Volume operator

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 ψ of quantum General Relativity. I.e. ⟨ ψ , V ( R ) ψ ⟩ 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.