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Hi,
I have a question about imposing constraints in order to obtain theories of gravity from gauge algebras.
Let's take as a warming-up Poincare gravity. The procedure is as follows:
* Gauge the Poincare group with generators P (translations) and M (rotations) to obtain the vielbein and spin connection as gauge fields
* Impose curvature (R) constraints
*Obtain Einstein gravity
The curvature constraint is R(P)=0. This constraint does two things:
* It makes the spin connection a dependent field, which we want: the only propagating degree of freedom should be the vielbein!
* It enables one to rewrite the P-transformation on the vielbein (which is the only independent field left) as a general coordinate transformation minus a Lorentz transformation, which we want: a theory of gravity has as "gauge transformations" local Lorentz transformations and general coordinate transformations; no P-transformations are present.
My question is basically: these two reasons are completely different from each-other, but somehow I suspect there is a link between them. What is this link?
My question becomes more clear in the conformal case. Here we start with the conformal group. The procedure is then as follows:
*Gauge the conformal group with generators P (translations),M (Lorentz transformations ),K (special conformal transformations),D (dilatations)
*Impose curvature constraints
*Gauge away the gauge field belonging to the dilatations
*Write down the action of a conformal scalar
*Obtain the Hilbert action from this action via our curvature constraints
Again, the curvature constraints allow us to make certain gauge fields dependent and to remove the P-transformations from the remaining independent fields (the vielbein and the dilatation-gauge field). So again I wonder: what is the link between these two demands?
I hope my question is clear :)
I have a question about imposing constraints in order to obtain theories of gravity from gauge algebras.
Let's take as a warming-up Poincare gravity. The procedure is as follows:
* Gauge the Poincare group with generators P (translations) and M (rotations) to obtain the vielbein and spin connection as gauge fields
* Impose curvature (R) constraints
*Obtain Einstein gravity
The curvature constraint is R(P)=0. This constraint does two things:
* It makes the spin connection a dependent field, which we want: the only propagating degree of freedom should be the vielbein!
* It enables one to rewrite the P-transformation on the vielbein (which is the only independent field left) as a general coordinate transformation minus a Lorentz transformation, which we want: a theory of gravity has as "gauge transformations" local Lorentz transformations and general coordinate transformations; no P-transformations are present.
My question is basically: these two reasons are completely different from each-other, but somehow I suspect there is a link between them. What is this link?
My question becomes more clear in the conformal case. Here we start with the conformal group. The procedure is then as follows:
*Gauge the conformal group with generators P (translations),M (Lorentz transformations ),K (special conformal transformations),D (dilatations)
*Impose curvature constraints
*Gauge away the gauge field belonging to the dilatations
*Write down the action of a conformal scalar
*Obtain the Hilbert action from this action via our curvature constraints
Again, the curvature constraints allow us to make certain gauge fields dependent and to remove the P-transformations from the remaining independent fields (the vielbein and the dilatation-gauge field). So again I wonder: what is the link between these two demands?
I hope my question is clear :)
