In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an antiunitary transformation, is a bijective
antilinear map
:
between two
complex Hilbert spaces such that
:
for all
and
in
, where the horizontal bar represents the
complex conjugate. If additionally one has
then
is called an antiunitary operator.
Antiunitary operators are important in quantum theory 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the invariance transformations of complex Hilbert space
leave the absolute value of scalar product invariant:
:
for all
and
in
.
Due to
Wigner's theorem these transformations can either be
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
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
*
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 definition of the
adjoint operator
is changed to compensate the complex conjugation, becoming
* The adjoint of an antiunitary
is also antiunitary and
(This is not to be confused with the definition of
unitary operators
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
, as the antiunitary operator
is not complex linear.)
Examples
* The complex conjugate operator
is an antiunitary operator on the complex plane.
* The operator
where
is the second
Pauli matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in c ...
and
is the complex conjugate operator, is 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
:
For
, the operator
acts on two-dimensional complex Hilbert space. It is defined by
:
Note that for
:
so such
may not be further decomposed into {{nowrap,
'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.
References
*Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412
*Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol1, no5, 1960, pp.414–416
See also
*
Unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
*
Wigner's Theorem
*
Particle physics and representation theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this con ...
Linear algebra
Functional analysis