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The basic idea is, the photon's wavefunction should be symmetric under exchange, however the spin part is anti-symmetric and the space part is symmetric and therefore forbidden.

I have trouble understanding the argument about the space part:

Since the photons conserve linear momentum in the particle rest frame and space is isotropic, they must be emitted in spherical waves.

Why the space is isotropic? Is isotropy an intrinsic property of original particle or just because the final particles are identical?

I guess the right answer is the latter one, because ##\rho^+ \to \pi^+ \pi^0## the final pions lie in ##P## wave and it doesn't bothered by the Bose-Einstein statistics. (Compare this with ##\rho^0\to \pi^0 \pi^0##, which is forbidden.)

However, I still believe the isotropy is an intrinsic property of the original particle. I'm looking for an explanation more mathematically, or a definition of isotropy in the language of group theory.

Any suggestions?