In a string of recent posts I have developed a gedankenexperiment that involves neutral pions in an optical cavity decaying to photons. I finally figured out how to handle the photons in the cavity courtesy of a book on quantum optics by Mandel and Wolf. So let’s imagine we have a single neutral pion at rest inside an optical cavity. 99% of the time, a neutral pion will decay to two photons. In order to simplify the math a bit, let’s work with a simple 1-D cavity with mirrors on each end acting as the “walls” of the potential well (this is one of the examples discussed in Mandel and Wolf). As Mandel and Wolf describe, the normal modes for light in an empty optical cavity are found by solving the Helmholtz equation and setting appropriate boundary conditions. In the 1-D case, the energy eigenvalues for the light are given as

where *L* is the length of the “box,” i.e. the width of the cavity. Now, according to special relativity, the four-momentum must be conserved for the decay process. As such, we have

.

This actually tells us right away that the energies of the emitted photons should be about 67.5 MeV since the rest pass a neutral pion is roughly 135 MeV. But wait a second. As we saw earlier, since they’ll be in a box, these photons will be quantized. How do we interpret this, then? Is 67.5 MeV equal to *E* for a *particular* value of *n*, or does it simply tell us that the ratio of *n*/*L* is limited? Either way, something non-local is going on because, at the instant of decay, the photons must immediately know that they are in a box.

Now here’s another peculiar thing. The most logical way to interpret the 67.5 MeV is to say that it is one of the eigenvalues for the photon in the cavity. Thus, if we plug it in for , we find that *n*/*L* is equal to some number. This number is positive and real, but – crucially – *not necessarily an integer*. But we know that *n has* to be an integer. There are only two possible conclusions that we can reach from this:

*L*cannot be any value, i.e. paradoxically, the size of the box is somehow*predetermined*by the fact that it contains a neutral pion initially, which is utterly absurd, or- if the box is the wrong size, the
*neutral pion simply cannot decay via this decay mode*. But, once again, this implies that*the pion somehow knows that it is in a box to begin with*! In fact it also implies that the pion*knows something about the size of the box*!

While the second option squares better with our ability to arbitrarily choose a box size, it does, nevertheless, imply that there is something non-local going on. The only explanation I can think of is that the pion is somehow entangled with the box. But then, if it is, how did it get that way? Suppose I were able to isolate a neutral pion in a vacuum and then bring two halves of a spherical cavity together around it. How does it become entangled with the box if it never comes in contact with the box (or anything else, for that matter)? The only explanation I can think of is that the neutral pion itself is a non-local object.

Hi Ian,

This is basically another case of frustrated spontaneous emission, which is when an atom can’t decay because it’s in a cavity where those photon modes can’t exist. The photon modes are more naturally thought of as waves than the pion modes you mention, but in quantum field theory it’s all waves (at least when we aren’t looking! :-), so it’s really the same thing. They’re both *extended* objects (whether that makes them *nonlocal* objects is obviously up for debate.)

Now, there is a way to “explain” this (semiclassically) without having the pion mode stretch all the way to the mirrors… That’s because spontaneous emission can be thought of as stimulated emission by the zero-point-field. In this perspective, it’s not that the pion “knows” about the mirrors, it’s that the mirrors prevent the relevant zero-point photon modes from existing in the first place, which means they can’t trigger the spontaneous emission. (Does that count as being “entangled” with the box? Not sure.)

Of course, this gets a lot trickier if the mirrors are moving; what matters is the location of the mirrors in the future, when the photon will arrive there. In this case, the picture in the previous paragraph falls apart somewhat, and one is tempted to imagine a non-local connection between the walls and the pion. But as I’m a huge fan of locality, I’ll also plug the retrocasual option: the photon mode has to be consistent with the future as well as with the past.

I think this is a good question to raise, to show conflicting paradigms (quantum emission from dedicated sources versus the contextual properties of “cavities.”) Ken Wharton clearly has good insight into the problem, but I’m still not sure how this affects what can happen or if the box matters after all.

Aha! Indeed, Ken has unlocked the riddle. I still think there are issues here, even from a field theoretic standpoint. But I am glad that I can now put a concrete name to this phenomenon. Thanks!