Approximate Killing symmetries in non-perturbative quantum gravity

Abstract

We study the notion of approximate Killing vector fields in several toy models of non-perturbative two-dimensional quantum gravity. Using the framework of discrete exterior calculus, we show how to formulate quantum observables related to such approximate Killing vector fields. Using these methods, we aim to investigate symmetry properties of the space–time geometry produced by the quantum gravitational model at hand. Since we expect quantum fluctuations to dominate at small scales, our goal is to construct a scale-dependent notion of symmetry that might be used to determine whether the emergent (semi-)classical geometry admits any approximate Killing symmetries. We have evaluated one particular choice of such an observable on three ensembles of discrete geometry. We find that the method is useful in the setting where fluctuations are small, but that more work is needed before these ideas can be applied in the deep quantum regime.

Generalised spectral dimensions in non-perturbative quantum gravity

“…As is often the case for generalisations of continuum objects to discrete manifolds, there is no unique choice for the definition of discrete k-form fields and discrete k-tensor fields. A specific definition for both objects has been studied in detail in [26]. The k-form fields were already discussed above and are associated to the k-simplices.…”

“…Noting that the tangent space is not well defined on the boundary of a n-simplex, it is unnatural to define X k on the k-simplex σ k for the cases 0 < k < n. Instead, it will be defined on the interior of the n-simplices, where there exists a unique definition of a tangent space. In [26] it was discussed how defining discrete vector fields on the interior of simplices has certain advantageous properties. It is important to note that discrete k-tensor fields defined in this fashion do not admit a duality relation to the discrete k-form fields, because the number of k-simplices for 0 < k < n is different from the number of n-simplices for simplicial manifolds T. However, there exists a surjective function 13…”

“…For the case k = 1, X k (T) defines what is known as discrete vector fields, for which an example is given in figure 3. Various studies show that discrete vector fields possess more favourable properties than discrete one-form fields when the goal is to approximate continuum vector fields or oneform fields [25,26,29]. In view of these deliberations, the question arises if a discrete notion of diffusion can be defined both in terms of discrete k-form fields and discrete k-tensor fields.…”

Perturbations of the almost Killing equation and their implications

“…Alternately, one can show that even on a Minkowskian background, the AKE admits wavelike solutions for which the components Qµν ≫ 0, so that they cannot be considered as approximate Killing vectors by any means. 2 For further discussion of this point, see [13] and [14].…”

“…The literature contains several approaches for defining generalized Killing vectors and symmetries. Specific examples include Matzner's Eigenvector approach [1], which has recently been of interest for studying quantum geometries in Causal Dynamical Triangulations [2], symmetry-seeking coordinates [3], affine collineations [4] and the almost Killing equation (henceforth AKE) [5,6]. The latter approach, the generalized Killing vectors defined by the AKE, forms the subject of this paper.…”