Unitary transformation

Unitary transformation

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In mathematics, a unitary transformation may be informally defined as a transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

 that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

More precisely, a unitary transformation is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 between two Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s. In other words, a unitary transformation is a bijective function


where and are Hilbert spaces, such that


for all and in . A unitary transformation is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

, as one can see by setting in this formula.

In the case when and are the same space, a unitary transformation is an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

 of that Hilbert space, and then it is also called a 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...

.

A closely related notion is that of antiunitary transformation, which is a bijective function


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...

.

See also

  • Time reversal
    Time reversal
    Time reversal may refer to:* In physics, T-symmetry - the study of thermodynamics and the symmetry of certain physical laws where the concept of time is reversed — ie...

  • antiunitary
  • Unitary group
    Unitary group
    In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...

  • 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...

  • Unitary matrix
  • 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....