Devil's curve

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.$⟨\psi ,V(R)\psi ⟩$ 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.

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