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| {{for|the Poincaré group (fundamental group) of a topological space|Fundamental group}}
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| {{Group theory sidebar |Topological}}
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| {{Lie groups |Other}}
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| In [[physics]] and [[mathematics]], the '''Poincaré group''', named after [[Henri Poincaré]],<ref>*{{Citation
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| |author=Poincaré, Henri
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| |year=1905/6
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| |title=[[s:fr:Sur la dynamique de l’électron (juillet)|Sur la dynamique de l’électron]]
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| |journal=Rendiconti del Circolo matematico di Palermo
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| |volume=21
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| |pages=129–176}}
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| *Wikisource translation: [[s:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]]</ref> is the [[Group (mathematics)|group]] of [[isometry|isometries]] of [[Minkowski spacetime]], introduced by [[Hermann Minkowski]].<ref>{{Citation
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| |author=Minkowski, Hermann
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| |year=1907/8
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| |title=[[s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern|Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern]]
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| |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
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| |pages=53–111}}
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| *Wikisource translation: [[s:The Fundamental Equations for Electromagnetic Processes in Moving Bodies|The Fundamental Equations for Electromagnetic Processes in Moving Bodies]].</ref><ref>{{Citation
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| |author=Minkowski, Hermann
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| |year=1908/9
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| |title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]
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| |journal=Physikalische Zeitschrift
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| |volume=10
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| |pages=75–88}}</ref> It is a [[Non-abelian group|Non-abelian]] [[Lie group]] with 10 generators, of fundamental importance in physics.
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| == Basic explanation ==
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| An [[isometry]] is a way in which the contents of spacetime could be shifted that would not affect the [[proper time]] along a [[trajectory]] between [[event (relativity)|event]]s. For example, if everything was postponed by two hours including 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 was shifted five miles to the west, you would also see no change in the interval. It turns out that the length of a rod is also unaffected by such a shift.
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| If one ignores the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation (by a fixed angle) around any of the three spatial axes, or a [[lorentz transformation|boost]] in any of the three spatial directions, altogether 1 + 3 + 3 + 3 = 10.
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| If one combines such isometries together (implementing one and then the other), the result is also such an isometry (although, in general, a linear combination of the ten basic ones detailed). These isometries form a [[group (mathematics)|group]]. That is, there is an identity (no shift, everything stays where it was), and inverses (move everything back to where it was), and it obeys the [[associative law]]. The name of this specific group is the "''Poincaré group''".
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| ==Technical explanation==
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| The Poincaré group is the [[group (mathematics)|group]] of [[isometry|isometries]] of [[Minkowski spacetime]]. It is a 10-dimensional [[compact space|noncompact]] [[Lie group]]. The [[abelian group]] of [[Translation (geometry)|translations]] is a [[normal subgroup]], while the [[Lorentz group]] is also a subgroup, the [[group action#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,
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| :<math>\mathbf{R}^{1,3} \rtimes SO(1,3) \,.</math>
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| 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.
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| 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]].
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| 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.
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| The '''Poincaré algebra''' is the [[Lie algebra]] of the Poincaré 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 [[Matrix exponential|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=http://books.google.co.uk/books?id=7VLMj4AvvicC&pg=PA273&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=LF9uUa7XNoLw0gX914GACA&ved=0CEEQ6AEwAg#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref><ref>{{cite book|isbn=1-13950-4320|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=http://books.google.co.uk/books?id=hRavtAW5EFcC&pg=PA11&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=LF9uUa7XNoLw0gX914GACA&ved=0CEYQ6AEwAw#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}</ref>
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|
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| {{Equation box 1
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| |indent =:
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| |equation = <math>~[P_\mu, P_\nu] = 0\,</math>
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| :<math>~\frac{ 1 }{ i }~[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\,</math>
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| :<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>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}
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| where {{mvar|P}} is the [[Lie group#The Lie algebra associated to a Lie group|generator]] of translations, {{mvar|M}} is the generator of Lorentz transformations, and {{mvar|η }} is the Minkowski metric (see [[sign convention]]).
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| 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
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| :<math>[J_m,P_n] = i \epsilon_{mnk} P_k ~,</math>
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| :<math>[J_i,P_0] = 0 ~,</math>
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| :<math>[K_i,P_k] = i \eta_{ik} P_0 ~,</math>
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| :<math>[K_i,P_0] = -i P_i ~,</math>
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| :<math>[J_m,J_n] = i \epsilon_{mnk} J_k ~,</math>
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| :<math>[J_m,K_n] = i \epsilon_{mnk} K_k ~,</math>
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| :<math>[K_m,K_n] = -i \epsilon_{mnk} J_k ~,</math>
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| where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". Note the important 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]].
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| 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.
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| 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 numbers]] {{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]] quantum number. Many quantum field theories in practice do violate parity and charge conjugation. In those cases, the {{mvar|P}} and the {{mvar|C}} are forfeited. Since [[CPT symmetry|''CPT'']] is an invariance of every [[quantum field theory]], a time reversal quantum number could easily be constructed out of those given.
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| 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.
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| ==Poincaré symmetry==
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| '''Poincaré symmetry''' is the full symmetry of [[special relativity]] and includes | |
| *'''[[translation (physics)|translations]]''' (i.e., displacements) in time and space, '''''P'''''. These form the [[abelian group|abelian]] [[Lie group]] of translations on space-time.
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| *'''[[rotation]]s''' in space (this forms the non-Abelian [[Lie group]] of 3-dimensional rotations, with generators '''''J''''')
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| *'''[[Lorentz Boost|boosts]]''', i.e., transformations connecting two uniformly moving bodies, with generators '''''K'''''.
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| The last two symmetries, '''''J''''' and '''''K''''', together make up the '''[[Lorentz group]]''' (see [[Lorentz invariance]]).
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| 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. Objects which are invariant under this group are said to possess '''Poincaré invariance''' or '''relativistic invariance'''.
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| ==See also==
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| * [[Euclidean group]]
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| * [[Representation theory of the Poincaré group]]
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| * [[Wigner's classification]]
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| * [[Symmetry in quantum mechanics]]
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| * [[Center of mass (relativistic)]]
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| * [[Pauli–Lubanski pseudovector]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book|title=Group Theory in Physics| author= Wu-Ki Tung| year= 1985|publisher=World Scientific Publishing| isbn=9971-966-57-3 }}
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| *{{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 }}
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| *{{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=http://books.google.co.uk/books?id=nnuW_kVJ500C&pg=PA62&dq=pauli-lubanski+pseudovector&hl=en&sa=X&ei=Wl1uUd75NtCZ0QWOp4HwDw&ved=0CDsQ6AEwAQ#v=onepage&q=pauli-lubanski%20pseudovector&f=false}}
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| {{DEFAULTSORT:Poincare Group}}
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| [[Category:Lie groups]]
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| [[Category:Particle physics]]
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| [[Category:Quantum field theory]]
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| [[Category:Theory of relativity]]
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| [[Category:Symmetry]]
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