Antiunitary

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In mathematics, an antiunitary transformation, is a bijective function

U:H_1\to H_2\,

between two complex Hilbert spaces such that

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

for all x and y in H1, where the horizontal bar represents the complex conjugate. If additionally one has H1 = H2 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 symmetry. Their fundamental importance in quantum physics is further demonstrated by Wigner's Theorem.

[edit] Decomposition of a unitary 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 Wθ, 0\le\theta\le\pi. The operator W0:C − > C is just simple complex conjugation on C

W_0(z)=\overline{z}

For 0<\theta\le\pi, the operation Wθ acts on two-dimensional complex Hilbert space. It is defined by

W_\theta((z_1,z_2))=(e^{i\theta/2}\overline{z_2},e^{-i\theta/2}\overline{z_1}).

Note that for 0<\theta\le\pi

Wθ(Wθ((z1,z2))) = (eiθz1,e − iθz2),

so such Wθ may not be further decomposed into W0's, which square to the identity map.

Note that the above decomposition of unitary operators constrasts 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.

[edit] References

  • Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409-412

[edit] See also