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 that the transition probability between two quantum states must be invariant under a symmetry transformation, and this transformation is either unitary or antiunitary. Time reversal symmetry in quantum mechanics is a prime example of an antiunitary symmetry. If a system is invariant under time reversal, its evolution can be described by an antiunitary operator.

The concept of antiunitarity is crucial for understanding phenomena like Kramers degeneracy, which arises in systems with time-reversal symmetry in the presence of an odd number of spin-1/2 particles. While unitary operators represent transformations that preserve the quantum mechanical phase, antiunitary operators incorporate a reversal of phase or orientation, often linked to physical operations like time reversal or charge conjugation.