A left-handed simplicial action for euclidean general relativity

TL;DR

This work presents a left-handed, SU(2)-based simplicial formulation of Euclidean GR that discretizes Plebanski's action on a derived complex, introducing discrete variables for the self-dual 2-forms, connections, and a spin-2 field. It proves that, under a uniformly refining sequence of simplicial (and hypercubic) decompositions and with smooth continuum fields, the simplicial action converges to the Plebanski action, and the discrete field equations reproduce the Plebanski equations in the continuum limit, unlike Regge-based formulations. The results support the viability of a covariant path integral (sum-over-histories) for loop-quantized GR using these fundamental variables, and they offer practical advantages for numerical relativity due to simpler discrete equations. The hypercubic analogue further suggests potential computational efficiency, while the work clarifies that the continuum limit aligns with Plebanski GR rather than Ashtekar canonical theory in degenerate regimes, guiding quantization strategies toward Reisenberger-type constraints.

Abstract

An action for simplicial euclidean general relativity involving only left-handed fields is presented. The simplicial theory is shown to converge to continuum general relativity in the Plebanski formulation as the simplicial complex is refined. This contrasts with the Regge model for which Miller and Brewin have shown that the full field equations are much more restrictive than Einstein's in the continuum limit. The action and field equations of the proposed model are also significantly simpler then those of the Regge model when written directly in terms of their fundamental variables. An entirely analogous hypercubic lattice theory, which approximates Plebanski's form of general relativity is also presented.

Figures (4)

• Figure 1: Panel a) shows the corner cell $a_P$ associated with the vertex $P$ of a 3-simplex. Notice that the intersection of $a_P$ with any of the triangular faces incident on $P$ is itself the two dimensional corner cell of $P$ in the triangular face, and that the six corners of $a_P$ are located at the centers of the simplices incident on $P$, defining, topologically, a cube. Panel b) shows the cell $P^*$ dual to the vertex $P$ in a two dimensional simplicial complex. The boundaries of other dual cells are indicated by dashed lines. The cells and subcells of this realization of the dual complex generally are not flat where they meet a boundary between simplices.
• Figure 2: Panel a) illustrates the definitions of $s(\sigma\nu)$ and the edges $l(\nu\tau)$ and $r(\tau\sigma)$. In the middle lies the center of a 2-simplex $\sigma$. The corners are the centers of all the 4-simplices $\nu, \nu', ...$ that share $\sigma$. The curve $C_\nu C_{\tau_1} C_{\nu'}$ connecting $C_\nu$ and $C_{\nu'}$ is the edge $\tau_1^*$ in $\Delta^*$ which is dual to $\tau_1$. Similarly $\cup_\nu s(\sigma\nu)$ is the 2-cell $\sigma^* < \Delta^*$ dual to $\sigma$. One may think of $s(\sigma\nu)$ as the wedge of $\sigma^*$ in $\nu$: $s(\sigma\nu) = \sigma^* \cap \nu$. Likewise, $l(\nu\tau) = \tau^*\cap \nu$ and the radial edge $r(\tau\sigma) = \sigma^*\cap\tau$. Panel b) shows the analogous structure in a 3-dimensional complex with a 3-simplex playing the role of $\nu$, 2-simplices as $\tau_1$ and $\tau_2$, and a 1-simplex as $\sigma$.
• Figure 3: The dual 2-cell $\sigma^* < \Delta^*$ is shown with its uniform orientation indicated by an arrow circling in a positive sense. The routes by which the $w_{\sigma\nu_n}$ are parallel transported from $C_{\nu_n}$ to $C_\sigma$ to form the $u_{\sigma\nu_n}$ are indicated by bold lines.
• Figure 4: This is the analogue of Fig. \ref{['cells']} for a hypercubic lattice. Panel a) illustrates the definitions of $s(\sigma\nu)$ and the edges $l(\nu\tau)$ and $r(\tau\sigma)$. In the middle lies the center of a 2-cube $\sigma$. The corners are the centers of all the 4-cubes $\nu, \nu', ...$ that share $\sigma$. The curve $C_\nu C_{\tau_1} C_{\nu'}$ connecting $C_\nu$ and $C_{\nu'}$ is the edge $\tau_1^*$ in the lattice $\Box^*$ dual to $\Box$, which is dual to $\tau_1$. Similarly $\cup_\nu s(\sigma\nu)$ is the 2-cell $\sigma^* < \Box^*$ dual to $\sigma$. Panel b) shows the analogous structure in a 3-dimensional cubic lattice with a 3-cube playing the role of $\nu$, 2-cubes as $\tau_1$ and $\tau_2$, and a 1-cube as $\sigma$.