In mathematics, an antiunitary transformation, is a bijective antilinear map
Contents
- Invariance transformations
- Geometric Interpretation
- Properties
- Examples
- Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries
- References
between two complex Hilbert spaces such that
for all
Antiunitary operators are important in Quantum Theory because they are used to represent certain symmetries, such as time-reversal 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
for all
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 corresponds (up to translation) to unitaries and antiunitaries, respectively.
Properties
Examples
where
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
For
Note that for
so such
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.