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Wikipedia / Antiunitary operator

In mathematics, an antiunitary transformation is a bijective antilinear map

between two complex Hilbert spaces such that

, where the horizontal bar represents the complex conjugate is called an antiunitary operator.

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

Invariance transformations

In quantum mechanics leave the absolute value of scalar product invariant:

.

Due to Wigner's theorem these transformations can either 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

Examples

  • is an antiunitary operator on the complex plane.

  • is the second Pauli matrix.

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

acts on two-dimensional complex Hilbert space. It is defined by

may not be further decomposed into 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. 409412

  • Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp.414416

See also