Geometrical model of massive spinning particle in four-dimensional Minkowski space

Kaparulin, D. S.; Lyakhovich, Alex; Retuntsev, I. A.

doi:10.1088/1742-6596/1337/1/012005

Cited by 3 publications

(6 citation statements)

References 26 publications

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“…We have associated the timelike spinning particle trajectories as curves on parabolic cylinders. We have shown that these curves are solutions to the fourth-order differential equation (28). Equations ( 36), (40) determine the particle momentum and total angular momentum in terms of derivatives of trajectory.…”

Section: World Paths On World Sheetsmentioning

confidence: 94%

“…Let us now find explicit solution for α and representation for resultant (28). Introduce special notation for combinations of derivatives of curvature and torsion,…”

Section: World Paths On World Sheetsmentioning

confidence: 99%

“…Equation (28) has two obvious gauge symmetries: reparametrization and translations along the cylinder axis,…”

Section: World Paths On World Sheetsmentioning

confidence: 99%

“…In d = 4, the same problem has been solved in [27]. An alternative derivation of equations of motion for cylindrical curves has been given in [28]. In all the cases, the differential equations involve invariants of trajectory up to the fourth order in derivatives in a very complicated combination, and they are known only in implicit form.…”

Section: Introductionmentioning

confidence: 99%

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On the world sheet of continuous helicity particle

Kaparulin,

Lyakhovich,

Retuntsev

2021

Preprint

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We consider the class of spinning particle theories, whose quantization corresponds to the continuous helicity representation of the Poincare group. The classical trajectories of the particle are shown to lie on the parabolic cylinder with a lightlike axis irrespectively to any specifics of the model. The space-time position of the cylinder is determined by the values of momentum and total angular momentum. The value of helicity determines the focal distance of parabolic cylinder. Assuming that all the world lines lying on one and the same cylinder are connected by gauge transformations, we derive the geometrical equations of motion for the particle. The timelike world paths are shown to be solutions to a single relation involving the invariants of trajectory up to fourth order in derivatives. Geometrical equation of motion is non-Lagragian, but it admits equivalent variational principle in the extended set of dynamical variables. The lightlike paths are also admissible on the cylinder, but they do not represent the classical trajectories of this spinning particle. The classical trajectories of massless particle (with zero helicity) are shown to lie on hyperplanes, whose spacetime position depends on momentum and total angular momentum.

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Section: World Paths On World Sheetsmentioning

confidence: 94%

“…Let us now find explicit solution for α and representation for resultant (28). Introduce special notation for combinations of derivatives of curvature and torsion,…”

Section: World Paths On World Sheetsmentioning

confidence: 99%

“…Equation (28) has two obvious gauge symmetries: reparametrization and translations along the cylinder axis,…”

Section: World Paths On World Sheetsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the world sheet of continuous helicity particle

Kaparulin,

Lyakhovich,

Retuntsev

2021

Preprint

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“…This substructure is of indefinite physical character within a model for fundamental spin; as such, we shall pursue an alternative model. Beyond Clifford algebras, there are models which seek to avoid the 'internal space' common to models of spin: Savasta et al [30,31] associate spin not with Lorentz symmetry but the ( )  SO 4, symmetry present between three-velocities and rates of change of proper time in Minkowski space; Kaparulin et al [32] relate spin to geometric qualities of a particle worldline, also in Minkowski space; and Bühler [33] connects spin to polarisations of a wave in Euclidean three-space via the Lie group ( )  SL 2, . However, none of these models align with our stated goals of directly utilising only the ( )  SO 3, symmetry of real Euclidean three-space.…”

Section: Algebraic Theories In Physicsmentioning

confidence: 99%

An algebraic theory of non-relativistic spin

Bradshaw

2024

Phys. Scr.

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In this paper we present a new, elementary derivation of non-relativistic spin using exclusively real algebraic methods. To do this, we formulate a novel method to decompose the domain of a real endomorphism according to its algebraic properties. We reveal non-commutative multipole tensors as the primary physically meaningful observables of spin, and indicate that spin is fundamentally geometric in nature. In so doing, we demonstrate that neither dynamics nor complex numbers are essential to the fundamental description of spin.

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A Relationship Between Spin and Geometry

Bradshaw

2024

Adv. Appl. Clifford Algebras

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In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.

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Geometrical model of massive spinning particle in four-dimensional Minkowski space

Cited by 3 publications

References 26 publications

On the world sheet of continuous helicity particle

On the world sheet of continuous helicity particle

An algebraic theory of non-relativistic spin

A Relationship Between Spin and Geometry

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