Loop quantum gravity and General relativity

In summary, the conversation discusses the concept of Lorentz invariance in relation to SL(2,C) and SU(2) subgroups. It is mentioned that SU(2) is a double-cover of the 3D spatial rotation group and that performing a Lorentz boost results in a different rotation group. The topic of changing basis using SL(2,C) transformations is also brought up, along with the question of how to derive that the determinant of an sl(2,C) matrix is equal to 1. The conversation ends with the suggestion to email Rovelli for clarification.
  • #1
Heidi
413
40
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
 
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  • #2
Heidi said:
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).

Heidi said:
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.
 
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  • #4
Heidi said:
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?
 
  • #5
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?
 
  • #6
Heidi said:
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.
 
  • #7
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}
 
  • #8
Heidi said:
[...] 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.
 
  • #9
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
 
  • #10
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.
 

Related to Loop quantum gravity and General relativity

1. What is the difference between Loop quantum gravity and General relativity?

Loop quantum gravity and General relativity are two different theories that attempt to explain the nature of gravity. General relativity is a classical theory of gravity that describes the force of gravity as the curvature of space-time caused by massive objects. Loop quantum gravity, on the other hand, is a quantum theory of gravity that attempts to describe gravity at the smallest scales, using the principles of quantum mechanics.

2. How does Loop quantum gravity address the issue of singularities in General relativity?

In General relativity, singularities are points where the curvature of space-time becomes infinite. Loop quantum gravity proposes that at the smallest scales, space-time is made up of discrete, quantized units called "loops." This allows for a more consistent and well-behaved description of space-time, avoiding the issue of singularities.

3. Can Loop quantum gravity and General relativity be unified?

Many scientists are working towards a theory of quantum gravity that would unify Loop quantum gravity and General relativity. However, at this point, there is no widely accepted theory that successfully unifies these two theories.

4. How does Loop quantum gravity relate to other theories, such as string theory?

Loop quantum gravity and string theory are both theories that attempt to explain the nature of gravity at the quantum level. However, they approach the problem in different ways and have different mathematical frameworks. Some scientists are exploring the possibility of combining these theories to create a more comprehensive theory of quantum gravity.

5. What are the potential implications of Loop quantum gravity and General relativity?

If Loop quantum gravity and General relativity can be successfully unified, it could lead to a better understanding of the fundamental nature of the universe. It could also potentially help to resolve some of the unanswered questions in physics, such as the nature of black holes and the origin of the universe.

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