Hamiltonian formulation of general relativity

[phnjs]

I am currently studying loop quantum gravity, and therefore GR in Ashtekar variables (A,E). I see the vector constraint E^a_iF^i_{ab}=0 is said to generate spatial diffeomorphisms (where F is the Yang-Mills field strength in terms of A), but how can I show this? How do spatial diffeomorphisms act on these variables?

 

[Nonlinearity]

The Ashtekar fields are by definition invariant under SU(2) and diff, which pretty well answers your question.

 

[f-h]

The fields aren't invariant, they transform according to the transformation generated by the constraints.

Quoting from Section 4.3.4 of Rovellis Quantum Gravity, the variation of the connection under a diffeomorphism generated by f is given by: df A^i_a = f^b @_b A^i_a + A^i_b @_a f^b
If you act with this on the Action it generates after some fiddling (and after using the internal degrees of freedom constraint) the Vector constraint (remember that E^a_i = dS/dA^i_a)

Unfortunately I don't have a good enough grasp on this to give a good intuitive/geometrical argument yet... Maybe somebody else can give a more elucidating explanaition.

 

[Careful]

In the ADM and Ashtekhar framework: calculate the poisson brackets of the phase space variables with the constraints and evaluate the result on shell. The latter equals the Lie derivative of the phase space variable with respect to the corresponding vectorfield. In other words, the action of the constraints on the dynamical variables equals the infinitesimal ``push forward´´ under the corresponding vectorfield when the Einstein equations of motion are satisfied.

In the loop framework: the connection/vielbein Ashtekhar variables are integrated to the parallel transport/flux variables which are again in the context of abstract spin networks generalized to abstract holonomies and fluxes. In this case, there is no proper implementation of the diffeomorphism algabra (sorry for the typo), since a one parameter group of diffeomorphisms will act discontinuously on the space of spin networks, hence no derivative can be taken. ADDENDUM You still have an implementation of the diffeomorphism group obviously which allows you to define a rigging map, that is a map from the spin networks to the diff invariant states (simply suitable group averaging).

Cheers,

Careful

 

[marcus]

Welcome phnjs and Nonlinearity!

Welcome to the newcomers, who started this thread. Very glad to have questions and discussion from new people.

phnjs, you asked about the "EF = 0" socalled diffeomorphism constraint. Smolin was just discussing that today in his video lecture #13, at about slide #7.

https://www.physicsforums.com/showthread.php?t=107445

If you go to #13 of the Smolin Lectures you will see that one of the menu bar options at the top is "slide list"

if you go to slide #7 and say "play from slide"
then it will start the lecture about 20 or 30 minutes into the hour (roughly, IIRC something like 20 or 30 minutes plus or minus)

and he will be explaining the diffeomorphism constraint

or better, start at slide #6, so you get some of the lead-up as preparation.

I should warn you this is the CLASSICAL treatment in #13
he is studying ashtekar variables and that classical formalism
but maybe it is better to learn about that first

and then later, I think in #14 or some later lecture, he will discuss quantizing it.

If you do try watching some of the lecture, please let me know if it is what you were asking about---and responsive to your original question. If I am way off base I want to know.

 

[phnjs]

Thanks for all your advice, which collectively has resolved my problem. I hope that soon I’ll be able to say I understand LQG!

Marcus, thanks for the welcome! I agree with you that the classical approach is the best place to start. Those lectures are a useful tool thank you, when I find the time I’ll watch them all. They do answer my question. I did study a course on theoretical physics (Part III, Cambridge) so I already knew some concepts in quantum gravity, however the loop approach/Ashtekar variables are quite new to me.

phnjs