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Lorentz group

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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

Named after Hendrik Lorentz. (Not to be confused with Ludvik Lorenz, whose name is attached to the Lorenz gauge.)

Last revised on December 18, 2017 at 00:32:00. See the history of this page for a list of all contributions to it.