nLab
Lorentz group

Contents

Definition

The Lorentz group is the orthogonal group for an invariant bilinear form of signature (+++)(-+++\cdots), O(d1,1)O(d-1,1).

In physics, in the theory of relativity the Lorentz group acts canonically as the group of linear isometries of Minkowski spacetime preserving a chosen basepoint. This is called the action by Lorentz transformations.

The elements in the Lorentz group in the image of the special orthogonal group SO(d1)O(d1,1)SO(d-1) \hookrightarrow O(d-1,1) are rotations in space. The further elements in the special Lorentz group SO(d1,1)SO(d-1,1), which mathematically are “hyperbolic rotations” in a space-time plane, are called boosts in physics.

One distinguishes the following further subgroups of the Lorentz group O(d1,1)O(d-1,1):

  • the proper Lorentz group SO(d1,1)SO(d-1,1) is the subgroup of elements which have determinant +1 (as elements SO(d1,1)GL(d)SO(d-1,1)\hookrightarrow GL(d) of the general linear group);

  • the proper orthochronous (or restricted) Lorentz group SO +(d1,1)SO(d1,1)SO^+(d-1,1) \hookrightarrow SO(d-1,1) is the further subgroup of elements which do not act by reflection along the timelike axis.

Properties

Connected components

As a smooth manifold, the Lorentz group O(d1,1)O(d-1,1) has four connected components. The connected component of the identity is the the proper orthochronous group SO +(3,1)SO^+(3,1). The other three components are

  1. SO +(d1,1)PSO^+(d-1,1)\cdot P

  2. SO +(d1,1)TSO^+(d-1,1)\cdot T

  3. SO +(d1,1)PTSO^+(d-1,1)\cdot P T,

where, as matrices

Pdiag(1,1,1,,1) P \coloneqq diag(1,-1,-1, \cdots, -1)

is the operation of point reflection at the origin in space, where

Tdiag(1,1,1,,1) T \coloneqq diag(-1,1,1, \cdots, 1)

is the operation of reflection in time and hence where

PT=TP=diag(1,1,,1) P T = T P = diag(-1,-1, \cdots, -1)

is point reflection in spacetime.

Spin cover

While the proper orthochronous Lorentz group SO +(d1,1)SO^+(d-1,1) is connected, it is not simply connected. Its universal double cover is the Lorentzian spin group Spin(d1,1)Spin(d-1,1)

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group

References

Revised on June 29, 2017 10:35:08 by Daniel Luckhardt (134.76.82.216)