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The Poincaré group, named after
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
(1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
that is of importance as a model in our understanding of the most basic fundamentals of
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
.


Overview

The Poincaré group consists of all
coordinate transformations In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
of Minkowski space that do not change the spacetime interval between events. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. In total, there are ten
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a " boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s being produced as the composition of an even number of reflections. In
classical physics Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features
shear mapping In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance function, signed distance from a given straight line, line parallel (geometry), paral ...
s to relate co-moving frames of reference. In
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, i.e. under the effects of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article.


Poincaré symmetry

Poincaré symmetry is the full symmetry of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. It includes: * ''
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s'' (displacements) in time and space, forming the abelian Lie group of spacetime translations (''P''); * ''
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s'' in space, forming the non-abelian Lie group of three-dimensional rotations (''J''); * '' boosts'', transformations connecting two uniformly moving bodies (''K''). The last two symmetries, ''J'' and ''K'', together make the Lorentz group (see also '' Lorentz invariance''); the semi-direct product of the spacetime translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic invariance. 10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws: * 1 for the energy – associated with translations through time * 3 for the momentum – associated with translations through spatial dimensions * 3 for the angular momentum – associated with rotations between spatial dimensions * 3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time


Poincaré group

The Poincaré group is the group of Minkowski spacetime isometries. It is a ten-dimensional noncompact
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
. The four-dimensional abelian group of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
, while the six-dimensional Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the
affine group In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real nu ...
which includes all translations and Lorentz transformations. More precisely, it is a semidirect product of the spacetime translations group and the Lorentz group, : \mathbf^ \rtimes \operatorname(1, 3) \,, with group multiplication : (\alpha, f) \cdot (\beta, g) = (\alpha + f \cdot \beta,\; f \cdot g). Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a group contraction of the de Sitter group , as the de Sitter radius goes to infinity. Its positive energy unitary irreducible representations are indexed by
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
(nonnegative number) and spin (
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
or half integer) and are associated with particles in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
(see Wigner's classification). 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. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, the universal cover of the Poincaré group : \mathbf^ \rtimes \operatorname(2, \mathbf), which may be identified with the double cover : \mathbf^ \rtimes \operatorname(1, 3), is more important, because representations of \operatorname(1, 3) are not able to describe fields with spin 1/2; i.e. fermions. Here \operatorname(2,\mathbf) is the group of complex 2 \times 2 matrices with unit determinant, isomorphic to the Lorentz-signature spin group \operatorname(1, 3).


Poincaré algebra

The Poincaré algebra is the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper (\det\Lambda = 1), orthochronous (_0 \geq 1) part of the Lorentz subgroup (its
identity component In mathematics, specifically group theory, the identity component of a group (mathematics) , group ''G'' (also known as its unity component) refers to several closely related notions of the largest connected space , connected subgroup of ''G'' co ...
), \mathrm(1, 3)_+^\uparrow, is connected to the identity and is thus provided by the
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
\exp\left(ia_\mu P^\mu\right)\exp\left(\frac\omega_ M^\right) of this
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
. In component form, the Poincaré algebra is given by the commutation relations: where P is the generator of translations, M is the generator of Lorentz transformations, and \eta is the (+,-,-,-) Minkowski metric (see Sign convention). The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, J_i = \frac\epsilon_ M^, and boosts, K_i = M_. In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as : \begin[] [J_m, P_n] &= i \epsilon_ P_k ~, \\[] [J_i, P_0] &= 0 ~, \\[] [K_i, P_k] &= i \eta_ P_0 ~, \\[] [K_i, P_0] &= -i P_i ~, \\[] [J_m, J_n] &= i \epsilon_ J_k ~, \\[] [J_m, K_n] &= i \epsilon_ K_k ~, \\[] [K_m, K_n] &= -i \epsilon_ J_k ~, \end where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification [J_m + iK_m,\, J_n -iK_n] = 0 permits reduction of the Lorentz subalgebra to \mathfrak(2) \oplus \mathfrak(2) and efficient treatment of its associated Representation theory of the Lorentz group, representations. In terms of the physical parameters, we have : \begin \left mathcal H, p_i\right&= 0 \\ \left mathcal H, L_i\right&= 0 \\ \left mathcal H, K_i\right&= i\hbar cp_i \\ \left _i, p_j\right&= 0 \\ \left _i, L_j\right&= i\hbar\epsilon_p_k \\ \left _i, K_j\right&= \fracc\mathcal H\delta_ \\ \left _i, L_j\right&= i\hbar\epsilon_L_k \\ \left _i, K_j\right&= i\hbar\epsilon_K_k \\ \left _i, K_j\right&= -i\hbar\epsilon_L_k \end The Casimir invariants of this algebra are P_\mu P^\mu and W_\mu W^\mu where W_\mu is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group. The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a c ...
s fall in representations of this group. These are usually specified by the ''four-momentum'' squared of each particle (i.e. its mass squared) and the intrinsic
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s J^, where J is the spin quantum number, P is the parity and C is the charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantum field theories; where this occurs, P and C are forfeited. Since CPT symmetry is invariant in quantum field theory, a time-reversal quantum number may be constructed from those given. As a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted.


Other dimensions

The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The -dimensional Poincaré group is analogously defined by the semi-direct product : \operatorname(1, d - 1) := \mathbf^ \rtimes \operatorname(1, d - 1) with the analogous multiplication : (\alpha, f) \cdot (\beta, g) = (\alpha + f \cdot \beta,\; f \cdot g). The Lie algebra retains its form, with indices and now taking values between and . The alternative representation in terms of and has no analogue in higher dimensions.


See also

* Euclidean group * Galilean group * Representation theory of the Poincaré group * Wigner's classification * Symmetry in quantum mechanics * Pauli–Lubanski pseudovector *
Particle physics and representation theory There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this con ...
* Continuous spin particle * super-Poincaré algebra


Notes


References

* * * {{DEFAULTSORT:Poincare Group Lie groups Group Quantum field theory Theory of relativity Symmetry