Recently I've heard the statement that by including Dirac fermions into the Einstein-Hilbert action we make torsion be non-zero, so that is one of problem of quantum gravity. How to prove that explicitly? Intuitively it's somehow connected with the form of Dirac action in curved spacetime (which includes vierbein), but I don't know how to demonstrate it directly.
Maybe it can be done by assuming Christoffel symbols and metric as independent quantities and then by variation of action by Christoffel symbol? In result I'll get some equation for Christoffel symbol, then I'll add to one equation the another one with indices permutation for getting equation on torsion tensor. If the tensor-free part of equation will Benin-zero, then torsion isn't zero. Is this thinking right?