Recalling that a generic spin state can be written as a superposition of states with definite m, we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a half-integer, the values of m are also all half-integers, giving (−1) = −1 for all m, and hence upon rotation by 2 π the state picks up a minus sign. This fact is a crucial element of the proof of the spin–statistics theorem.