming that QM is incomplete, because otherwise it is absurd in a particular way. The experiment involves two particles, originally conjoined, and sent off in opposite directions. Each particle gets measured along one particular axis (x, y, or z), and each result can either be +1/2 or -1/2. ...

I think I explained the EPR experiment poorly the day before. Something about how you end up with measurements that cannot be explained classically. Because in the classical case, it is only possible to see certain distributions of outcomes, and they're violated in the actual experiment. And if I showed you the math, you'd understand….

But there's a much more simple way to explain it: the outcome of a measurement here can depend on the choice of what to measure there, even though light itself could not travel fast enough to tell the particle here "how to behave" (i.e., how it should be measured). To show this, you'd need to (a) demonstrate that the outcome of measurement here and the choice of measurement there are correlated, and (b) do it using an experiment where the two measurements happen almost simultaneously. You might be thinking: can't the correlation be explained by them having shared some information when they started out (together)? This is called a "hidden variable" (i.e., some influence that we just haven't discovered). The only other alternative is what they dubbed "spooky action at a distance." That was clearly wrong, so there must be hidden variables, they said.

What JS Bell showed was that even hidden variables have their limits: they could not produce the outcome that QM would predict, for his particular experiment. If it's not hidden variables, then it must be spooky action, right? Actually, no. The preferred interpretation is that there is no action. It's just that the joint state of the system is described by a mathematical entity that doesn't have well-defined values on both sub-systems (the spin of the particles). In other words, it's simply meaningless to say that a particle even has a well-defined spin (either plus or minus ½) in a given direction, assuming it was last measured along some other direction (called a "non-commuting" direction). This is the famous Heisenberg Uncertainty Principle. It also is the wave-particle duality: if you know where the light "is," it cannot behave like a wave. And when it acts like a wave, you cannot know where it is -- because it is not in any particular place. It's not just that we don't know where it is.

It's like the EPR pair above: the second particle does not have a well-defined spin in the chosen direction. If it did, its result wouldn't depend on the choice of measurement of the first particle. If it has a well-defined spin now, it can only be because the other one "gave" it one now, after having been measured.

So the natural question is: what constitutes a measurement? Because whatever it is "prevents" particles from acting like waves. And this is where the story really starts to get fun….

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