Poincaré group

The Poincaré group ( named after the French mathematician and physicist Henri Poincaré ) is a special group in mathematics that has applications found in physics.

Historical

The Poincaré group emerged historically for the first time in the study of invariance of electrodynamics by Poincaré, Lorentz and others, and played a crucial role in the formulation of special relativity. In particular, the Poincaré group was named after the formalization of the theory of relativity by Hermann Minkowski to an important mathematical structure in all relativistic theories, including quantum electrodynamics.

Geometric definition

The Poincaré group is the affine invariance of the pseudo -Euclidean Minkowski space, in particular, the Minkowski space with respect to the Poincaré group is a homogeneous space whose geometry defines it in terms of the Erlangen program. It differs from the Lorentz group, which is the linear invariance of Minkowski space, by the addition of translations. Therefore, it is similar in structure to the Euclidean group in three dimensional space that contains all geometric Kongruenzabbildungen. In fact, the Euclidean group is included as a subgroup in the Poincaré group. The main difference, however, is that the Poincaré group is not the lengths and angle gets in three-dimensional space, but the lengths and angles defined with respect to the indefinite scalar product in Minkowski pseudo space. In particular, it receives so-called proper time intervals in special relativity theory.

Algebraic definition

The Poincaré group is the semidirect product of the Lorentz group and the group of translations in the. Each element of the Poincaré group is therefore as a couple

Displayed, and the group is multiplied by

Optionally, wherein the Lorentz transformation in their natural effect acts as automorphism on.

Other properties

The Poincaré group is a 10 -dimensional noncompact Lie group. It is an example of a non- semisimple group.

The Lie algebra to the Poincaré group is defined by the following relations:

The four infinitesimal generator of the translations and the six infinitesimal generators are the Lorentz transformations.

• Lie group
• Theory of Lie groups
• Special Theory of Relativity