3.2.5 Poincaré Group
The Poincaré group was first defined by Minkowski (1908) as the group of Minkowski space-time isometries. It is a ten-generator non-Abelian Lie group of fundamental importance in physics.
A Minkowski space-time isometry has the property that the interval between events is left invariant. In Minkowski space (i.e. ignoring the effects of gravity), there are ten degrees of freedom of the isometries, which may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a “boost” in any of the three spatial directions (three degrees). Composition of transformations is the operator of the Poincaré group, with proper rotations being produced as the composition of an even number of reflections.
In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features shear mappings to relate co-moving frames of reference.
Poincaré symmetry is the full symmetry of Special Relativity. It includes: translations, or displacements, in time and space (P), forming the abelian Lie group of translations on space-time; rotations in space, forming the non-Abelian Lie group of three-dimensional rotations (J); boosts, transformations connecting two uniformly moving bodies (K).
The last two symmetries, J and K, together make the Lorentz group; the semi-direct product of the translations group and the Lorentz group then produce the Poincaré group. Objects which are invariant under this group are then said to possess Poincaré invariance or relativistic invariance.
The Poincaré group is the group of Minkowski space-time isometries. It is a ten-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup, while the Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the affine group which includes all translations and Lorentz transformations. More precisely, it is a semidirect product of the translations and the Lorentz group.