Lorentz Group  

In General > s.a. Racah Coefficients; tests of lorentz invariance.
* For 4D spacetime: The 6D Lie group O(3,1) of transformations on \(\mathbb R\)4 (or \(\mathbb C\)4 for the complex Lorentz group) which leaves the form η = diag(–1,1,1,1) invariant:

L:= {Λ ∈ GL(4, \(\mathbb R\)) | ΛT η Λ = η} .

* Connected components: The real Lorentz group has 4 connected components,

L+u:= {Λ ∈ L | det Λ = 1, Λ00 > 0}
L+d:= {Λ ∈ L | det Λ = 1, Λ00 < 0}
Lu:= {Λ ∈ L | det Λ = –1, Λ00 > 0}
Ld:= {Λ ∈ L | det Λ = –1, Λ00 < 0} ;

The subgroup of proper Lorentz transformations L+ = SO(3,1); Examples of specific transformations are PLu, PTL+d, TLd.
* Restricted Lorentz group: The restricted, or proper orthochronous Lorentz group is L+u; Its double covering is SL(2, \(\mathbb C\)) (or SL(2, \(\mathbb C\)) × SL(2, \(\mathbb C\)), in the complex case); Any element Λ ∈ L+u can be decomposed as

Λ = Λ1 Λ2 Λ3 ,   with Λ1, Λ3 rotations and Λ2 a boost along the z direction.

@ General references: Oblak a1508-ln [and conformal transformations of the sphere, undergraduate-level].
@ Properties of the group: Ungar AJP(92)sep; Schmidt IJTP(98)gq/95 [non-compactness].
@ Properties of transformations: Urbantke FPL(03)mp/02 [as hyperplane/line reflections]; Jadczyk & Szulga EJLA(16)-a1611.
@ Lie algebra: Coll & San José Martínez GRG(02)gq/01 [generators]; Hanson GRG(12)-a1103 [orthogonal decomposition]; > s.a. orthogonal lie groups.
@ Related topics: Singh AJP(86)feb; Toller mp/03 [homogeneous spaces]; Girelli & Livine gq/04 [as deformed Galileo group]; Simon et al IJTP(06)qp [in terms of Hamilton's "turns"]; Kerner a0901 [from \(\mathbb Z\)3- graded cubic algebra].
> Online resources: see Wikipedia page.

Representations > s.a. 4-spinors; CPT theorem; poincaré group [inhomogeneous Lorentz group]; special-relativistic kinematics.
* Result: Every irreducible representation of SL(2, \(\mathbb C\)) is equivalent to D(j/2, k/2) = {Aam}, with j, k ∈ \(\mathbb N\), which acts on tensors

Ta... bc'... d' = T(a... b)(c'... d')   by   Ta... bc'... d'  \(\mapsto\)  Aam Abn A*c'p' A*d'q' Tm... np'... q' .

* Relationships: The irr's of SU(2) are equivalent to D(j/2, 0) = D(j/2).
* Unitary representations: It has no finite-dimensional uirr's; Hence, we must use infinite-dimensional representations.
* Real D-dimensional:

Jab = ηbc xa (∂/∂xc) – ηac xb (∂/∂xc) .

@ General references: Naimark 64; Gopala Rao et al JPA(95), JPA(95), JPA(95).
@ Unitary: Dirac PRS(45); Mukunda & Simon JMP(95); Kubieniec JMP(05) [uirr, proper orthochronous], JMP(05) [supplementary series]
@ Transformations of specific quantities: Jordan et al PRA(06)qp/05 [spin density matrices]; > s.a. thermal radiation; electromagnetic-field dynamics.
@ Special types of representations: Fredsted JMP(01) [exponentiated spin-1/2 and 1 representations]; Mashhoon AdP(09)-a0908 [non-local, and accelerated observers]; Hanson a1201 [exponential of the spin representation]; Atehortua et al a1210 [non-linear, and DSR]; Sellaroli a1509 [infinite-dimensional]; Varlamov IJTP(16)-a1602 [interlocking representations and classification of relativistic wave equations]; Kocik a1604 [Cromlech, menhirs and celestial sphere].
@ Related topics: Mukunda & Radhakrishnan JMP(73) [3D]; Manogue & Schray JMP(93)ht [10D, in terms of octonions]; Varlamov mp/02, JPA(06)mp/05 [and relativistic spherical functions]; Newman & Price AJP(10)jan [complex formulation].

Variations > s.a. modified lorentz group and symmetry violations.
@ Discrete version: Lorente & Kramer JPA(99) [on hypercubic lattice]; Levi et al PRD(04)ht/03 [Lorentz invariance]; Tarakanov a1301 [discrete subgroups]; Arrighi et al NJP(14) [for quantum walks and quantum cellular automata].
@ Emergence in a discrete setting: Livine & Oriti JHEP(04)gq; Sengupta CQG(14) + Bojowald CQG+(14) [in a 2D model for lqg].

Lorentz Invariance in Physics > s.a. poincaré group.
* Derivation: The structure of the Lorentz transformations follows from the absence of privileged inertial reference frames and the group structure of the transformations; It is not necessary to assume the existence of an invariant speed.
@ General references: in Will 93, ch2 [evidence]; Rodrigues & Sharif FP(01) [local, in general relativity]; Wolf et al gq/03-proc [tests]; Lämmerzahl AdP(05); Afshordi a1511 [in high-energy physics].
@ Origin of Lorentz symmetry: Froggatt & Nielsen hp/02-proc [derivation in quantum field theory]; Korbel ht/04 [quantum]; Albrecht & Iglesias PRD(15)-a1003 [from a random Hamiltonian]; Shanahan FP(14)-a1401 [and matter waves]; Höhn & Müller NJP(16)-a1412 [operational approach, from quantum communication]; Raasakka a1705 [from transformations between local thermal states in Local Quantum Physics].
@ As an emergent symmetry: Bednik et al JHEP(13)-a1305 [in strongly coupled theories]; Khoury et al CQG(14)-a1305, IJMPD(14)-a1405-GRF; Kharuk & Sibiryakov TMP(16)-a1505 [and chiral fermions]; Roy et al JHEP(16)-a1510 [near fermionic quantum critical points]; > s.a. Einstein-Aether Theory.
@ For specific systems: Kim cm/96-proc [in condensed matter]; Chen PRL(14)-a1404 [chiral theory, spin-1/2 particle with definite helicity]; Bisio et al FP(17)-a1707 [from a quantum walk]; > s.a. cellular automaton; electricity; relativistic quantum mechanics.
@ Related topics: Chkareuli et al PRL(01)hp [and origin of gauge symmetries]; Peres & Terno JMO(02)qp/01 [of open systems]; Szabó FPL(04) [not fundamental]; Rodrigues et al IJGMP(05)mp [and ambiguity of curvature/torsion]; Casadio PLB(13)-a1303 [Lorentz invariance in a quantum field theory with Planck-scale cutoff]; Shanahan FP(14) [and wave properties of matter]; Pelissetto & Testa AJP(15)apr-a1504 [without the invariant speed assumption, elementary proof].
> Related topics: see affine connections; cellular automaton; finsler geometry; hamiltonian systems; history of relativistic physics; lorentz-group phenomenology; probabilities; quantum-gravity phenomenology; special relativity.


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