Group theory in GR and quantum gravity

[alexh110]

In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions:

Is the gauge group for gravity defined as the set of all possible Weyl tensors on a general 4D Riemann manifold? Which abstract group maps onto this set? Is it GL(4) or a subgroup of GL(4)? How do you derive the number of gravitational force bosons from the gauge group structure?

Which abstract groups represent all possible Riemann curvature tensors, and all possible metric tensors?

What is the equivalent of the Lorentz group for GR?
I.e. the group of transformations between all possible reference frames.

How is all of this connected with the conformal group? What is the purpose of conformal invariance?

 

[dextercioby]

...if you thought those questions were too difficult,lemme ask you,guys,something easier to answer.
Classical gravity,described by Hilbert-Einstein action,is the only gauge theory for which i can't develop(and that's because i don't know)the Hamiltonian formalism at a classical level.
Will someone be so kind to give me any ideas??

 

[selfAdjoint]

Originally posted by dextercioby
...if you thought those questions were too difficult,lemme ask you,guys,something easier to answer.
Classical gravity,described by Hilbert-Einstein action,is the only gauge theory for which i can't develop(and that's because i don't know)the Hamiltonian formalism at a classical level.
Will someone be so kind to give me any ideas??

In order to do Hamiltonian mechanics in GR you have to split time from space, which means you are not even Lorentz covariant, let alone generally covariant. Nevertheless there are efforts to do this. MTW references two works by Dirac:

"Fixation of coordinates in the Hamiltonian theory of Gravitation" Phys Rev 114, 924-930 (and citations therein)

Lectures on Quantum Mechanics, Belfer Graduate School of Science Monograph Series Number two, Yeshiva University, New York, 1964 (and citations)

In spite of the title, the latter book apparently has a description of Dirac's Hamiltonian theory of Gravitation.

You might also google on the ADM formalism (Arnowitt, Deser, & Misner).