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, ) | a R⁴,   L} ,   with   (a, ) (a', '):= (a + 'a, ') .

* Topology: It has 4 connected components (from those of L), while the complex P has 2 components; It is doubly connected, a 2 rotation (as a curve) cannot be deformed to the identity, but a 4 rotation can; Its (2-fold) covering group is ISL(2,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. lorentz invariance; maxwell theory.
* And physics: In Minkowski space they give rise to tensor fields (representations of ISL(2,C) give rise to spinor fields).
* In 1+1 dimensions: The Poincaré group has only 1D finite irr's (> see group representations).
@ References: in Wald 84, 13.1 [short]; Mirman 95 [massless fields]; Burdik et al NPPS(01)ht-in [field theory Lagrangians].
@ General articles: Wigner AM(39), AM(47); Bargmann AM(47); Bargmann & Wigner PNAS(48); Bargmann AM(54); Wigner 59; Halpern 68.
@ Continuous spin: Brink et al JMP(02)ht; Khan & Ramond JMP(05)ht/04 [from higher dimensions].
@ Semigroup representations: Bohm et al PLA(00)ht/99, ht/99-in.
@ Other: Brooke & Schroeck JMP(96) [m = 0, any s]; Brink ht/05-in [non-linear representations, susy].

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,C); The latter can always be decomposed into irrep's.
* Classification: Irrep's of ISL(2,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).

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; [, P^a] = ^a_b P^b, where := (i/2) _bc J^bc.
* 2+1 dimensions: The commutation relations are [J_a, J_b] = _abc J^c; [P^a, P^b] = 0; [J_a, P_b] = _abc P^c, where J_a:= _abc J^bc.

Other References > s.a. [lie algebra]; CPT.
@ And position operator in quantum theory: Aldaya et al JPA(93).
@ And relativistic field theory: Savvidou JMP(02)gq/01, CQG(01)gq [2 actions]; Froggatt & Nielsen AdP(05)ht [emergence of Poincaré invariance].
@ Super-Poincaré algebra: McKeon NPB(00) [2D, 3D, 4D, 5D].
@ Extended Poincaré group: de Mello & Rivelles JMP(04)mp/02 [2D, representations]; Lindesay mp/03, mp/03.
@ Quantum field theory with twisted Poincaré invariance: Joung & Mourad JHEP(07); Balachandran et al a0708; Abe a0709 [correspondence with regular quantum field theory].
@ Deformations: Bimonte et al ht/97-in [quantum Poincaré group]; ; Bacry JPA(93); Heuson MPLA(98) [modified uncertainty, etc]; Bruno et al PLB(01)ht; Blohmann m.QA/01-PhD, CMP(03)m.QA/01 [spin reps of q-deformed algebra]; Lukierski ht/04-in, ht/04-in [and DSR]; Bacry RPMP(04) ["physical" deformations]; Camacho & Camacho-Galván GRG(05)gq [-Poincaré group and quantum theory]; > s.a. modified lorentz symmetry.

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