On Paravectors and Their Associated Algebras

Vaz, Jayme

doi:10.1007/s00006-019-0948-1

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…To depart from this scenario, we will use a different name, introduce the idea of weighted points, and resort to the concept of a paravector. A paravector is the sum of a scalar and a vector [9,12,17]. Although this is equivalent to introducing a new dimension, it is in fact more than this; we will identify the paravectors within the multivector structure of a Clifford algebra, in such a way that we are able to identify their subspace using the automorphisms of the Clifford algebra.…”

Section: Representation Of Pointsmentioning

confidence: 99%

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

Vaz¹,

Mann²

2019

Preprint

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We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra Cℓ 3,3 of the quadratic space R 3,3 . We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspective projection can be written in this Clifford algebra as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation. An important point is that the expression for the operations of reflection and rotation in Cℓ 3,3 preserve the subspaces that can be associated with the algebras Cℓ 3,0 and Cℓ 0,3 , so that reflection and rotation can be expressed in terms of Cℓ 3,0 or Cℓ 0,3 , as well-known. However, all other operations mix those subspaces in such a way that they need to be expressed in terms of the full Clifford algebra Cℓ 3,3 . An essential aspect of our formulation is the representation of points in terms of objects called paravectors. Paravectors have been used previously to represents points in terms of an algebra closely related to the Clifford algebra Cℓ 3,3 . We compare these different approaches.

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Section: Representation Of Pointsmentioning

confidence: 99%

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

Vaz¹,

Mann²

2019

Preprint

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On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space

Vaz

Mann

2020

Adv. Appl. Clifford Algebras

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From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Cafaro,

Bahreyni,

Rossetti

2024

Symmetry

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Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.

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On Paravectors and Their Associated Algebras

Cited by 3 publications

References 18 publications

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

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