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In mathematics, an antiunitary transformation, is a bijective antilinear map
between two complex Hilbert spaces such that
for all x {\displaystyle x} and y {\displaystyle y} in H 1 {\displaystyle H_{1}} , where the horizontal bar represents the complex conjugate. If additionally one has H 1 = H 2 {\displaystyle H_{1}=H_{2}} then U {\displaystyle U} is called an antiunitary operator.
Antiunitary operators are important in quantum theory 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.