A lattice worldsheet sum for 4-d Euclidean general relativity

TL;DR

The paper advances a lattice-regularized, spin-network-based approach to four-dimensional Euclidean quantum gravity, aiming for exact diffeomorphism invariance and Planck-scale discreteness realized through SU(2) spin spectra. It constructs a local lattice gauge theory where spacetime is a 4D cellular complex and the interior connection is integrated to yield a covariant sum over spin worldsheets, interpretable as discrete geometries. A concrete Euclidean GR model is proposed on a simplicial complex, formulated as a formal quantization of a Plebanski-based action with a metricity constraint implemented via a Gaussian-like operator; the spin-worldsheet formulation reveals how geometry emerges from the intertwiner- and spin-label structure, with BF theory recovered in a topological limit. This framework potentially provides a UV-complete setting for gravity (and matter) by summing over discrete spacetime geometries at the Planck scale, while reproducing classical GR in suitable limits.

Abstract

A lattice model for four dimensional Euclidean quantum general relativity is proposed for a simplicial spacetime. It is shown how this model can be expressed in terms of a sum over worldsheets of spin networks, and an interpretation of these worldsheets as spacetime geometries is given, based on the geometry defined by spin networks in canonical loop quantized GR. The spacetime geometry has a Planck scale discreteness which arises "naturally" from the discrete spectrum of spins of SU(2) representations (and not from the use of a spacetime lattice). The lattice model of the dynamics is a formal quantization of the classical lattice model of \cite{Rei97a}, which reproduces, in a continuum limit, Euclidean general relativity.

Figures (8)

• Figure 1: The heavy black lines show the edges of the dual boundary $[\partial\mu]^*$ of a 3-cell $\mu$. In a 4-cell, which is difficult to draw, the boundary is a three dimensional cellular complex, and the edges of the dual boundary connect the centers of the cells of this complex.
• Figure 2: The three panels show the spin worldsheets inside a cell spanning various types of spin networks on the cell boundary. (The spin networks are indicated by heavy lines). Since the four dimensional situation is hard to draw three dimesional analogs are shown. Panel a) shows spin network consisting of a single unknotted loop of spin $j$, which is spanned by a disk of spin $j$. Panel b) shows a spin network with two vertices and three edges. It is spanned by three faces with the topology of disks which are joined at a trivalent branch line running between the two vertices via the center of the cell. Each face is bounded by one of the edges, and carries the spin of that edge. The two halves of the branch line, on either side of the center, each carry the intertwiner label of the adjacent spin network vertex. These can in general be distinct. Panel c) shows a spin network with four vertices and six edges. The four branch lines each start at a spin network vertex and end at the center of the cell, which serves as a branch point, or worldsheet vertex.
• Figure 3: A spin network consisting of two linked loops, and the spin worldsheet that spans it are shown. Since loops cannot be linked in the topological 2-sphere that is the boundary of a 3-cell, I cannot illustrate this type of worldsheet with a three dimensional analog. Instead I have tried to represent the four dimensional situation directly. In depicting knots in a plane one uses breaks in the lines to indicate the parts of the lines that are pushed down below the plane because there is a crossing. Here a broken surface in three dimensions is used to represent a surface in four dimensions, where the breaks indicate regions that are pushed into the fourth dimension, off the 3-space which the viewer is visualizing (see Carter for more on this and other techniques of four dimensional visualization). We see that the worldsheet consists of two disks, each spanning a loop, which intersect at a single point - the center of the cell. No attempt has been made to show the 4-cell. The 3-space which the picture images is a 3-surface that cuts through the 4-cell in such a way that it contains most of the spin worldsheet.
• Figure 4: The diagram shows a lattice spin worldsheet in a three dimensional spacetime. The worldsheet consists of four unbranched components, three of which meet along a branch line, while the fourth forms an isolated bubble. The three connected unbranched components also meet the boundary of the spacetime $\Pi$ on a spin network $\Gamma$, which is drawn with heavy lines. A possible assignment of spins to the unbranched components and the edges of $\Gamma$ have been written in. Note that the branch line carries no intertwiner label since it is only trivalent. To keep the figure simple only the boundary $\partial\Pi$ of the spacetime cellular complex has been shown, and that has been chosen to be a simple box, even though $\partial\Pi$ can, in fact, be quite irregular. The relationship of the spin worldsheet to the individual cells of $\Pi$ is illustrated in Figs. \ref{['cellworldsheet1']} and \ref{['cellworldsheet2']}. The spin worldsheet shown is made up of quadrangles. This is in fact generally true, as will be explained further on in the text. However, the quadrangles will generally not form a rectangular grid. Some possible features of worldsheets that are not illustrated are self intersections at points and non-orientable components.
• Figure 5: Panel a) shows the corner cell $c_P$ associated with the vertex $P$ of a 3-simplex $\mu$. Notice that the intersection of $c_P$ with any of the triangular faces of $\mu$ that are incident on $P$ is itself the two dimensional corner cell of $P$ in the face in question. Note also that $c_P$ is diffeomorphic to a cube, and each of the subsimplices of $\mu$ that touch $P$ (including $\mu$ and $P$) contain one corner of $c_P$. These features are shared by corner cells in any cell (not necessarily a simplex). Panel b) shows a two dimensional example of the construction of corner cells within a generic polygonal cell $\mu$. The heavy line along the boundary $\partial\mu$ shows the dual boundary cell $P^*_{\partial\mu}$, formed by the union of the two corner cells of $P$ in $\partial\mu$. The images of $P^*_{\partial\mu}$ in the concentric, sucessively smaller images of $\partial\mu$ are also indicated by heavy lines. Together these sweep out the corner cell of $P$ in $\mu$. The 1-cells $\lambda_1$ and $\lambda_2$ of $\partial\mu$ incident on $P$, mentioned in the text, and their centers are labeled.