Heisenberg group

In mathematics, the Heisenberg group is a specific group of matrices and generalizations of them. Every Heisenberg group has a topological structure and is a Lie group .

The Heisenberg group was introduced by Hermann Weyl to explain the equivalence of the Heisenberg picture and the Schrödinger picture in quantum mechanics .

definition

Upper 3 × 3 triangular matrices of the form

${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}}$

with entries , and , which can come from (any) commutative ring , form a group under the usual matrix multiplication , the so-called Heisenberg group. The entries often come from the ring of real numbers or that of whole numbers . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$

properties

You can understand the Heisenberg group with entries from as a central extension of the group , which you can see best when you click through ${\displaystyle \mathbb {R} }$ ${\displaystyle (\mathbb {R} \times \mathbb {R} ,+)}$ ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle (a,b,c)\cdot (a',b',c')=(a+a',b+b',c+c'+ab')}$

defines a group multiplication and

${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}1&a'&c'\\0&1&b'\\0&0&1\\\end{pmatrix}}={\begin{pmatrix}1&a+a'&c+c'+ab'\\0&1&b+b'\\0&0&1\\\end{pmatrix}}}$

observed.

Lie algebra

The Lie algebra of the Heisenberg group is the Heisenberg algebra .

application

In quantum mechanics , the Heisenberg group has the function of a symmetry group .

Generalizations

There are higher-dimensional generalized Heisenberg groups. As a matrix group, the -th Heisenberg group consists of the square upper triangular matrices the size of the shape ${\displaystyle n}$ ${\displaystyle n+2}$

${\displaystyle {\begin{pmatrix}1&a&c\\0&E_{n}&b\\0&0&1\\\end{pmatrix}}}$

where a row vector is the length , a column vector is the length and the - is the identity matrix . ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle b}$ ${\displaystyle n}$ ${\displaystyle E_{n}}$ ${\displaystyle n\times n}$