Oriented matroids—combinatorial structures underlying loop quantum gravity

Abstract

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator (Ashtekar A and Lewandowski J 1998 Adv. Theor. Math. Phys. 1 388) in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in three-dimensional Riemannian space and can be represented by sign strings containing relative orientations of embedded edges. We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid (Ziegler G M 1998 Electron. J. Comb.; Björner A et al 1999 Oriented Matroids (Cambridge: Cambridge University Press)). Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs. Hence, the mathematical methods developed for oriented matroids can be applied to the difficult combinatorics of embedded graphs underlying the construction of LQG. As a first application we revisit the analysis of Brunnemann and Rideout (2008 Class. Quantum Grav. 25 065001 and 065002), and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3 (Ziegler G M 1998 Electron. J. Comb.; Björner A et al 1999 Oriented Matroids (Cambridge: Cambridge University Press)), and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest non-zero eigenvalue of the volume spectrum does not grow as one increases the maximum spin jmax at the vertex, for any orientation of the edge tangents. This indicates that, in contrast to the area operator, considering large jmax does not necessarily imply large volume eigenvalues. In addition we give an outlook to possible starting points for rewriting the combinatorics of LQG in terms of oriented matroids.