“Pion stars,” spherical cavities, and more on the nature of mass
In a recent post on information and the nature of mass, I presented a gedankenexperiment that involved a spherical, thin-shelled, perfectly internally reflective optical cavity that initially contains neutral pions that then decay to photons. Using special relativistic arguments I demonstrated how the total mass of the system should remain unchanged in the process. So, for instance, if the cavity were large enough such that some massive object could orbit (due to gravity), the object’s orbit should not change after the photons decay. This experiment has continued to gnaw away at my brain and I began to think of the quantum mechanical aspects of the problem.
Let’s idealize this a bit in order to simplify things and start by only considering a single neutral pion (initially at rest) in the cavity. Quantum mechanically we can treat the problem as a spherical “particle in a box” problem. Since the neutral pion is initially non-relativistic since it is at rest, the three-dimensional Schrödinger equation can be validly applied in order to find the energy eigenstates of the associated pion. This is a fairly standard procedure and, in fact, is often included in elementary quantum mechanics texts as a problem. One simply solves the radial Schrödinger equation and tacks on the spherical harmonics. Solutions generally take the form of Bessel functions (note, however, that the probability densities depend only on r since the distributions are spherically symmetric).
But what happens when the pion decays to the photons? Photons are highly relativistic and applying the Schrödinger equation here is a bit suspect (though note that it can be used to find approximate solutions, e.g. see Slater’s classic Quantum Theory of Matter). We might try to apply the Klein-Gordon equation in the limit that . But it isn’t clear that this would give the correct answer either. Iwo Bialynicki-Birula has written several papers on photon wavefunctions (a nice summary paper is here). Tom Moore also does a nice job “explaining” the two-slit experiment with wavefunctions in his book Six Ideas That Shaped Physics, Unit Q. In the latter, Moore uses the free-particle approximation and really just treats the stream of photons as a fluctuating semi-classical E&M field. I think for the present problem a semi-classical approach such as this might be a bit oversimple.
At any rate, this raises a number of intriguing questions. Presumably, conservation of energy would require that there be some kind of correspondence between the energy eigenstates of the pion and that of the photon pair (aside from the well-known phase correspondence for such a decay that is due to the conservation of charge conjugation, i.e. ). How would such a restriction affect the solutions to the photon eigenvalue problem? In the gedankenexperiment for mass, we invoked special relativity in order to explain that the mass remains unchanged. Since this also tells us that the energy of the system is related to the mass, does this quantization of energy levels necessarily quantize the mass of the system? Could this tell us a little something about quantum gravity in the process?
One way to answer these questions in an ensemble manner (i.e. for many pions and many photons) is to consider the pions as a degenerate Fermi gas, i.e. to consider the internal pressure of a cavity as high enough that you essentially have a “pion star.” After the pions have completely decayed to photons, the “star” is now modeled as a photon gas. What is interesting about this approach is that it can be shown that the total energy of the pions in the cavity is dependent on the number of pions in the cavity but is not dependent on the volume, while the opposite is true for the total energy of the photons. One could tie volume to the total energy of the fermions via the Fermi energy, but that is slightly circular reasoning in my book. Either way, if we set the total energies equal to one another (for simplicity assume the temperature is the same in both the pion and photon cases and is very low, ), the number of pions is linearly proportional to the volume, which actually makes sense.
But that doesn’t really tell us much. For our purposes here, the more fruitful approach might be to start with a single neutral pion that then decays to a pair of photons, , and see what that implies, first. Then we can look at the ensemble characteristics to see how the situation differs. In any case, I have not gone into this in much detail yet, but am working on it. I think it would be an intriguing idea to pursue, though, particularly if we can tie together the quantum and relativistic mass considerations. Any seriously legitimate contributions given in the responses to this post will, of course, lead to co-authorship (should you choose). Any attempt to steal these ideas and beat me to publication will result in a serious ass-kicking (assuming the ideas are even worthy of publication).