Antiunitary operator
In mathematics, an antiunitary transformation is a bijective antilinear map between two complex Hilbert spaces such that for allAntiunitary operators are important in quantum mechanics because they are used to represent certain symmetries, such as time reversal.[1] Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.In quantum mechanics, the invariance transformations of complex Hilbert spaceDue to Wigner's theorem these transformations can either be unitary or antiunitary.Congruences of the plane form two distinct classes.The first conserves the orientation and is generated by translations and rotations.The second does not conserve the orientation and is obtained from the first class by applying a reflection.On the complex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively.In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.