In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an antiunitary transformation, is a bijective antilinear map


between two complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 Hilbert spaces such that


for all and in , where the horizontal bar represents the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

. If additionally one has 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

Wigner's theorem

Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....

.

Invariance transformations

In Quantum mechanics, the invariance transformations of complex Hilbert space leave the absolute value of scalar product invariant:


for all and in .
Due to Wigner's Theorem

Wigner's theorem

Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....

 these transformations fall into two categories, they can be unitary

Unitary

Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...

 or antiunitary.

Geometric Interpretation

Congruences

Congruence (geometry)

In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

 of the plane form two distinguish 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 corresponds (up to translation) to unitaries and antiunitaries, respectively.

Properties

• holds for all elements of the Hilbert space and an antiunitary .
• When is antiunitary then is unitary. This follows from
• For unitary operator the operator , where is complex conjugate operator, is antiunitary. The reverse is also true, for antiunitary the operator is unitary.
• For antiunitary the adjoint operator is also antiunitary and

Examples

• The complex conjugate operator is an antiunitary operator on the complex plane.
• The operator
  where is the second Pauli matrix
  Pauli matrices
  The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
  
   and is the complex conjugate operator, is an antiunitary. It satisfies .
  

  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 , . The operator is just simple complex conjugation on C
  
  
  For , the operation acts on two-dimensional complex Hilbert space. It is defined by
  
  
  Note that for
  
  
  so such may not be further decomposed into '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.
  

  See also

  • Unitary operator
    Unitary operator
    In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
    
    
  • Wigner's Theorem
    Wigner's theorem
    Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....
    
    
  • Particle physics and representation theory
    Particle physics and representation theory
    In physics, the connection between particle physics and representation theory is a natural connection, first noted by Eugene Wigner, between the properties of elementary particles and the representation theory of Lie groups and Lie algebras...
    
    
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