Volume operator

A quantum field theory of general relativity provides operators that measure the geometry of spacetime. 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.