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## Main Question or Discussion Point

Today in class, by the existence of an operator that exchanges the states of two indistinguishable particles, we attempted to derive the existence of fermions and bosons & how this relates to the symmetries of multiparticle wave functions.

The argument given in my textbook is: define an exchange operator P. "Clearly" P^2 = I. Therefore, the eigenvalues of P are +1 and -1. Systems of identical particles are eigenvectors of an exchange operator, so they are therefore either symmetric or antisymmetric under exchange of particles.

On the other hand, I don't see why we need P^2 = I. We can have P^2 = e^(i theta) for any value of theta because the only requirement we have is that exchanging twice leaves us with a state that is physically indistinguishable.

So what gives? Is the argument given in my book completely bogus? Can someone direct me to another book that has an explanation of bosons & fermions? (I'm looking at Townsend by the way)

The argument given in my textbook is: define an exchange operator P. "Clearly" P^2 = I. Therefore, the eigenvalues of P are +1 and -1. Systems of identical particles are eigenvectors of an exchange operator, so they are therefore either symmetric or antisymmetric under exchange of particles.

On the other hand, I don't see why we need P^2 = I. We can have P^2 = e^(i theta) for any value of theta because the only requirement we have is that exchanging twice leaves us with a state that is physically indistinguishable.

So what gives? Is the argument given in my book completely bogus? Can someone direct me to another book that has an explanation of bosons & fermions? (I'm looking at Townsend by the way)