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
4 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].

@ __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 a1607.

@ __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].

**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;
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 *κ*-Poincaré algebra, dual DSR]; Barcaroli et al a1703 [*κ*-Poincaré dispersion relations in curved spacetime]; > s.a.
doubly-special relativity; modified
lorentz
symmetry.

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