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

This paper proposes a fractional dimensional framework for physics that utilizes fractional derivative operators to model space as a superposition between integer dimensions, thereby offering a two-channel representation of classical motion that unifies local and non-local (memory) dynamics through a "space-dimension-time" coupling.

Original authors: Ali Dorostkar

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ali Dorostkar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a car drive down a straight road. In the world of classical physics (the kind taught in high school), the car's path is simple: it's either moving at a steady speed, or it's speeding up. We describe this using whole numbers: 1st derivative for speed, 2nd derivative for acceleration. It's like counting steps: one step, two steps, three steps.

This paper, written by Ali Dorostkar, suggests that the universe is actually a bit more like a dimmer switch than a simple on/off light. It proposes that between the "whole number" steps of reality, there is a continuous, fuzzy space where things can exist in "fractional" states.

Here is the core idea broken down into simple analogies:

1. The "Two-Channel" Radio

The author suggests that every movement in the universe is actually a broadcast on two radio channels at the same time:

Channel A (The Local Channel): This is the "now." It's what we see with our eyes. It's the car moving right here, right now. It has no memory of the past.
Channel B (The Memory Channel): This is the "history." It's the car remembering where it was a second ago, or a minute ago. It's the "ghost" of the past influencing the present.

In standard physics, we usually only listen to Channel A. This paper argues that Channel B is always there, but it's hidden. The "fractional dimension" is the knob that controls how much of Channel B we are tuning into.

2. The "Rotating Compass" Analogy

Think of a compass.

North (0 degrees) is a standard integer dimension (like a straight line).
East (90 degrees) is another integer dimension (like a flat plane).

In this paper, the author says reality isn't just North or East. It can be North-North-East. The "fractional dimension" is the angle of that compass needle.

When the needle points exactly North (Angle = 0), the physics looks "local" (no memory).
When the needle points North-North-East (Angle = 45 degrees), the physics starts to "remember" the past. The movement isn't just about where you are now, but where you were.

The paper uses a mathematical tool called a Fractional Derivative to describe this rotation. It's like a magic lens that lets you see the "in-between" spaces between the standard dimensions.

3. The "Slippery Slope" of Motion

The paper does something surprising: it takes simple, boring motions (like a car driving at a constant speed) and shows they can be described in a complex, "memory-filled" way.

The Standard View: A car drives at 60 mph. Simple.
The Paper's View: That same car driving at 60 mph is actually the result of a complex dance between the "now" and the "past." The "memory" of the car is constantly shifting its internal "dimension" (the angle of the compass) to keep the speed steady.

The author found a specific pattern: for a car to move at a constant speed, its "memory dimension" has to change in a very specific, linear way over time. It's as if the car is constantly adjusting its internal "memory dial" to keep the outside world looking smooth and simple.

4. The "Euler Number" Surprise

One of the most fascinating findings in the paper is a mathematical coincidence. When the author calculated how this "memory dial" changes for both constant speed and constant acceleration, the numbers kept pointing to $e$ (Euler's number, roughly 2.718).

Think of $e$ as the "heartbeat" of growth and decay in nature. The paper suggests that this number isn't just a math trick; it's the fundamental "slope" of how memory works in the geometry of space-time. It's like finding that every time you roll a ball down a hill, it follows a path dictated by the same secret code.

5. Why Does This Matter? (The "Why")

You might ask, "If the car looks the same, why do we need this?"

It explains the "In-Between": Standard physics struggles with things that are neither fully solid nor fully fluid, or systems that have "long memories" (like rubber bands or the stock market). This framework gives a geometric home to those "in-between" states.
It unifies the rules: The paper suggests that the same rules that govern a swinging pendulum (Quantum Mechanics) and a falling apple (Classical Mechanics) might just be different settings on the same "memory dial."
It redefines "Time": Instead of time being a straight line where the past is gone, this view suggests time is a "thick" layer. The present moment is actually a blend of the immediate now and a fading echo of the past.

The Big Picture

Imagine the universe as a piece of music.

Old Physics hears only the melody (the notes being played right now).
This Paper hears the melody plus the echo in the concert hall.

The author is saying that the "echo" (the memory) isn't just noise; it's a fundamental part of the music. By turning the "fractional dimension" knob, we can understand how the echo shapes the melody, allowing us to see the hidden geometry of how things move, remember, and change.

In short: The universe isn't just a series of snapshots (frames of a movie). It's a continuous film where the past is still slightly visible in the present, and this paper provides the mathematical map to navigate that blurry, beautiful space between the frames.

1. Problem Statement

The paper addresses fundamental limitations in classical physics and the current application of fractional calculus:

Limitations of Classical Physics: Newtonian mechanics assumes integer-dimensional space and local-in-time dynamics, failing to inherently account for memory effects or non-locality without ad-hoc additions.
Limitations of Current Fractional Models: While fractional calculus is widely used to model memory (e.g., in viscoelasticity), the fractional order ( $\alpha$ ) is typically treated as a fixed phenomenological parameter tuned to data. It lacks a direct geometric or dynamical interpretation and is not derived from first principles.
The Gap: There is no unified framework where the fractional order is treated as a dynamic, time-dependent geometric descriptor of motion that emerges naturally from kinematic consistency, rather than an external fitting parameter.

2. Methodology

The author proposes a Two-Channel Kinematic Framework based on fractional dimensional space. The core methodology involves:

Geometric Interpretation of Fractional Dimensions: Space is viewed not just as integer dimensions ( $n$ ) but as a continuous spectrum including fractional dimensions ( $n < \alpha < n+1$ ). Basis vectors in fractional space are conceptualized as rotations of integer basis vectors, mathematically realized via fractional derivative (FDr) operators.
Two-Channel Decomposition: Any physical trajectory $x(t)$ $x (t)$ is decomposed into a convex superposition of two channels:
1. Local Channel: Governed by integer-order calculus (memoryless, standard Newtonian).
2. Nonlocal (Memory) Channel: Governed by variable-order fractional operators ( $\alpha(t)$ ).
  $x(t) = C_1 x_{loc}(t) + C_2 x_{mem}(t)$
  where $C_1 + C_2 = 1$ .
Variable-Order Trajectory ( $\alpha(t)$ ): Instead of a fixed $\alpha$ , the fractional order is promoted to a time-dependent trajectory $\alpha(t)$ . This trajectory is not defined by an independent equation of motion but is implicitly determined by kinematic consistency conditions (i.e., the fractional operator must reproduce the observed physical velocity or acceleration).
Normalized Operators: The study utilizes normalized fractional derivatives to ensure dimensional consistency, allowing the fractional operator to act as a "memory slope."

3. Key Contributions

A. Theoretical Framework: Geometric Memory

Axiomatic Foundation: The paper establishes axioms for physics in fractional dimensions, including the quantization of dimensions based on photon properties and the observer-dependent nature of dimensionality.
Space-Dimension-Time (SDT): A new conceptual model where space and time are coupled through a "dimensional platform," suggesting that dimension itself is a dynamic variable.
Two-Channel Schrödinger Equation: The framework is extended to quantum mechanics, proposing a unified evolution equation where the time derivative is a convex combination of a local derivative and a variable-order fractional operator, linking the path integral (sum over histories) to geometric memory.

B. Kinematic Results for Classical Motion

The paper derives specific constraints on the order trajectory $\alpha(t)$ for standard Newtonian motions:

Uniform Motion: For a body moving at constant velocity, the memory slope must be constant. The non-trivial solution for the order trajectory $\alpha_v(t)$ $α_{v} (t)$ is found to be a linear function of time with a universal asymptotic slope related to Euler's number $e$ $e$ .
- Result: $\alpha_v(t) \approx -e \cdot t + \text{const}$ (asymptotically).
Uniform Acceleration: For constant acceleration, the memory slope must be affine (linear) in time. The order trajectory $\alpha_a(t)$ $α_{a} (t)$ exhibits a linear trend with a universal slope of $-e$ $- e$ in the asymptotic limit.
- Result: The constant $e$ emerges as a structural invariant of the Gamma-function-based order equation, not a dynamical constant.
Harmonic Oscillator: The framework demonstrates that sinusoidal motion can be sustained by a time-dependent fractional order without altering the restoring force, providing an alternative geometric representation of oscillatory dynamics.

C. Applications to Other Fields

Electromagnetism: The wave equation and Doppler effect are re-derived. The Doppler shift is interpreted as a linear trajectory in fractional dimension, while gravitational redshift is linked to a quadratic trajectory in fractional dimension.
Viscoelasticity: The paper reframes viscoelastic memory not just as a dynamical consequence of microstructure but as a structural property of temporal response, where the present state structurally contains the past via the organization of time.

4. Key Results and Findings

Duality of Representation: A single observable Newtonian trajectory can be represented in two ways:
1. As a purely local, memoryless system ( $\alpha = 1$ ).
2. As a nonlocal system with memory, where the fractional order $\alpha(t)$ evolves dynamically to satisfy kinematic constraints.
Universal Asymptotic Slope: Numerical and asymptotic analysis reveals that for both uniform motion and uniform acceleration, the variable order $\alpha(t)$ tends toward a linear function with a slope of $-e$ (Euler's number). This suggests a deep spectral structure in fractional-dimensional kinematics.
Memory as Geometry: Acceleration is geometrically interpreted as the curvature of the memory channel. Even if the physical trajectory is purely Newtonian, the internal "memory channel" possesses non-trivial structure.
Extension to Quantum Mechanics: The two-channel approach successfully maps the Schrödinger equation, suggesting that quantum non-locality (path integrals) can be viewed as a geometric memory effect encoded in a variable-order time operator.

5. Significance and Future Directions

Paradigm Shift: The paper moves fractional calculus from a phenomenological tool (fitting data) to a geometric representation theory. It posits that memory is an intrinsic geometric feature of motion, not an external force.
Unification: It offers a unified language to describe classical mechanics, wave propagation, and quantum evolution through the lens of variable-order fractional dimensions.
Open Problems:
- The current formulation derives $\alpha(t)$ from kinematic constraints. A major future goal is to derive $\alpha(t)$ from first-principles physical mechanisms (e.g., via a variational principle or microscopic dynamics).
- The paper proposes an extended action functional where $\alpha(t)$ is a dynamical field, opening the door to new theories of "memory-geometry" interactions.
- Future work aims to apply this to anomalous diffusion, effective mass theories, and the quantization of dimension.

Conclusion

This study presents a novel geometric memory framework where the fractional order is a dynamic variable representing the "memory slope" of a system. By decomposing motion into local and nonlocal channels, the author demonstrates that classical Newtonian trajectories can be exactly reproduced by variable-order fractional operators. The discovery of the universal slope $-e$ in the order trajectory suggests a fundamental link between the Gamma function's analytic structure and the geometry of physical motion, offering a new perspective on the nature of time, memory, and dimensionality in physics.