Heisenberg group

As Heisenberg group is referred to them a certain group of matrices and generalizations in mathematics. Each Heisenberg group has a topological structure and is a Lie group.

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

Definition

Upper 3x3 triangular matrices of the form

With entries, and that can come from a ( arbitrary) commutative ring form a group under the usual matrix multiplication, called the Heisenberg group. The entries often are taken from the ring of real numbers or the integers.

Properties

It is the Heisenberg group with entries from a central extension of the group understand what can best be seen when looking at by

Defines a group multiplication and

Observed.

Lie algebra

The Lie algebra of the Heisenberg group is the Heisenberg algebra

Application

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

Generalizations

There are higher-dimensional generalized Heisenberg groups. As Matrizengruppe is the n-th Heisenberg group of the upper triangular square of size n 2 of the form

Wherein the length of a row vector, a column vector of length and the unit matrix.

• Lie group
• Theory of Lie groups