In mathematics, an antiunitary transformation is a bijective antilinear map

between two complex Hilbert spaces such that

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.

In quantum mechanics, the invariance transformations of complex Hilbert space ${\displaystyle H}$ leave the absolute value of scalar product invariant:

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H}$.

Due to Wigner's theorem these transformations can either be unitary or antiunitary.

Congruences of the plane form two distinct classes. The first conserves the orientation and is generated by translations and rotations. The second does not conserve the orientation and is obtained from the first class by applying a reflection. On the complex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively.