Loop quantum gravity and General relativity

[Heidi]

Hi PFs,
I am reading this paper written by carlo Rovelli:
https://arxiv.org/abs/1010.1939
there are many things that i fail to understand, but i would like to begin with a simple thing.
Rovelli write that:
It is locally Lorentz invariant at each vertex, in the sense that the vertex amplitude (21) is SL2C invariant: if we choose a different SU2 subgroup of SL2C (in physical terms, if we perform a local Lorentz transformation), the amplitude does not change.
How to derive that such another su2 subgroup defines a lorentz transformation?
It may be obvious , sorry for my level

 

[strangerep]

Rovelli write that:
It is locally Lorentz invariant at each vertex, in the sense that the vertex amplitude (21) is SL2C invariant: if we choose a different SU2 subgroup of SL2C (in physical terms, if we perform a local Lorentz transformation), the amplitude does not change.
How to derive that such another su2 subgroup defines a lorentz transformation?

SL(2,C) is a double-cover of the Lorentz group. (For each Lorentz transformation there are 2 possible corresponding SL(2,C) transformations.)

Similarly, SU(2) is a double-cover of the 3D spatial rotation group, which is a subgroup of the Lorentz group. If one performs a Lorentz boost, one also gets a different rotation group (albeit isomorphic to the original).

It may be obvious , sorry for my level

You seem not to be familiar with the structure of the Lorentz group, all this double-cover stuff, and its implications for the associated group representations. Ballentine ch7 does a good job of explaining this for the rotation group. That's would probably be a useful start.

 

[Heidi]

Thanks Strangerep,
take two matrices S and T in the same SU2 representation and a given basis on whic they act.
S defines a map on the vector space. Is there (are there) Sl(2,C) transformations on the basis such that the map is given now by T in the new basis?

Reference: https://www.physicsforums.com/threads/loop-quantum-gravity-and-general-relativity.1017124/

 

[strangerep]

take two matrices S and T in the same SU2 representation and a given basis on whic they act.
S defines a map on the vector space. Is there (are there) Sl(2,C) transformations on the basis such that the map is given now by T in the new basis?

Heh, this is getting a bit close to homework, or textbook exercises, for which we normally require that an attempt-at-solution be given.

But if you're merely asking how to transform $S \in SU(2)$ into $T \in SU(2)$, wouldn't left-multiplying $S$ by $T S^{-1}$ do the job?

 

[Heidi]

This is not a summer homework. I read first A new look in loop quantum gravity end a found this paper in its references.
Rovelli writes the changing S to T on a vertex means physically to do a lorentz transformation. so i think that there is a change of frame matrix C so that
S = C T C^{-1}
where C is a sl(2,C) matrix
How to derive that det C = 1?

 

[malawi_glenn]

where C is a sl(2,C) matrix
How to derive that det C = 1?

If C is a sl(2,C) matrix, its determinant = 1 by definition.

 

[Heidi]

It does not work with S = id
P id P^{-1} = id so this is not the way Rovelli assign a Sl(2,c) matrix to a couple of SU2 matrix
maybe like strangerep answer with ST^{-1}

 

[strangerep]

[...] this is not the way Rovelli assign a Sl(2,c) matrix to a couple of SU2 matrix [...]

Maybe you should try emailing Rovelli direct, and ask him.

 

[Heidi]

I see tha rovelli does not talk about two matrices but about 2 subsets of matrices;maybe with two invariant three d space associated with a lorentz rotation

 

[Heidi]

I think that i understand now. A SU2 subgroup is associated to the spatial rotations with a given time direction remaining unchanged. another subgroup is associated to another timedirection (in the light cone) so choosing another subgroup corresponds to make a Lorentz transformation.