Computing the graph-changing dynamics of loop quantum gravity

TL;DR

This work tackles the graph-changing dynamics of the canonical Hamiltonian constraint in Loop Quantum Gravity (LQG) by presenting the first numerical implementation that acts on 3- and 4-valent spin networks without fixed-graph truncations. It introduces ghost functions encoding spin-network data as ordered lists and defines the action of the constraint operator $C_s$ via a linear functional on these ghost states, e.g., $C_s[f(s_i)]=\sum_j c_j(s_i) f(s_j)$. Volumes are computed for perturbatively transformed 4-valent networks and contrasted with a graph-preserving implementation, revealing that graph-changing dynamics can decrease the volume for small lapse $N$ while graph-preserving dynamics may increase it at larger $|N|$, signaling qualitative differences. An infinite family of solutions to the Hamiltonian constraint is identified, constructed within the algebraic dual of the spin-network span, providing a route to physical states without matter. The results demonstrate that graph-preserving truncations miss key features of the true dynamics and open avenues for broader applications, including cosmology, black-hole evolution, and extensions to other graph-changing systems.

Abstract

In loop quantum gravity (LQG), states of the gravitational field are represented by labeled graphs called spin networks. Their dynamics can be described by a Hamiltonian constraint, { which acts on the spin network states modifying both spins and graphs.} Fixed-graph approximations of the dynamics have been extensively studied, but its full graph-changing action so far remains elusive. The latter, alongside the solutions of its constraint, are arguably the missing features { in canonical LQG to access phenomenology in all its richness}. Here, we discuss a recently developed numerical tool that, for the first time, implements graph-changing dynamics via the Hamiltonian constraint. We explain how it is used to find new solutions to that constraint and to show that some quantum geometric observables behave differently than in the graph-preserving truncation. We also point out that these new numerical methods can find applications in other domains.

Figures (4)

• Figure 1: The dipole model as an example of a spin network. The corresponding spin network consists of two $4$-valent nodes sharing each of their four links. Its dual is formed by two tetrahedra with faces that are pairwise glued (in four dimensions). These tetrahedra represent quanta of volume in a discretized geometry. The unitary formed by exponentiating the Hamiltonian with the lapse $N$ as a perturbative parameter transforms the dipole model differently when graph-changing (left) or graph-preserving (right) dynamics are considered. All figures in this work are based on figures of Ref. Companion.
• Figure 2: A $4$-valent spin network node (center), with its assigned ghost function above, is mapped under the action of the Hamiltonian into six modified structures which contain inner loops. Ghost functions containing the lists encoding such spin networks are given. Double arrows emphasize the reversible character of the Hamiltonian, and the numbers within them highlight the location of the added loop. All figures in this work are based on figures of Ref. Companion.
• Figure 3: Pseudocode exemplifying the implementation of the Hamiltonian. The code first checks whether an inner loop is present. If absent, it generates a linear combination of graphs with inner loops introduced in all six locations, with spin $1/2$ on the newly created link. If present, a series of steps is followed for each possible location (the case for location 1 is explicitly shown, while for other locations similar rules are implied by the dashed-line continuation of the diagram to the right). Namely, coupling a new loop at the innermost-loop location merely alters spins without affecting the graphs. If the spin of the connecting link becomes zero, it is removed, and the inner-loop data in the corresponding list is shifted to the left by four entries. Also, inner loops are introduced (deeper) in all other positions, but if a loop was inserted at position $3$ right before adding one at position $1$ (these loops share no links), it either removes its extra link or changes spins. The diagram displays examples of the simplest spin networks (orange graphs) for which the rules apply. All figures in this work are based on figures of Ref. Companion.
• Figure 4: Lapse dependence of the dimensionless volume expectation value. The curves are shown for two spin networks with $j_1 = j_2 = j_3 = j_4 =1/2$, $\varepsilon = 0$, and $i=0$ (red curves) or $i=1$ (green curves). We compare graph-changing (solid) and graph-preserving (dashed) Hamiltonians, for which expectation values are calculated up to third and fourth order in $N$, respectively. Inset: curves for the volume variance. All figures in this work are based on figures of Ref. Companion.