Causal Atomic Resolution

If spacetime degenerates into a nonmanifold structure near the Planck-scale, the number of fundamental elements involved in even the smallest-scale interactions currently observable might be very large: possibly on the order of Avogadro’s number.   This statement applies not only to pure causal theory, but also to dynamical triangulations, spin-foams in loop quantum gravity, and various other models of fundamental spacetime structure.  Organizing the information associated with such complex interactions is an important preamble to making detailed calculations and predictions.

Causal atomic resolution is an attractive way of organizing such information in the context of pure causal theory.   A causal atom is a special type of subgraph of a causal graph, that contains the intersection of its past and future.   An example is the Alexandrov subset given by intersecting the past of one element with the future of another.   Alexandrov subsets are generalizations of intervals in linearly ordered sets, and causal atoms are a further generalization.  They are also analogous to convex subsets in Euclidean space.  A causal atomic decomposition of a causal graph is a simplified graph whose vertices are causal atoms of the original graph.   A causal atomic resolution is a sequence of causal atomic decompositions that progressively simplify the original graph by combining causal atoms at each level of the sequence into larger atoms at the next level.   Causal atomic resolution imposes a fractal structure on causal graphs.   It organizes information in a manner analogous to multiresolution analysis in the theory of wavelets in functional analysis.

Causal atomic resolution is similar to the theory of categorification in abstract algebra.   Categorification replaces set-theoretic constructs with category-theoretic constructs, and vice versa (decategorification).   For instance, objects may be promoted to categories, or categories demoted to objects.  Elements may be promoted to objects, or objects demoted to elements.  Morphisms may be promoted to functors, or functors demoted to morphisms, and so on.  Categorification may be generalized in a graph-theoretic setting.  Every category possesses an underlying directed graph, but not every directed graph is a category.   A general system of “objects” and “morphisms” without the category-theoretic requirements of composition, associativity, and identity may be called a graphegory, and the graphegoric generalization of categorifcation may be called graphication.  Causal atomic resolution is a special case of graphication, in which elements are promoted to directed graphs, rather than objects of some other graphegory.

In the quantum causal theory of spacetime microstructure, causal atomic resolution may be combined with power space and path integral methods to carry the theory in promising and unexplored directions.   This approach features holism, self-similarity, and iterated structure.  Graphication appears here in two different ways: as the origin of quantum-theoretic structure, and as a computational tool for organizing information at different scales.

Causal atomic resolution has the potential for important applications to existing areas of computer science, economics, biology, physics, and mathematics.  These areas include computational complexity, network theory, neuroscience, self-organization, quantum information theory, and of course quantum gravity, abstract algebra, and graph theory.   The potential cross-pollination between the concrete and the abstract afforded by causal atomic resolution makes it an interesting topic for further study.

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