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?