nLab antiunitary operator

Definition

An anti-unitary operator $\Theta$ on a Hilbert space $\mathscr{H}$ is an anti-linear map $\mathscr{H} \longrightarrow \mathscr{H}$ which preserves the Hermitian inner product $\langle-,-\rangle$ up to complex conjugation:

$$\langle \Theta - ,\, \Theta - \rangle = \overline{ \langle - ,\, - \rangle } \,.$$

This means equivalently, with respect to the Hermitian conjugate $(-)^\dagger$ of anti-linear maps (see there), that an anti-linear map is anti-unitary iff

$$\Theta^\dagger = \Theta^{-1} \,,$$

which characterization is structurally identical to that of ordinary unitary operators.

Related concepts

• antilinear map, anti-dual linear space

• time-reversal symmetry

• Wigner's theorem

References

• Armin Uhlmann, around (112) in: Anti- (Conjugate) Linearity, Sci. China Phys. Mech. Astron. 59 (2016) 630301 [arXiv:1507.06545, doi:10.1007/s11433-015-5777-1]

See also:

• Wikipedia, Antiunitary operator

and see the references at Wigner's theorem.