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 [1] in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in 3-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 [2,3]. 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 [4,5], and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3 [2, 3], and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest nonzero 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.

Transit sets of -point crossover operators

“…They reflect properties such as linear dependencies, facial relationship, convexity, duality, and have bearing on solutions of associated optimization problems. Beyond their connection with many areas of mathematics, the theory of oriented matroids has in recent years found applications in diverse areas of science and technology, including metabolic network analysis [Gagneur and Klamt(2004), Müller and Bockmayr(2013), Müller et al(2014)Müller, Regensburger, andSteuer, Reimers(2014)], electronic circuits [Chaiken(1996)], geographic information science [Stell and Webster(2007)], and quantum gravity [Brunnemann and Rideout(2010)].…”

New insights in quantum geometry

“…How to effectively characterize the realizable sign configurations has been a question that has hampered the numerical analysis of the volume spectrum for some time. Recently however, Brunnemann and Rideout have found in matroid theory the tool to answer all the questions pertaining to these signs [13]. Matroids are combinatorial structures generalizing linear dependencies in a real vector space, and [13] describes in detail how they can be used to describe the diffeomorphism invariance classes of tangential structure of vertices.…”

“…These results will be quite important when the action of the Hamilton constraint (of which the AL volume operator is an important ingredient) is studied in detail. Additionally, as pointed out in [13], matroids can also be used to describe the combinatorics of edge composition in graphs, thus opening another potential application in LQG.…”

“…I have picked four areas of development for a review, namely (i) U(N) structure on intertwiner spaces and interpretation in terms of polygons and twisted geometries [6,7,8,9,10,11]. (ii) Sign factors and volume spectra [12,13]. (iii) Quantum geometry with a classical spatial background [14,15].…”

“…From[13] with kind permisson: Histograms for each sign configuration at the 6vertex, with incoming spins up to j max = 13/2. The right figure shows the small eigenvalue region.…”