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. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), Nature) is the science of Nature in the broadest sense. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, an abelian group is a commutative group, i. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In mathematics, groups are often used to describe symmetries of objects. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ...

Another way of putting it is the Poincaré group is a group extension of the Lorentz group by a vector representation of it. In mathematics, for G a group or algebra over a field, or other algebraic structure, G′ is an extension of G if there is an exact sequence . See also central extension, extension problem, field extension. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In mathematics, a group representation is a way of viewing a group in some more concrete way. ...

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 mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... Fig. ...

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. An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

In component form, the Lie algebra of the Poincaré group satisfies In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

• [P_μ,P_ν] = 0

• [M_μν,P_ρ] = η_μρP_ν − η_νρP_μ

• [M_μν,M_ρσ] = η_μρM_νσ − η_μσM_νρ − η_νρM_μν + η_νσM_μρ

where P is the generator of translation and M is the generator of Lorentz transformations. See sign convention. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...

See also

• Euclidean group
• Henri Poincaré
• hyper generalized orthogonal Lie algebra
• Lorentz group
• Wigner's classification