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The problem of quantum gravity is impossibly hard because it is too general.  Quantum gravity problem is not trivial but not nearly as hard if we consider a static spacetime such as Einstein static model $\mathbf{R}\times S^3(a)$.  Quantum field theory on this specific spacetime is essentially a solved problem.  In fact, even Euclidean quantum field theory following Osterwalder-Schrader quantization and Wightman axioms has been worked out in this case ( jaffee-ritter-qft1jaffee-ritter-qft2 osterwalder-schrader-2osterwalder-schrader-1 ).  In presentations of quantum field theory on curved spacetimes the Einstein static spacetime appears often as a relatively simple example (see Fewster lecturenote-3908 (1)).