Flux formulation of loop quantum gravity: Classical framework

TL;DR

This work develops a BF vacuum-based classical framework for loop quantum gravity, shifting the focus from holonomies to integrated fluxes and establishing a continuum holonomy-flux algebra via a modified projective limit. It introduces integrated flux observables labeled by co-paths, defines their d=2 and d=3 constructions through shadow graphs, and proves that the resulting Poisson algebra is closed even when flux surfaces intersect. The discrete phase spaces are connected by embedding and projection maps that are constrained by flatness and curvature data, leading to a reduced, symplectic continuum structure and a modified projective limit that respects curvature constraints. The formalism provides a natural route to coarse graining and dynamics, with a clear geometric interpretation of curvature excitations and curvature-induced torsion, and offers a dual perspective to the Ashtekar–Lewandowski representation that could facilitate spin foam and Regge-type dynamics.

Abstract

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.

Figures (13)

• Figure 1: Alexander 2-4 move obtained by placing a vertex in the edge connecting the vertices $v_1$ and $v_4$, and then connecting this new vertex to $v_2$ and $v_3$. We have illustrated the behavior of two possible choices for the root (red and green nodes). If we choose the flag of the upper triangle to consist of the vertex $v_1$ and the edge connecting $v_1$ and $v_2$, one can see that the triangle which inherits these flag simplices in the refined triangulation is chosen to be the new root. For the bottom triangle the flag is given by $v_1$ and the edge connecting $v_1$ and $v_3$.
• Figure 2: Example of a 1-dimensional co-path $\pi$ (solid red) which consists of the four edges $e_1,\ldots,e_4$, and its shadow graph $\Gamma_\pi$ (dashed red) which consists of the links $l_5,\ldots,l_{12}$ connecting the source nodes of the links $l_1,\ldots,l_4$ while staying as close as possible to $\pi$.
• Figure 3: Construction of a 2-dimensional integrated simplicial flux $\mathbf{X}_\pi$ according to definition \ref{['2d integrated flux']}. The holonomy $g_{rl_1(0)}=h_{12}^{-1}h_{11}^{-1}h_{10}h_9$ goes from the root $r$ to the node $l_1(0)$ along the tree (of portion of which is represented in dashed brown). The holonomy $g_{l_1(0)l_i(0)}$ goes from the node $l_1(0)$ to the node $l_i(0)$ along the shadow graph $\Gamma_\pi$ (dashed red). For example, we have $g_{l_1(0)l_2(0)}=h_5h_4^{-1}$, and $g_{l_1(0)l_3(0)}=h_8h_7h_6h_5h_4^{-1}$.
• Figure 4: Composition of the two integrated fluxes $\mathbf{X}_{\pi_1}$ and $\mathbf{X}_{\pi_2}$ associated to the co-paths $\pi_1=e^1_2\circ e^1_1$ (solid red) and $\pi_2=e^2_2\circ e^2_1$ (solid blue). The integrated flux $\mathbf{X}_{\pi_1}$ is given by $\mathbf{X}_{\pi_1}=g^{-1}_{rl^1_1(0)}\left(X_{l^1_1}+g^{-1}_{l^1_1(0)l^1_2(0)}X_{l^1_2}g_{l^1_1(0)l^1_2(0)}\right)g_{rl^1_1(0)}$, where $g_{l^1_1(0)l^1_2(0)}=h_5h_4h_3^{-1}$ and $g_{rl^1_1(0)}=h_{11}^{-1}h_{10}^{-1}h_9h_8$. The integrated flux $\mathbf{X}_{\pi_2}$ is given by $\mathbf{X}_{\pi_2}=g^{-1}_{rl^2_1(0)}\left(X_{l^2_1}+g^{-1}_{l^2_1(0)l^2_2(0)}X_{l^2_2}g_{l^2_1(0)l^2_2(0)}\right)g_{rl^2_1(0)}$, where $g_{l^2_1(0)l^2_2(0)}=h_6^{-1}$ and $g_{rl^2_1(0)}$ is the holonomy along the tree from the root node to the node $l^2_1(0)$ (and is not represented for the sake of clarity). To compose the two fluxes, one has to take $\mathbf{X}_{\pi_2}$, undo its parallel transport to the root using $g_{rl^2_1(0)}$, transport it to the frame $l^1_2(0)$ using the holonomy $h_{7}$ along the path $\gamma_{12}$ connecting $l^1_2(0)$ to $l^2_1(0)$, transport it to the frame $l^1_1(0)$ using the holonomy $g_{l^1_1(0)l^1_2(0)}$, transport it to the root using the holonomy $g_{rl^1_1(0)}$, and finally add it with $\mathbf{X}_{\pi_1}$.
• Figure 5: A piece of 3-dimensional triangulation consisting of six tetrahedra glued together, and a choice of surface path $\pi$ consisting of six triangles (red). Because of the orientation of the edges, these six triangles all have a counter-clockwise orientation. The dual graph to this piece of triangulation consists of six 4-valent nodes, and the links dual to the triangles of $\pi$ are labelled by $l_1,\ldots,l_6$. The tetrahedron dual to the source $l_1(0)$ has been chosen as the reference frame in which the fluxes have to be transported and added. The shadow graph (dashed red) connects the source nodes of the links dual to the triangles of $\pi$, and a choice of shadow tree (dashed brown) enables to define uniquely the parallel transport between the frame of each flux and the frame $l_1(0)$. Notice that there are five other possible choices for the shadow tree.

Theorems & Definitions (21)

• Definition 2.1: Simplices
• Definition 2.2: Simplicial complex
• Definition 2.3: $k$-skeleton
• Definition 2.4: Star
• Definition 2.5: Triangulation
• Definition 2.6: Geometric triangulation
• Definition 2.7: Dual complex
• Definition 2.8: Spanning tree
• Definition 2.9: Refining Alexander moves
• Definition 2.10: Flagged structure