Any physical state of two identical particles must be either symmetric or anti-symmetric with respect to the exchange of the particles. Any physical state can be as a linear combination of basis vectors of the state space. The state space of a two particle system therefore is not E E , but either the subspace or symmetric or the subspace of anti-symmetric states. It is restricted . This extends naturally to an N -particle system.
Every elementary particle is either a fermion or a boson.
A state of many identical particles is totally anti-symmetric if the particles are fermions, and it is totally symmetric if the particles are bosons. Assume that two particles are placed at random into one of two boxes. Look at the definition of a determinant. . . We have the Pauli exclusion principle . Two different tensor product vectors, which have the same occupation numbers, are equal to each other within a factor of 1 after applying S or A .