A quantum field theory of General Relativity provides operators that measure the geometry of space time. The volume operator of a region is defined as the operator that yields the expectation value of a volume measurement of the region , given a state of quantum General Relativity. I.e. is the expectation value for the volume of . Loop Quantum Gravity, for example, provides volume operators, area operators and length operators for regions, surfaces and path respectively.
- Carlo Rovelli and Lee Smolin, "Discreteness of Area and Volume in Quantum Gravity", Nuclear Physics B 442, 593 (1995).