Browsing by Author "Pasechnik, Roman"

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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.