One must then have P ^ ^ 2 ψ ψ ( r ) = e i ϕ ϕ ψ ψ ( r ) , since an overall phase is unobservable. The operator P ^ ^ 2 , which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases e i ϕ ϕ . If P ^ ^ 2 is an element e i Q of a continuous U(1) symmetry group of phase rotations, then e − − i Q is part of this U(1) and so is also a symmetry. In particular, we can define P ^ ^ ′ ≡ ≡ P ^ ^ e − − i Q / 2 , which is also a symmetry, and so we can choose to call P ^ ^ ′ our parity operator, instead of P ^ ^ . Note that P ^ ^ ′ 2 = 1 and so P ^ ^ ′ has eigenvalues ± ± 1 . Wave functions with eigenvalue + 1 under a parity transformation are even functions, while eigenvalue − − 1 corresponds to odd functions. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ± ± 1 .