Poincaré group - Citizendia

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In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element For the Mechanical engineering and Architecture usage see Isometric projection. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity It is a 10-dimensional noncompact Lie group. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. That is, the full Poincaré group is the affine group of the Lorentz group, the semidirect product of the translations and the Lorentz transformations: $\mathbf{R}^{1,3} \rtimes O(1,3)$ . In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

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, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group. In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

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 fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

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. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group

The Poincaré algebra is the Lie algebra of the Poincaré group. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie 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 translation, $M$ is the generator of Lorentz transformations and $η$ is the Minkowski metric (see sign convention). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Physics, a sign convention is a choice of the signs (plus or minus of a set of quantities in a case where the choice of sign is arbitrary

The Poincaré group is the full symmetry group of any relativistic field theory. In Physics, a field is a Physical quantity associated to each point of Spacetime. As a result, all elementary particles fall in representations of this group. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made These are usually specified by the four-momentum of each particle (i. e. its mass) and the intrinsic quantum numbers J^PC, where J is the spin quantum number, P is the parity and C is the charge conjugation quantum number. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Physics, C-symmetry means the symmetry of physical laws under a charge -conjugation transformation. 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. CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In quantum field theory (QFT the forces between particles are mediated by other particles

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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, translation is movement that changes the position of an object as opposed to Rotation. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to

The last two symmetries together make up the Lorentz group (see Lorentz invariance). In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally 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. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can Things which are invariant under this group are said to have Poincaré invariance or relativistic invariance.

Poincaré symmetry

See also