Antiunitary Symmetries, First Look

Canonical treatment: Antiunitary Symmetries owns Wigner’s theorem, operator transformations, physical examples, exercises, and references. This Toolkit page retains the conjugate-linear prerequisites.

A map $A$ on a complex Hilbert space is antilinear when

$$A(a\psi+b\phi)=a^*A\psi+b^*A\phi.$$

It is antiunitary when it is bijective and, in the physics inner-product convention,

$$\langle A\psi\mid A\phi\rangle = \langle\psi\mid\phi\rangle^*.$$

Thus it preserves norms and transition probabilities while conjugating complex scalars. In particular $AiA^{-1}=-i$.

Complex Conjugation and Factorization

After choosing an orthonormal basis, complex conjugation

$$K\sum_n c_n\lvert n\rangle = \sum_n c_n^*\lvert n\rangle$$

is antiunitary and obeys $K^2=I$. The operator $K$ depends on the chosen basis; antiunitarity itself does not.

Every antiunitary operator can be written

$$A=UK,$$

where $U$ is unitary relative to that basis. Conversely, a unitary operator times $K$ is antiunitary. The factorization changes when the reference basis changes, while the map $A$ remains the same.

Type and Composition Ledger

First map	Second map	Composition
linear unitary	linear unitary	linear unitary
linear unitary	antiunitary	antiunitary
antiunitary	linear unitary	antiunitary
antiunitary	antiunitary	linear unitary

The square $A^2$ is therefore unitary. Its phase or action on an irreducible sector can carry physical information, but that interpretation belongs to the canonical symmetry page and the time-reversal chapter.

Routing

• Quantum Symmetries explains ray and transition-probability preservation.
• Time Reversal gives the principal physical application.
• Antiunitary Time Reversal develops the dynamical transformation.
• Rays and Global Phase supplies the projective-state viewpoint.