Uncovering the Role of K in Canonical Variables of Loop Quantum Gravity

In Summary, K is an independent variable in the canonical formulation of QG. It is a function of A and E. It can be computed by solving the algebra of constraints.
  • #1
zwicky
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In LQG the canonical variables are a SU(2) connection A, and the "electric" field E, such that they form a canonical pair, i.e., {A,E}=1. But the constraints that generates diffeomorphisms contains also the variable K (extrinsic curvature). My question is, is this variable an independent one? if yes, which one is its canonically conjugate momentum? How can I compute the algebra of constraints with K? Is this a composed variable in the sense that K=K[A,E]?

I know this questions seems to be out of topic because is a textbook question, but anyway, if someone can help, I'll be grateful.

Z.
 
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  • #2
zwicky said:
In LQG the canonical variables are a SU(2) connection A, and the "electric" field E, such that they form a canonical pair, i.e., {A,E}=1. But the constraints that generates diffeomorphisms contains also the variable K (extrinsic curvature). My question is, is this variable an independent one? if yes, which one is its canonically conjugate momentum? How can I compute the algebra of constraints with K? Is this a composed variable in the sense that K=K[A,E]?

I know this questions seems to be out of topic because is a textbook question, but anyway, if someone can help, I'll be grateful.

Z.

Lecture 5 in this list of video lectures might be useful to you by providing an alternative perspective.
https://www.physicsforums.com/showpost.php?p=3860461&postcount=20
This however does not directly answer your question. Hopefully others will want to answer from within the context of canonical Dirac approach to QG.
 
  • #3
marcus said:
Lecture 5 in this list of video lectures might be useful to you by providing an alternative perspective.
https://www.physicsforums.com/showpost.php?p=3860461&postcount=20
This however does not directly answer your question. Hopefully others will want to answer from within the context of canonical Dirac approach to QG.

Thank you for the reply.
 
  • #4
Zwicky, given that no one who is more focused on the canonical approach has stepped in, let me say a bit more.

You might have a look at slide #36 of Rovelli's Perimeter colloquium talk
http://arxiv.org/12040059 [Broken]
You can download the slides PDF and scroll to #36.
The discussion starts shortly before minute 40 of the talk, if you want to watch and listen just to that section you can drag the time button to, say, minute 36 and get an idea of what leads up to it.

The slide heading is How does GR come in? e is the tetrad, ω is the full 4d connection,
F is the curvature of the connection ω.
S[e,ω] is the Holst action, which is the action on which the spinfoam or "covariant" approach is based. This is the current formulation (Pirsa 12040059) of Loop that I'm trying to concentrate on (and follow that series of 14 talks that goes with the colloquium.)

Hopefully someone who is more interested in the older canonical approach will eventually step in and respond directly to your question.

If you do get interested in this approach to QG, be sure to distinguish between the
"Hamiltonian", which plays such an important role in the canonical formulation, and the classical Hamilton function which in a sense is the classical precursor of quantum transition amplitudes.

I guess in a few minutes we should have lecture 8 of the introductory lecture series.
Lecture 8 http://pirsa.org/12040029/
(Well, I guessed wrong. There was some delay in posting video and slides PDF.
So I have no idea when to expect them.)
 
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  • #5
Thank you for taking your time Marcus.
I agree with you that my question is related with the "old" canonical formalism, and now the tendency is to start directly with spin foam formalism with a 4d connection (tK would be related with the 0-component of \omega, 0-component with respect of group indices). But my questions is concerned to the standard formulation of canonical gravity in terms of Ashtekar-Barbero variables. I checked Ashtekar, Lewandowski notes, Thiemann book, and many lectures notes available on-line, and in each one of them is not clear what happens with K, part of this term is hidden into the new connection A=\omega+\gamma*K, but the Hamiltonian constraint depends on K!, to me its seems like K has to be a function of A and E, but how? I'm sure that I am missing something basic here, but I just what to understand this.

But I will take a look at the full lectures. Thanks for that!

Zwicky.
 

1. What are canonical variables in LQG?

Canonical variables in LQG (Loop Quantum Gravity) are a set of mathematical variables used to describe the fundamental building blocks of spacetime in this quantum theory of gravity. They include the triad variables, which describe the spatial geometry, and the connection variables, which describe the curvature of spacetime.

2. How are canonical variables used in LQG?

Canonical variables are used in LQG to construct a quantum theory of gravity by quantizing the classical theory of General Relativity. This involves promoting the classical variables to operators, which act on a quantum state of spacetime, and using mathematical equations to describe the dynamics of these variables.

3. What is the significance of canonical variables in LQG?

Canonical variables play a crucial role in LQG as they provide a link between the classical theory of General Relativity and the quantum theory of gravity. They allow for a consistent and mathematically rigorous description of the fundamental building blocks of spacetime at the quantum level.

4. Are there any challenges associated with using canonical variables in LQG?

Yes, there are several challenges associated with using canonical variables in LQG. One of the main challenges is finding a way to incorporate the principles of quantum mechanics into the theory of General Relativity. This has led to the development of various approaches, such as spin networks and spin foams, to deal with the mathematical complexities involved.

5. How does LQG differ from other theories of quantum gravity in terms of canonical variables?

LQG differs from other theories of quantum gravity, such as string theory, in terms of its use of canonical variables. While LQG uses the triad and connection variables, string theory uses different variables, such as strings and branes, to describe the fundamental building blocks of spacetime. Additionally, LQG is a background-independent theory, meaning that it does not require a fixed background spacetime, unlike string theory which relies on a fixed background.

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