A New Frontier in Dimensional Analysis: The Emergence of a Science of Unit Systems and Constants

James Rogers, SE Ohio, 24 Nov 2024 1504

Abstract:

In this paper, we propose a revolutionary approach to dimensional analysis, one that transcends traditional methods and introduces a novel framework for understanding physical constants. By analyzing the fundamental constants of nature through the lens of unit systems, we argue that these constants are not inherent properties of the universe, but rather emerge from the human-defined structures of measurement. This introspective analysis of unit systems, which we call Unit System Science, provides new insights into the nature of physical laws and their connection to the measurement frameworks we use. This new approach shifts the role of constants from being absolute quantities to manifestations of dimensional relationships, opening the door to a deeper understanding of both physical reality and the way we model it.

This framework is currently demonstrated in a python program that shows how Unit System Science can reframe and simplify constants and compares them to the current accepted values for the constants. The file is here:

https://github.com/BuckRogers1965/RedefineUnitsForPlancksConstant/blob/main/simplified_plancks_constants.py

1. Introduction

For centuries, scientists have relied on dimensional analysis to bridge the gap between units and the physical quantities they represent. In traditional dimensional analysis, constants like the gravitational constant $G$, Planck's constant $h$, and the permittivity of free space ${ϵ}_0$ are treated as universal values that govern the behavior of the physical world. However, these constants are often viewed as mysterious or arbitrary numbers, whose origins are not always fully understood.

This paper suggests that a deeper investigation of these constants, when placed within the context of the unit systems we use, reveals a new and profound insight: the physical constants are not inherently fundamental but are instead the products of the way we construct and scale our units. By formalizing this relationship, we present a new branch of science, which we term Unit System Science, that introspectively analyzes how our choice of units shapes the constants that define physical reality. This approach, we argue, has the potential to fundamentally alter our understanding of both the constants themselves and the physical laws they underpin.

2. The Traditional View of Physical Constants

In the conventional framework of physics, constants such as $G$, $h$, and ${ϵ}_0$ are viewed as intrinsic properties of the universe. These constants are treated as universal quantities that are independent of the unit systems we use to measure them. For example,  Planck's constant $h$ serves as the foundation for quantum mechanics, and G is foundational to all gravity formulas.

While these constants are crucial to the functioning of our physical theories, their precise origin remains somewhat obscure. They are often treated as given values in the equations that define physical laws, without necessarily explaining why these constants have the values they do. The existence of these constants in various equations suggests a deeper, more profound relationship, but this relationship is rarely explored at the level of unit systems themselves.

3. Unit Systems as Generators of Constants

We propose that instead of treating constants as inherent features of nature, we should view them as the emergent products of the relationships encoded in our unit systems. Constants such as $h$, $G$, and ${ϵ}_0$ arise from scaling factors that are defined by how we choose to measure length, mass, time, temperature, and other fundamental properties.

To illustrate this, consider the following relationships from our framework:

$$h = \alpha^3 \beta / c$$ $$G = \alpha^3 / \beta$$ $$k = \alpha^3 \beta / \gamma$$ $$\epsilon_0 = \delta^2 / (\alpha^3 \beta)$$

where $\alpha$, $\beta$, $\gamma$, and $\delta$ are the scaling factors for length, mass, temperature, and charge, respectively. These scaling factors are human-defined, and their interplay within the unit system gives rise to the values of the constants we observe in nature.

They have the values of :

alpha: float  = 1.53843945498419101549e-06 

beta: float   = 5.45551186133462110058e-08 

gamma: float  = 1.43877687750393716548e-02 

delta: float  = 1.32621132205611221308e-18 

An example of how this can simplify the formulas, Planck's charge constant reduced to:

q_P = δ √2

Thus, constants like $h$, $G$, and ${ϵ}_0$ are not immutable properties of the universe. They are manifestations of how we construct our system of units and reflect the relationships between different units of measurement. This insight leads to a broader conceptualization of dimensional analysis, which we now call Unit System Science.

4. Unit System Science: A New Branch of Science

Unit System Science is the study of how human-defined units—such as meters, kilograms, seconds, and amperes—affect the constants of nature that we use to describe the physical world. It is a discipline that explores the emergent properties of units and how these units define the fundamental laws of physics. This new field introduces the following key principles:

1. Constants as Emergent Properties: Constants like $h$, $G$, and ${ϵ}_0$ emerge from the scaling relationships between different units. They are not fundamental in the traditional sense but are the result of how we define our measurement systems.

2. Dimensional Relationships: The relationship between different physical quantities and their corresponding units is the foundation of physical constants. By formalizing these relationships, we gain a deeper understanding of the constants and their place in our physical theories.

3. Reevaluating the Role of Units: Unit System Science encourages a deeper appreciation of how our choices of units shape the very structure of physical laws. This includes understanding the implications of defining new units or adjusting existing ones.

4. A New Perspective on Physical Laws: By recognizing that constants emerge from unit systems, we can reevaluate the equations and physical laws that rely on them. This shift in perspective can lead to a deeper understanding of fundamental physics, potentially offering new insights into unification or the nature of physical reality.

5. Practical Implications and Future Directions

The implications of Unit System Science are far-reaching. This new approach has the potential to change how we view the constants of nature and how we design physical theories. A few potential avenues for future exploration include:

• Reinterpreting the Constants of Nature: If constants like $G$ and $h$ are not fundamental but are the result of our unit choices, we may be able to redefine them in ways that lead to new theoretical insights. For example, we might uncover hidden relationships between physical phenomena that were previously obscured by our unit system.

• Unit System Redesign: Unit System Science might inspire the creation of new unit systems that are more efficient or more reflective of the underlying nature of the universe. By understanding how the scaling of units influences physical constants, we can propose new systems that simplify or unify the description of physical laws.

• Unification of Physical Forces: This framework could potentially offer new paths toward the unification of fundamental forces by recognizing the underlying relationships between the units and constants that describe each force. For example, the strong force, gravity, and electromagnetism could be reinterpreted through the lens of Unit System Science.

6. Conclusion

Unit System Science offers a new frontier in dimensional analysis and provides a groundbreaking way of understanding the constants that define the laws of nature. By recognizing that these constants are not inherent properties of the universe but emerge from the human-defined structure of measurement systems, we open the door to a more profound understanding of physical reality. This introspective analysis of unit systems has the potential to reshape our conception of physics and could lead to new theoretical breakthroughs in the study of fundamental forces, constants, and the structure of the universe itself.

References:

• Feynman, R.P., Leighton, R.B., & Sands, M. (1963). The Feynman Lectures on Physics. Addison-Wesley.
• Planck, M. (1900). On the Theory of Radiation. Annalen der Physik.
• Einstein, A. (1905). On the Electrodynamics of Moving Bodies. Annalen der Physik.
  
  
  

Appendix A, a snapshot of the functions from the program.

    # Calculate Planck units using simplified formulas

    h_calc = (constants.alpha**3 * constants.beta) / constants.c

    h_bar_calc = (constants.alpha**3 * constants.beta) /(2 * np.pi * constants.c)

    G_calc = constants.alpha**3 / constants.beta

    length_calc = (constants.alpha**3) / ((2 * np.pi)**(1/2) * constants.c**2)

    time_calc = (constants.alpha**3) / ((2 * np.pi)**(1/2) * constants.c**3)

    mass_calc = constants.beta / (2 * np.pi)**(1/2)

    charge_calc = constants.delta *(2)**(1/2)

    temp_calc = (constants.c**2 * constants.gamma) / ((2 * np.pi)**(1/2) * constants.alpha**3)

    alphafine_calc = constants.e**2  / (2 *(constants.delta**2))

    angular_momentum_calc = h_calc / (2 * np.pi)

    energy_calc = ((constants.beta**2 * constants.c**4)/(2 * np.pi))**(1/2)

    force_calc = constants.beta * constants.c**4 / constants.alpha**3

    power_calc = constants.beta * constants.c**5 / constants.alpha**3

    density_calc = (constants.beta * constants.c**6 * 2 * np.pi)/(constants.alpha**9)

    area_calc = length_calc**2

    volume_calc = length_calc**3

    acceleration_calc = constants.c**4 / constants.alpha**3

    pressure_calc = (constants.beta * constants.c**8 *2*np.pi) / ( constants.alpha**9)

    boltzmann_calc = (constants.alpha**3 * constants.beta) / constants.gamma

    e0_calc = constants.delta**2 / (constants.alpha**3 * constants.beta) 

    Kb_calc=2*np.pi**5*((constants.alpha**3*constants.beta)/constants.gamma**4)*constants.c/15   

    alphafine_calc = constants.e**2  / (2 *(constants.delta**2))

    r_gas_calc =  constants.beta *constants.alpha**3  * constants.mol / (constants.gamma )

    RK_calc =   (constants.beta * constants.alpha**3) / (constants.e**2 * constants.c)