Quantum propagation in Smolin’s weak coupling limit of 4D Euclidean gravity

Two desirable properties of a quantum dynamics for loop quantum gravity (LQG) are that its generators provide an anomalyfree representation of the classical constraint algebra and that physical states which lie in the kernel of these generators encode propagation. A physical state in LQG is expected to be a sum over graphical $SU(2)$ spin network states. By propagation we mean that a quantum perturbation at one vertex of a spin network state propagates to vertices which are “many links away” thus yielding a new spin network state which is related to the old one by this propagation. A physical state encodes propagation if its spin network summands are related by propagation. Here we study propagation in an LQG quantization of Smolin’s weak coupling limit of Euclidean gravity based on graphical $U(1{)}^3$ “charge” network states. Building on our earlier work on anomalyfree quantum constraint actions for this system, we analyse the extent to which physical states encode propagation. In particular, we show that a slight modification of the constraint actions constructed in our previous work leads to physical states which encode robust propagation. Under appropriate conditions, this propagation merges, separates and entangles vertices of charge network states. The “electric” diffeomorphism constraints introduced in previous work play a key role in our considerations. The main import of our work is that there are choices of quantum constraint constructions through LQG methods which are consistent with vigorous propagation thus providing a counterpoint to Smolin’s early observations on the difficulties of propagation in the context of LQG type operator constructions. Whether the choices considered in this work are physically appropriate is an open question worthy of further study.

Figure 1

Fig. 1 shows an undeformed GR vertex $v$ of a chargenet $c$ with its $I$th and $J$th edges as labeled. The vertex is deformed along its $I$th edge in Fig. 1 wherein the displaced vertex $v_I$ and the $C^0$ kink, ${\tilde{v}}_J$ on the $J$th edge are labeled. Figure 1 shows the result of a Hamiltonian type deformation obtained by multiplying the chargenet holonomies obtained by coloring the edges of Fig. 1 by flipped images of charges on their counterparts in $c$, Fig. 1 by negative of these flipped charges and Fig. 1 by the charges on $c$. If the edges of Fig. 1 are colored by the charges on their counterparts in $c$ then one obtains an electric diffemorphism deformation.

Figure 2

In Fig. 2 the vertex structure of Fig. 2 is deformed along its $K$th edge and the displaced vertex $v_K$ and the $C^0$ kink ${\tilde{v}}_J$ on the $J$th edge are as labeled. Figure 2 shows the result of a Hamiltonian type deformation obtained by multiplying the 3 chargenet holonomies obtained by coloring the edges of Fig. 2 by the flipped images of the charges on their counterparts in $c$, the edges of Fig. 2 by the negative of these flipped charges and the edges of Fig. 2 by the charges on $c$. If the edges of Fig. 2 are colored by the charges on their counterparts in $c$ then one obtains an electric diffemorphism deformation.

Figure 3

Fig. 3 shows the simple 2 vertex chargenet of interest. Figure 3 shows the result of a Hamiltonian constraint deformation of the chargenet of Fig. 3 along its $I$th edge with the displaced vertex $v_I$ and the $C^0$ kink ${\tilde{v}}_J$ as labeled.

Figure 4

Figure 4 shows an undeformed GR vertex $v$ of a chargenet $c$ with its $I$th and $J$th edges as labeled. The vertex is deformed along its $I$th edge in Fig. 4 wherein the displaced vertex $v_I$ and the $C^0$ kink, ${\tilde{v}}_J$ on the $J$th edge are labeled. Figure 4 shows the result of an $H_{\pm }$ type deformation obtained by multiplying the chargenet holonomies obtained by coloring the edges of Fig. 4 by flipped images of charges on their counterparts in $c$, Fig. 4 by negative of these flipped charges and Fig. 4 by the charges on $c$. If the flip is positive, the deformation is generated by $H_{+}$ and if negative, by $H_{-}$. As result the charges on this deformed chargenet on the deformed edges are the sum of the $\pm$-flipped and unflipped charges and the segments from $v$ to the kinks ${\tilde{v}}_J$ carry the negative of the $\pm$-flipped images of their charges in $c$.

Figure 5

The figures show the sequence of ket set elements Fig. 5 to Fig. 5 which encode propagation from vertex A to vertex B as described in the main text. The ket set is the minimal one containing Fig. 5. It underlies a physical state subject to the additional physical conditions of Sec. 3b.

Figure 6

Fig. 6 shows a best case graph structure for the purposes of putative propagation based on $N\to N$ deformations.

Figure 7

Figure 7 shows an undeformed GR vertex $v$ of a chargenet $c$. Figure 7 shows the conical deformation of the 3 preferred edges $e_{J^i},i=1$, 2, 3 along the $I$th edge, the displaced vertex $v_I$ and the 3 $C^0$ kinks, ${\tilde{v}}_{J^i},i=1$, 2, 3 are as labeled. Figure 7 shows the remaining edges being pulled exactly along the $I$th edge. The deformations of Figs. 7 and 7 combine to yield the $N\to 4$ deformation shown in Fig. 7. Figure 7 shows the result of a Hamiltonian type deformation obtained by multiplying the chargenet holonomies obtained by coloring the edges of Figs. 7 and 7 by the flipped images of charges on their counterparts in $c$, of Fig. 7 by negative of these flipped charges and Fig. 7 by the charges on $c$. If the edges of Figs. 7 and 7 are colored by the charges on their counterparts in $c$ then they combine to yield the holonomy of Fig. 7, this being the result of an electric diffemorphism deformation.

Figure 8

The figures show the sequence of ket set elements Fig. 8 to Fig. 8 which encode propagation from vertex A to vertex B of different valence as described in the main text. The ket set is the minimal one containing Fig. 8 appropriate to the $N\to 4$ deformation.

Figure 9

The figures show the sequence of ket set elements Fig. 9 to Fig. 9 which encode propagation from vertex A to vertex C as described in the main text. The unperturbed chargenet in Fig. 9 is based on a graph which is dual to a triangulation of the Cauchy slice. The ket set is the minimal one which contains Fig. 9 and is appropriate to the $N\to 4$ deformation.