Antiunitary operator

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

${\displaystyle |\langle Tx,Ty\rangle |=|\langle x,y\rangle |}$

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

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

• ${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }$ 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
  $${\displaystyle \left\langle U^{2}x,U^{2}y\right\rangle ={\overline {\langle Ux,Uy\rangle }}=\langle x,y\rangle .}$$

  
• For unitary operator ${\displaystyle V}$ the operator ${\displaystyle VK}$, where ${\displaystyle K}$ is complex conjugate operator, is antiunitary. The reverse is also true, for antiunitary ${\displaystyle U}$ the operator ${\displaystyle UK}$ is unitary.
• For antiunitary ${\displaystyle U}$ the definition of the adjoint operator ${\displaystyle U^{*}}$ is changed to compensate the complex conjugation, becoming
  $${\displaystyle \langle Ux,y\rangle ={\overline {\left\langle x,U^{*}y\right\rangle }}.}$$

  
• The adjoint of an antiunitary ${\displaystyle U}$ is also antiunitary and
  $${\displaystyle UU^{*}=U^{*}U=1.}$$

  
  (This is not to be confused with the definition of unitary operators, as the antiunitary operator ${\displaystyle U}$ is not complex linear.)

• The complex conjugate operator ${\displaystyle K,}$ ${\displaystyle Kz={\overline {z}},}$ is an antiunitary operator on the complex plane.
• The operator
  $${\displaystyle U=i\sigma _{y}K={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}K,}$$

  
  where ${\displaystyle \sigma _{y}}$ is the second Pauli matrix and ${\displaystyle K}$ is the complex conjugate 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 ${\displaystyle W_{\theta }}$, ${\displaystyle 0\leq \theta \leq \pi }$. 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 ${\displaystyle 0<\theta \leq \pi }$, 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 ${\displaystyle 0<\theta \leq \pi }$

${\displaystyle W_{\theta }\left(W_{\theta }\left(\left(z_{1},z_{2}\right)\right)\right)=\left(e^{i\theta }z_{1},e^{-i\theta }z_{2}\right),}$

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 Vol1, no5, 1960, pp.414–416

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