Emergence Of Topology Research Articles

Connectomes as holographic states

We use the topological quantum field theory description of states in Chern-Simons theory to discuss the relation between spacetime connectivity and entanglement, exploring the paradigm entanglement=topology. We define a special class of states in Chern-Simons with properties similar to those of holographic states. While the holographic states are dual to classical geometries, these connectome states represent classical topologies, which satisfy a discrete analog of the Ryu-Takayanagi formula and characteristic inequalities for the entanglement entropy. Generic states are linear combinations of connectomes, and the theory also has nonperturbative states which are global spacetime defects formed by a large number of quantum fluctuations. Topological presentation of quantum states and emergence of topology from entanglement may be useful for building a generalization to geometry, that is quantum gravity. Thinking of further quantum gravity comparisons we discuss replica wormholes and conclude that similar objects exist beyond gravitational theories. The topological theory perspective suggests that the sum over all wormholes is always factorizable, even though the individual ones might not be.

On greedy network formation

Greedy navigability is a central issue in the theory of networks. However, the exogenous nature of network models do not allow for describing how greedy routable-networks emerge in reality. In turn, network formation games focus on the very emergence proess, but the applied shortest-path based cost functions exclude navigational aspects. This paper takes a frst step towards incorporating both emergence (missing in algorithmic network models) and greedy navigability (missing in network formation games) into a single framework, and proposes the Greedy Network Formation Game. Our first contribution is the game definition, where we assume a hidden metric space underneath the network, and, instead of usual shortest path metric, we use the length of greedy paths as the measure of communiation cost between players. Our main finding is that greedy-routable small worlds do not emerge on constant dimensional Eulidean grids. This simply means that the emergence of topologies on which w eunderstood the priniples of greedy forwarding cannot be explained endogenously. We also present a very brief outlook on how the situation hanges in the hyperbolic space.

HUYGENS' PRINCIPLE, PHYSICS AND COMPUTERS

We discuss the emergence of topology from a consideration of set extensions in General Systems Theory. Boundaries arise in a natural way, separating independent elements or regions of the system. Our aim is a unification of Etter theory, Kron's method of Tearing and Jessel's formulation of Huygens' Principle. This should make explicit the equivalence between the objective, structural, holographic and the subjective, relative definitions of information, sought in Bowden (1994b), reprinted in this Special Issue. It connects the abstract generalisations of Schrodinger's equation and Bom's rule derived in probabilistic Etter theory with the real world of electrical and other physical phenomena in General Physical Systems Theory. This paper can be considered as a continuation of Bowden (1990; 1994a) and as a response to Bowden (1994b), reprinted in this issue. We review the ideas behind Kron's Method of Tearing and Jessel's Principle of Secondary sources (both special cases of the above theory) and their equivalence. We follow Hiley's argument in Hiley (1996) to show how Schrodinger's equation can be thought of as specifying the evolution of (a series of) tearings in continuous space. These can be shown on a commutative diagram as a series of similarity transforms. We compare this with Etter's derivation (Etter, 1998). We describe briefly a recently published derivation of Maxwell's equations from a non-commutative algebra and show how they fit onto a related commutative diagram. Finally we make some comments on applications of the general theory to computer systems. This paper is a series of vignettes of work in progress. It is designed to point the direction of work to come in Constructive Physics.