Is the Conservation of Angular Momentum in Quantum Mechanics Only Probabilistic?

[fermi]

I have a symmetry question in QM that's been puzzling me. To state the puzzle, I use a simplified example of a neutral pion in an otherwise empty universe. Since the pion is spinless, and since it possesses no electric or magnetic dipole (or higher) moments, this little set-up is spherically symmetric. Furthermore, the underlying dynamics of the electromagnetic decay also gives us a spherically symmetric amplitude for the decay of the pion into two photons. In other words, the dynamics and the initial conditions are both spherically symmetric. And yet, when the pion decays into two photons, this symmetry appears to be broken down to only an axial symmetry around the line of flight for the back-to-back photons. Now, one can argue that if one observes many (millions) such decays, the symmetry will be restored, and the final photons will be distributed uniformly over all angles. This is indeed so, but it does not take anything away from what happens to a single pion.

Symmetries and conservation laws are related. The rotational symmetry implies the conservation of angular momentum. In the example above, the angular momentum is strictly conserved in every single pion decay event. Yet the the spherical symmetry which this conservation law is based on appears to be only probabilistic, and observable only with many pions. That bothers me a bit. Can you give an argument why this should be so?

 

[StatusX]

The wavefunction of the pion + photon system will remain spherically symmetric until it is observed. The collapse of the wavefunction spontaneously breaks this symmetry, in a way similar to how a pencil balancing on its end breaks rotational symmetry when it falls in a particular direction. This has nothing to do with conservation laws, which depend only on symmetries of the underlying equations, not of the particular state.

 

[conway]

I agree with StatusX as to the point when the symmetry is broken, but I don't think his answer really deals with the problem as presented by Fermi. Because I don't think the pion-photon system can really be spherically symmetric. Maxwell's equations don't have spherically symmetric radiating solutions, and I don't see how the photon system can do anything (prior to the moment of detection) that doesn't come from Maxwell's equations.

 

[Bob S]

The neutral pion at rest (usually) decays into two back-to-back (180 degrees apart) 67.49 MeV photons. When Lorentz-transformed to a moving frame, the photon energy is transformed (usually increased) and the angular distribution folds forward into a cone. In fact, If both photons have the same energy for a decaying 10 GeV pi zero, they would be about 5 GeV each.