# Poincare vs Lorentz Group

1. Feb 7, 2012

### waterfall

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. Feb 7, 2012

### niklaus

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.

3. Feb 7, 2012

### waterfall

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?

4. Feb 7, 2012

### lugita15

All known laws of physics are Poincare-invariant.

5. Feb 7, 2012

### Matterwave

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.

6. Feb 7, 2012

### lugita15

I meant locally.

7. Feb 7, 2012

### George Jones

Staff Emeritus
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.

8. Feb 9, 2012

### haushofer

Then you should mean the Lorentz group, not the Poincare group. GR is not invariant under the local Poincare group.

9. Feb 11, 2012

### lugita15

Isn't GR invariant under the local diffeomorphism group, which is much bigger than the local Poincare group?

10. Feb 11, 2012