The Role of Fractional Dimension in Study Physics: A Two-Channel Representation with Geometric Memory

Abstract

In this study, we explore the field of physics through the lens of fractional dimensionality. We propose that space is not confined to integer dimensions alone, but can also be understood as a superposition of spaces that exist between these integer dimensions. The concept of fractional dimensional space arises from the idea that the space between integer dimensions is filled, which occurs through the application of a fractional derivative operator (the local part) that rotates the integer dimension to encompass all spaces between two integers. It examines how fractional dimensional frameworks can enhance our understanding of classical mechanics, particularly regarding the duality of memory versus no-memory behavior, or local versus non-local dynamics. In the lens of fractional dimension, motion in classical physics can be analyzed through two distinct solutions. When the fractional dimensional trajectory takes values of 1 or 2 corresponding to the first and second integer derivatives, it represents linear and accelerated systems, respectively, yielding trivial solutions via differentiation. However, if the fractional dimensional trajectory evolves as a linear function of time in fractional dimensional space or follows a nonlinear path, surprisingly it results in a non-trivial solution for linear and accelerated systems, respectively. This approach offers a broader framework for describing motion, extending to memory (non-local) effect beyond traditional local integer-order differentiation . Moreover, we propose that the coupling of space and time, commonly referred to as space-time, is better understood as space-dimension-time within this framework, where the dimension serves as an interconnecting platform.