Math 2.0

Topological groups, Lie groups Classical Lie groups Example

Poincaré group

Builds on Lorentz groupSemidirect productTranslation

Example

Let O(1,3)O(1, 3) be the Lorentz group, acting on Minkowski space ℝ1,3=ℝ4\R^{1,3} = \R^4. The Poincaré group is the set of pairs (Λ,a)(\Lambda, a) with Λ∈O(1,3)\Lambda \in O(1, 3) and a∈ℝ4a \in \R^4, acting on ℝ4\R^4 by (Λ,a)⋅x=Λx+a,(\Lambda, a) \cdot x = \Lambda x + a, a Lorentz transformation followed by a translation. Composing two such maps gives the group law (Λ,a)(Λ′,a′)=(ΛΛ′, Λa′+a).(\Lambda, a)(\Lambda', a') = (\Lambda \Lambda', \ \Lambda a' + a). Check: (Λ,a)⋅((Λ′,a′)⋅x)=Λ(Λ′x+a′)+a=ΛΛ′x+(Λa′+a)(\Lambda, a) \cdot ((\Lambda', a') \cdot x) = \Lambda(\Lambda' x + a') + a = \Lambda\Lambda' x + (\Lambda a' + a). The identity is (I,0)(I, 0) and the inverse of (Λ,a)(\Lambda, a) is (Λ−1,−Λ−1a)(\Lambda^{-1}, -\Lambda^{-1} a).

So the Poincaré group is the semidirect product ℝ4⋊O(1,3)\R^4 \rtimes O(1, 3), a Lie group of dimension 4+6=104 + 6 = 10.

Remarks

The Poincaré group is the group of all isometries of Minkowski space, that is, the bijections ff of ℝ4\R^4 with Q(f(x)−f(y))=Q(x−y)Q(f(x) - f(y)) = Q(x - y) for the Minkowski form QQ. The translations form an abelian normal subgroup, so the Poincaré group is not semisimple. Often the name refers to the identity component ℝ4⋊SO+(1,3)\R^4 \rtimes SO^+(1, 3). Its irreducible unitary representations, studied by Wigner, are labelled by mass and spin in physics.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptQuadratic form, concept, no evidence levelQuadratic formconceptReal projective space, object, no evidence levelReal projective spaceobjectRiemann sphere, concept, no evidence levelRiemann sphereconceptSemidirect product, concept, no evidence levelSemidirect productconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptTranslation, concept, no evidence levelTranslationconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectIndefinite orthogonal group, example, no evidence levelIndefinite orthogonal…exampleSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)exampleSL(2,C), example, no evidence levelSL(2,C)exampleLorentz group, example, no evidence levelLorentz groupexamplePoincaré group, example, no evidence levelPoincaré groupexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Determinant concept
  3. General linear group object
  4. Group concept
  5. Hermitian matrix concept
  6. Identity matrix object
  7. Lie group concept
  8. Matrix definition
  9. Möbius transformation concept
  10. Quadratic form concept
  11. Real projective space object
  12. Riemann sphere concept
  13. Semidirect product concept
  14. Signature of a quadratic form concept
  15. Translation concept
  16. Vector space definition
  17. Ring definition
  18. Square matrix definition
  19. Transpose definition
  20. Linear map definition
  21. Unit of a ring definition
  22. Conjugate transpose definition
  23. Multilinear map definition
  24. Orthogonal matrix definition
  25. Division ring definition
  26. Alternating multilinear map definition
  27. Orthogonal group object
  28. Unitary matrix definition
  29. The quaternions example
  30. Determinant definition
  31. Unitary group object
  32. Indefinite orthogonal group example
  33. Special linear group object
  34. Special orthogonal group object
  35. Special unitary group SU(n) example
  36. Rotation group SO(3) example
  37. SU(2) example
  38. SL(2,C) example
  39. Lorentz group example
  40. Poincaré group example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincaré_group
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MSC22E43, 83A05, 22E70
Instance ofLie group
UsesLorentz group, Semidirect product, Translation
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