In physics, a pregeometry is a hypothetical structure from which the geometry of the universe develops. Some cosmological models feature a pregeometric universe before the Big Bang. The term was championed by John Archibald Wheeler in the 1960s and 1970s as a possible route to a theory of quantum gravity. Since quantum mechanics allowed a metric to fluctuate, it was argued that the merging of gravity with quantum mechanics required a set of more fundamental rules regarding connectivity that were independent of topology and dimensionality. 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.
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[1] provided a survey and critique of pregeometry or near-pregeometry proposals up to that time. A summary of these is given below:
Discrete spacetime by Hill
A proposal anticipating Wheeler's pregeometry, though assuming some geometric notions embedded in quantum mechanics and special relativity. A subgroup of Lorentz transformations with only rational coefficients is deployed. Energy and momentum variables are restricted to a certain set of rational numbers. Quantum wave functions work out to be a special case semi-periodical functions though the nature of wave functions is ambiguous since the energy-momentum space cannot be uniquely interpreted.
Discrete-space structure by Dadic and Pisk
Spacetime as an unlabeled graph whose topological structure entirely characterizes the graph. Spatial points are related to vertices. Operators define the creation or annihilation of lines which develop into a Fock space 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.
Pregeometric graph by Wilson
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 the metrical foundation of space-time.
Number theory pregeometry by Volovich
Spacetime as a non-Archimedean geometry over a field of rational numbers and a finite Galois field where rational numbers themselves undergo quantum fluctuations.
Causal sets by Bombelli, Lee, Meyer and Sorkin
All of spacetime at very small scales is a causal set consisting of locally finite set of elements with a partial order 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 causal set 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.
Random graphs by Antonsen
Spacetime is described by dynamical graphs 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.
Bootstrap universe by Cahill and Klinger
An iterative map composed of monads and the relations between them becomes a tree-graph 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.
Axiomatic pregeometry by Perez, Bergliaffa, Romero and Vucetich
An assortment of ontological presuppositions describes spacetime a result of relations between objectively existing entities. From presuppositions emerges the topology and metric of Minkowski spacetime.
Cellular networks by Requardt
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 a deeper external-parameter "clock-time" and the graphs lead to a natural metrical structure.
Simplicial quantum gravity by Lehto, Nielsen and Ninomiya
Spacetime is described as having a deeper pregeometric structure based on three dynamical variables, vertices of an abstract simplicial complex, 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 lattice quantization is formulated resulting in a quantum gravity description of spacetime.
Quantum automaton universe by Jaroszkiewicz and Eakins
Event states (elementary or entangled) are provided topological relationships via tests (Hermitian operators) endowing the event states with evolution, irreversible acquisition of information, and a quantum arrow of time. 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.
Rational-number spacetime by Horzela, Kapuscik, Kempczynski and Uzes
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.
Further readingedit
Some additional or related pregeometry proposals are:
Akama, Keiichi. "An Attempt at Pregeometry: Gravity with Composite Metric"[2]
Requardt, Mandred; Roy, Sisir. "(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces"[3]
Sidoni, Lorenzo. "Horizon thermodynamics in pregeometry"[4]
Referencesedit
^Meschini; et al. (August 2006). "Geometry, pregeometry and beyond". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 36 (3): 435–464. arXiv:gr-qc/0411053. Bibcode:2005SHPMP..36..435M. doi:10.1016/j.shpsb.2005.01.002. S2CID 55663184.
^Akama, Keiichi (1978). "An Attempt at Pregeometry: Gravity with Composite Metric" (PDF). Progress of Theoretical Physics. 60 (6): 1900–1909. doi:10.1143/PTP.60.1900. Retrieved 30 October 2013.
^Requardt, Mandred; Roy, Sisir (2001). "(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces". Classical and Quantum Gravity. 18 (15): 3039–3057. arXiv:gr-qc/0011076. Bibcode:2001CQGra..18.3039R. doi:10.1088/0264-9381/18/15/317. S2CID 14941099.
^Sidoni, Lorenzo (2013). "Horizon thermodynamics in pregeometry". Journal of Physics: Conference Series. 410: 012140. arXiv:1211.2731. doi:10.1088/1742-6596/410/1/012140. S2CID 118590032.
Misner, Thorne, and Wheeler ("MTW"), Gravitation (1971) ISBN978-0-7167-0344-0 §44.4 "Not geometry, but pregeometry as the magic building material", §44.5 "Pregeometry as the calculus of prepositions"
April 11, 2024
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In physics a pregeometry is a hypothetical structure from which the geometry of the universe develops Some cosmological models feature a pregeometric universe before the Big Bang The term was championed by John Archibald Wheeler in the 1960s and 1970s as a possible route to a theory of quantum gravity Since quantum mechanics allowed a metric to fluctuate it was argued that the merging of gravity with quantum mechanics required a set of more fundamental rules regarding connectivity that were independent of topology and dimensionality 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 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 1 provided a survey and critique of pregeometry or near pregeometry proposals up to that time A summary of these is given below Discrete spacetime by Hill A proposal anticipating Wheeler s pregeometry though assuming some geometric notions embedded in quantum mechanics and special relativity A subgroup of Lorentz transformations with only rational coefficients is deployed Energy and momentum variables are restricted to a certain set of rational numbers Quantum wave functions work out to be a special case semi periodical functions though the nature of wave functions is ambiguous since the energy momentum space cannot be uniquely interpreted Discrete space structure by Dadic and Pisk Spacetime as an unlabeled graph whose topological structure entirely characterizes the graph Spatial points are related to vertices Operators define the creation or annihilation of lines which develop into a Fock space 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 Pregeometric graph by Wilson 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 the metrical foundation of space time Number theory pregeometry by Volovich Spacetime as a non Archimedean geometry over a field of rational numbers and a finite Galois field where rational numbers themselves undergo quantum fluctuations Causal sets by Bombelli Lee Meyer and Sorkin All of spacetime at very small scales is a causal set consisting of locally finite set of elements with a partial order 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 causal set 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 Random graphs by Antonsen Spacetime is described by dynamical graphs 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 Bootstrap universe by Cahill and Klinger An iterative map composed of monads and the relations between them becomes a tree graph 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 Axiomatic pregeometry by Perez Bergliaffa Romero and Vucetich An assortment of ontological presuppositions describes spacetime a result of relations between objectively existing entities From presuppositions emerges the topology and metric of Minkowski spacetime Cellular networks by Requardt 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 a deeper external parameter clock time and the graphs lead to a natural metrical structure Simplicial quantum gravity by Lehto Nielsen and Ninomiya Spacetime is described as having a deeper pregeometric structure based on three dynamical variables vertices of an abstract simplicial complex 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 lattice quantization is formulated resulting in a quantum gravity description of spacetime Quantum automaton universe by Jaroszkiewicz and Eakins Event states elementary or entangled are provided topological relationships via tests Hermitian operators endowing the event states with evolution irreversible acquisition of information and a quantum arrow of time 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 Rational number spacetime by Horzela Kapuscik Kempczynski and Uzes 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 Further reading editSome additional or related pregeometry proposals are Akama Keiichi An Attempt at Pregeometry Gravity with Composite Metric 2 Requardt Mandred Roy Sisir Quantum Space Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces 3 Sidoni Lorenzo Horizon thermodynamics in pregeometry 4 References edit Meschini et al August 2006 Geometry pregeometry and beyond Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics 36 3 435 464 arXiv gr qc 0411053 Bibcode 2005SHPMP 36 435M doi 10 1016 j shpsb 2005 01 002 S2CID 55663184 Akama Keiichi 1978 An Attempt at Pregeometry Gravity with Composite Metric PDF Progress of Theoretical Physics 60 6 1900 1909 doi 10 1143 PTP 60 1900 Retrieved 30 October 2013 Requardt Mandred Roy Sisir 2001 Quantum Space Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces Classical and Quantum Gravity 18 15 3039 3057 arXiv gr qc 0011076 Bibcode 2001CQGra 18 3039R doi 10 1088 0264 9381 18 15 317 S2CID 14941099 Sidoni Lorenzo 2013 Horizon thermodynamics in pregeometry Journal of Physics Conference Series 410 012140 arXiv 1211 2731 doi 10 1088 1742 6596 410 1 012140 S2CID 118590032 Misner Thorne and Wheeler MTW Gravitation 1971 ISBN 978 0 7167 0344 0 44 4 Not geometry but pregeometry as the magic building material 44 5 Pregeometry as the calculus of prepositions Retrieved from https en wikipedia org w index php title Pregeometry physics amp oldid 1168433733, wikipedia, wiki, book, books, library,
