In mathematics, an antiunitary transformation is a bijective antilinear map between two complex Hilbert spaces such that for all x and y in H_1, where the horizontal bar represents the complex conjugate. If additionally one has H_1 = H_2 then U is called an antiunitary operator. Antiunitary operators are important in quantum mechanics 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 H leave the absolute value of scalar product invariant: for all x and y in H. 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

Examples

Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries

An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries W_\theta,. The operator is just simple complex conjugation on \mathbb{C} For, the operator W_\theta acts on two-dimensional complex Hilbert space. It is defined by Note that for so such W_\theta may not be further decomposed into W_0's, which square to the identity map. Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. 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.