Antiunitary operator

Definition

An antiunitary operator between complex Hilbert spaces $\mathcal H$ and $\mathcal K$ is a conjugate-linear bijection $A:\mathcal H\to\mathcal K$ such that

\langle Ax,Ay\rangle_{\mathcal K} =\overline{\langle x,y\rangle_{\mathcal H}} \qquad(x,y\in\mathcal H),

when the inner product is linear in its first argument. Equivalently, $A$ is a surjective conjugate-linear isometry. This differs from a unitary operator, which is linear.

Basic properties

Every antiunitary operator preserves norms and orthogonality. Its inverse and adjoint are antiunitary, while the product of two antiunitaries is unitary. After choosing an orthonormal basis, any antiunitary on one Hilbert space can be written

A=UK,

where $U$ is unitary and $K$ is coordinatewise complex conjugation. The factorization depends on the chosen conjugation; antiunitarity itself does not.

Because $A$ is conjugate-linear,

A(\lambda x)=\overline{\lambda}\,Ax.

It is therefore not a complex-linear element of $\mathcal B(\mathcal H,\mathcal K)$ , even though it is continuous and real-linear.

Quantum symmetries

Wigner's theorem says that a bijection of rays preserving transition probabilities is induced by either a unitary or an antiunitary operator, unique up to a phase. The antiunitary possibility is essential for time-reversal and related discrete symmetries. Such symmetries need not belong to the identity component of a continuously acting gauge group.

Convention warning

The displayed conjugation identity is valid whether the Hilbert-space inner product is taken linear in its first or its second argument; what changes is which slot is linear when manipulating the formula. The term antilinear isometry does not by itself imply surjectivity; antiunitary does.

References

Eugene P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959, Chapter 20.
V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer, 1985, Chapter II. Publisher record.