Antiunitary operator

In mathematics, an antiunitary transformation, is a bijective antilinear map

between two complex Hilbert spaces such that

for all x and y in H₁, where the horizontal bar represents the complex conjugate. If additionally one has H₁ = H₂ 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.

Decomposition of a unitary 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_θ, . The operator is just simple complex conjugation on C

For , the operation W_θ acts on two-dimensional complex Hilbert space. It is defined by

Note that for

so such W_θ may not be further decomposed into W₀'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.

References

• Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412

See also

• Unitary operator
• Wigner's Theorem
• Particle physics and representation theory