Geometrical Incompatibility of Quantum Measurements
  The problem of quantization of general relativity is considered in the framework of noncommutative differential geometry. Operator analogues for interval, scalar curvature, values of the Einstein tensor are proposed. Quantum measurements of these observables lead to a paradox: different procedures of measurements can supply non equivalent geometrical pictures of space-time. A concrete example of such situation is provided. 
Georgy Parfionov, Yury Romashev