Poincaré Group  

In General > s.a. lorentz group.
$ Def: The inhomogeneous Lorentz group ISO(3, 1) of symmetries of Minkowski space; It has the structure of a semidirect product of the vector representation D (1/2,1/2) of the Lorentz group (i.e., the Euclidean group), and the (homogeneous) Lorentz group,

P = {(a, Λ) | a ∈ \(\mathbb R\)4, Λ ∈ L} ,  with  (a, Λ) (a', Λ'):= (a + Λ'a, ΛΛ') .

* Topology: It has four connected components (from those of L), while the complex P has 2 components; It is doubly connected, a 2π rotation (as a curve) cannot be deformed to the identity, but a 4π rotation can; Its (2-fold) covering group is ISL(2, \(\mathbb C\)).
* And gauge theory: A gauge theory of the Poincaré group can be built only in 2+1 dimensions, since otherwise there is no invariant, non-degenerate metric on the Lie algebra; > s.a. gauge theory of gravity.
@ References: Kim & Noz 86.

Representations > s.a. dirac equation; lorentz invariance; maxwell theory.
* And physics: In Minkowski space they give rise to tensor fields (representations of ISL(2, \(\mathbb C\)) give rise to spinor fields).
* In 1+1 dimensions: The Poincaré group has only 1D finite irreducible representations (> see group representations).
@ General references: Wigner AM(39), AM(47); Bargmann AM(47); Bargmann & Wigner PNAS(48); Bargmann AM(54); Wigner 59; Halpern 68; in Wald 84, §13.1 [short]; Mirman 95 [massless fields]; Burdik et al NPPS(01)ht [field theory Lagrangians]; Straumann a0809-conf [rev]; Nisticò JPCS(19)-a1901 [new ones].
@ Continuous spin: Brink et al JMP(02)ht; Khan & Ramond JMP(05)ht/04 [from higher dimensions]; Schuster & Toro JHEP(13)-a1302 [evidence for consistent interactions], JHEP(13)-a1302 [local, covariant gauge-field action]; Font et al FdP(14)-a1302 [and perturbative string theory]; Rivelles EPJC(17)-a1607; Najafizadeh JHEP(20)-a1912 [supersymmetric gauge theory]; Buchbinder et al NPB(20)-a2005 [Lagrangian]; > s.a. spinning particles.
@ Semigroup representations: Bohm et al PLA(00)ht/99, ht/99-proc.
@ Related topics: Brooke & Schroeck JMP(96) [m = 0, any s]; Brink ht/05-conf [non-linear representations, supersymmetry]; Kaźmierczak a1009 [non-trivial realization of the space-time translations in field theory]; Pedro a1307 [Majorana spinor representation], a1309 [real representations]; Csáki et al a2010 [multi-particle representations]; Buchbinder et al PLB(21)-a2011 [massless, in 6D]; Bermúdez a2105 [for classical relativistic dynamics].

Special Representations
* Identity (trivial) representation: Physically, it gives the vacuum.
* On a Hilbert space: An element of P acting on quantum states must be either unitary or antiunitary (without loss of generality, from preservation of transition amplitudes); Elements connected to the identity will be unitary.
* Unitary up to a phase: They can be reduced to those up to a sign, and these to the (true) unitary representations of the universal covering group ISL(2, \(\mathbb C\)); The latter can always be decomposed into irrep's.
* Classification: Irrep's of ISL(2, \(\mathbb C\)) can be labelled by the Casimir operators, m2 and s2, of the Lie algebra,
(a) m2 < 0: > see tachyons,
(b) m2 = 0, translations all represented by I: not very significant,
(c) m2 = 0, not all translations represented by I: either (1) helicity 0, 1/2, 1, ..., or (2) "continuous spin",
(d) m2 > 0: S2 = s(s+1), s = 0, 1/2, 1, ...;
A realization of (a) and (b) as spacetime fields appears not to exist; The useful ones seem to be just (c1) and (d).
@ References: Longo et al CMP(15)-a1505 [infinite-spin representations].

Lie Algebra
* Generators: Pa, Jab, where a, b = 1, ..., d, with commutation relations

[Jab, Jcd] = SO(n−1) relations;    [Pa, Pb] = 0;    [Pa, Jbc] = i gab Pc − i gac Pb .

* 1+1 dimensions: The commutation relations are [Pa, Pb] = 0; [Λ, Pa] = εab Pb, where Λ:= (i/2) εbc Jbc.
* 2+1 dimensions: The commutation relations are [Ja, Jb] = εabc Jc; [Pa, Pb] = 0; [Ja, Pb] = εabc Pc, where Ja:= \({1\over2}\)εabc Jbc.

Other References > s.a. lie algebra / categories in physics [Poincaré 2-group]; CPT symmetry; Drinfel'd Doubles; Position [tests of position invariance].
@ And position operator in quantum theory: Aldaya et al JPA(93).
@ And field theory: Savvidou JMP(02)gq/01, CQG(01)gq [2 actions]; Froggatt & Nielsen AdP(05)ht [emergence of Poincaré invariance]; D'Olivo & Socolovsky a1104 [Poincaré gauge invariance of general relativity and Einstein-Cartan gravity]; > s.a. dirac fields.
@ Super-Poincaré algebra / group: McKeon NPB(00) [2D, 3D, 4D, 5D]; Antoniadis et al JMP(11)-a1103 [extension].
@ Extensions: de Mello & Rivelles JMP(04)mp/02 [2D, representations]; Lindesay mp/03, mp/03; Bonanos & Gomis JPA(10)-a0812 [infinite sequence of extensions]; Rausch de Traubenberg IJGMP(12) [cubic extension]; Fuentealba et al JHEP(15)-a1505 [with half-integer spin generators]; László JPA(17)-a1512 [non-SUSY]; Llosa a1512 [transformations between accelerated frames].
@ Quantum field theory with twisted Poincaré invariance: Joung & Mourad JHEP(07); Balachandran et al PRD(08)-a0708; Abe PRD(08)-a0709 [correspondence with regular quantum field theory].
@ Deformations: Bimonte et al ht/97-proc [quantum Poincaré group]; Bacry JPA(93); Heuson MPLA(98) [modified uncertainty, etc]; Bruno et al PLB(01)ht; Blohmann PhD(01)m.QA, CMP(03)m.QA/01 [spin representations of q-deformed algebra]; Lukierski ht/04-proc, in(05)ht/04 [and DSR]; Bacry RPMP(04) ["physical" deformations]; Camacho & Camacho-Galván GRG(05)gq [κ-Poincaré group and quantum theory]; Girelli & Livine CQG(10)-a1001 [and group field theory]; Amelino-Camelia et al PRL(11)-a1006 [and worldlines, locality]; Magpantay PRD(11)-a1011 [dual \(\kappa\)-Poincaré algebra, dual DSR]; Barcaroli et al PRD(17)-a1703 [κ-Poincaré dispersion relations in curved spacetime]; Kuznetsova & Toppan EPJC(19)-a1803 [lightlike]; Gubitosi & Heefer PRD(19)-a1903 [κ-Poincaré model and relative locality]; > s.a. doubly-special relativity; modified lorentz symmetry.

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