Emergent space-time via a geometric renormalization method

Abstract

We present a purely geometric renormalization scheme for metric spaces (including uncolored graphs), which consists of a coarse graining and a rescaling operation on such spaces. The coarse graining is based on the concept of quasi-isometry, which yields a sequence of discrete coarse grained spaces each having a continuum limit under the rescaling operation. We provide criteria under which such sequences do converge within a superspace of metric spaces, or may constitute the basin of attraction of a common continuum limit, which hopefully, may represent our space-time continuum.We discuss some of the properties of these coarse grained spaces as well as their continuum limits, such as scale invariance and metric similarity, and show that different layers of spacetime can carry different distance functions while being homeomorphic.Important tools in this analysis are the Gromov-Hausdorff distance functional for general metric spaces and the growth degree of graphs or networks. The whole construction is in the spirit of the Wilsonian renormalization group.Furthermore we introduce a physically relevant notion of dimension on the spaces of interest in our analysis, which e.g. for regular lattices reduces to the ordinary lattice dimension. We show that this dimension is stable under the proposed coarse graining procedure as long as the latter is sufficiently local, i.e. quasi-isometric, and discuss the conditions under which this dimension is an integer. We comment on the possibility that the limit space may turn out to be fractal in case the dimension is non-integer. At the end of the paper we briefly mention the possibility that our network carries a translocal far-order which leads to the concept of wormhole spaces and a scale dependent dimension if the coarse graining procedure is no longer

On the implications of the Bekenstein bound for black hole evaporation

“…In this picture the particles of the Standard Model are regarded as analogous to quasiparticles, arising together with the classical background geometry from the interactions between the fundamental degrees of freedom. We have not provided here a framework for such dynamical emergence (see [12][13][14][15][17][18][19][20][21][22][23] for some examples along these lines); however we assume that the unitary evolution -which is a central requirement of the emergent quantum theory -is preserved down to the fundamental level. The unitary evolution inevitably leads to a reshuffling of the fundamental degrees of freedom and this is reflected on the emergent level as an entanglement between quantum fields and geometry.…”

“…In quantum graphity [15,16], fundamental degrees of freedom and their interactions are represented by a complete graph with dynamical structure. For more approaches, see, e.g., [17][18][19][20][21][22][23].…”

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Emergence of spacetime in a restricted spin-foam model

“…Already memory cost in defining such a matrix is very high, not to mention the computational cost in computing its complete spectrum. 4 Instead of diagonalizing the entire matrix one can study the return probability of a discrete geometry via a random walker, similar to the studies in causal dynamical triangulations [18]. The random walker randomly jumps from lattice site to lattice site, where the jump probabilities are related to the entries of ∆.…”

“…For simplicity we assume the lattice to be of equal size in all dimensional directions 4. Since ∆ is proportional to the adjacency matrix many of its entries are empty.…”