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Poincare vs Lorentz Group

  1. Feb 7, 2012 #1
    The words Poincare and Lorentz sound pretty elegant. I think they are French words like Loreal Or Laurent.

    I know Poincare has to do with spacetime translation and Lorentz with rotations symmetry. But how come one commonly heard about Lorentz symmetry in Special Relativity and General Relativity and rarely of Poincare symmetry. Yet in Quantum Field Theory, Poincare invariance seems to be more used than lorentz invariance. Would anyone happen to know why? Thanks.
  2. jcsd
  3. Feb 7, 2012 #2
    Poincaré was French, Lorentz was Dutch...

    The Lorentz group is a subgroup of the Poincaré group. In addition to the Lorentz transformations, the Poincaré group also contains translations in space and time. The reason you don't hear about them much is that translations are a bit boring compared to the rest. Invariance under translations implies the conservation of energy and momentum, but that is already the case in classical mechanics, going to Minkowski space doesn't really affect that.
  4. Feb 7, 2012 #3
    So everything obeys Poincare invariance... but it's still mentioned especially in QFT because maybe there is some exception? Which doesn't obey Poincare invariance or the translation symmetry?
  5. Feb 7, 2012 #4
    All known laws of physics are Poincare-invariant.
  6. Feb 7, 2012 #5


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    That's not true in GR obviously when you no longer have flat space-times. The isometries of the metric is no longer the Poincare group.

    One should note that objects such as the Dirac spinor are representations of the Lorentz group (extended by parity), and not the Poincare group. I don't remember how to fix it to encompass the full Poincare group symmetries, but Wigner was the one to pioneer that I believe.
  7. Feb 7, 2012 #6
    I meant locally.
  8. Feb 7, 2012 #7

    George Jones

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    Four-dimensional Dirac spinor S space is a carrier space for a non-unitary representation of the (universal cover) of the Lorentz group, including, I think parity. The infinite-dimensional space of Dirac spinor fields (i.e. functions that map Minkoswki spacetime M -> S) that satisfy the free particle Dirac equation is a carrier space for a unitary representation of the (universal cover) of the Poincare group.
  9. Feb 9, 2012 #8


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    Then you should mean the Lorentz group, not the Poincare group. GR is not invariant under the local Poincare group.
  10. Feb 11, 2012 #9
    Isn't GR invariant under the local diffeomorphism group, which is much bigger than the local Poincare group?
  11. Feb 11, 2012 #10
    Poincare group gauge theory of gravitation is briefly discussed here:

    Vanishing Vierbein in Gauge Theories of Gravitation


    (see also references therein)

    It leads naturally to torsion, which can be eliminated by additional constraints, or allowe to couple to matter fields - according to the choice.
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