Poincare vs Lorentz Group

In summary: The vanishing Vierbein in gauge theories of gravity leads to torsion. If additional constraints are placed on the theory, it can be eliminated. Alternatively, torsion can be coupled to matter fields.
  • #1
waterfall
381
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.
 
Physics news on Phys.org
  • #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.
 
  • #3
niklaus said:
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.

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
waterfall said:
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?
All known laws of physics are Poincare-invariant.
 
  • #5
lugita15 said:
All known laws of physics are Poincare-invariant.

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
Matterwave said:
That's not true in GR obviously when you no longer have flat space-times.
I meant locally.
 
  • #7
Matterwave said:
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.

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
lugita15 said:
I meant locally.

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

Vanishing Vierbein in Gauge Theories of Gravitation

http://arxiv.org/abs/gr-qc/9909060

(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.
 

What is the difference between the Poincare Group and the Lorentz Group?

The Poincare Group and the Lorentz Group are both mathematical groups that describe the symmetries of space and time. The main difference between them is that the Poincare Group includes translations in space and time, while the Lorentz Group does not.

What are the applications of the Poincare and Lorentz Groups in physics?

The Poincare and Lorentz Groups are important in the study of special relativity, which describes the behavior of objects moving at very high speeds. They also have applications in quantum field theory and particle physics.

Can the Poincare and Lorentz Groups be used interchangeably?

No, the Poincare and Lorentz Groups are not interchangeable. While they are both related to symmetries in space and time, they have different mathematical structures and describe different physical phenomena. The Poincare Group is a larger group that includes translations, while the Lorentz Group is a subgroup of the Poincare Group.

What is the significance of the Poincare and Lorentz Groups in Einstein's theory of relativity?

The Poincare and Lorentz Groups play a crucial role in Einstein's theory of relativity. They are used to describe the symmetries of space and time that are necessary for the laws of physics to remain the same in all reference frames. This is a fundamental principle of relativity.

Are there any experimental tests that have been conducted to distinguish between the Poincare and Lorentz Groups?

Yes, there have been several experimental tests to distinguish between the Poincare and Lorentz Groups. One example is the Michelson-Morley experiment, which tested for the existence of the luminiferous ether and provided evidence for the Lorentz Group. Other experiments, such as the Kennedy-Thorndike experiment, have also been conducted to test the predictions of the Lorentz Group.

Similar threads

  • Special and General Relativity
Replies
7
Views
1K
  • Special and General Relativity
Replies
29
Views
2K
  • Special and General Relativity
Replies
7
Views
973
  • Special and General Relativity
Replies
9
Views
4K
  • Special and General Relativity
Replies
7
Views
1K
  • Special and General Relativity
3
Replies
83
Views
3K
  • Quantum Physics
Replies
10
Views
2K
Replies
2
Views
3K
  • Quantum Physics
3
Replies
87
Views
4K
  • Special and General Relativity
Replies
4
Views
2K
Back
Top