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 U is called an antiunitary operator.
Antiunitary operators are important in quantum theory because they are used to represent certain symmetries, such as time reversal and charge conjugation symmetry. 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 fall into two categories, they can be unitary or antiunitary.

Geometric Interpretation

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 corresponds to unitaries and antiunitaries, respectively.

Properties

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,. The operator is just simple complex conjugation on
For, the operator acts on two-dimensional complex Hilbert space. It is defined by
Note that for
so such may not be further decomposed into '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, but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.