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Group theory
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Group theory

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. It is a 10-dimensional 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:

\mathbf{R}^{1,3} \rtimes O(1,3).\,

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:

  • [P_\mu, P_\nu] = 0\,
  • [M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\,
  • [M_{\mu\nu}, M_{\rho\sigma}] = \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\nu\sigma} M_{\mu\rho}\,

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 four-momentum of each particle (i.e. its mass) and the intrinsic quantum numbers JPC, 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 space-time)
  • rotations in space (this forms the non-Abelian Lie group of 3-dimensional 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 semi-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.

See also

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