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

[zwicky]

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.

 

[marcus]

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.

 

[zwicky]

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.

 

[marcus]

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
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.)

 

[zwicky]

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.