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10 pages, 603 KiB

Open AccessArticle

Are Quantum-Classical Hybrids Compatible with Ontological Cellular Automata?

by Hans-Thomas Elze

Universe 2022, 8(4), 207; https://doi.org/10.3390/universe8040207 - 26 Mar 2022

Cited by 4 | Viewed by 1885

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of ‘classical’ two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a ‘classical’ chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI. Full article

(This article belongs to the Special Issue The Quantum & The Gravity)

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16 pages, 401 KiB

Open AccessArticle

Development of Task-Space Nonholonomic Motion Planning Algorithm Based on Lie-Algebraic Method

by Arkadiusz Mielczarek and Ignacy Dulęba

Appl. Sci. 2021, 11(21), 10245; https://doi.org/10.3390/app112110245 - 1 Nov 2021

Cited by 2 | Viewed by 1859

Abstract

In this paper, a Lie-algebraic nonholonomic motion planning technique, originally designed to work in a configuration space, was extended to plan a motion within a task-space resulting from an output function considered. In both planning spaces, a generalized Campbell–Baker–Hausdorff–Dynkin formula was utilized to transform a motion planning into an inverse kinematic task known for serial manipulators. A complete, general-purpose Lie-algebraic algorithm is provided for a local motion planning of nonholonomic systems with or without output functions. Similarities and differences in motion planning within configuration and task spaces were highlighted. It appears that motion planning in a task-space can simplify a planning task and also gives an opportunity to optimize a motion of nonholonomic systems. Unfortunately, in this planning there is no way to avoid working in a configuration space. The auxiliary objective of the paper is to verify, through simulations, an impact of initial parameters on the efficiency of the planning algorithm, and to provide some hints on how to set the parameters correctly. Full article

(This article belongs to the Special Issue Advances in Robot Motion and Control—In Memory of Professor Krzysztof Kozlowski)

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15 pages, 277 KiB

Open AccessArticle

Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

by Thomas Berry and Matt Visser

Physics 2021, 3(2), 352-366; https://doi.org/10.3390/physics3020024 - 13 May 2021

Cited by 3 | Viewed by 3362

Abstract

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary

4 \times 4

Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae. Full article

(This article belongs to the Section High Energy Physics)

19 pages, 402 KiB

Open AccessArticle

Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

by Daniel Condurache and Ioan-Adrian Ciureanu

Mathematics 2020, 8(7), 1185; https://doi.org/10.3390/math8071185 - 19 Jul 2020

Cited by 10 | Viewed by 5678

Abstract

The paper proposes, for the first time, a closed form of the Baker–Campbell–Hausdorff–Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements [...] Read more.

S E (3)

and the properties of its Lie algebra

s e (3)

are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group

S E (3)

and the Lie group of the orthogonal dual tensors. Full article

(This article belongs to the Special Issue New Formulations in the Applied Mechanics to Robotics)

11 pages, 841 KiB

Open AccessArticle

A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion

by Xiaomin Duan, Huafei Sun and Xinyu Zhao

Entropy 2019, 21(5), 531; https://doi.org/10.3390/e21050531 - 25 May 2019

Cited by 2 | Viewed by 3632

Abstract

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group [...] Read more.

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group structures of

S E (3)

reflects the characteristics of change-points. Once a change-point occurs, the distance between the current point and the Riemannian mean of its neighbor points should be a local maximum. A gradient descent algorithm is proposed to calculate the Riemannian mean. Using the Baker–Campbell–Hausdorff formula, the first-order approximation of the Riemannian mean is taken as the initial value of the iterative procedure. The performance of our method was evaluated by numerical examples and manipulator experiments. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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10 pages, 242 KiB

Open AccessFeature PaperArticle

Explicit Baker–Campbell–Hausdorff Expansions

by Alexander Van-Brunt and Matt Visser

Mathematics 2018, 6(8), 135; https://doi.org/10.3390/math6080135 - 8 Aug 2018

Cited by 15 | Viewed by 8132

Abstract

The Baker–Campbell–Hausdorff (BCH) expansion is a general purpose tool of use in many branches of mathematics and theoretical physics. Only in some special cases can the expansion be evaluated in closed form. In an earlier article we demonstrated that whenever [...] Read more.

[X, Y] = u X + v Y + c I

, BCH expansion reduces to the tractable closed-form expression

Z (X, Y) = \ln (e^{X} e^{Y}) = X + Y + f (u, v) [X, Y],

where

f (u, v) = f (v, u)

is explicitly given by the the function

f (u, v) = \frac{(u - v) e^{u + v} - (u e^{u} - v e^{v})}{u v (e^{u} - e^{v})} = \frac{(u - v) - (u e^{- v} - v e^{- u})}{u v (e^{- v} - e^{- u})} .

This result is much more general than those usually presented for either the Heisenberg commutator,

[P, Q] = - i ℏ I

, or the creation-destruction commutator,

[a, a^{†}] = I

. In the current article, we provide an explicit and pedagogical exposition and further generalize and extend this result, primarily by relaxing the input assumptions. Under suitable conditions, to be discussed more fully in the text, and taking

L_{A} B = [A, B]

as usual, we obtain the explicit result

\ln (e^{X} e^{Y}) = X + Y + \frac{I}{e^{- L_{X}} - e^{+ L_{Y}}} (\frac{I - e^{- L_{X}}}{L_{X}} + \frac{I - e^{+ L_{Y}}}{L_{Y}}) [X, Y] .

We then indicate some potential applications. Full article

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