Antiunitary

In mathematics, an antiunitary transformation, is a bijective function

:$U:H\_1\; o\; H\_2,$

between two complex Hilbert spaces such that

:$langle\; Ux,\; Uy\; angle\; =\; overline\{langle\; x,\; y\; angle\}=langle\; y,\; x\; angle$

for all $x$ and $y$ in $H\_1$, where the horizontal bar represents the complex conjugate. If additionally one has $H\_1\; =\; H\_2$ 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 symmetry. Their fundamental importance in quantum physics is further demonstrated by Wigner's Theorem.

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 $W\_\; heta$, $0le\; hetalepi$. The operator $W\_0:C->C$ is just simple complex conjugation on C

:$W\_0(z)=overline\{z\}$

For , the operation $W\_\; heta$ acts on two-dimensional complex Hilbert space. It is defined by

:$W\_\; heta((z\_1,z\_2))=(e^\{i\; heta/2\}overline\{z\_2\},e^\{-i\; heta/2\}overline\{z\_1\})$.

Note that for

:$W\_\; heta(W\_\; heta((z\_1,z\_2)))=(e^\{i\; heta\}z\_1,e^\{-i\; heta\}z\_2)$,

so such $W\_\; heta$ may not be further decomposed into $W\_0$'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.

References

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

ee also

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