Some Quantum Mechanical Properties of the Wolfram Model

Gorard, Jonathan

doi:10.25088/complexsystems.29.2.537

Cited by 25 publications

(37 citation statements)

References 54 publications

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“…The Wolfram model [32] [31] [15] [14] is an explicitly computational framework based upon abstract rewriting that attempts to formalize the fundamental mathematical structures underlying space, time and matter. The model seeks to capture the ways in which simple abstract rewriting rules may be composed, so as to yield more complex structures (suggestively titled "universes") that admit certain emergent "laws of physics".…”

Section: Introductionmentioning

confidence: 99%

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Arsiwalla¹,

Gorard²,

Elshatlawy³

2021

Preprint

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We investigate the algebraic and compositional properties of multiway (non-deterministic) abstract rewriting systems, which are the archetypical structures underlying the formalism of the so-called Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where these homotopic maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck's homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of multiway rewriting systems may potentially be used for making formal connections to homotopy spaces upon which models of physics can be instantiated.

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Section: Introductionmentioning

confidence: 99%

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Arsiwalla¹,

Gorard²,

Elshatlawy³

2021

Preprint

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“…On the other hand, the Wolfram model [6] [7] is an intrinsically discrete spacetime formalism based on hypergraph transformation dynamics [8] [9], in which a causal graph representing the conformal structure of spacetime is generated algorithmically by means of an abstract rewriting system defined over hypergraphs; much work has already been done in developing the underlying mathematical formulations of both general relativity and quantum mechanics in the context of the Wolfram model, and in investigating connections to related formalisms such as causal set theory [10] and categorical quantum mechanics [11]. However, one notable feature of this formalism for our present purposes is that the natural formulation of the Einstein field equations in Wolfram model systems is as a discrete Cauchy problem, in which the initial hypergraph defines the Cauchy initial data, and the hypergraph transformation rules define the time evolution dynamics; it is already known that one can use the Ollivier-Ricci constructions [12] of both scalar and sectional curvatures in Wolfram model hypergraphs so as to recover a discrete form of the Einstein-Hilbert action which subsequently reduces to the standard Benincasa-Dowker action over causal sets [13] for particular classes of rules (specifically those satisfying a weak ergodicity hypothesis, as well as an asymptotic dimension preservation condition).…”

Section: Introductionmentioning

confidence: 99%

Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution

Gorard¹

2021

Preprint

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Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which Cauchy initial data is specified on a three-dimensional spatial hypersurface, and then evolved forwards in time. The Wolfram model offers an inherently discrete formulation of the Einstein field equations as an a priori Cauchy problem, in which Cauchy initial data is specified on a single spatial hypergraph, and then evolved by means of hypergraph substitution rules, with the resulting causal network corresponding to the conformal structure of spacetime. This article introduces a new numerical general relativity code based upon the conformal and covariant Z4 (CCZ4) formulation with constraint-violation damping, with the option to reduce to the standard BSSN formalism if desired, with Cauchy data defined over hypergraphs; the code incorporates an unstructured generalization of the adaptive mesh refinement technique proposed by Berger and Colella, in which the topology of the hypergraph is refined or coarsened based upon local conformal curvature terms. We validate this code numerically against a variety of standard spacetimes, including Schwarzschild black holes, Kerr black holes, maximally extended Schwarzschild black holes, and binary black hole mergers (both rotating and non-rotating), and explicitly illustrate the relationship between the discrete hypergraph topology and the continuous Riemannian geometry that is being approximated. Finally, we compare the results produced by this code to the results obtained by means of pure Wolfram model evolution (without the underlying PDE system), using a hypergraph substitution rule that provably satisfies the Einstein field equations *

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“…In a similar vein, the Wolfram model [12] [13] (formally introduced in Section 2 of the present article) is a discrete spacetime formalism based on hypergraph transformation dynamics [14] [15], in which a causal set is effectively generated algorithmically via an abstract rewriting system defined over hypergraphs. In addition to defining a purely algorithmic dynamics for causal sets, the presence of the underlying hypergraph/rewriting structure (in addition to the usual causal structure) leads to multiway systems produced by Wolfram model evolution that exhibit an apparently far richer variety of mathematical features than one observes in conventional causal set dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…In much the same way as relativistic observers may be interpreted as foliating a causal network into discrete spacelike hypersurfaces, we now consider a quantum mechanical analog in which "observers" foliate a multiway evolution graph into what we shall term "branchlike hypersurfaces" (which we shall represent as "branchial graphs" -analogous to the discrete Cauchy surfaces obtained by foliating a causal network), as described in [15] and [65]:…”

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confidence: 99%

“…This formally justifies our interpretation of each branchial graph as being a subspace of a finite-dimensional Hilbert space, and leads, moreover, to a rigorous derivation of the limiting metric on branchlike hypersurfaces. Specifically, just as the points on discrete spacelike hypersurfaces can be interpreted as points on a continuous Riemannian manifold (with the combinatorial distance metric converging to a smooth Riemannian metric), points on branchlike hypersurfaces can be shown to correspond to points in a continuous projective Hilbert space, with the combinatorial distance on branchial graphs converging to a smooth Fubini-Study metric on CP n [15][65]:…”

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confidence: 99%

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Algorithmic Causal Sets and the Wolfram Model

Gorard

2020

Preprint

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The formal relationship between two differing approaches to the description of spacetime as an intrinsically discrete mathematical structure, namely causal set theory and the Wolfram model, is studied, and it is demonstrated that the hypergraph rewriting approach of the Wolfram model can effectively be interpreted as providing an underlying algorithmic dynamics for causal set evolution. We show how causal invariance of the hypergraph rewriting system can be used to infer conformal invariance of the induced causal partial order, in a manner that is provably compatible with the measure-theoretic arguments of Bombelli, Henson and Sorkin. We then illustrate how many of the local dimension estimation algorithms developed in the context of the Wolfram model may be reformulated as generalizations of the midpoint scaling estimator on causal sets, and are compatible with the generalized Myrheim-Meyer estimators, as well as exploring how the presence of the underlying hypergraph structure yields a significantly more robust technique for estimating spacelike distances when compared against several standard distance and predistance estimator functions in causal set theory. We finally demonstrate how the Benincasa-Dowker action on causal sets can be recovered as a special case of the discrete Einstein-Hilbert action over Wolfram model systems (with ergodicity assumptions in the hypergraph replaced by Poisson distribution assumptions in the causal set), and also how both classical and quantum sequential growth dynamics can be recovered as special cases of Wolfram model multiway evolution with an appropriate choice of discrete measure.

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Some Quantum Mechanical Properties of the Wolfram Model

Cited by 25 publications

References 54 publications

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution

Algorithmic Causal Sets and the Wolfram Model

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