Unitary group

In mathematics, the unitary group on a complex Hilbert space the group of all unitary complex linear maps referred to above. Unitary groups and their sub-groups play a central role in quantum physics, where they are used for the description of symmetries of the wave function.

Properties

In the general case, the unitary group is a Banach - Lie group with the supremum norm. We can provide the unitary group with the weak operator topology. This fall, restricted to the unitary group, with the strong operator topology together. For finite dimensional Hilbert spaces induced by the supremum norm topology and the operator topology coincide.

The unitary group to a Hilbert space of finite dimension is a real dimension of the Lie group and is designated by. The group is a subgroup of the general linear group and can be concretely realized by the set of unitary matrices with matrix multiplication as the group operation. For a given form the unitary matrices with determinant 1 with a designated subset of the special unitary group.

Example

The simplest addition to the trivial group is unitary group, called the circle group, the group of linear transformations of the complex numbers, which let the square modulus unchanged, with the concatenation as the group operation. The group is abelian and can be implemented concretely by the set of functions, each multiplied by a given complex number by a phase factor, which is a real number:

The figure describes a rotation of the complex plane by the angle. This group is topologically isomorphic to the group with the multiplication of complex numbers as the group operation.

The center of any of, said n-dimensional unit matrix is, and therefore isomorphic to

• Linear Algebra
• Lie group
• Group Theory