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[[File:Henri Poincare.jpg|thumb|180px|Henri Poincaré]]
{{for|the Poincaré group (fundamental group) of a topological space|Fundamental group}}
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The '''Poincaré group''', named after [[Henri Poincaré]] (1906),<ref>{{citation |author=Poincaré, Henri |date=<!--1905–6--> |title=Sur la dynamique de l'électron |journal=Rendiconti del Circolo Matematico di Palermo |volume=21 |pages=129–176 |doi=10.1007/bf03013466|title-link=s:fr:Sur la dynamique de l'électron (juillet) |hdl=2027/uiug.30112063899089 |url=https://zenodo.org/record/1428444/files/article.pdf }} ([[Wikisource]] translation: [[s:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]]). The group defined in this paper would now be described as the homogeneous Lorentz group with scalar multipliers. </ref> was first defined by [[Hermann Minkowski]] (1908) as the [[Group (mathematics)|group]] of [[Minkowski_space#Lorentz transformations and symmetry|Minkowski spacetime isometries]].<ref>{{citation |author=Minkowski, Hermann |date=<!--1907–8--> |title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |pages=53–111|title-link=s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern }} (Wikisource translation: [[s:The Fundamental Equations for Electromagnetic Processes in Moving Bodies|The Fundamental Equations for Electromagnetic Processes in Moving Bodies]]).</ref><ref>{{citation |author=Minkowski, Hermann |date=<!--1908–9--> |title=Raum und Zeit |journal=Physikalische Zeitschrift |volume=10 |pages=75–88|title-link=s:de:Raum und Zeit (Minkowski) }}</ref> It is a ten-dimensional [[Non-abelian group|non-abelian]] [[Lie group]] of fundamental importance in [[physics]].

== Overview ==
A [[Minkowski_space#Lorentz transformations and symmetry|Minkowski spacetime isometry]] has the property that the interval between [[event (relativity)|event]]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]] 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]]), there are ten degrees of freedom of the [[Minkowski_space#Lorentz transformations and symmetry|isometries]], 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 "[[Lorentz transformation|boost]]" in any of the three spatial directions (three degrees). Composition of transformations is the operator of the Poincaré group, with proper rotations being produced as the composition of an even number of reflections.

In [[classical physics]], the [[Galilean group]] is a comparable ten-parameter group that acts on [[absolute time and space]]. Instead of boosts, it features [[shear mapping]]s to relate co-moving frames of reference.

== Poincaré symmetry ==
'''Poincaré symmetry''' is the full symmetry of [[special relativity]]. It includes:
* ''[[Translation (physics)|translation]]s'' (displacements) in time and space ('''''P'''''), forming the [[Abelian group|abelian]] [[Lie group]] of translations on space-time;
* ''[[rotation]]s'' in space, forming the non-Abelian Lie group of [[three-dimensional rotation]]s ('''''J''''');
* ''[[Lorentz transformation|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 translations group and the Lorentz group then produce the Poincaré group. Objects which are invariant under this group are then said to possess '''Poincaré invariance''' or '''relativistic invariance'''.

== Poincaré group ==
The Poincaré group is the group of Minkowski spacetime [[Minkowski_space#Lorentz transformations and symmetry|isometries]]. It is a ten-dimensional [[Compact space|noncompact]] [[Lie group]]. The [[abelian group]] of [[Translation (geometry)|translation]]s is a [[normal subgroup]], while the [[Lorentz group]] is also a subgroup, the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of the origin. The Poincaré group itself is the minimal subgroup of the [[affine group]] which includes all translations and [[Lorentz transformation]]s. More precisely, it is a [[semidirect product]] of the translations and the Lorentz group, 
:<math>\mathbf{R}^{1,3} \rtimes \mathrm{O}(1,3) \,,</math>
with group multiplication
:<math>(\alpha, f) \cdot (\beta, g) = (\alpha + f \cdot \beta, \; f \cdot g)</math>.<ref name=":0">{{Cite book|url=https://books.google.de/books?id=mVovDwAAQBAJ|title=BMS Particles in Three Dimensions|last=Oblak|first=Blagoje|date=2017-08-01|publisher=Springer|isbn=9783319618784|location=|pages=80|language=en}}</ref>

Another way of putting this is that the Poincaré group is a [[group extension]] of the [[Lorentz group]] by a vector [[Group representation|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 SO(4,1) ~ Sp(2,2), as the [[de Sitter space|de Sitter radius]] goes to infinity.

Its positive energy unitary irreducible [[Representation of a Lie group|representations]] are indexed by [[mass]] (nonnegative number) and [[Spin (physics)|spin]] ([[integer]] or half integer) and are associated with particles in [[quantum mechanics]] (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]], the universal cover of the Poincaré group
:<math>\mathbf{R}^{1,3} \rtimes \mathrm{SL}(2,\mathbf{C}), </math>
which may be identified with the double cover
:<math>\mathbf{R}^{1,3} \rtimes \mathrm{Spin}(1,3), </math>
is more important, because representations of <math> \mathrm{SO}(1,3)</math> are not able to describe fields with spin 1/2, i.e. [[fermions]].  Here <math> \mathrm{SL}(2,\mathbf{C})</math> is the group of complex  <math>2 \times 2 </math> matrices with unit determinant, isomorphic to the [[Spin group#Indefinite signature|Lorentz-signature spin group]] <math>\mathrm{Spin}(1,3)</math>.

==Poincaré algebra==
{{Lie groups |Other}}
The '''Poincaré algebra''' is the [[Lie algebra]] of the Poincaré group. It is a [[Lie algebra extension#By semidirect sum|Lie algebra extension]] of the Lie algebra of the Lorentz group. More specifically, the proper (det''Λ''=1), [[Lorentz group#Connected components|orthochronous]] ({{math|''Λ''<sup>0</sup><sub>0</sub>≥1}}) part of the Lorentz subgroup (its [[identity component]]), SO<sup>+</sup>(1, 3),  is connected to the identity and is thus provided by the [[Exponential_map_(Lie_theory)|exponentiation]]  {{math|exp(''ia<sub>μ</sub>P<sup>μ</sup>'') exp(''iω<sub>μν</sub>M<sup>μν</sup>''/2)}} of this [[Lie algebra]].  In component form, the Poincaré algebra is given by the commutation relations:<ref>{{cite book |title=General Principles of Quantum Field Theory |author=N.N. Bogolubov| publisher=Springer |edition=2nd |isbn=0-7923-0540-X |year=1989 |page=272 |url=https://books.google.com/books?id=7VLMj4AvvicC&pg=PA273&dq=pauli-lubanski+pseudovector#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref><ref>{{cite book |isbn=978-1-13950-4324 |author=T. Ohlsson |title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory |publisher=Cambridge University Press |year=2011 |page=10 |url=https://books.google.com/books?id=hRavtAW5EFcC&pg=PA11&dq=pauli-lubanski+pseudovector#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref>
{{Equation box 1
 |indent=: |cellpadding=6 |border |border colour=#0073CF |bgcolor=#F9FFF7
 | equation = <math>~[P_\mu, P_\nu] = 0\,</math>
: <math>~\frac{ 1 }{ i }~[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\,</math>
: <math>~\frac{ 1 }{ i }~[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}\, ,</math>
}}
where {{mvar|P}} is the [[Lie group#The Lie algebra associated with a Lie group|generator]] of translations, {{mvar|M}} is the generator of Lorentz transformations, and {{mvar|η }} is the (+,−,−,−) Minkowski metric (see [[Sign convention]]).

The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, {{math|''J<sub>i</sub>'' {{=}} ''ϵ<sub>imn</sub>M<sup>mn</sup>''/2}}, and boosts, {{math|''K<sub>i</sub>'' {{=}} ''M<sub>i0</sub>''}}.  In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as
: <math>[J_m,P_n] = i \epsilon_{mnk} P_k ~,</math>
: <math>[J_i,P_0] =  0 ~,</math>
: <math>[K_i,P_k] = i \eta_{ik} P_0 ~,</math>
: <math>[K_i,P_0] = -i P_i ~,</math>
: <math>[J_m,J_n] = i \epsilon_{mnk} J_k ~,</math>
: <math>[J_m,K_n] = i \epsilon_{mnk} K_k ~,</math>
: <math>[K_m,K_n] = -i \epsilon_{mnk} J_k  ~,</math>

where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification {{math|[''J<sub>m</sub>+i K<sub>m</sub> ,  J<sub>n</sub>−i K<sub>n</sub>''] {{=}} 0}},  which permits reduction of the Lorentz subalgebra to '''su(2)'''⊕'''su(2)''' and efficient treatment of its associated [[Representation theory of the Lorentz group|representations]], is important. In terms of the physical parameters, we have

<math>\left[\mathcal H,p_i\right]=0</math>

<math>\left[\mathcal H,L_i\right]=0</math>

<math>\left[\mathcal H,K_i\right]=i\hbar cp_i</math>

<math>\left[p_i,p_j\right]=0</math>

<math>\left[p_i,L_j\right]=i\hbar\epsilon_{ijk}p_k</math>

<math>\left[p_i,K_j\right]=\frac{i\hbar}c\mathcal H\delta_{ij}</math>

<math>\left[L_i,L_j\right]=i\hbar\epsilon_{ijk}L_k</math>

<math>\left[L_i,K_j\right]=i\hbar\epsilon_{ijk}K_k</math>

<math>\left[K_i,K_j\right]=-i\hbar\epsilon_{ijk}L_k</math>

The [[Casimir invariant]]s of this algebra are {{math|''P''<sub>''μ''</sub>''P''<sup>''μ''</sup>}} and {{math|''W''<sub>''μ''</sub> ''W''<sup>''μ''</sup>}} where {{math|''W''<sub>''μ''</sub>}} 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]]s fall in [[Wigner's classification|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]]s {{math|''J<sup>PC</sup>''}}, where {{mvar|J}} is the [[Spin (physics)|spin]] quantum number, {{mvar|P}} is the [[Parity (physics)|parity]] and {{mvar|C}} is the [[Charge conjugation|charge-conjugation]] quantum number. In practice, charge conjugation and parity are violated by many [[Quantum field theory|quantum field theories]]; where this occurs, {{mvar|P}} and {{mvar|C}} are forfeited. Since [[CPT symmetry]] is [[Invariant (physics)|invariant]] in quantum field theory, a [[T-parity|time-reversal quantum number]] may be constructed from those given.

As a [[topological space]], 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

:<math>\mathrm{IO}(1,d-1) := \mathbf{R}^{1,d-1} \rtimes \mathrm{O}(1,d-1) </math>

with the analogous multiplication

:<math>(\alpha, f) \cdot (\beta, g) = (\alpha + f \cdot \beta, \; f \cdot g)</math>.<ref name=":0" />

The Lie algebra retains its form, with indices {{mvar|µ}} and {{mvar|ν}} now taking values between {{math|0}} and {{math|d-1}}. The alternative representation in terms of {{math|''J<sub>i</sub>''}} and {{math|''K<sub>i</sub>''}} has no analogue in higher dimensions.

==Super-Poincaré algebra==
A related observation is that the [[representations of the Lorentz group]] include a pair of inequivalent two-dimensional complex [[spinor]] representations <math>2</math> and <math>\overline{2}</math> whose [[tensor product]] <math>2\otimes\overline{2}=3\oplus1</math> is the [[adjoint representation]]. 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]]s, e.g. a [[pion]] being composed of a [[quark]]-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]]. The mathematical appeal of this idea is that one is working with the [[fundamental representation]]s, instead of the adjoint representations.  The physical appeal of this idea is that the fundamental representations correspond to [[fermion]]s, which are seen in nature. So far, however, the implied [[supersymmetry]] here, of a symmetry between spatial and fermionic directions, cannot be 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]]
* [[Wigner's classification]]
* [[Symmetry in quantum mechanics]]
* [[Center of mass (relativistic)]]
* [[Pauli–Lubanski pseudovector]]
* [[Particle physics and representation theory]]

==Notes==
{{reflist|30em}}

== References ==
{{wikibooks|Associative Composition Algebra|Homographies|Poincaré group}}
* {{cite book |title=Group Theory in Physics |author=Wu-Ki Tung |year=1985 |publisher=World Scientific Publishing |isbn=9971-966-57-3}}
* {{cite book |title=The Quantum Theory of Fields |volume=1 |last=Weinberg |first=Steven |year=1995 |publisher=Cambridge University press |location=Cambridge |isbn=978-0-521-55001-7 |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev }}
* {{cite book |title=Quantum Field Theory |author=L.H. Ryder |publisher=Cambridge University Press |edition=2nd |isbn=0-52147-8146 |year=1996 |page=62 |url=https://books.google.com/books?id=nnuW_kVJ500C&pg=PA62&dq=pauli-lubanski+pseudovector#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}

{{DEFAULTSORT:Poincare Group}}
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