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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.
Its positive energy unitary irreducible representations are indexed by massMass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass Inertial mass (nonnegative number) and spinIn physics, spin is an intrinsic angular momentum associated with microscopic particles. It is a purely quantum mechanical phenomenon without any analogy in classical mechanics. Whereas classical angular momentum arises from the rotation of an extended ob ( integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st or half integer), and are associated with particles in quantum mechanicswavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n 1,2,3,. and angular momentum (increasing across: s p d . Brighter areas correspond to higher probability density for a position measurement. The angular mom.
In accordance with the Erlangen programAn influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen uber neuere geometrische Forschungen''. This Erlangen Program (Klein at the time was at Erlangen) proposed a new kind of solu, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as an homogeneous spaceIn 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 o for the group.
In component form, the Lie algebraIn 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. Lie algebras were introduced to study the concept of infi of the Poincaré group satisfies
where P is the generator of translation and M is the generator of Lorentz transformations. See sign convention.