Browsing by Author "Jorgensen, Palle E. T."

Now showing 1 - 6 of 6

Extending Dirac and Faddeev-Jackiw Formalisms To Fractal First Α-Order Lagrangian Systems
(Springer Basel Ag, 2025) Golmankhaneh, Alireza Khalili; Sevli, Hamdullah; Tavares, Dina; Jorgensen, Palle E. T.
This paper presents the foundational concepts of fractal calculus before generalizing the Dirac Constraint Formalism and the Faddeev-Jackiw Formalism for first alpha-order Lagrangian systems in fractal spaces with non-integer dimensions. We provide a detailed analysis of the generalization process, highlighting the theoretical framework and key results, including the extended structure of the constraint systems in these Lagrangian formulations. Specific examples are discussed to demonstrate the practical application of the generalized formalism and to validate the consistency of our results. Moreover, graphical visualizations are included to enhance clarity, offering a visual interpretation of the findings and illustrating the relationship between the theory and its real-world implications.
Formulation and Quantization of Field Equations on Fractal Space-Time
(Springer/plenum Publishers, 2025) Golmankhaneh, Alireza Khalili; Pasechnik, Roman; Jorgensen, Palle E. T.; Li, Shuming
This paper explores the framework of fractal calculus and its application to classical and quantum field theories. We begin with a brief overview of the fundamental concepts of fractal calculus. Building on this foundation, we introduce the formulation of the classical scalar field within a fractal space. The study is then extended to the quantization of the fractal field, where we examine how fractal geometry influences the quantization process. As a key example, we consider the fractal version of the Klein-Gordon equation and analyze how the fractal dimension affects the behavior of the field. Graphical representations are provided to illustrate the impact of fractal dimensions on the solutions. The paper concludes with a summary of the results and their potential implications for future research in fractal field theory.
Fractal Calculus of Variations for Problems With Constraints
(World Scientific Publ Co Pte Ltd, 2025) Golmankhaneh, Alireza Khalili; Cattani, Carlo; Pasechnik, Roman; Furuichi, Shigeru; Jorgensen, Palle E. T.
In this paper, we present a summary of fractal calculus and propose the use of Lagrange multipliers for both fractal calculus and fractal variational calculus with constraints. We examine the application of these methods across various branches of physics. By employing fractal variational calculus with constraints, we derive fundamental equations such as the fractal mechanical wave equation, the fractal Schr & ouml;dinger equation in quantum mechanics, Maxwell's equations in fractal electromagnetism, and the Lagrange equation for constraints in fractal classical mechanics. Several examples are provided to illustrate these concepts in detail.
Fractal Frenet Equations for Fractal Curves: A Fractal Calculus Approach
(Springer int Publ Ag, 2025) Golmankhaneh, Alireza Khalili; Jorgensen, Palle E. T.; Prodanov, Dimiter
The formulation of Fractal Frenet equations, which are differential equations intended to characterize the geometric behavior of vector fields along fractal curves, is presented in this study. It offers a framework for calculating the length of such irregular curves by introducing a fractal analogue of arc length. The notion of a fractal unit tangent vector, which characterizes the local direction of the curve, and the fractal curvature vector, which depicts the bending behavior at each point, are two examples of fundamental geometric ideas that are extended to the fractal environment. Furthermore, the concept of fractal torsion is established to describe the three-dimensional spatial twisting of fractal curves.
Fractal Green Function Theory
(Elsevier, 2026) Golmankhaneh, Alireza Khalili; Cattani, Carlo; Kalita, Hemanta; Furuichi, Shigeru; Jorgensen, Palle E. T.
This paper provides a comprehensive study of fractal calculus and its application to differential equations within fractal spaces. It begins with a review of fractal calculus, covering fundamental definitions and measures related to fractal sets. The necessary preliminaries for understanding fractal Green's functions are introduced, laying the groundwork for further exploration. We develop the fractal Green's function for inhomogeneous fractal differential equations and extend this to the fractal Helmholtz equation. The application of the fractal Green's function to the Schr & ouml;dinger equation is also investigated, focusing on the fractal Schr & ouml;dinger-type differential equation with a fractal mesonic potential. Additionally, the scattering amplitude is derived within the fractal Born approximation, offering insights into scattering phenomena in fractal spaces. The findings highlight the significant impact of fractal geometry on classical and quantum mechanics and present new methods for addressing problems in fractal environments.
Fractal Quantum Nambu Mechanics
(Springer, 2025) Khalili Golmankhaneh, Alireza; Pasechnik, Roman; Jorgensen, Palle E. T.; Li, Shuming
This paper develops a comprehensive framework for the extension of classical and quantum mechanics to fractal settings. We begin by summarizing the classical formulation of Fractal Nambu Mechanics and then introduce its quantization. The Fractal Hamilton-Jacobi Theory is established to describe dynamical systems evolving over fractal time and space, followed by a fractal generalization of the quantum Hamilton-Jacobi framework. We further formulate the Fractal Nambu-Hamilton-Jacobi Theory and propose its quantum counterpart-the Quantum Fractal Nambu-Hamilton-Jacobi Theory. These constructions demonstrate how the structure of Nambu mechanics, when combined with local fractal calculus, can provide new insights into systems with multiple invariants and non-smooth geometric evolution.