Parity forms the abelian group Z 2 due to the relation P ^ ^ 2 = 1 ^ ^ . All Abelian groups have only one-dimensional irreducible representations. For Z 2 , there are two irreducible representations: one is even under parity, P ^ ^ ϕ ϕ = + ϕ ϕ , the other is odd, P ^ ^ ϕ ϕ = − − ϕ ϕ . These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.