Antiunitary operator

In mathematics, an antiunitary transformation is a bijective antilinear map

${\displaystyle U:H_{1}\to H_{2}\,}$

between two complex Hilbert spaces such that

${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}}$

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1}}$, where the horizontal bar represents the complex conjugate. If additionally one has ${\displaystyle H_{1}=H_{2}}$ then ${\displaystyle U}$ is called an antiunitary operator.

Antiunitary operators are important in quantum mechanics because they are used to represent certain symmetries, such as time reversal. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.