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
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
:
between two
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
Hilbert spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
such that
:
for all
and
in
, where the horizontal bar represents the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. 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
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 are represented on the Hilbert sp ...
.
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
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 are represented on the Hilbert sp ...
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 semigrou ...
or antiunitary.
Geometric Interpretation
Congruences
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
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
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and ...
, 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 ...
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 co ...
*
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 are represented on the Hilbert sp ...
*
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