Lorentz Group

In General > s.a. lorentz invariance in physics; Racah Coefficients.
* For 4D spacetime: The 6D Lie group SO(3,1), i.e., the group of transformations on B⁴ (or C⁴ for the complex Lorentz group) which leaves the form = diag(–1,1,1,1) invariant:

L:= {  GL(4, R) | ^T   = } .

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

L+u:= { L | det = 1, 00 > 0} L+d:= { L | det =1, 00 < 0}

L–u:= { L | det = –1, 00 > 0} L–d:= { L | det = –1, 00 < 0} ;

For example, P L_–^u, PT L₊^d, T L_–^d.
* Restricted Lorentz group: The restricted, or proper orthochronous Lorentz group is L₊^u; Its double covering is SL(2, C) (or SL(2, C) × SL(2, C), in the complex case); Any element L₊^u can be decomposed as

= _{1 2 3} ,   with ₁, ₃ rotations and ₂ a boost along the z direction .

@ Properties of group: Ungar AJP(92); Schmidt IJTP(98)gq/95 [non-compactness].
@ Properties of transformations: Urbantke FPL(03)mp/02 [as hyperplane/line reflections].
@ Related topics: Singh AJP(86); Toller mp/03 [homogeneous spaces]; Girelli & Livine gq/04 [as deformed Galileo]; Simon et al qp/06 [in terms of Hamilton's "turns"].

Representations > s.a. CPT; poincaré [inhomogeneous Lorentz group]; special relativity.
* Result: Every irr of SL(2, C) is equivalent to D^{(j/2, k/2)} = {A_a^m}, with j, k N, which acts on tensors

T_{a... bc'... d'} = T_{(a... b)(c'... d')}   by   T_{a... bc'... d'} A_a^m A_bⁿ A*_c'^p' A*_d'^q' T_{m... 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:

J^ab = ^bc x^a (/x^c) – ^ac x^b (/x^c) .

@ 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. electromagnetism.
@ Related topics: Mukunda & Radhakrishnan JMP(73) [3D]; Manogue & Schray JMP(93)ht [10D, in terms of octonions]; Fredsted JMP(01) [exponentiated spin-1/2 and 1 representations]; Varlamov mp/02, JPA(06)mp/05 [and relativistic spherical functions].

Lie Algebra
* For SO(3,1): The generators are the rotations S_i and boosts K_i ,

satisfying [S_i, S_j] = _ijk S_k , [S_i, K_j] = _ijk K_k , and [K_i, K_j] = –_ijk S_k .
* For SO(2,1): The generators are T₀, T₁ and T₂, with commutators [T_i, T_j] = f_ij^k T_k = _ijk g^kl T_l, where ₀₁₂ = 1, and g_ij = diag(–1,1,1) = f_ik^l f_jl^k.
@ References: Coll & San José Martínez GRG(02)gq/01 [generators and composition].

Variations > s.a. modified lorentz group.
@ Discrete version: Lorente & Kramer JPA(99) [on hypercubic lattice]; Levi et al PRD(04)ht/03 [Lorentz invariance].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 13 jun 2008