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, *m*^{2} and
*s*^{2}, of the Lie algebra,

(a) *m*^{2} < 0: > see tachyons,

(b) *m*^{2} = 0, translations
all represented by I: not very significant,

(c) *m*^{2} = 0, not all translations
represented by I: either (1) helicity 0, 1/2, 1, ..., or (2) "continuous spin",

(d) *m*^{2} > 0: *S*^{2}
= *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__:
*P*^{a},
*J*^{ab}, where
*a*, *b* = 1, ..., *d*, with commutation relations

[*J*^{ab},
*J*^{cd}] = SO(*n*−1)
relations; [*P*^{a},
*P*^{b}] =
0; [*P*^{a},
*J*^{bc}]
= i *g*^{ab}
*P*^{c}
− i *g*^{ac}
*P*^{b} .

* __1+1 dimensions__:
The commutation relations are [*P*^{a},
*P*^{b}] = 0;
[Λ, *P*^{a}]
= *ε*^{a}_{b}
P^{b}, where
Λ:= (i/2) *ε*_{bc}
*J*^{bc}.

* __2+1 dimensions__:
The commutation relations are [*J _{a}*,

**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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