What Are the Key Groups in Gravity 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 group of all possible Weyl tensors on a general 4D Riemann manifold?
Is it a subgroup of GL(4)?
How do you derive the number of gravitational force bosons from the group structure?

What 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?
Is this a subgroup of GL(4)?

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

 

[jtolliver]

Is [the gauge group of gravity] a subgroup of GL(4)?

Only if there are no fermions, in which case it would be SO(3,1) (or rather the connected part of it). To account for fermions the double cover, SL(2,C) is used instead.

How do you derive the number of gravitational force bosons from the group structure?

normally it would be the number of generators of the group. here it is complicated somewhat by the fact that lorentz transformations do not leave graviton flavor invariant, so it is a matter of interpretation whether there are 6 types or just 1(depending on if you consider them distinct particles anyway, or consider them all to be the same particle because they are the same upto a lorentz transformation).

What groups represent all possible Riemann curvature tensors, and all possible metric tensors?

They don't seem to be endowed with a natural binary operation, so you should specify the group operation you are considering.

What is the equivalent of the Lorentz group for GR?
I.e. the group of transformations between all possible reference frames?
Is this a subgroup of GL(4)?

The equivalent group is the group of continuous transformations that locally look like lorentz transformations(essentially a lorentz transformation for each point in space-time such that the mapping from space-time to lorentz transformations is continuous).

 

[alexh110]

I've been told elsewhere that the gauge group for gravity is the general diffeomorphism group (i.e. in terms of transformations of the metric field tensor), which is what you're describing at the end there. But how does this tally with it being SL (2,C)? As I recall, SL (2,C) is related to the Lorentz group: so perhaps you are looking only at infinitessimal local transformations of the field? I suppose this is analagous to what you do in Yang-Mills gauge theory.

Also shouldn't the gauge group have representations which contain the particle spectrum to which gravity couples. I wondered how this works in the case of gravity, both in terms of representations of SL (2,C), and the general diffeomorphism group?

it is a matter of interpretation whether there are 6 types or just 1(depending on if you consider them distinct particles anyway, or consider them all to be the same particle because they are the same upto a lorentz transformation).

I'm not quite clear where the 6 comes from? I'm guessing it is the number of generators of SL (2,C)?
Do you mean that there would be 6 types of graviton if you had a flat space-time manifold; but for a curved manifold there is only one?

 

[Haelfix]

It depends how you're formulating your theory.

Typically you first have to decide what we mean by a 'gauge group of gravity' eg some mapping from spacetime to a lie group.

Now, this can be done in several ways.. It can be done with the lorentz group, the full poincare group (inhomogenous lorentz group), or even SU(2).

To read of gravity and how it effects masses, you probably want to work with the universal covering of the Poincare group as the connection ... SL(2,C). The reason this is wanted is b/c it is easier to deal with than using projective representations and imposing superselection rules on SL(2,C)/Z2 (eg the poincare group).

Now, typically the full diffeomorphism group is really a symmetry of spacetime, not of its gauge bundles. You can change this, in some theories, but that gets confusing.