Unitary group

In mathematics , the designated unitary group on a complex Hilbert space , the group of all the unitary complex linear maps over . Unitary groups and their subgroups play a central role in quantum physics , where they are used to describe symmetries of the wave function . ${\displaystyle \mathrm {U} (H)}$ ${\displaystyle H}$ ${\displaystyle H}$

properties

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

The unitary group over a finite-dimensional Hilbert space of dimension is a real Lie group of dimension and is denoted by. The group is a subgroup of the general linear group and can be realized concretely through the set of unitary matrices with the matrix multiplication as a group operation. For a given , the unitary matrices with determinant 1 form a subgroup of denoted by , the special unitary group . ${\displaystyle H}$ ${\displaystyle n}$ ${\displaystyle n^{2}}$ ${\displaystyle \mathrm {U} (n)}$ ${\displaystyle \mathrm {U} (n)}$ ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$ ${\displaystyle n\times n}$ ${\displaystyle n}$ ${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle \mathrm {U} (n)}$

example

The simplest unitary group next to the trivial group is U (1) , the so-called circle group , the group of linear mappings of complex numbers that leave the square of the absolute value unchanged, with the concatenation as a group operation. The group is Abelian and can be specifically implemented through the set of functions that each multiply a given complex number by a phase factor, where a real number is: ${\displaystyle \mathrm {U} (0)}$ ${\displaystyle u_{\alpha }}$ ${\displaystyle e^{\mathrm {i} \alpha }}$ ${\displaystyle \alpha }$

${\displaystyle u_{\alpha }(z)=e^{\mathrm {i} \alpha }\;z}$

The figure describes a rotation of the complex number plane by the angle . This group is topologically isomorphic to the group with the multiplication of complex numbers as a group operation. ${\displaystyle u_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \{z\in \mathbb {C} ;\,|z|=1\}}$

The center of for any is , wherein the n -dimensional unit matrix was, and therefore isomorphic to ${\displaystyle \mathrm {U} (n)}$ ${\displaystyle n}$ ${\displaystyle \{z\cdot 1_{n};\,z\in \mathbb {C} ,\,|z|=1\}}$ ${\displaystyle 1_{n}}$ ${\displaystyle \mathrm {U} (1)}$