Quantum Theory of Geometry II: Volume operators

TL;DR

This work develops a background-free quantum geometry by constructing a volume operator for a 3D region $R$ using two regularization routes on the space of generalized connections: an intrinsic scheme that preserves differential structure and a Rovelli–Smolin-type scheme that is more topological. Both approaches yield self-adjoint, discretely spectrummed operators on the kinematic Hilbert space ${\cal H}$, yet differ by a quantization ambiguity tied to regulator choices. To restore diffeomorphism covariance and remove background memory, the intrinsic scheme is complemented by an averaging procedure over background structures, resulting in a final volume operator ${\hat V}_R$ that is unique up to an overall constant ${\kappa_o}$. The paper analyzes the operator’s gauge and diffeomorphism properties, derives its intrinsic formulation, and discusses the spectrum and simple vertex examples, highlighting implications for quantum dynamics and the Hamiltonian constraint in non-perturbative quantum gravity.

Abstract

A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.

Figures (3)

• Figure 1: Illustration of the action of $J^i_{xe}$ operator on $\Psi_\gamma$.
• Figure 2: The figure illustrates the way cells $C, C', C"$ (the dashed lines) and 2-surfaces $S_a, {S'}_a, {S"}_a$ (the bold faced lines) of a partition ${\cal P}$ are adapted to a neighborhood of a vertex $v$ of a graph $\gamma$ according to the conditions $i)-iii)$ above. For simplicity, one dimension has been dropped.
• Figure 3: The size change of $C_v$ and $C$ does not change the fact, that $e_1,e_2$ intersect the neighboring cell $C$.