antiunitaries
An antiunitary operator is a linear operator on a Hilbert space that is composed of a unitary operator and complex conjugation. Mathematically, an antiunitary operator $A$ satisfies the property $A(\alpha \psi + \beta \phi) = \alpha^ A \psi + \beta^ A \phi$ for all complex numbers $\alpha, \beta$ and all vectors $\psi, \phi$ in the Hilbert space, and it preserves the inner product up to complex conjugation: $\langle A\psi, A\phi \rangle = \langle \psi, \phi \rangle^$. This means that while unitaries preserve the inner product, antiunitaries reverse the order of conjugation when applying the inner product.
Antiunitary operators are fundamental in quantum mechanics, particularly in the study of symmetries. Wigner's theorem states
The concept of antiunitarity is crucial for understanding phenomena like Kramers degeneracy, which arises in systems