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.^[1] Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.