Inmathematics, anantiunitary transformation is abijectiveantilinear map

${\displaystyle U:H_1\to H_2}$

between twocomplexHilbert spaces such that

${\displaystyle ⟨Ux,Uy⟩=\overline{⟨x,y⟩}}$

for all${\displaystyle x}$ and${\displaystyle y}$ in${\displaystyle H_1}$, where the horizontal bar represents thecomplex conjugate. If additionally one has${\displaystyle H_1=H_2}$ then${\displaystyle U}$ is called anantiunitary operator.

Antiunitary operators are important inquantum mechanics because they are used to represent certainsymmetries, such astime reversal.^[1] Their fundamental importance in quantum physics is further demonstrated byWigner's theorem.

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

${\displaystyle |⟨Tx,Ty⟩|=|⟨x,y⟩|}$ 

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

Due toWigner's theorem these transformations can either beunitary 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 thecomplex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively.

• ${\displaystyle ⟨Ux,Uy⟩=\overline{⟨x,y⟩}=⟨y,x⟩}$  holds for all elements${\displaystyle x,y}$  of the Hilbert space and an antiunitary${\displaystyle U}$ .
• When${\displaystyle U}$  is antiunitary then${\displaystyle U^2}$  is unitary. This follows from$${\displaystyle ⟨U^{2}x,U^{2}y⟩=\overline{⟨Ux,Uy⟩}=⟨x,y⟩.}$$ 
• For unitary operator${\displaystyle V}$  the operator${\displaystyle VK}$ , where${\displaystyle K}$  is complex conjugation (with respect to some orthogonal basis), is antiunitary. The reverse is also true, for antiunitary${\displaystyle U}$  the operator${\displaystyle UK}$  is unitary.
• For antiunitary${\displaystyle U}$  the definition of theadjoint operator${\displaystyle U^{\ast }}$  is changed to compensate the complex conjugation, becoming$${\displaystyle ⟨Ux,y⟩=\overline{⟨x,U^{\ast }y⟩}.}$$ 
• The adjoint of an antiunitary${\displaystyle U}$  is also antiunitary and$${\displaystyle U{U}^{\ast }=U^{\ast }U=1.}$$  (This is not to be confused with the definition ofunitary operators, as the antiunitary operator${\displaystyle U}$  is notcomplex linear.)

• The complex conjugation operator${\displaystyle K,}$ ${\displaystyle Kz=\overline{z},}$  is an antiunitary operator on the complex plane.
• The operator  where${\displaystyle {\sigma }_y}$  is the secondPauli matrix and${\displaystyle K}$  is the complex conjugation operator, is antiunitary. It satisfies${\displaystyle U^2=-1}$ .

An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries${\displaystyle W_{\theta }}$ ,${\displaystyle 0\le \theta \le \pi }$ . The operator${\displaystyle W_0:\mathbb{C}\to \mathbb{C}}$  is just simple complex conjugation on${\displaystyle \mathbb{C}}$ 

${\displaystyle W_0(z)=\overline{z}}$ 

For , the operator${\displaystyle W_{\theta }}$  acts on two-dimensional complex Hilbert space. It is defined by

${\displaystyle W_{\theta }\left(\left(z_1,z_2\right)\right)=\left(e^{\frac{i}{2}\theta }\overline{z_2},e^{-\frac{i}{2}\theta }\overline{z_1}\right).}$ 

Note that for 

${\displaystyle W_{\theta }\left(W_{\theta }\left(\left(z_1,z_2\right)\right)\right)=\left(e^{i\theta }{z}_1,e^{-i\theta }{z}_2\right),}$ 

so such${\displaystyle W_{\theta }}$  may not be further decomposed into${\displaystyle W_0}$ 's, which square to the identity map.

Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.

• Wigner, E. "Normal Form of Antiunitary Operators",Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412
• Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp.414–416

• Unitary operator
• Wigner's Theorem
• Particle physics and representation theory