The length operator in Loop Quantum Gravity

TL;DR

The paper tackles the problem of defining a length observable in Loop Quantum Gravity within the dual picture of quantum geometry where nodes encode volume and links encode area. It introduces an explicit construction of the elementary length operator by fluxizing the classical one-dimensional length, combining a two-hand operator with a regulated inverse volume, and then extends it to curves on a dual network. The length operator is shown to have a discrete spectrum and to exhibit nontrivial commutators with the volume operator, while semiclassical analyses on large-spin states reproduce the expected scaling of length with geometric area; these results bolster the interpretation of spin-network states as discretized geometries. The work lays groundwork for measuring extended curves and exploring continuous-metric constraints, signaling a path toward a coherent semiclassical limit and informing future studies of quantum-geometric observables in LQG.

Abstract

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity - the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.

Figures (6)

• Figure 1: A portion of a spin network graph and the associated dual picture of quantum geometry. The region $R_n$ is dual to the node $n$. Two adjacent regions are shown. The surfaces $S_1$ and $S_2$ are dual to the links $e_1$ and $e_2$. They identify a curve $\gamma$ on the boundary of $R_n$.
• Figure 2: (a) A cubic cell with the regularized quantity (\ref{['eq:Q']}) represented. (b) Action of the three-hand operator. The cubic cell is shown in gray. (c) Shrinking property of the three-hand operator.
• Figure 3: (a) A curve as the intersection of two surfaces. The decomposition in cubic cells is shown. (b) Cubic cell with the regularized quantity (\ref{['eq:regularized Y']}) shown. (c) Action of the two-hand operator on a spin network state. The cubic cell is shown in gray.
• Figure 4: Tikhonov regularization of the inverse. The plot shows the function $\frac{x}{x^2+\varepsilon^2}$ (full line) and the function $x^{-1}$ defined for $x>0$ (dashed line). The limit $\varepsilon\to 0$ defines an extension of the inverse of $x$ to the domain $x\geq 0$. Such extension vanishes in $x=0$ and coincides with $x^{-1}$ for $x>0$. This same property is shared by the eigenvalues of the operators $\widehat{V}$ and $\widehat{V^{-1}}$, as can be shown using spectral decomposition.
• Figure 5: (a) Eigenvalues of the volume operator in units $(8\pi \gamma L_P^2)^{3/2}$. A monochromatic four-valent node has been considered. The $(2j_0+1)$ eigenvalues are plotted as a function of the half-integer spin $j_0$. Degeneracy to be taken into account. The maximum eigenvalue scales as $j_0^{3/2}$. (b) Eigenvalues of the length operator for a wedge of monochromatic four-valent node. Units $(8\pi \gamma L_P^2)^{1/2}$ are used. The $(2j_0+1)$ eigenvalues are plotted as a function of the spin $j_0$. The maximum eigenvalue scales linearly in $j_0$.