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 $\psi$ 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.

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

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