Antiunitary operator
In mathematics, an antiunitary transformation is a bijective antilinear map between two complex Hilbert spaces such that for all and in , where the horizontal bar represents the complex conjugate. If additionally one has then is called an antiunitary operator. Antiunitary operators are important in quantum theory because they are used to represent certain symmetries, such as time reversal. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem. Invariance transformations In quantum mechanics, the invariance transformations of complex Hilbert space leave the absolute value of scalar product invariant: for all and in . Due to Wigner's theorem these transformations can either be unitary or antiunitary. Geometric Interpretation 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. Properties holds for all elements of the Hilbert space and an antiunitary . When is antiunitary then is unitary. This follows from For unitary operator the operator , where is complex conjugate operator, is antiunitary. The