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 symmetry. Their fundamental importance in quantum physics is further demonstrated by Wigner's Theorem.
In Quantum mechanics, the invariance transformations of complex Hilbert space leave the absolute value of scalar product invariant:
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.
- 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 reverse is also true, for antiunitary the operator is unitary.
- For antiunitary the definition of the adjoint operator is changed into
- The adjoint of an antiunitary is also antiunitary and
- (This is not to be confused with the definition of unitary operators, as the antiunitary operator is not complex linear.)
- The complex conjugate operator is an antiunitary operator on the complex plane.
- The operator
where is the second Pauli matrix and is the complex conjugate operator, is antiunitary. It satisfies .
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 C
For , the operation 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 (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.
- Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412
- Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol1, no5, 1960, pp.414–416