- 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 0< hetalepi, 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 0< hetalepi
: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
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