Poincaré group
   HOME

TheInfoList



OR:

The Poincaré group, named after Henri Poincaré (1906), was first defined by
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
(1908) as the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of Minkowski spacetime isometries. It is a ten-dimensional non-abelian
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
that is of importance as a model in our understanding of the most basic fundamentals of
physics Physics is the natural science that studies matter, its 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 which r ...
.


Overview

A Minkowski spacetime isometry has the property that the interval between
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
s is left invariant. 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 stop-watch 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 Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...
of an object is also unaffected by such a shift. A time or space reversal (a reflection) is also an isometry of this group. In Minkowski space (i.e. ignoring the effects of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
), there are ten degrees of freedom of the
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, which 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 proper rotations being produced as the composition of an even number of reflections. In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the
Galilean group In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
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 a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
s to relate co-moving frames of reference.


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 regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. It includes: * ''
translation Translation is the communication of the Meaning (linguistic), 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 ...
s'' (displacements) in time and space (''P''), forming the abelian
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of translations on space-time; * ''
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s'' in space, forming the non-Abelian Lie group of
three-dimensional rotation In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a t ...
s (''J''); * '' boosts'', transformations connecting two uniformly moving bodies (''K''). The last two symmetries, ''J'' and ''K'', together make the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
(see also
Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
); the
semi-direct product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
of the 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 Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, imply 10 conservation laws: 1 for the energy, 3 for the momentum, 3 for the angular momentum and 3 for the velocity of the center of mass.


Poincaré group

The Poincaré group is the group of Minkowski spacetime
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
. It is a ten-dimensional noncompact
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. The
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
of
translation Translation is the communication of the Meaning (linguistic), 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 ...
s is a normal subgroup, while the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
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 over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Rela ...
which includes all translations and
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s. More precisely, it is a semidirect product of the translations 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 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 :1\to N\;\ove ...
of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
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 In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a ...
of the de Sitter group SO(4,1) ~ Sp(2,2), as the de Sitter radius goes to infinity. Its positive energy unitary irreducible
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
are indexed by
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
(nonnegative number) and spin (
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
or half integer) and are associated with particles in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(see
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
). In accordance with the
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
, 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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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 In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
. 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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the Poincaré group. It is a
Lie algebra extension In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by ta ...
of the Lie algebra of the Lorentz group. More specifically, the proper (\det\Lambda = 1),
orthochronous In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
(_0 \geq 1) part of the Lorentz subgroup (its
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...
), SO(1, 3)_+^\uparrow, is connected to the identity and is thus provided by the
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
\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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. In component form, the Poincaré algebra is given by the commutation relations: where P is the
generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...
of translations, M is the generator of Lorentz transformations, and \eta is the (+,-,-,-) Minkowski metric (see
Sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
). 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 invariant In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
s of this algebra are P_\mu P^\mu and W_\mu W^\mu where W_\mu is the
Pauli–Lubanski pseudovector In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański, It describ ...
; they serve as labels for the representations of the group. The Poincaré group is the full symmetry group of any
relativistic field theory In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point ...
. 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. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
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 numbers 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
; where this occurs, P and C are forfeited. Since
CPT symmetry Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, 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 ''d''-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.


Super-Poincaré algebra

A related observation is that the
representations of the Lorentz group The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representatio ...
include a pair of inequivalent two-dimensional complex
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
representations 2 and \overline whose
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
2\otimes\overline = 3\oplus1 is the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
. One may identify this last bit with four-dimensional Minkowski space itself (as opposed to identifying it with a spin-1 particle, as would normally be done for a pair of
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, e.g. a
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
being composed of a
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
-anti-quark pair). This strongly suggests that it might be possible to extend the Poincaré algebra to also include spinors. This leads directly to the notion of the
super-Poincaré algebra In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symme ...
. The mathematical appeal of this idea is that one is working with the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
s, instead of the adjoint representations. The physical appeal of this idea is that the fundamental representations correspond to
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, which are seen in nature. So far, however, the implied
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
here, of a symmetry between spatial and fermionic directions, has not been seen experimentally in nature. The experimental issue can roughly be stated as the question: if we live in the adjoint representation (Minkowski spacetime), then where is the fundamental representation hiding?


See also

* Euclidean group *
Representation theory of the Poincaré group In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theor ...
*
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
*
Symmetry in quantum mechanics Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the ...
*
Center of mass (relativistic) In physics, relativistic center of mass refers to the mathematical and physical concepts that define the center of mass of a system of particles in relativistic mechanics and relativistic quantum mechanics. Introduction In non-relativistic physi ...
*
Pauli–Lubanski pseudovector In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański, It describ ...
*
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 In theoretical physics, a continuous spin particle (CSP), sometimes called an infinite spin particle, is a massless particle never observed before in nature. This particle is one of Poincaré group's massless representations which, along with ordi ...


Notes


References

* * * {{DEFAULTSORT:Poincare Group Lie groups
Group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
Quantum field theory Theory of relativity Symmetry