- Poincaré group
In
physics andmathematics , the Poincaré group, named afterHenri Poincaré , is the group of isometries ofMinkowski spacetime . It is a 10-dimensional noncompactLie group . Theabelian group of translations is anormal subgroup while theLorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is theaffine group of the Lorentz group, thesemidirect product of the translations and theLorentz transformation s: .Another way of putting it is the Poincaré group is a
group extension of theLorentz 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 inquantum mechanics .In accordance with the
Erlangen program , the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as ahomogeneous 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 is the generator of translation, is the generator of Lorentz transformations and is the Minkowski metric (see
sign convention ).The Poincaré group is the full symmetry group of any
relativistic field theory . As a result, allelementary particle s fall in representations of this group. These are usually specified by the "four-momentum" of each particle (i.e. its mass) and the intrinsicquantum numbers JPC, where J is the spin quantum number, P is theparity and C is thecharge 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 everyquantum 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 abelianLie group of translations on space-time)
*rotation s in space (this forms the non-AbelianLie group of 3-dimensional rotations)
*boosts, ie, transformations connecting two uniformly moving bodies.The last two symmetries together make up theLorentz group (seeLorentz invariance ). These are generators of aLie group called the Poincaré group which is asemi-direct 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.ee also
*
Euclidean group
*Representation theory of the Poincaré group
*Wigner's classification
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