When Frequency IS Energy: An Alternative Unit System

 J. Rogers, SE Ohio

Abstract

We explore an alternative formulation of physical units where frequency and energy are harmonized into a single scale, eliminating the need for Planck's constant h in quantum mechanical equations. In this framework, h emerges instead in classical mechanics as the conversion factor between natural frequency-energy and the artificial "Joule" scale. This thought experiment demonstrates that the appearance of fundamental constants depends entirely on coordinate choices, not on deep physical principles. Constants like h are Jacobian elements in unit transformations, not fundamental truths about nature.

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1. Introduction: The Arbitrary Nature of Unit Systems

Since the 2019 redefinition of SI base units, we explicitly recognize that fundamental constants are defined by committee vote rather than measured from nature. The values of c, h, k_B, and e are now exact by definition, chosen to maintain continuity with previous measurement standards.

This raises a profound question: If we can choose these constants by fiat, what does "fundamental" mean?

We propose that fundamental constants are better understood as off-diagonal Jacobian elements in the transformation between human-chosen measurement coordinates and the natural dimensionless ratios that govern physics. These are the natural ratios of Newton.  The Planck Jacobians that were calculated in 1899 with h, not hbar, were the bridge between measurement and natural ratios.  The choice of which constants appear as "fundamental" is arbitrary—a consequence of historical accident rather than physical necessity.

To demonstrate this, we construct an alternative unit system where frequency and energy are the same scale, showing that h would then appear in classical mechanics rather than quantum mechanics.

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2. The Standard SI Framework

2.1 Current Unit Structure

In SI units, we independently define:

• Length: meter (m)
• Time: second (s)
• Mass: kilogram (kg)
• Energy: joule (J = kg⋅m²/s²)
• Frequency: hertz (Hz = 1/s)

Because energy and frequency have different dimensions in SI, we need a conversion factor:

E = hf

where h = 6.62607015 × 10⁻³⁴ J⋅s is Planck's constant (exactly, by definition since 2019).

2.2 Where h Appears

Quantum Mechanics:

• E = hf (photon energy)
• λ = h/(mv) (de Broglie wavelength)
• ΔE⋅Δt ≥ h/4pi (uncertainty principle - order of magnitude)

Classical Mechanics:

• E = ½mv² (no h)
• E = mgh (no h)
• F = ma (no h)

The narrative: "h is the fundamental quantum of action, appearing only in quantum mechanics because quantum effects are fundamentally different from classical physics."

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3. Alternative Framework: Defining Energy as Frequency

3.1 Our Choice of Unit Structure

We choose to define a modified unit system:

• Length: meter (m)
• Time: second (s)
• Mass: kilogram (kg)
• Frequency: hertz (Hz = 1/s)
• Energy: hertz (Hz) - we simply define energy to be measured in Hz

In this system, energy and frequency are the same by our choice of definition.

3.2 The New "Fundamental Constant"

Because mass and frequency have different dimensions, we need a conversion factor:

m = f × (m_P⋅t_P)

where m_P⋅t_P = h/c² is the mass-frequency conversion constant.

This is not a new constant—just the same Jacobian expressed differently.

3.3 Where h NOW Appears

Quantum Mechanics:

• E = f (no h!)
• p = f/c (no h!)

Classical Mechanics:

• E_kinetic = (½mv²) / h
• E_potential = (mgh_gravity) / h
• E_spring = (½kx²) / h

Every classical energy formula now has h in it!

To express quantum energy in the legacy "Joule" unit:

E_Joules = E_frequency × h = f × h

The new narrative: "h is the classical energy conversion factor. It appears in every classical energy formula because classical mechanics uses the Joule scale instead of natural frequency. Quantum mechanics works directly in frequency without needing h."

This is purely our choice. We could equally have defined energy in Hz from the beginning. There's no law of nature forcing us to use Joules.

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4. Detailed Comparison

4.1 Photon Energy

Standard Framework:

E = hf

"A photon of frequency f has energy equal to Planck's constant times its frequency."

Alternative Framework:  What if we discovered quantum mechanics first, then we could have defined energy in terms of frequency directly.  

E = f

"A photon of frequency f has energy f. Period."

To express in Joules (if needed):

E_Joules = h × f

4.2 Kinetic Energy

Standard Framework:

E = ½mv²

No constants needed—mass and velocity directly give energy in Joules.

Alternative Framework (energy in Hz):

Classical kinetic energy would be:

E = (½mv²) / h

h appears directly in the classical kinetic energy formula.

Students learning classical mechanics would see:

• E_kinetic = (½mv²) / h
• E_potential = (mgh_gravity) / h
• Every classical energy formula has h in it

They would ask: "Why do we need Planck's constant in classical mechanics?"

Answer: "Because we're measuring energy in its natural units (frequency), and h converts from the artificial Joule scale."

4.3 Mass-Energy Equivalence

Standard Framework:

E = mc²

"Mass converts to energy with c² as the conversion factor."

Alternative Framework (energy in Hz):

E = (mc²) / h

h appears directly in the mass-energy formula.

The famous equation becomes E = mc²/h when energy is measured in frequency!

Or equivalently:

f = mc² / h

The rest mass of a particle has an equivalent frequency, and h is the conversion factor.  We would have classified this as just another classical formula.

4.4 Quantum Harmonic Oscillator

Standard Framework:

E_n = hf(n + ½)

"Energy levels are quantized in units of hf."

Alternative Framework:

E_n = f(n + ½)

"Energy levels are simply f(n + ½) in frequency units."

To express in Joules:

E_n,Joules = h × f(n + ½)

The quantization is the same—the factor (n + ½) appears in both. Only the units change.

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5. Implications for Physical Understanding

5.1 "Quantum" is Not About h

In the alternative framework, quantum mechanics does not require h. The quantization (discrete energy levels, wave-particle duality, uncertainty relations) all emerge from the mathematics regardless of units.

h is not "the quantum of action"—it's a unit conversion factor that happens to appear in quantum formulas only because we chose energy and frequency as separate scales.

5.2 Classical Energy Becomes "Weird"

In the frequency-energy framework, classical mechanics looks peculiar:

• Why does ½mv² / h give energy?
• Why does mgh / h give potential energy?
• Why is h in every single classical energy formula?
• What's the physical significance of dividing by h?

Students would ask: "Why does classical mechanics need Planck's constant in every energy equation?"

The answer: "Because we chose to define energy in Joules (kg⋅m²/s²) rather than using the natural frequency scale (Hz). The constant h converts from Joules to the natural frequency-energy. We could have made a different choice."

Every classical mechanics textbook would have h appearing in kinetic energy, potential energy, spring energy, rotational energy—everywhere energy appears.

5.3 The Jacobian Perspective

All fundamental constants are off-diagonal Jacobian elements in coordinate transformations:

Standard SI Jacobians:

c = l_P/t_P         (space-time coupling)
h = m_P⋅l_P²/t_P    (action scale)
G = l_P³/(m_P⋅t_P²) (gravitational coupling)
k_B = m_P⋅l_P²/(t_P²⋅T_P) (energy-temperature coupling)

Alternative Jacobians (frequency-energy harmonized):

c = l_P/t_P             (unchanged)
m_P⋅t_P = h/c²          (mass-frequency coupling)
G = l_P³/(m_P⋅t_P²)     (unchanged)
k_B = 1/(T_P⋅t_P)       (frequency-temperature coupling)

Same physics, different coordinate choices.

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6. Newton's Natural Philosophy

6.1 Ratios, Not Constants

Isaac Newton in Principia Mathematica worked with ratios and proportions, not absolute units:

"I frame no hypotheses... it is enough that gravity really exists and acts according to the laws I have explained."

Newton's gravity law:

F ∝ m₁⋅m₂/r²

The proportionality constant G only appears when we force this into SI units with independent mass, length, and time scales.

In natural units:

F_nat = (m₁_nat /r_nat) × (m₂_nat / r_nat)

No G—it's just 1 (identity).  Just two time fields each interacting locally. 

6.2 Return to Natural Philosophy

Modern physics forgot Newton's insight: the physics lives in the dimensionless ratios.

Constants like c, ℏ, G are not fundamental truths—they're conversion factors needed because we chose misaligned coordinate axes for historical/practical reasons.

Our framework:

Newton's natural ratios (dimensionless)
+ Planck Jacobians (l_P, m_P, t_P, T_P)
= Modern physics in SI units

Planck showed us the natural scales. Einstein unified space and time. But we kept treating the conversion factors as if they were fundamental.

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7. The Measurement Committee Problem

7.1 Constants by Fiat

Since May 20, 2019, SI units are defined by fixing exact values for:

• c = 299,792,458 m/s (exactly)
• h = 6.62607015 × 10⁻³⁴ J⋅s (exactly)
• e = 1.602176634 × 10⁻¹⁹ C (exactly)
• k_B = 1.380649 × 10⁻²³ J/K (exactly)

These values were chosen by the CODATA committee to maintain continuity with previous measurement standards.

They were literally set by committee vote.

The jacobians that define the constants for the units now define our measurement scales as the operational truth of our rullers. 

7.2 The Philosophical Problem

If fundamental constants can be chosen by committee, what does "fundamental" mean?

Answer: They're not fundamental—they're definitions of our unit system.

The meter is defined such that c = 299,792,458 m/s. We could equally define the meter such that c = 1 m/s, but then everyday distances would have different numerical values.

The physics (dimensionless ratios) is independent of our choice.

The ratio of Earth's orbital radius to the Sun's radius is ~215. This is true regardless of whether we measure in meters, Planck lengths, or "Smoots."

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8. Practical Example: The Hydrogen Atom

8.1 Standard Framework

The energy levels of hydrogen:

E_n = -13.6 eV / n²

where 13.6 eV involves h in the denominator when derived from first principles.

8.2 Alternative Framework

In frequency-energy units:

f_n = 13.6 eV / (h⋅n²) = 3.29 × 10¹⁵ Hz / n²

8.3 Where h Truly Belongs

h appears in the hydrogen energy formula only because we're converting:

• Electron mass (kg) → natural mass
• Charge (Coulombs) → natural charge
• Energy (Joules or Hz) → dimensionless ratio

In the alternative framework where E = f:

• The energy levels are naturally in Hz
• No h needed for quantum mechanics
• h only appears if we convert back to the classical definition of Joules

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9. Why This Matters

9.1 Clarifying What's Fundamental

Not fundamental:

• The numerical value of c, h, G, k_B
• Whether h appears in quantum or classical formulas
• The choice of Joules vs Hz for energy

Truly fundamental:

• Dimensionless ratios (α, pi, mass ratios, length ratios, etc.)
• Geometric relationships (F ∝ m₁m₂/r²)
• Topological properties (quantization patterns)

9.2 Avoiding Pseudoprofundity

Much ink has been spilled on questions like:

• "Why does nature choose c = 3×10⁸ m/s?"
• "Why is h so small?"
• "Why is the fine structure ratio 1/137?"

The first two questions are meaningless—c and h are unit conversion factors whose numerical values depend on our arbitrary choice of meters, kilograms, and seconds.

Only the third question is meaningful—α ≈ 1/137 is a dimensionless ratio, independent of units.

9.3 Multiverse Pseudoscience

Some theorists propose that:

• Constants like c, ℏ, α "could have been different"
• We live in a "goldilocks universe" where they happen to allow life
• The multiverse explains fine-tuning

But this conflates two categories:

Jacobians (not fundamental):

• c, h, G, k_B
• Their values are whatever we define them to be
• In natural units, they're all 1
• Can't "be different" in any physical sense

Dimensionless ratios (actually fundamental):

• α ≈ 1/137
• m_proton/m_electron ≈ 1836
• These genuinely characterize the universe
• These are what could "be different" (if anything can)

The multiverse conversation should focus exclusively on dimensionless ratios.

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10. Pedagogical Benefits

10.1 Teaching Natural Units Early

We propose introducing natural units before SI units in physics education:

Step 1: Teach physics in dimensionless ratios

• Forces as geometric relationships
• Energy as frequency
• Mass as frequency × (mass-time constant)

Step 2: Introduce SI units as a coordinate choice

• "For historical reasons, we measure length in meters..."
• "This requires conversion factors c, h, etc."

Step 3: Show that constants depend on coordinate choice

• In one frame, h appears in quantum mechanics
• In another frame, h appears in classical mechanics
• Neither is "more true"

10.2 Demystifying Quantum Mechanics

Students often think quantum mechanics is "weird" because h appears everywhere.

Alternative narrative: "Quantum mechanics is natural frequency-based physics. Classical mechanics looks weird because we insist on using Joules, requiring conversion factors."

This inverts the usual perspective and clarifies that quantum mechanics isn't ontologically special—it's just physics in natural units.

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11. Connection to Running Alpha Research

11.1 Why Dimensionless Matters

The geometric model succeeds because:

• It works with dimensionless energy scale (γ, rapidity)
• The coupling factor k is a dimensionless ratio
• No Feynman diagrams, virtual particles, or renormalization needed

The physics lives in dimensionless relationships.

If we'd been thinking in terms of natural frequency-energy from the start, the geometric interpretation might have been obvious:

"As collision frequency increases, the geometric coupling between electromagnetic and gravitational intensities strengthens according to a simple power law."

No need for virtual particles—just geometry and dimensionless ratios.

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12. Conclusion

We have demonstrated that the appearance of Planck's constant h in physical formulas depends entirely on coordinate choices, not on fundamental physics.

In a unit system where frequency IS energy:

• Quantum mechanics requires no h
• Classical mechanics requires h to convert to Joules
• The physics (dimensionless ratios) remains unchanged

Key insights:

1. Constants are Jacobians: c, h, G, k_B are off-diagonal elements in coordinate transformations, not fundamental truths.

2. Units are arbitrary: Since 2019, we literally define constants by committee vote. There's nothing "natural" about Joules or seconds.

3. Ratios are fundamental: The actual physics lives in dimensionless quantities like α ≈ 1/137 and mass ratios.

4. Newton was right: Natural philosophy should work with ratios and proportions, not absolute scales.

5. Paradigm matters: Modern physics got confused by treating measurement artifacts (constants) as fundamental truths.

The question is not "Why does quantum mechanics need h?"

The question is "Why did we choose to define energy in Joules, requiring h as a conversion factor?"

The answer: historical accident and practical convenience for everyday engineering.

We could rebuild physics education to define energy in Hz from the start, making quantum mechanics the "natural" framework and classical mechanics the one requiring conversion factors.

Or, more radically: We could teach physics entirely in dimensionless ratios, introducing unit systems only as practical conveniences for engineering applications.

Either way, we must stop pretending that Jacobian elements are fundamental truths about reality. Energy units are our choice, not nature's.

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Appendix A: Formula Translations

Physical Quantity	Standard (SI)	Alternative (f = E)
Photon energy	E = hf	E = f
De Broglie wavelength	λ = h/p	λ = h/p
Kinetic energy	E = ½mv²	E = ½mv²/h
Mass-energy	E = mc²	f = mc^2/h
Uncertainty	ΔE⋅Δt ≥ ℏ/2	Δf⋅Δt ≥ 1/4pi

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Appendix B: The Planck Jacobian Matrix

The transformation between SI and natural units can be written as a matrix:

[Length]     [l_P    0      0      0   ] [1]
[Mass  ]  =  [0      m_P    0      0   ] [1]
[Time  ]     [0      0      t_P    0   ] [1]
[Temp  ]     [0      0      0      T_P ] [1]

The off-diagonal Jacobians arise from cross-terms:

c = l_P/t_P
h = m_P⋅l_P²/t_P
G = l_P³/(m_P⋅t_P²)
k_B = m_P⋅l_P²/(t_P²⋅T_P)

These are not independent—they're functions of the four diagonal scales (l_P, m_P, t_P, T_P).

There are no free parameters in the Jacobian. Once you choose four base scales, all "fundamental constants" are determined.

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References

1. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica
2. Planck, M. (1899). "Über irreversible Strahlungsvorgänge"
3. BIPM (2019). "The International System of Units (SI)", 9th edition
4. CODATA (2018). "Recommended Values of Fundamental Physical Constants"
5. Einstein, A. (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?"
6. Duff, M.J. (2004). "Comment on time-variation of fundamental constants"

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This paper demonstrates that the appearance of fundamental constants in physical formulas is a consequence of coordinate choices, not fundamental physics. By harmonizing frequency and energy, we invert the usual presentation: quantum mechanics becomes "natural" while classical mechanics requires conversion factors. This pedagogical exercise reveals that constants like h are Jacobian elements in unit transformations—measurement artifacts, not fundamental truths about nature.