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28 pages, 847 KB

Open AccessArticle

The Standard Model Symmetry and Qubit Entanglement

by Jochen Szangolies

Entropy 2025, 27(6), 569; https://doi.org/10.3390/e27060569 - 27 May 2025

Viewed by 1960

Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza–Klein proposal aims to recover gauge symmetries by means of dimensional reduction in higher-dimensional gravitational theories. Integrating both of these, gravitational and gauge degrees of freedom in

3 + 1

dimensions may be obtained upon dimensional reduction in higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with

5 + 1

- and

9 + 1

-dimensional spacetimes, respectively, which are reduced to

3 + 1

dimensions upon singling out a preferred complex direction. Depending on the interpretation of the residual symmetry, either the Standard Model gauge group,

S U (3) \times S U (2) \times U (1) / Z_{6}

, or the symmetry of Minkowski spacetime together with the gauge symmetry of a right-handed ‘half-generation’ of fermions can be recovered. Thus, there seems to be a natural way to accommodate the chirality of the weak force in the given construction. This motivates a picture in which spacetime emerges from the area law contribution to the entanglement entropy, while gauge and matter degrees of freedom are obtained due to area-law-violating terms. Furthermore, we highlight the possibility of using this construction in quantum simulations of Standard Model fields. Full article

(This article belongs to the Special Issue Foundational Aspects of Gauge Field Theory)

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26 pages, 2722 KB

Open AccessArticle

Microsaccades, Drifts, Hopf Bundle and Neurogeometry

by Dmitri Alekseevsky

J. Imaging 2022, 8(3), 76; https://doi.org/10.3390/jimaging8030076 - 17 Mar 2022

Cited by 2 | Viewed by 3449

Abstract

The first part of the paper contains a short review of the image processing in early vision is static, when the eyes and the stimulus are stable, and in dynamics, when the eyes participate in fixation eye movements. In the second part, we give an interpretation of Donders’ and Listing’s law in terms of the Hopf fibration of the 3-sphere over the 2-sphere. In particular, it is shown that the configuration space of the eye ball (when the head is fixed) is the 2-dimensional hemisphere

S_{L}^{+}

, called Listing hemisphere, and saccades are described as geodesic segments of

S_{L}^{+}

with respect to the standard round metric. We study fixation eye movements (drift and microsaccades) in terms of this model and discuss the role of fixation eye movements in vision. A model of fixation eye movements is proposed that gives an explanation of presaccadic shift of receptive fields. Full article

(This article belongs to the Special Issue Mathematical Modeling of Human Vision and Its Application to Image Processing)

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22 pages, 4681 KB

Open AccessFeature PaperArticle

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

by José Ignacio Extreminana-Aldana, José Manuel Gutiérrez-Jiménez, Luis Javier Hernández-Paricio and María Teresa Rivas-Rodríguéz

Mathematics 2021, 9(16), 1914; https://doi.org/10.3390/math9161914 - 11 Aug 2021

Cited by 1 | Viewed by 2091

Abstract

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

32 pages, 461 KB

Open AccessArticle

The Role of Spin(9) in Octonionic Geometry

by Maurizio Parton and Paolo Piccinni

Axioms 2018, 7(4), 72; https://doi.org/10.3390/axioms7040072 - 12 Oct 2018

Cited by 4 | Viewed by 4631

Abstract

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical [...] Read more.

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical 8-form and its analogies with quaternionic geometry as well as the role of

Spin (9)

both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel

Spin (9)

manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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