Simplifying the Fine-Structure Constant with a New Geometric Constant

Abstract

The fine-structure constant, $\alpha$, is traditionally expressed in terms of the elementary charge $e$, Planck’s reduced constant $\hbar$, the speed of light $c$, and the permittivity of free space $\epsilon_0$. This paper explores an alternative expression for $\alpha$ by introducing a new geometric constant in place of $\hbar$. This approach reveals a more streamlined form for $\alpha$ that may offer insights into the fundamental nature of electromagnetic interactions and the structure of spacetime.

This paper is written with the understanding that the meter is redefined to make hc = K = 2*10^-25 J m.  This would change every constant with m in it as a unit. 

1. Introduction

The fine-structure constant $\alpha$ is a dimensionless constant central to electromagnetism, quantum mechanics, and relativity, governing the strength of the electromagnetic interaction between elementary charged particles. Traditionally defined as:

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c},$$

this expression depends on multiple constants, each representing fundamental aspects of physical interactions. This paper proposes a reformulation of $\alpha$ using a newly defined geometric constant, simplifying its expression and potentially revealing new geometric insights.

2. Background and Motivation

2.1 The Standard Definition of the Fine-Structure Constant

In conventional physics, the fine-structure constant is given by:

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c},$$

where:

• $e$: Elementary charge,
• $\epsilon_0$: Vacuum permittivity,
• $\hbar$: Reduced Planck’s constant,
• $c$: Speed of light in a vacuum.

2.2 Introducing the New Geometric Constant

This work proposes an alternative formulation by replacing $\hbar$ with a new geometric constant that provides an equivalent value but offers a simpler expression for $\alpha$. We define the geometric constant as:

$$\hbar = \frac{2 \times 10^{-25}}{2 \pi c}.$$

By substituting this expression into the formula for $\alpha$, we aim to achieve a more direct relationship.

3. Derivation of the Simplified Fine-Structure Constant

Step 1: Substitute the New Geometric Constant

Starting with the standard formula for $\alpha$:

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c},$$

we substitute $\hbar$ with the new constant:

$$\hbar = \frac{2 \times 10^{-25}}{2 \pi c}.$$

Thus,

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \cdot \frac{2 \times 10^{-25}}{2 \pi c} \cdot c}.$$

Step 2: Simplify the Expression

Upon substitution, the expression for $\alpha$ becomes:

$$\alpha = \frac{e^2}{4 \epsilon_0 \cdot 10^{-25}}.$$

Step 3: Final Simplified Expression

After simplification, the fine-structure constant $\alpha$ is represented by:

$$\alpha = \frac{e^2}{4 \epsilon_0 \cdot 10^{-25}}.$$

This final form removes $\hbar$ and $c$ from the expression, revealing a streamlined relationship where $\alpha$ depends only on $e$, $\epsilon_0$, and a single geometric constant $10^{-25}$.

4. Implications of the Simplified Expression

4.1 Geometric Interpretation

This simplification suggests a fundamental geometric relationship underlying $\alpha$, reducing the complexity of constants traditionally associated with electromagnetic interactions. The dependency on a single geometric constant hints that $\alpha$ may be inherently tied to spacetime structure rather than requiring independent factors like $\hbar$ and $c$.

4.2 Potential Applications and Insights

By reducing $\alpha$ to a form based only on $e$, $\epsilon_0$, and a single geometric term, this reformulation could have implications in fields relying on quantum electrodynamics (QED), cosmology, and theoretical physics. It may open new pathways for re-examining fundamental forces through purely geometric and charge-based perspectives.

5. Conclusion

This paper presents a novel formulation for the fine-structure constant by incorporating a geometric constant in place of $\hbar$. The resulting expression for $\alpha$ simplifies its dependency on fundamental constants and emphasizes a direct, geometric relationship that could further our understanding of the fine-structure constant’s role in physics. Future research may explore whether similar simplifications apply to other fundamental constants, potentially revealing new structural insights into the nature of spacetime and interactions in the universe.