How Nonassociative Geometry Describes a Discrete Spacetime

Nesterov, Alexander I.; Mata, H.

doi:10.3389/fphy.2019.00032

Cited by 6 publications

(8 citation statements)

References 82 publications

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“…One may see that this problem also appears from the fact that the equation ( 36) is actually well defined operation if and only if it vanishes on the annihilator of multiplication map Tm : T (2) G → G defined by (31) (36), we get l * g (r * h ) −1 µ g = ν h which in the lack of associativity is a contradiction with the composability condition (35).…”

Section: Cotangent Bundle Of a Quasiloopoidmentioning

confidence: 99%

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Nonassociative analogs of Lie groupoids

Grabowski,

Ravanpak

2021

Preprint

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We introduce nonassociative geometric objects generalising naturally Lie groupoids, called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids is canonically a smooth loopoid (which is nontrivial in case of loopoids) and this is untrue for the cotangent bundles. After providing a few natural constructions, we show how the Lie-like functor associates with these objects skew-algebroids and almost Lie algebroids, respectively, and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.

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Section: Cotangent Bundle Of a Quasiloopoidmentioning

confidence: 99%

“…Contrary to the case of smooth loops (e.g. [4,29,26,35,36,40]) , the literature in this subject oriented on 'nonassociative Lie groupoids' is not very extensive. Besides some aspects contained in the Sabinin's monograph [38], we can indicate our short introductory note [11].…”

Section: Introductionmentioning

confidence: 99%

Nonassociative analogs of Lie groupoids

Grabowski,

Ravanpak

2021

Preprint

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“…In [40][41][42][43][44][45][46], we proposed a new unified algebraic approach, based on nonassociative geometry, for describing both continuum and discrete spacetimes. In our model of spacetime, time is quantized, and a random/stochastic process governs the evolution of spacetime geometry.…”

Section: Introductionmentioning

confidence: 99%

“…In our model of spacetime, time is quantized, and a random/stochastic process governs the evolution of spacetime geometry. As a result, we obtained a partially ordered set of events with spacetime geometry encoded in the nonassociative structure of spacetime [46]. Among advanced models that propose discreteness, three are related to our work: causal sets [47][48][49], causal dynamical triangulations [32,[50][51][52][53][54][55][56] and complex quantum network manifolds (CQNMs) [39,57,58].…”

Section: Introductionmentioning

confidence: 99%

Spacetime as a Complex Network and the Cosmological Constant Problem

Nesterov

2023

Universe

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We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from “atoms” of spacetime and entangled states forming n-dimensional simplices (n=1,2,3). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that, for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the Universe’s topology. The “foamy” structure, analogous to Wheeler’s “spacetime foam”, significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the Universe.

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“…More recent works have produced notions of discrete spacetimes (see Refs. [9,10]) and consequent questions regarding how they produce our apparent continuum.…”

Section: Introductionmentioning

confidence: 99%

General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

Manighalam

Kon

2020

Quantum Inf Process

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Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

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How Nonassociative Geometry Describes a Discrete Spacetime

Cited by 6 publications

References 82 publications

Nonassociative analogs of Lie groupoids

Nonassociative analogs of Lie groupoids

Spacetime as a Complex Network and the Cosmological Constant Problem

General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

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