and vectors sort out into either category depends on whether the number of dimensions of space is either an odd or even number. The categories of odd or even given below for the parity transformation is a different, but intimately related issue. The answers given below are correct for 3 spatial dimensions. In a 2 dimensional space, for example, when constrained to remain on the surface of a planet, some of the variables switch sides. Odd Classical variables whose signs flip when inverted in space inversion are predominantly vectors. They include: Even Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include: Quantum mechanics Possible eigenvalues In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, , is a unitary operator, in general acting on a state as follows: . One must then have , since an overall phase is unobservable. The operator , which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases . If is an element of a continuous U(1) symmetry group of phase rotations, then is part of this U(1) and so is also a symmetry. In particular, we can define , which is also a symmetry, and so we can choose to call our parity operator, instead of . Note that and so has eigenvalues . 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 . For electronic wavefunctions, even states are usually indicated by a subscript g for gerade (German: even) and odd states by a subscript u for ungerade (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled and the next-closest (higher) energy level is labelled . The wave functions of a particle moving into an external potential, which is centrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions. The law of conservation of parity of particles states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of