are considered, where Ω Ω ^ ^ denotes a unitary operator. Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse is the Hermitian conjugate Ω Ω ^ ^ − − 1 = Ω Ω ^ ^ † † . The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are: