From Wikipedia, the free encyclopedia
Group
theory 

Group theory
Finite groups and classification of
finite simple groups 
Cyclic group
Z_{n}
Symmetric
group, S_{n}
Dihedral
group, D_{n}
Alternating
group A_{n}
Mathieu groups
M_{11}, M_{12}, M_{22}, M_{23},
M_{24}
Conway groups
Co_{1}, Co_{2}, Co_{3}
Janko groups J_{1}, J_{2}, J_{3}, J_{4}
Fischer groups
F_{22}, F_{23}, F_{24}
Baby
Monster group B
Monster group
M


This box: view • talk • edit

In physics and mathematics, the
Poincaré group, named after Henri
Poincaré, is the group of isometries of Minkowski
spacetime. It is a 10dimensional noncompact Lie group. The abelian group of translations is a normal subgroup
while the Lorentz
group is a subgroup, the stabilizer of a point. That is, the full
Poincaré group is the affine group of the Lorentz group, the semidirect
product of the translations and the Lorentz transformations:
Another way of putting it is the Poincaré group is a group extension
of the Lorentz
group by a vector representation of it.
Its positive energy unitary irreducible representations are
indexed by mass (nonnegative
number) and spin (integer or half integer), and are associated
with particles in quantum mechanics.
In accordance with the Erlangen program, the geometry of
Minkowski space is defined by the Poincaré group: Minkowski space
is considered as a homogeneous space for the group.
The Poincaré algebra is the Lie algebra of the
Poincaré group. In component form, the Poincaré algebra is given by
the commutation relations:
where P is the generator of translations,
M is the generator of Lorentz
transformations and η is the Minkowski
metric (see sign
convention).
The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all
elementary particles fall in
representations of this group. These are usually specified by the
fourmomentum of each particle (i.e. its mass) and the
intrinsic quantum numbers J^{PC}, where J is
the spin
quantum number, P is the parity and C is the charge conjugation quantum number. Many
quantum field theories do violate parity and charge conjugation. In
those case, we drop the P and the C. Since CPT is an invariance of every quantum
field theory, a time reversal quantum number could easily be
constructed out of those given.
Poincaré
symmetry
Poincaré symmetry is the full symmetry of special
relativity and includes
 translations (ie,
displacements) in time and space (these form the abelian Lie group of translations
on spacetime)
 rotations in space (this forms the
nonAbelian Lie group of
3dimensional rotations)
 boosts, ie, transformations
connecting two uniformly moving bodies.
The last two symmetries together make up the Lorentz
group (see Lorentz
invariance). These are generators of a Lie group called the Poincaré
group which is a semidirect
product of the group of translations and the Lorentz group.
Things which are invariant under this group are said to have
Poincaré invariance or relativistic
invariance.
See also