The definition of a quantum canonical transformation is thus an arbitrary unitary change of basis on the space of all state vectors. U is an arbitrary unitary matrix, a complex rotation in phase space, U † † = U − − 1 . These transformations leave the sum of the absolute square of the wavefunction components invariant, while they take states which are multiples of each other (including states which are imaginary multiples of each other) to states which are the same multiple of each other.