Pregeometry (physics)

Last updated

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

Some additional or related pregeometry proposals are:

Related Research Articles

<span class="mw-page-title-main">Dimension</span> Property of a mathematical space

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

<span class="mw-page-title-main">Differential geometry</span> Branch of mathematics dealing with functions and geometric structures on differentiable manifolds

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars.

Physical causality is a physical relationship between causes and effects. It is considered to be fundamental to all natural sciences and behavioural sciences, especially physics. Causality is also a topic studied from the perspectives of philosophy, statistics and logic. Causality means that an effect can not occur from a cause that is not in the back (past) light cone of that event. Similarly, a cause can not have an effect outside its front (future) light cone.

In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949, who discovered a solution to the equations of general relativity (GR) allowing CTCs known as the Gödel metric; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of quantum gravity which would replace GR, an idea which Stephen Hawking labeled the chronology protection conjecture. Others note that if every closed timelike curve in a given space-time passes through an event horizon, a property which can be called chronological censorship, then that space-time with event horizons excised would still be causally well behaved and an observer might not be able to detect the causal violation.

<span class="mw-page-title-main">Discrete geometry</span> Branch of geometry that studies combinatorial properties and constructive methods

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

<span class="mw-page-title-main">Spin foam</span> Topological structure used in a description of quantum gravity

In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop quantum gravity as a version of quantum foam.

Background independence is a condition in theoretical physics that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. In particular this means that it must be possible not to refer to a specific coordinate system—the theory must be coordinate-free. In addition, the different spacetime configurations should be obtained as different solutions of the underlying equations.

In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to the Planck length. At these distances, quantum mechanics has a profound effect on physical phenomena.

In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961.

<span class="mw-page-title-main">Causal dynamical triangulation</span> Hypothetical approach to quantum gravity with emergent spacetime

Causal dynamical triangulation, theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent.

Rafael Dolnick Sorkin is an American physicist. He is professor emeritus of physics at Syracuse University and the Perimeter Institute for Theoretical Physics, and a Fellow of the American Physical Society. He is best known as initiator and main proponent of the causal sets approach to quantum gravity.

In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold. It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is

<span class="mw-page-title-main">Causal sets</span> Approach to quantum gravity using discrete spacetime

The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

<span class="mw-page-title-main">Clique complex</span> Abstract simplicial complex describing a graphs cliques

Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques of an undirected graph.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

In theoretical physics, the problem of time is a conceptual conflict between general relativity and quantum mechanics in that quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as malleable and relative. This problem raises the question of what time really is in a physical sense and whether it is truly a real, distinct phenomenon. It also involves the related question of why time seems to flow in a single direction, despite the fact that no known physical laws at the microscopic level seem to require a single direction. For macroscopic systems the directionality of time is directly linked to first principles such as the second law of thermodynamics.

<span class="mw-page-title-main">Causal fermion systems</span> Candidate unified theory of physics

The theory of causal fermion systems is an approach to describe fundamental physics. It provides a unification of the weak, the strong and the electromagnetic forces with gravity at the level of classical field theory. Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory. Therefore, it is a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting spacetime manifold, the general concept is to derive spacetime as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when spacetime no longer has a manifold structure on the microscopic scale. As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity.

References

  1. 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.
  2. 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.
  3. 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.
  4. 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.