A geometric renormalization group in discrete quantum space–time

Requardt, Manfred

doi:10.1063/1.1619579

Cited by 28 publications

(57 citation statements)

References 42 publications

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“…This procedure also establishes a connection to Connes' noncommutative geometry (for details see for example [23], [30] or [31]). In section 4 of [32] we showed that the evolving dynamics belongs to the same general class of graph transformations or dynamics, as is the case for spin network dynamics or causal set dynamics. I.e., we discuss in this paper primarily the continuum limit of the "space-like slices" in S − T .…”

Section: Main Strategy: the Big Picturementioning

confidence: 90%

Emergent space-time via a geometric renormalization method

Rastgoo

Requardt

2016

Phys. Rev. D

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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

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Section: Main Strategy: the Big Picturementioning

confidence: 90%

Emergent space-time via a geometric renormalization method

Rastgoo

Requardt

2016

Phys. Rev. D

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“…We formulated such a concept in [14] and showed its usefulness. In [7] we used it among other concepts, to characterize the nature of what we called a geometric renormalization process, i.e., a coarse-graining and rescaling process which, hopefully, allows us to construct a continuum limit, the supposed fixed point of the process, from a discrete underlying substratum. The idea has some vague similarities to the real-space renormalization group of statistical mechanics, but it represents, as far as we can see, a technically much more complicated enterprise.…”

Section: Remark 17mentioning

confidence: 99%

“…Lemma 4.10 in [14], the Discussion in Sect. VII of [7], and Theorem 6.8 in [8]). More specifically, edge deletions are called k-local if, in the transition from G to G , only edges are deleted in G so that for the corresponding pairs of nodes (x, y), it holds that y ∈ B G (x, k) with respect to G .…”

Section: Remark 25mentioning

confidence: 99%

“…That is, applying s to a group element, g, may send it to an element being far away from g (for large l) in a naive metrical (Z 3 )-sense. We note that we studied possibly related phenomena in [7,8], where we analyzed the possibility that space-dimension changes under, what we called, the coarsegraining or renormalization process. It turned out that only a very peculiar and non-local network wiring (with respect to some coarse-grained metric), called by us critical network states, allows for such a dimensional reduction under renormalization.…”

Section: Theorem 35 For the Heisenberg Group We Havementioning

confidence: 99%

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The Continuum Limit of Discrete Geometries

Requardt

2006

Int. J. Geom. Methods Mod. Phys.

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In various areas of modern physics and in particular in quantum gravity or foundational space-time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space-time and, in particular, its dimension.Keywords: Gromov-Hausdorff Distance, discrete metric spaces, continuum limit, geometric growth at infinity, dimension. 285 Int. J. Geom. Methods Mod. Phys. 2006.03:285-313. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/07/15. For personal use only.

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“…We have been promoting a discrete network approach to quantum space-time physics in recent years (see for example [15] or [16] and further references given there) which is assumed to underly our more macroscopic continuum physics on the Planck-scale. Our present analysis shows that these discrete model systems, perhaps contrary to naive wisdom, are in fact quite rich as to their structural properties.…”

Section: Introductionmentioning

confidence: 99%