A quantum field theory of general relativity provides operators that measure the geometry of spacetime. The volume operator ${\displaystyle V(R)}$ of a region ${\displaystyle R}$ is defined as the operator that yields the expectation value of a volume measurement of the region ${\displaystyle R}$, given a state ${\displaystyle \psi }$ of quantum General Relativity. I.e.${\displaystyle \langle \psi ,V(R)\psi \rangle }$ is the expectation value for the volume of ${\displaystyle R}$. Loop Quantum Gravity, for example, provides volume operators, area operators and length operators for regions, surfaces and path respectively.

Sources

• Carlo Rovelli and Lee Smolin, "Discreteness of Area and Volume in Quantum Gravity", Nuclear Physics B 442, 593 (1995).
• Abhay Ashtekar and Jerzy Lewandowski, Quantum Theory of Geometry II: Volume operators