Antiunitary operator

In mathematics, an antiunitary transformation, is a bijective antilinear map

${\displaystyle U:H_{1}\to H_{2}\,}$

between two complex Hilbert spaces such that

${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}}$

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 U is called an antiunitary operator.

Antiunitary operators are important in quantum theory because they are used to represent certain symmetries, such as time reversal and charge conjugation symmetry. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.

Invariance transformations

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 fall into two categories, they can be unitary or antiunitary.

Geometric Interpretation

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.

Properties

• ${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }$ 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.)

Examples

• 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 ${\displaystyle U^{2}=-1}$.

Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries

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 ${\displaystyle W_{0}:\mathbb {C} \rightarrow \mathbb {C} }$ 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.

References

• 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

See also

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