Copy of a post in another thread:
Treating GR as an ordinary field theory encountered in the SM is practically useless,because the quantum theory obtained is not renormalizable.
Hilbert-Einstein action describes at quantum level a theory of SELFINTERRACTING gravitons,quanta of gravitational field i.e.particles with spin two.
Other attempts have been made of finding something else instead of the HE action.The linearized theory of gravity (developed by Einstein in 1916) is basically "good" when it comes to quantum behavior (i analyzed this theory using standard BRST antiparanthesis-antifield forrmalism (developed frankly by Batalin&Vilkovisky,but it's usually called Lagrangian BRST (cf.the Hamiltonian approach found independently by Becchi,Rouet,Storra and Tyiutin))),but it has the disadvantage of working with the gauge-fixed Pauli-Fierz action which describes the FREE (i.e.NONSELFINTERRACTING) GRAVITONS.In fact,these gravitons interract with ghost fields.Other attempt was the so-called Weyl gravity (i.e. gravity based not on the Riemann curvature tensor (of the curved manifold called spacetime),but on the conformal Weyl tensor,un ugly (still 4-th order) tensor).This theory is excellent at quantum level (i.e.renormalizable) but the classical (nonrelativistic) limit of the unquantized action gives you 4-th order LODE of motion (cf.2-nd order LODE of motion in the Newtonian limit of Einstein GR).
A step forward was made by Elie Cartan who developed the so called "Einstein-Cartan GR" which used other fields (called vierbeins,usually seen as vielbeins) for describing the gravitational field.This theory is good because it allows coupling with spinor and scalar matter fields in a theory called SUPERGRAVITY.If I'm not mistaking,these theories of Sugra (apud Supergravity),though allow an unifying theory of all 4 fundamental interractions,are,at quantum evel,still nonrenormalizable.I mean,if they were (it doesn't matter how many supemultiplets of particles it envolved),why would ST and LQG be alive today??
And then,ST,the final (??) frontiere.It is said to give a satisfactory behavior of gravitational interraction at the quantum level (this time there are no point-particles like in SM anf SUGRA,but strings,10-dimensional objects).
This a plainy simple and incomplete (probably incorrectas well,at least in its final lines) review on the the work that's been done by theorists worldwide in the last 70 years or so in the field of Quantum Gravidynamics (the name i give for the theory of QG in agreement with common names used in the SM).
Daniel.
