Poincaré group

In physics and mathematics, the Poincaré group 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 semidirect product of the translations and the Lorentz transformations.

Related Topics:
Physics - Mathematics - Group - Isometries - Minkowski spacetime - Noncompact - Lie group - Abelian group - Translations - Normal subgroup - Lorentz group - Stabilizer - Semidirect product - Lorentz transformation

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Another way of putting it is the Poincaré group is a group extension of the Lorentz group by a vector representation of it.

Related Topics:
Group extension - Lorentz group - Representation

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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.

Related Topics:
Representations - Mass - Spin - Integer - Quantum mechanics

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In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as an homogeneous space for the group.

Related Topics:
Erlangen program - Homogeneous space

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In component form, the Lie algebra of the Poincaré group satisfies

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• $[P\_mu,\; P\_\; u]\; =\; 0$
• $[M\_\{mu\; u\},\; P\_\; ho]\; =\; eta\_\{mu\; ho\}\; P\_\; u\; -\; eta\_\{\; u\; ho\}\; P\_mu$
• $[M\_\{mu\; u\},\; M\_\{\; hosigma\}]\; =\; eta\_\{mu\; ho\}\; M\_\{\; usigma\}\; -\; eta\_\{musigma\}\; M\_\{\; u\; ho\}\; -\; eta\_\{\; u\; ho\}\; M\_\{mu\; u\}\; +\; eta\_\{\; usigma\}\; M\_\{mu\; ho\}$

where P is the generator of translation and M is the generator of Lorentz transformations. See sign convention.

Related Topics:
Generator - Sign convention

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Poincaré group: See also ►