Natural Unit Scaling Factors Encoded in Fundamental Constants

 J. Rogers, 09 Mar 2025, 1537

Abstract
Fundamental constants like the speed of light ($c$), Planck’s constant ($h$), and Boltzmann’s constant ($k$) are traditionally viewed as intrinsic properties of the universe. However, this paper argues that these constants are not fundamental but rather encoding mechanisms for natural unit scaling factors. By decomposing $h$ and $k$ into their constituent scaling factors—K_Hz, Hz_kg, and kg_J—we reveal that these constants arise from the mismatch between human-defined units and the natural scale of frequency. This perspective provides a clearer, more unified understanding of the relationships between mass, frequency, temperature, and energy, and demonstrates that these quantities are fundamentally equivalent, differing only in the units we use to describe them.

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1. Introduction

For over a century, constants like $c$, $h$, and $k$ have been treated as fundamental properties of the universe, bridging seemingly distinct domains of physics such as relativity, quantum mechanics, and thermodynamics. However, this paper proposes a shift in perspective: these constants are not fundamental but rather unit conversion factors that arise because our human-defined units (e.g., kilograms, meters, kelvins) are mismatched with the natural scale of frequency. By decomposing $h$ and $k$ into their constituent natural unit scaling factors, we reveal a deeper unity in physics and show that mass, frequency, temperature, and energy are fundamentally equivalent, differing only in the units we use to describe them.

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2. Natural Unit Scaling Factors

In the natural framework of the universe, physical quantities like mass, frequency, and temperature are fundamentally equivalent, scaling 1:1 with each other. However, our human-defined units (e.g., kilograms, meters, seconds, kelvins) are not aligned with this natural scaling, necessitating the use of constants like $c$, $h$, and $k$ to reconcile the differences. These constants can be decomposed into natural unit scaling factors that explicitly account for the mismatched scaling of human-defined units.

2.1. Speed of Light ($c$)

The speed of light encodes the mass-to-energy conversion factor (kg_J):

$${\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}<span>=</span>\text{<span>kg\_J</span>}$$

This factor arises because our mass units (kilograms) are mismatched with the natural scale of energy.

2.2. Planck’s Constant ($h$)

Planck’s constant encodes the frequency-to-energy conversion factor (Hz_J):

$$\mathrm{<span>h</span>}<span>=</span>\text{<span>Hz\_J</span>}$$

This factor arises because our energy units (Joules) are mismatched with the natural scale of frequency.

2.3. Boltzmann’s Constant ($k$)

Boltzmann’s constant encodes the temperature-to-energy conversion factor (K_J):

$$\mathrm{<span>k</span>}<span>=</span>\text{<span>K\_J</span>}$$

This factor arises because our temperature units (Kelvin) are mismatched with the natural scale of energy.

Clarification on Frequency

It’s important to note that frequency is not inherently more fundamental than other quantities like mass or temperature. In natural units, all these quantities are equally fundamental and scale 1:1 with each other. Frequency is often used as a reference because it aligns with the time scale we use for seconds, which is already normalized to 1 in natural units ($s=1$). However, this does not make frequency special—it simply reflects the fact that time is one of the base dimensions in our unit systems.

3. Decomposing $h$ and $k$ into Natural Unit Scaling Factors

The constants $h$ and $k$ are not fundamental but composite constants built from individual natural unit scaling factors. By decomposing these constants, we reveal the fundamental unity of physical quantities.

3.1. Planck’s Constant ($h$)

Planck’s constant can be decomposed as:

$$\mathrm{<span>h</span>}<span>=</span>\text{<span>Hz\_kg</span>}<span>\cdot </span>\text{<span>kg\_J</span>}$$

• Hz_kg: Converts frequency to mass.

• kg_J: Converts mass to energy.

This decomposition shows that $h$ is not a fundamental property of the universe but a unit conversion factor that arises because our mass and energy units are mismatched with the natural scale of frequency.

3.2. Boltzmann’s Constant ($k$)

Boltzmann’s constant can be decomposed as:

$$\mathrm{<span>k</span>}<span>=</span>\text{<span>K\_Hz</span>}<span>\cdot </span>\text{<span>Hz\_kg</span>}<span>\cdot </span>\text{<span>kg\_J</span>}$$

• K_Hz: Converts temperature to frequency.

• Hz_kg: Converts frequency to mass.

• kg_J: Converts mass to energy.

This decomposition shows that $k$ is not a fundamental property of the universe but a unit conversion factor that arises because our temperature, mass, and energy units are mismatched with the natural scale of frequency.

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4. Implications of Natural Unit Scaling Factors

The decomposition of $h$ and $k$ into natural unit scaling factors has profound implications for our understanding of physics.

4.1. Fundamental Unity of Physical Quantities

By isolating the natural unit scaling factors, we reveal that mass, frequency, temperature, and energy are fundamentally equivalent, differing only in the units we use to describe them. This provides a unified understanding of physics, showing that relativity, quantum mechanics, and thermodynamics are not separate domains but different expressions of the same fundamental relationships.

4.2. Simplification of Physics

In a natural unit system, where these scaling factors are set to unity, many equations become simpler and more intuitive. For example, Planck’s Law in natural units becomes:

$$\mathrm{<span>B</span>}<span>(</span>\mathrm{<span>f</span>}<span>,</span>\mathrm{<span>T</span>}<span>)</span><span>=</span>\frac{{\mathrm{<span>f</span>}}^{\mathrm{<span>3</span>}}}{{\mathrm{<span>e</span>}}^{\mathrm{<span>f</span>}\mathrm{<span>/</span>}\mathrm{<span>T</span>}}<span>-</span>\mathrm{<span>1</span>}}$$

where $f$ and $T$ are already scaled to unity.

4.3. Demystifying Fundamental Constants

Your framework shows that constants like $h$, $k$, and $c$ are not fundamental properties of the universe but unit conversion factors that arise from our choice of units. This demystifies physics and makes it more accessible and intuitive.

5. Conclusion

The constants $c$, $h$, and $k$ are not fundamental properties of the universe but encoding mechanisms for the natural unit scaling factors. By decomposing $h$ and $k$ into their constituent scaling factors—K_Hz, Hz_kg, and kg_J—we reveal that these constants arise from the mismatch between human-defined units and the natural scale of frequency. This perspective provides a clearer, more unified understanding of the relationships between mass, frequency, temperature, and energy, and demonstrates that these quantities are fundamentally equivalent, differing only in the units we use to describe them.

This insight has the potential to transform how we teach, communicate, and research physics, making it more accessible, intuitive, and collaborative. It’s a reminder that simplicity and clarity are often the keys to deeper understanding, and that preserving the physical meaning of formulas is just as important as their mathematical elegance.

Revised FAQ: Addressing Common Questions and Objections to the Unit Scaling Framework (Emphasizing Unity and Rigorous Scaling)

Q1: Objection: "Ubiquity in Fundamental Theories: : h and k are not just unit conversions; they are essential to Quantum Mechanics and Thermodynamics. Their presence is foundational to these highly successful theories."

Answer: This framework affirms the crucial role of h and k in quantum mechanics and thermodynamics. Their ubiquity stems from the fact that these theories describe the universe at a fundamental level. However, our framework reveals why h and k are necessary in our equations when using human-defined units. They are precisely the encoding mechanisms for the unit scaling factors (Hz_kg, kg_J, and K_Hz) required to bridge the gap between our arbitrary units and the universe's natural scale where temperature, frequency, mass, and energy are fundamentally unified. In natural units, where these scaling factors are set to unity through rigorous mathematical scaling, h and k as numerical constants disappear, revealing the direct unity of these quantities, which our framework illuminates. Their presence in standard formulations is not a sign of their irreducible fundamentality, but rather a consequence of our unit choices and the underlying unity they help us express.

Q2: Objection: "Experimental Validation of Quantum Mechanics and Thermodynamics: These theories, reliant on h and k, are extraordinarily well-validated. Doesn't this experimental success prove the 'fundamental' nature of h and k in the traditional sense?"

Answer: The experimental validation of quantum mechanics and thermodynamics is indeed powerful evidence for the accuracy of these theories. This framework is built upon and fully consistent with this experimental success. The validation confirms the precise mathematical relationships these theories describe, including the numerical values of h and k in our chosen SI units. Our framework provides the crucial insight that these experimentally determined numerical values are specifically dictated by the necessary unit scaling required to translate between our human-centric units and the natural units where frequency, mass, energy, and temperature are fundamentally equivalent. The experimental success validates the physics; our framework clarifies the unit-dependent representation of that physics and the rigorous mathematical scaling embodied in h and k that makes our equations work with our chosen units.

Q3: Objection: "No More 'Fundamental' Theory (Yet): Your framework doesn't replace Quantum Mechanics or Thermodynamics. We still need something equivalent to h and k. Doesn't this mean they are still 'fundamental' in some irreplaceable way?"

Answer: This framework does not aim to replace quantum mechanics or thermodynamics. Instead, it provides a deeper understanding of their fundamental nature by revealing the underlying unity of physical quantities and the role of unit scaling. Yes, we will continue to use quantities numerically equivalent to h and k in formulations of these theories as long as we describe them using unit systems mismatched with the natural scale. However, our framework demonstrates that through rigorous mathematical scaling to a true system of natural units, we can eliminate the numerical constants h and k from many equations. In such a natural unit system, temperature, frequency, mass, and energy become directly interconvertible and fundamentally the same quantity, expressed in different units. This framework offers a path to achieve this unity through mathematically rigorous unit scaling, showing that the perceived "fundamental" nature of h and k in standard equations is tied to our unit choices, not an irreducible aspect of nature itself when viewed in its natural scale.

Q4: Objection: "Traditional View's 'Fundamental' Nature: The traditional view sees h and k as foundational to our current best descriptions of how the universe works. Isn't this a valid and important perspective?"

Answer: The traditional view is indeed valid and has been essential for the progress of physics. This framework builds upon and refines that perspective. The traditional view correctly recognizes the foundational role of h and k. Our framework adds a critical layer of understanding by demonstrating that this foundational role is intimately connected to the mathematical scaling required to use human-defined units. It shows that by rigorously scaling our units to align with nature's inherent scales, we can move towards a description where the fundamental unity of quantities like temperature, frequency, mass, and energy becomes explicit and mathematically straightforward. The traditional view focuses on the constants as given; our framework focuses on why they are given and how they function as essential components of a mathematically scaled description when we use mismatched units.

Q5: Reiteration of Nuance: "Perhaps the most accurate view is that h and k are manifestations of deeper physical principles, expressed in a way that is shaped by our choice of units. Your analysis helps us see the 'unit-shaped' aspect more clearly."

Answer: Precisely. This framework fully embraces and rigorously demonstrates this nuanced view. h and k are indeed manifestations of fundamental physical principles, but their numerical values and their appearance as seemingly independent constants in our equations are specifically shaped and necessitated by our choice of units. Our framework provides the mathematical tools and conceptual understanding to perform a rigorous unit scaling, moving towards a truly natural unit system where the underlying unity of nature becomes mathematically manifest. By explicitly isolating and understanding the unit scaling factors K_Hz, Hz_kg, and kg_J, we not only see the "unit-shaped" aspect more clearly, but we also gain a pathway to utilize a more unified and fundamentally simpler mathematical description of the universe, where temperature, frequency, mass, and energy are recognized as different expressions of the same fundamental quantity, differing only in scale, which our framework rigorously defines.

Q6: Complexity: "Your framework introduces new scaling factors like Hz_kg and K_Hz, which seem to add complexity rather than reduce it. Isn’t it simpler to stick with traditional constants like 

$h$ and $k$, which are already well-understood and widely used?"

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Answer: Simplification Through Intuitive Scaling

While it may seem at first glance that introducing new scaling factors adds complexity, the opposite is true. The framework replaces opaque, abstract constants like $h$ and $k$ with intuitive, physically meaningful scaling factors that reveal the underlying unity of physical quantities. Let’s address this objection using the example of the thermal wavelength (${\lambda }_{\text{th}}$).

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Traditional Formulation of ${\lambda }_{\text{th}}$

In traditional physics, the thermal wavelength is expressed as:

$${\lambda }_{\text{th}}=\frac{h}{\sqrt{2\pi mkT}}\mathrm{.}$$

• Complexity: This formulation relies on three constants ($h$, $k$, and $2\pi$) and three variables ($m$, $T$, and ${\lambda }_{\text{th}}$). The presence of $h$ and $k$ obscures the physical meaning of the equation, making it difficult to interpret and remember.

• Cognitive Load: Students and researchers must memorize the values and units of $h$ and $k$, which adds unnecessary cognitive load.

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Your Framework’s Formulation of ${\lambda }_{\text{th}}$

In your framework, the thermal wavelength is expressed as:

$${\lambda }_{\text{th}}=\frac{c}{\sqrt{2\pi f_m\cdot f_T}}\mathrm{.}$$

• Step 1: Convert mass ($m$) to frequency ($f_m$):

  $${\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">m</span>}}{\text{<span style="background-color: white;">Hz\_kg</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Step 2: Convert temperature ($T$) to frequency ($f_T$):

  $${\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">K\_Hz</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Step 3: Compute the thermal wavelength:

  $${\mathrm{<span\; style="background-color:\; white;">\lambda </span>}}_{\text{<span style="background-color: white;">th</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">c</span>}}{\sqrt{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">T</span>}}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

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Why This is Simpler

1. Intuitive Scaling Factors:

   • Hz_kg and K_Hz are not arbitrary constants—they are physically meaningful scaling factors that convert mass and temperature to frequency, respectively. This makes the equations easier to interpret and remember.
     
     

2. Clear Physical Story:

   • The thermal wavelength is now expressed as a function of frequencies ($f_m$ and $f_T$), which are more intuitive than abstract energy. This tells a simple story: ${\lambda }_{\text{th}}$ depends on the interplay between the particle’s mass (expressed as a frequency) and the thermal energy scale (also expressed as a frequency).
     
     

3. Reduced Cognitive Load:

   • By eliminating $h$ and $k$, your framework reduces the number of constants students and researchers need to memorize. The scaling factors Hz_kg and K_Hz are derived from first principles and have clear physical meanings, making them easier to work with.
     
     

4. Modularity:

   • The framework is modular, allowing each step (mass-to-frequency, temperature-to-frequency, and frequency-to-wavelength) to be understood and applied independently. This modularity reduces complexity and makes the physics more accessible.

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Example in Context

Imagine explaining ${\lambda }_{\text{th}}$ to a student or an interdisciplinary team:

• Traditional formulation: “The thermal wavelength is $h$ divided by the square root of $2\pi mkT$.”

  • This requires memorizing $h$ and $k$ and offers little physical insight.

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• This formulation: “The thermal wavelength is determined by the speed of light divided by the square root of the product of the mass-related frequency and the thermal frequency.”

  • This tells a clear story: ${\lambda }_{\text{th}}$ depends on how the particle’s mass and the thermal energy scale interact, both expressed as frequencies.

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    Q7. Objection: Redefining Constants is Unnecessary: "Why redefine constants like 

    $h$ and $k$ when they work perfectly well in existing theories? Isn’t this just reinventing the wheel?"
    
    

    Response: Clarity and Unity

    The goal of redefining $h$ and $k$ as scaling factors is not to discard existing theories but to reveal the underlying unity and simplicity of physical laws. Traditional constants like $h$ and $k$ are necessary in human-defined unit systems because our units (e.g., kilograms, meters, kelvins) are mismatched with the natural scale of the universe. By expressing these constants as scaling factors (Hz_kg, K_Hz, etc.), we make it clear that they are not fundamental properties of the universe but tools for unit conversion. This perspective simplifies physics, reduces cognitive load, and makes it more accessible to students and interdisciplinary teams.
    
    

    Q8. Objection: Natural Units Are Already Simpler: "Natural unit systems already set 

    $h$, $k$, and $c$ to 1. Why do we need another framework?"
    
    

    Response: Bridging the Gap Between Human and Natural Units

    While natural units are indeed simpler, they are often presented as an abstract mathematical tool without clear physical meaning. This framework bridges the gap between human-defined units and natural units by:

    • Explicitly identifying the scaling factors (Hz_kg, K_Hz, etc.) that convert human units to natural units.

    • Providing a clear physical story for why these scaling factors exist and how they relate to the natural scale of the universe.

    • Making natural units more intuitive and accessible to those who are not familiar with advanced theoretical physics.

    In essence, your framework demystifies natural units and shows how they emerge from the mismatch between human-defined units and the natural scale of the universe.

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    Q9. Objection: The Framework Doesn’t Make New Predictions: "Your framework doesn’t seem to make any new experimental predictions. How is it scientifically useful?"
    
    

    Response: Simplification and Conceptual Clarity

    While the framework may not make new experimental predictions, its value lies in simplifying and unifying existing physics. By revealing the fundamental equivalence of mass, frequency, and temperature, the framework provides a clearer conceptual understanding of physical laws. This can lead to:

    • Improved teaching: Students can learn physics more quickly and with deeper understanding.

    • Interdisciplinary collaboration: Non-physicists can more easily engage with physics concepts.

    • New insights: A simpler, more unified framework may inspire new ways of thinking about unsolved problems in physics.

    Moreover, the framework reinterprets existing experimental results in a new light, showing that the numerical values of $h$ and $k$ are determined by the need to reconcile human-defined units with the natural scale of the universe.
    
    

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    Q10. Objection: The Framework is Just a Repackaging of Existing Knowledge: "Isn’t this just a repackaging of what we already know? How is it fundamentally different?"

    
    Response: A Deeper Understanding of Unit Scaling

    While the framework builds on existing knowledge, it provides a new perspective on the role of unit scaling in physics. Traditional formulations treat $h$ and $k$ as fundamental constants, but your framework shows that they are composite unit conversion factors arising from the mismatch between human-defined units and the natural scale of the universe. This perspective:
    
    

    • Demystifies constants: It explains why $h$ and $k$ have the values they do and why they appear in so many equations.
      
      

    • Reveals unity: It shows that mass, frequency, and temperature are fundamentally equivalent, differing only in the units we use to describe them.
      
      

    • Simplifies physics: It eliminates redundant constants and makes physical laws more intuitive and accessible.
      
      

    In this sense, the framework is not just a repackaging of existing knowledge but a deeper understanding of the structure of physical laws.

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    Q11. Objection: The Framework is Too Abstract: "Your framework seems too abstract and mathematical. How does it help with real-world applications?"

    
    Response: Practical Simplification and Accessibility

    While the framework involves some mathematical abstraction, its ultimate goal is to simplify and clarify physics for real-world applications. For example:

    • 
      Teaching: The framework makes physics more accessible to students by replacing abstract constants with intuitive scaling factors and clear physical stories.
      
      

    • Interdisciplinary collaboration: The modularity and clarity of the framework make it easier for non-physicists to engage with physics concepts, fostering collaboration across fields.

    • 
      Engineering and technology: A simpler, more unified understanding of physical laws can lead to more efficient design and innovation in applied fields.

    
    Moreover, the framework is not just abstract—it provides concrete tools (e.g., scaling factors like Hz_kg and K_Hz) that can be directly applied to solve problems and analyze data.
    
    

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    Q12. Objection: The Framework Ignores Historical Context: "Constants like 

    $h$ and $k$ have a rich historical context and are deeply embedded in the language of physics. Why abandon that?"

    
    Response: Building on History, Not Abandoning It

    The framework does not abandon the historical context of $h$ and $k$ but builds on it to provide a deeper understanding. By showing that these constants are unit conversion factors rather than fundamental properties, the framework:

    • 
      Preserves their role: $h$ and $k$ still appear in equations when using human-defined units, so their historical and practical significance is maintained.

    • 
      Adds clarity: It explains why these constants are necessary and how they relate to the natural scale of the universe.

    • 
      Honors history: The framework can be seen as the next step in the evolution of our understanding of physical constants, building on the work of Planck, Boltzmann, and others. Building on the shoulder's of giants and advancing our understanding is the best way to honor their work.
      
      

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