All Topics  
Poincaré group

 

 
   Email Print
   Bookmark   Link
 

 

Poincaré group
 
 
  Topics Poincaré group Topic Home

In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 and mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Poincaré group, named after Henri Poincaré
Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
, is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of isometries
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 of Minkowski spacetime. It is a 10-dimensional noncompact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
 Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
. The abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 of translations
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 is a normal subgroup
Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
 while the Lorentz group
Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
 is a subgroup, the stabilizer
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 of a point. That is, the full Poincaré group is the affine group
Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself....
 of the Lorentz group, the semidirect product
Semidirect product

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 be put together from two subgroups, one of which is a normal subgroup....
 of the translations and the Lorentz transformation
Lorentz transformation

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 other....
s:



Another way of putting it is the Poincaré group is a group extension
Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
 of the Lorentz group
Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
 by a vector representation
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
 of it.

Its positive energy unitary irreducible representations
Representation of a Lie group

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry....
 are indexed by mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 (nonnegative number) and spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 (integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 or half integer), and are associated with particles in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
.

In accordance with the Erlangen program
Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen ?ber neuere geometrische Forschungen....
, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space
Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a Group G is a non-empty manifold or topological space X on which G acts continuous function by symmetry in a transitivity way....
 for the group.

The Poincaré algebra is the Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 of the Poincaré group.






Discussion
Ask a question about 'Poincaré group'
Start a new discussion about 'Poincaré group'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 and mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Poincaré group, named after Henri Poincaré
Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
, is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of isometries
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 of Minkowski spacetime. It is a 10-dimensional noncompact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
 Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
. The abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 of translations
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 is a normal subgroup
Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
 while the Lorentz group
Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
 is a subgroup, the stabilizer
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 of a point. That is, the full Poincaré group is the affine group
Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself....
 of the Lorentz group, the semidirect product
Semidirect product

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 be put together from two subgroups, one of which is a normal subgroup....
 of the translations and the Lorentz transformation
Lorentz transformation

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 other....
s:



Another way of putting it is the Poincaré group is a group extension
Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
 of the Lorentz group
Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
 by a vector representation
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
 of it.

Its positive energy unitary irreducible representations
Representation of a Lie group

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry....
 are indexed by mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 (nonnegative number) and spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 (integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 or half integer), and are associated with particles in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
.

In accordance with the Erlangen program
Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen ?ber neuere geometrische Forschungen....
, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space
Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a Group G is a non-empty manifold or topological space X on which G acts continuous function by symmetry in a transitivity way....
 for the group.

The Poincaré algebra is the Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 of the Poincaré group. In component form, the Poincaré algebra is given by the commutation relations:



where is the generator
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 of translations, is the generator of Lorentz transformations and is the Minkowski metric (see sign convention
Sign convention

In physics, a sign convention is a choice of the Negative and non-negative numberss 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. As a result, all elementary particle
Elementary particle

In particle physics, an elementary particle or fundamental particle is a wiktionary:particle not known to have substructure; that is, it is not known to be made up of smaller particles....
s 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
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 quantum number, P is the parity
Parity

Parity is a concept of equality of status or functional equivalence. It has several different specific definitions.* Parity , the name of the symmetry of interactions under spatial inversion...
 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
CPT symmetry

CPT symmetry is a fundamental Symmetry in physics of physical laws under transformation s that involve the inversions of electric charge, parity and time simultaneously....
 is an invariance of every quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, a time reversal quantum number could easily be constructed out of those given.

Poincaré symmetry


Poincaré symmetry is the full symmetry of special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 and includes
  • translations
    Translation (physics)

    In physics, translation is movement that changes the displacement of an object, as opposed to rotation. For example, according to Whittaker:...
     (ie, displacements) in time and space (these form the abelian
    Abelian group

    An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
     Lie group
    Lie group

    In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
     of translations on space-time)
  • rotation
    Rotation

    A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
    s     in space (this forms the non-Abelian Lie group
    Lie group

    In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
     of 3-dimensional rotations)
  • boosts, ie, transformations connecting two uniformly moving bodies.
The last two symmetries together make up the Lorentz group
Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
 (see Lorentz invariance). These are generators of a Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 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

  • Euclidean group
    Euclidean group

    In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
  • Representation theory of the Poincaré group
    Representation theory of the Poincaré group

    In mathematics, the representation theory of the Poincar? group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group....
  • Wigner's classification
    Wigner's classification

    In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy Irreducible representations of the Poincar? group, which have sharp mass eigenvalues....