Poincaré group

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

Historical

The Poincaré group appears historically for the first time in the investigation of the invariances of electrodynamics by Poincaré , Lorentz and others and played a decisive role in the formulation of the special theory of relativity . In particular, following the formalization of the theory of relativity by Hermann Minkowski , the Poincaré group became an important mathematical structure in all relativistic theories, including quantum electrodynamics .

Geometric definition

The Poincaré group is the affine invariance group of the pseudo-Euclidean Minkowski space , in particular the Minkowski space is a homogeneous space with respect to the Poincaré group , the geometry of which it defines in the sense of the Erlangen program . It differs from the Lorentz group , which is the linear invariance group of the Minkowski space, through the addition of translations . It is therefore similar in its structure to the Euclidean group in three-dimensional space , which contains all geometric congruence maps. In fact, the Euclidean group is included as a subgroup in the Poincaré group. The main difference, however, is that the Poincaré group does not receive the lengths and angles in three-dimensional space , but the lengths and angles defined in Minkowski space with respect to the indefinite pseudo-scalar product . In particular, it receives so-called proper time intervals in the special theory of relativity . ${\displaystyle \mathbb {R} ^{3+1}}$

Algebraic definition

The Poincaré group is the semi-direct product of the Lorentz group and the group of translations im . So each element of the Poincaré group is a pair ${\displaystyle O(3,1)}$ ${\displaystyle \mathbb {R} ^{3+1}}$

${\displaystyle (\Lambda ,\mathbf {a} ):\quad \Lambda \in O(3,1),\mathbf {a} \in \mathbb {R} ^{3+1}}$

representable, and the group multiplication is through

${\displaystyle (\Lambda ,\mathbf {a} )\cdot (\Lambda ',\mathbf {a} ')=(\Lambda \cdot \Lambda ',\mathbf {a} +\Lambda (\mathbf {a} '))}$

given, whereby the Lorentz transformation acts in its natural effect as an automorphism on . ${\displaystyle \Lambda }$ ${\displaystyle \mathbb {R} ^{3+1}}$

Other properties

The Poincaré group is a 10-dimensional non-compact Lie group . She is an example of a not semi-simple group .

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

${\displaystyle {\begin{aligned}{}[P_{\mu },P_{\nu }]&=0\\{}[M_{\mu \nu },P_{\kappa }]&=\eta _{\mu \kappa }P_{\nu }-\eta _{\nu \kappa }P_{\mu }\\{}[M_{\mu \nu },M_{\kappa \lambda }]&=\eta _{\mu \kappa }M_{\nu \lambda }-\eta _{\mu \lambda }M_{\nu \kappa }-\eta _{\nu \kappa }M_{\mu \lambda }+\eta _{\nu \lambda }M_{\mu \kappa }\end{aligned}}}$

where the four infinitesimal generators of the translations and the six infinitesimal generators of the Lorentz transformations . ${\displaystyle P_{\mu }}$ ${\displaystyle M_{\mu \nu }}$

The two Casimir operators of the Poincaré group that swap with all generators are

${\displaystyle {\begin{aligned}&P^{\alpha }P_{\alpha }\\&W^{\alpha }W_{\alpha }={\frac {1}{4}}\varepsilon ^{\alpha \beta \gamma \delta }{\varepsilon _{\alpha }}^{\nu \rho \sigma }P_{\beta }M_{\gamma \delta }P_{\nu }M_{\rho \sigma }\end{aligned}}}$

Physically, these are the square of the quadruple momentum and the square of the Pauli-Lubanski pseudo-vector . The factor is convention. ${\displaystyle P^{\alpha }P_{\alpha }}$ ${\displaystyle W^{\alpha }W_{\alpha }}$ ${\displaystyle 1/4}$

Individual evidence

1. ^ Henri Poincaré: Sur la dynamique de l'électron . In: Rendiconti del Circolo matematico di Palermo . tape 21 , 1906, pp. 129-176 (French, wikisource.org ).
2. ↑ Hermann Minkowski: The basic equations for the electromagnetic processes in moving bodies . In: News from the Society of Sciences in Göttingen . tape 1908 , 1908, pp. 52-111 ( wikisource.org ).
3. ↑ ^{a b} Jakob Schwichtenberg: Understanding modern physics through symmetry: A new approach to fundamental theories . 1st edition. Springer Spectrum, 2017, ISBN 978-3-662-53811-1 , pp. 93-95 .