Related papers: The Continuum Limit of Discrete Geometries
This article gives a review of a recent construction, the ambient cosmological metric, and its implications for the global geometry of the universe. According to this proposal, the universe is a bounding hypersurface carrying a conformal…
Motivated by quantum mechanical considerations we earlier suggested an alternative action for discretised quantum gravity which has a dimension of length. It is the so called "linear" action. The proposed action is a "square root" of the…
In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov-Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are…
In pregeometry a metric arises as a composite object at large distances. We investigate if its signature, which distinguishes between time and space, could be a result of the dynamics rather than being built in already in the formulation of…
Recent progress in quantum gravity and string theory has raised interest among scientists to whether or not nature behaves discretely at the Planck scale. There are two attitudes twoards this discretenes i.e. top-down and bottom-up…
We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf…
The attempt to gain a theoretical understanding of the concept of time in quantum mechanics has triggered significant progress towards the search for faster and more efficient quantum technologies. One of such advances consists in the…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
Current constraints on spatial curvature show that it is dynamically negligible: $|\Omega_{\rm K}| \lesssim 5 \times 10^{-3}$ (95% CL). Neglecting it as a cosmological parameter would be premature however, as more stringent constraints on…
Quantum gravitational effects suggest a minimal length, or spacetime interval, of order the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with…
Two examples, not connected at present, from author's papers (Nuovo Cim., 1992, v.105A, p.77 [hep-th/0207210] and GRG, 1999, v.31, p.1431 [gr-qc/0207017]) are considered here in which a physical model has discrete symmetries and additional…
The quantum geometry arising in Loop Quantum Gravity has been known to semi-classically lead to generalizations of length-geometries. There have been several attempts to interpret these so called twisted geometries and understand their role…
For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS)…
We present a new method for the quantization of totally constrained systems including general relativity. The method consists in constructing discretized theories that have a well defined and controlled continuum limit. The discrete…
We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The r\^ole of the metric is taken over by the time separation function, in terms of which all basic notions are…
We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…
The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for the validity of a mathematical description of fundamental physics in terms of continuous fields are a natural outcome of discrete…
In this paper, we propose a novel Quantum Spacetime Theory (QST) that reinterprets spacetime as an emergent structure, challenging the traditional block universe paradigm and aligning with research into emergent spacetime. Using a sphere…
Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms…
In this paper, a new structure is defined on a topological space that equips the space with a concept of distance in order to do that firstly, a generalization of quasi-pseudo-metric space named R.O-metric space is introduced, and some of…