Possible quantum kinematics. II. Nonminimal case

Gromov, Nikolay

doi:10.1063/1.3460841

Cited by 8 publications

(6 citation statements)

References 12 publications

(15 reference statements)

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“…A more challenging situation occurs if we have a degenerate metric and cycles are represented by parabolas [32] and respective FLT are based on dual numbers [4]. Besides theoretical interest such spaces are meaningful physical models [14,15,32]. The differential geometry loses its ground in the degenerate case and non-commutative/non-local effects appear [20, § 7.2].…”

Section: Discussion and Generalisationsmentioning

confidence: 99%

“…The differential geometry loses its ground in the degenerate case and non-commutative/non-local effects appear [20, § 7.2]. The presence of zero divisors among dual numbers prompts a projective approach [5; 20, § 4.5] to the cross ratio of four dual numbers, which replaces (14). Variational methods do not produce FLT-invariant family of geodesics in the degenerate case, instead geometry of cycles need to be employed [18].…”

Section: Discussion and Generalisationsmentioning

confidence: 99%

“…It brings us back to the initial equation ( 1) and is not possible for straight lines; or (ii) C, C = ±1 which was suggested in [17, § 4.2] and is useful, say, for tangency but is not possible for points. Recall, that the projective ambiguity is elegantly balanced in the cross ratio of four points: (14) (z 1 , z 2 ; z 3 ,…”

Section: Cycles Cross Ratiomentioning

confidence: 99%

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Cycles Cross Ratio: an Invitation

Kisil¹

2021

Preprint

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Section: Discussion and Generalisationsmentioning

confidence: 99%

Section: Discussion and Generalisationsmentioning

confidence: 99%

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Cycles Cross Ratio: an Invitation

Kisil¹

2021

Preprint

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“…It is common [15], [13], [55], [30], [29] to consider mainly Clifford algebras Cℓ(n, =)Cℓ(n, 0, 0,) of the Euclidean space or the algebra Cℓ(p, q, =)Cℓ(p, q, 0,) of the pseudo-Euclidean (Minkowski) spaces. However, Clifford algebras Cℓ(p, q, r,), r > 0 with nilpotent generators e 2 i = 0 correspond to interesting geometry [41], [35], [61], [52] and physics [25], [23], [24], [42], [43], [47] as well. Yet, the geometry with idempotent units in spaces with dimensionality n > 2 is still not sufficiently elaborated.…”

Section: Möbius-lie Geometry and The Cycle Librarymentioning

confidence: 99%

An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

Kisil

2019

ПМГЦ

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We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. ``to be orthogonal'', ``to be tangent'', etc.), as new objects in an extended M\"obius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of objects, e.g. loxodromes or continued fractions. The paper describes a method, which reduces a collection of conformally in\-vari\-ant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. To show its usefulness, the method is implemented as a {\CPP} library. It operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive {\Python} wrapper of the library is provided as well.

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“…The family P τ,t consists of the eigenvectors of the 4 × 4 matrix from (28) with suitably substituted entries. There is (up to a factor) exactly one τ -isotropic form in P τ,t , namely 1 τ 1 1 .…”

Section: 4mentioning

confidence: 99%

Möbius--Lie Geometry and Its Extension

Kisil¹

2019

GIQ

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These lectures review the classical Möbius-Lie geometry and recent work on its extension. The latter considers ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Möbius-Lie geometry. It is shown on examples, that such ensembles of cycles naturally parameterise many other conformally-invariant families of objects, two examples-the Poincaré extension and continued fractions are considered in detail. Further examples, e.g. loxodromes, wave fronts and integrable systems, are published elsewhere.The extended Möbius-Lie geometry is efficient due to a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. The algorithmic nature of the method allows to implement it as a C++ library, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two-and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well.

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Possible quantum kinematics. II. Nonminimal case

Cited by 8 publications

References 12 publications

Cycles Cross Ratio: an Invitation

Cycles Cross Ratio: an Invitation

An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

Möbius--Lie Geometry and Its Extension

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