233:, and a real-valued field associated with every pair of vertices; the abstract simplicial complex is set to correspond with a geometric simplicial complex and then geometric simplices are stitched together into a piecewise linear space. Developed further, triangulation, link distance, a piecewise linear manifold, and a spacetime metric arise. Further, a
253:. Information content in various ages of the universe modifies the tests so the universe acts as an automaton, modifying its structure. Causal set theory is then worked out within this quantum automaton framework to describe a spacetime that inherits the assumptions of geometry within standard quantum mechanics.
63:. Where geometry could describe the properties of a known surface, the physics of a hypothetical region with predefined properties, "pregeometry" might allow one to work with deeper underlying rules of physics that were not so strongly dependent on simplified classical assumptions about the properties of space.
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Space is described by a graph with densely entangled sub-clusters of nodes (with differential states) and bonds (either vanishing at 0 or directed at 1). Rules describe the evolution of the graph from a chaotic patternless pre-Big Bang condition to a stable spacetime in the present. Time emerges from
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Spacetime is described by a generalized graph consisting of a very large or infinite set of vertices paired with a very large or infinite set of edges. From that graph emerge various constructions such as vertices with multiple edges, loops, and directed edges. These in turn support formulations of
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No single proposal for pregeometry has gained wide consensus support in the physics community. Some notions related to pregeometry predate
Wheeler, other notions depart considerably from his outline of pregeometry but are still associated with it. A 2006 paper provided a survey and critique of
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linked to the notion of past and future in macroscopic spacetime and causality between point-events. Derived from the causal order is the differential structure and the conformal metric of a manifold. A probability is assigned to a
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framework. This discrete-space structure assumes the metric of spacetime and assumes composite geometric objects so it is not a pregeometric scheme in line with
Wheeler's original conception of pregeometry.
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with points (associated with vertices) and links (of unit length) that are created or annihilated according to probability calculations. The parameterization of graphs in a metaspace gives rise to time.
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An assortment of ontological presuppositions describes spacetime a result of relations between objectively existing entities. From presuppositions emerges the topology and metric of
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A preliminary investigation into how all events might be mapped with rational number coordinates and how this might help to better understand a discrete spacetime framework.
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of nodes and links. A definition of distance between any two monads is defined and from this and probabilistic mathematical tools emerges a three-dimensional space.
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becoming embedded in a manifold; thus there can be a transition from a discrete Planck scale fundamental unit of volume to a classical large scale continuous space.
394:
Requardt, Mandred; Roy, Sisir (2001). "(Quantum) Space-Time as a
Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces".
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allowed a metric to fluctuate, it was argued that the merging of gravity with quantum mechanics required a set of more fundamental rules regarding
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Requardt, Mandred; Roy, Sisir. "(Quantum) Space-Time as a
Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces"
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502:§44.4 "Not geometry, but pregeometry as the magic building material", §44.5 "Pregeometry as the calculus of prepositions"
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A proposal anticipating
Wheeler's pregeometry, though assuming some geometric notions embedded in quantum mechanics and
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Spacetime is described as having a deeper pregeometric structure based on three dynamical variables, vertices of an
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Studies in
History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
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pregeometry or near-pregeometry proposals up to that time. A summary of these is given below:
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a deeper external-parameter "clock-time" and the graphs lead to a natural metrical structure.
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whose topological structure entirely characterizes the graph. Spatial points are related to
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Event states (elementary or entangled) are provided topological relationships via tests (
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quantization is formulated resulting in a quantum gravity description of spacetime.
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feature a pregeometric universe before the Big Bang. The term was championed by
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work out to be a special case semi-periodical functions though the nature of
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is ambiguous since the energy-momentum space cannot be uniquely interpreted.
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Akama, Keiichi. "An
Attempt at Pregeometry: Gravity with Composite Metric"
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Meschini; et al. (August 2006). "Geometry, pregeometry and beyond".
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variables are restricted to a certain set of rational numbers. Quantum
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Rational-number spacetime by
Horzela, Kapuscik, Kempczynski and Uzes
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Sidoni, Lorenzo (2013). "Horizon thermodynamics in pregeometry".
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Axiomatic pregeometry by Perez, Bergliaffa, Romero and
Vucetich
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where rational numbers themselves undergo quantum fluctuations.
353:"An Attempt at Pregeometry: Gravity with Composite Metric"
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in the 1960s and 1970s as a possible route to a theory of
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Simplicial quantum gravity by Lehto, Nielsen and
Ninomiya
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Sidoni, Lorenzo. "Horizon thermodynamics in pregeometry"
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Structure from which the geometry of the universe arises
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geometry over a field of rational numbers and a finite
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Some additional or related pregeometry proposals are:
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Quantum automaton universe by Jaroszkiewicz and Eakins
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152:Causal sets by Bombelli, Lee, Meyer and Sorkin
155:All of spacetime at very small scales is a
27:is a hypothetical structure from which the
103:Discrete-space structure by Dadic and Pisk
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196:and the relations between them becomes a
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189:Bootstrap universe by Cahill and Klinger
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490:Misner, Thorne, and Wheeler ("MTW"),
449:Journal of Physics: Conference Series
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136:Number theory pregeometry by Volovich
86:coefficients is deployed. Energy and
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360:Progress of Theoretical Physics
471:10.1088/1742-6596/410/1/012140
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217:Cellular networks by Requardt
192:An iterative map composed of
123:Pregeometric graph by Wilson
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426:10.1088/0264-9381/18/15/317
330:10.1016/j.shpsb.2005.01.002
231:abstract simplicial complex
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180:Spacetime is described by
71:Discrete spacetime by Hill
177:Random graphs by Antonsen
131:foundation of space-time.
55:that were independent of
351:Akama, Keiichi (1978).
80:Lorentz transformations
159:consisting of locally
41:John Archibald Wheeler
251:quantum arrow of time
521:Mathematical physics
418:2001CQGra..18.3039R
373:10.1143/PTP.60.1900
322:2005SHPMP..36..435M
247:Hermitian operators
210:Minkowski spacetime
163:of elements with a
37:cosmological models
76:special relativity
500:978-0-7167-0344-0
402:(15): 3039–3057.
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145:Galois field
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53:connectivity
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492:Gravitation
25:pregeometry
510:Categories
455:: 012140.
379:30 October
285:References
198:tree-graph
170:causal set
161:finite set
157:causal set
116:Fock space
82:with only
479:118590032
462:1211.2731
516:Geometry
434:14941099
338:55663184
129:metrical
112:vertices
88:momentum
84:rational
57:topology
47:. Since
33:universe
29:geometry
494:(1971)
414:Bibcode
318:Bibcode
235:lattice
31:of the
21:physics
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194:monads
475:S2CID
457:arXiv
430:S2CID
404:arXiv
356:(PDF)
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308:arXiv
496:ISBN
381:2013
127:the
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