Volume operator

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A quantum field theory of General Relativity provides operators that measure the geometry of space time. The volume operator V(R) of a region R is defined as the operator that yields the expectation value of a volume measurement of the region R, given a state \psi of quantum General Relativity. I.e.\lang \psi, V(R) \psi \rang is the expectation value for the volume of R. Loop Quantum Gravity, for example, provides volume operators, area operators and length operators for regions, surfaces and path respectively.