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.$lang\; psi,\; V(R)\; psi\; ang$ 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.

[http://www.arxiv.org/abs/gr-qc/9711031 Article by A. Ashtekar and J. Lewandowski] (construction of a volume operator in Loop Quantum Gravity)