Tuesday, November 29, 2011

Einstein=>Bohm=>Feynmann=>Bell


Yesterday I said I had some insights to share concerning EPR, Bell, and entanglement. This whole questions is one of the central philosophical issues of quantum mechanics, and it is talked about EVERYWHERE. But there is something very wrong with the standard narrative. The story as it is normally told does not make complete sense to me. Something important is missing, and I haven't been able to put my finger on it. Let me explain.

The EPR paradox was, from the outset, a philosphical question which was not thought to have direct experimental consequences. Over time, this situation began to change, and ultimately with the publication of Bell's ground-breaking analysis 1964, experimenters were galvanized into action. Within eight years the first results had begun to appear (Clauser 1972) culminating with the famous results of Aspect in the 19880's. I'm recalling this from memory but I think I've pretty much got it right.

When Einstein put forward the paradox in 1935, there was no talk of entangled photons. His example was much more understandable. (There is some argument as to whether the published example was due more to Einstein, Podalsky, or Rosen but that's another question.) Essentially it deals with an unstable atom that spontaneously disintegrates for some reason. I believe a suitable example would be an atom in an excited state that emits a photon. According to Copenhagen, the emitted photon may be detected anywhere on the surface of an expanding sphere: the direction is random; and the atom which recoils will therefore be found travelling in a direction diametrically opposed to the photon, in order for conservation of momentum to be preserved. None of this seems peculiar in any way.

What Einstein pointed out was that strictly speaking, right up to the moment of detection the photon might have been found anywhere on that expanding sphere....and likewise the recoiling atom. It is only after the moment of detection of one particle that the direction of the other becomes determined.

This is very different from the obvious interpretation. The obvious determination is that at the moment of disintegration, the photon took off in one direction and the atom in the other. The "randomness" in question was merely our lack of knowledge of which directions the particles took. In this interpretation, there is nothing unusual in the fact that measuring the direction of one particle automatically tells us the direction of the other particle.

Einstein's argument was that this interpretation was not consistent with the theory of quantum mechanics. According to the theory, either of the two particles, up to the very moment of detection, truly had the capabilities of manifiesting their presence at any point on the surface of their respective spherical wavefronts! and it was only after either one of them made its presence felt in a detector, that the directional vector of its counterpart became fixed. In other words, a measurement on one side of the laboratory would instantly impact on the physical status of the particle on the other side of the laboratory. And this instant change of status took place no matter how big the room...even if it would seem that the information to get from A to B would have had to travel faster than the speed of light.

The catch was that no one believed at the time that this was any more than a philosophical question. How could anyone ever prove that the two particles hadn't shot off in their chosen directions at the very moment of disintegration? Certainly the theory said otherwise; but experimentally, all that anyone could ever do was to verify that the direction were indeed opposite to each other. No one knew a way to show that the particle which was in fact detected at A "might just as well" have been detected at B, but for random chance. People could shrug off the issue of faster-than-light comminucation with the answer that this was a mere artifact of the theoretical construct, not subject to experimental verification.

And so the matter rested until 1950. In that year, David Bohm proposed a different mechanism to settle the question. It was well known that in a helium atom, the two electrons had opposite spin orientations: if one was up, the other had to be down, and vice versa. Not everyone understood, as Bohm did, that the concepts of "up" and "down" were mere arbitrary labels that did not really describe physical reality. Which way was "up"? In fact there can be no answer to this question. It is a little bit more correct to say that whatever direction the spin of one electron is oriented, the other electron has its spin pointing the opposite way. But even this description does not do justice to the truth.

Bohm understood that the spins of the two electrons were so intimately "entangled" that there was absolutely no way to speak in a meaningful way about any preferred direction. The spins simply cancelled out to zero everywhere. It was a total system with zero spin, period. (I actually explain how two particles with spin can combine into a spin-zero state in this blogpost of May 2011 ).

Bohm proposed that if a helium atom could be made to expel its electrons, the spin-zero state of the electrons would be preserved. They would shoot off in opposite directions. If either electron were captured by a spin measuring apparatus such as a modified Stern-Gerlach setup, it would of necessity be detected in one of two possible spin states, depending on the orientation of the detector. This was not controversial. What Bohm pointed out, however, was that the other electron, once the first electron had been measured, was no longer free to have any old random spin: its spin, if measured would necessarily have to be precisely opposite the spin of the first one.

What made this different from Einstein's original example? For the first time, there was the tantalizing prospect that this effect might be experimentally observable. There was unfortunately a catch: no one knew a way to make a helium atom spontaneously expel both electrons. To this day, Bohm's experimnet has never been done; nor has any other experiment been done to measurement the entangled spin state of two electrons.

The next step in the historical narrative belongs, as far as I can determine, to Richard Feynmann. Feynmann is not much talked about in the general retelling of the story, but if the next step is not his then I don't know whose it is. All I know is that at some point, somebody realized that the mathematics of photons has in some way a one-to-one correspondence with the mathematics of electrons. And in 1962 Feynmann, lecturing an undergraduate class of second-year physics students at Caltech, proposed an equivalent experiment to Bohms using photons instead of electrons.

Feynmann's example was the self-annihilation of positronium, and his explanation is found in great detail in (of course!) the Feynmann Lectures on Physics, Voloume III. Positronium is an unstable "atom" made of one electron and one positron. In certain conditions, these particles will be found in a spin state identical to the spin-zero state of the electrons in a helium atom. Unlike helium, however, positronium is definitely unstabe, and it will absolutely decay in a matter of milliseconds. In fact, it is not so much a decay as a self-annihilation: and the products are not the two electrons, but rather two high-energy photons. The remarkable thing is that the photons inherit the spin-state of the positronium! The photons are spin-entangled.

If we can measure the polarization state of these decay products, we can experimentally observe the phenomenon of entanglement. Unfortunately, for technical reasons that I do not understand, this is not readily acheivable. Perhaps polarizers are not availabe for high-energy gamma rays as they are for visible light. What I do know is that, for whatever reason, the experiment has never been carried out. To this day, we have not measured the entanglement properties of the decay products of positronium.

Which just about brings us to the year 1964, when Bell published his ground-braking analysis. But that's a story for another day.

1 comment:

Anonymous said...

I am sorry, but the experiment was done by Dr. Wu, in 1950, and by Kasday et al. in 1975, and in 2003 by Damyanova, and 2016 by an undergraduate group of students.