J. Rogers, SE Ohio, 4 Apr 2025, 1931
Abstract: The search for a unified Theory of Everything (ToE) is often partly motivated by the desire to reconcile the distinct physical domains seemingly governed by the fundamental constants c (relativity), ħ (quantum mechanics), and k (thermodynamics). This paper argues that this specific motivation is misguided, stemming from a misunderstanding of the nature of dimensional constants within specific unit systems like the SI. We demonstrate that the non-unity numerical values of c, h, and k are primarily artifacts of historical, arbitrary definitions of the base SI units (meter, kilogram, Kelvin) relative to the SI second. These constants function as necessary scaling factors or conversion ratios between fundamentally equivalent physical quantities (Energy, Mass, Frequency, Temperature) within the SI framework. A straightforward rescaling of SI units, anchored to the SI second, reveals a "natural" unit system where c=1, h=1, k=1 emerge directly, dissolving the notion of distinct "scales" associated with these constants. Consequently, exotic physics (new dimensions, fields, particles) are not required to achieve this specific unification; a proper understanding of metrology suffices. This clarifies the true challenges for a ToE, which lie elsewhere (e.g., quantum gravity, dimensionless constants).
1. Introduction: The Conventional View of Fundamental Scales and Unification
Modern physics presents a picture where fundamental constants delineate distinct physical regimes. The speed of light, c, governs relativistic phenomena; the Planck constant, h (or ħ), dictates quantum behavior; the Boltzmann constant, k, connects microscopic energy to macroscopic temperature. Often, these are spoken of as defining the "relativistic scale," the "quantum scale," and the "thermal scale." While Quantum Field Theory successfully merges c and h, the unification of these with gravity (governed by G) and understanding their interplay remains a central goal, often motivating the search for a ToE like String Theory or Loop Quantum Gravity. A common, implicit assumption is that the vastly different numerical values of these constants (in SI units) reflect fundamentally different scales of reality that must be reconciled by deeper physical principles, potentially involving new physics at extreme energies or small distances.
This paper challenges the premise that c, h, and k themselves establish fundamentally distinct scales requiring unification via a ToE. We argue that their specific numerical values in SI are consequences of metrology, not deep physics, and that understanding this dissolves this particular motivation for seeking a ToE.
2. The SI System and the Emergence of c, h, k as Conversion Factors
The International System of Units (SI) provides our standard framework for measurement. Crucially, its base units – the second, meter, kilogram, Kelvin – were historically defined based on accessible phenomena or artifacts (atomic transitions, Earth's meridian, a platinum-iridium cylinder, the triple point of water). Physical laws relate quantities measured in these units, requiring proportionality constants.
Consider the fundamental equivalences between Energy (E), Mass (m), Frequency (f), and Temperature (T), represented physically by relationships like E=mc², E=hf, and E=kT. These state that these quantities are, in essence, different facets of the same underlying physical "stuff."
However, because the SI definitions for the Joule (derived from kg, m, s), the kilogram (historically, an artifact), the Hertz (derived from s), and the Kelvin (historically, water's properties) were not initially correlated to directly reflect these equivalences relative to a single base (like the second), conversion factors become necessary:
E = m * c²: c² is the conversion factor from SI kilograms to SI Joules.
E = h * f: h is the conversion factor from SI Hertz (s⁻¹) to SI Joules.
E = k * T: k is the conversion factor from SI Kelvin to SI Joules.
These constants (c, h, k) therefore acquire specific numerical values in SI units precisely because of the mismatch between the unit definitions and the underlying physical equivalences. They quantify how far the SI definitions of mass, time/frequency, and temperature are scaled relative to each other and to energy.
We can express these equivalences, anchored to 1 Hz in the SI system, as:
Equivalent to 1 Hz:
Energy = h_SI * (1 Hz) ≈ 6.626 × 10⁻³⁴ J
Mass = (h_SI / c_SI²) * (1 Hz) ≈ 7.372 × 10⁻⁵¹ kg (using m=hf/c²)
Temperature = (h_SI / k_SI) * (1 Hz) ≈ 4.799 × 10⁻¹¹ K (using T=hf/k)
This explicitly shows h, h/c², and h/k acting as the numerical ratios needed to translate 1 Hz into the corresponding SI values of Joules, kilograms, and Kelvin.
3. Demonstrating the Metrological Nature via Unit Rescaling
The artifactual nature of these constants' SI values becomes evident when we construct a "natural" unit system, not by imposing c=h=k=1 abstractly, but by rescaling the SI units based on the constants themselves, anchored to the invariant SI second.
Let S_s, S_m, S_kg, S_K be the scaling factors such that 1 new_unit = S * SI_unit. We keep the second fixed:
S_s = 1 (1 new_second = 1 SI second; 1 new_Hz = 1 SI Hz)
We require the new system to manifest the physical equivalences directly:
We want c_new = 1 new_m / new_s = 1. This requires 1 = c_SI / (S_m / S_s). Since S_s=1, we get S_m = c_SI.
Result: 1 new_meter = c_SI * meter_SI ≈ 3×10⁸ m.
We want the energy-frequency relation E=hf to become E=f numerically, meaning h_new = 1 new_J * new_s = 1. This requires 1 = h_SI / (S_J * S_s). Since S_s=1, we get S_J = h_SI.
Result: 1 new_Joule = h_SI * Joule_SI ≈ 6.6×10⁻³⁴ J.
We want the energy-temperature relation E=kT to become E=T numerically, meaning k_new = 1 new_J / new_K = 1. This requires 1 = k_SI / (S_J / S_K). Since S_J=h_SI, we get S_K = S_J * k_SI = h_SI * k_SI. Correction: S_K = S_J / (1 * k_new) = S_J / k_SI requires k_new=k_SI / (S_J/S_K) = 1, so S_K = S_J / k_SI = h_SI / k_SI.
Result: 1 new_Kelvin = (h_SI / k_SI) * Kelvin_SI ≈ 4.8×10⁻¹¹ K.
We want the mass-energy relation E=mc² to become E=m numerically (since c_new=1). This implies S_J = S_kg * S_m² / S_s² = S_kg * c_SI² / 1. Since S_J=h_SI, we get S_kg = h_SI / c_SI².
Result: 1 new_kg = (h_SI / c_SI²) * kg_SI ≈ 7.4×10⁻⁵¹ kg.
In this system, derived purely by scaling SI units based on the measured SI values of c, h, k relative to the fixed second, we achieve:
1 new_K = 1 new_Hz = 1 new_kg = 1 new_Joule (by definition relative to the second/Hz)
And consequently, c=1, h=1, k=1 when expressed in these new units.
This procedure demonstrates that the values 2.99...e8, 6.626...e-34, and 1.38...e-23 are precisely the factors needed to scale the meter, Joule (and thus kg), and Kelvin relative to the second to reveal the underlying 1=1=1=1 equivalence. They are accumulated conversion factors, not markers of fundamentally separate physical realities.
4. Implications: Revisiting the Need for Unification
The conventional argument suggests a ToE is needed to explain why the constants have their values and how their associated "scales" merge. The metrological perspective dissolves this specific problem statement for c, h, and k:
No "Scales" to Unify: The analysis shows these constants don't define distinct physical scales, but rather quantify the scaling between units for equivalent physical quantities within the SI system. There are no separate "c, h, k realms" to be merged by new physics.
Values Explained by Metrology: The numerical values arise directly from the ratio between the SI unit definitions and the underlying physical equivalences, anchored to the SI second. Their origin is metrological, not cosmological or rooted in undiscovered physics.
Unification Achieved by Rescaling: Achieving c=1, h=1, k=1 does not require a ToE; it requires only a consistent rescaling of measurement units, as demonstrated above. The 2019 SI redefinition, by fixing the numerical values of h, k, c, e, essentially operationalized this, defining the kg, K, m, A based on these fixed conversion factors relative to the second.
This does not imply that a ToE is unnecessary. Profound challenges remain:
Quantum Gravity: Unifying General Relativity (G, c) with Quantum Field Theory (ħ, c) is a critical, unsolved problem driven by mathematical incompatibility and the need to describe regimes where both are important (black holes, early universe). The metrological nature of c and h's SI values doesn't remove this challenge.
The Gravitational Constant G: This analysis focused on c, h, k. G introduces mass/energy as a source of spacetime curvature, a different kind of relationship. While Planck units combine G, ħ, c, the interpretation of G itself warrants separate discussion, though similar arguments about its SI value being unit-dependent apply.
Dimensionless Constants: Why does the fine-structure constant α ≈ 1/137 have its specific, unit-independent value? Why do particle mass ratios have their values? These are questions a ToE might address.
Origin of Equivalences: Why are E, m, f, T fundamentally equivalent? That remains a deep question about the nature of reality.
However, the specific problem of "unifying the scales of c, h, and k" is shown to be an illusion created by our system of units.
5. Conclusion
The belief that fundamental constants c, h, and k demarcate distinct physical scales requiring unification via a Theory of Everything is based on a misinterpretation of their role within specific measurement systems like SI. Their non-unity numerical values are not indicators of deep physical divisions but are rather metrological artifacts – conversion factors necessitated by the historical, arbitrary scaling of SI base units relative to the second and the underlying physical equivalences E~m~f~T.
A simple, consistent rescaling of SI units, anchored to the second, trivially yields a system where c=1, h=1, k=1, demonstrating that no new physics is required for this specific unification. The apparent "scales" dissolve when the unit system is chosen appropriately. While the quest for a true ToE remains vital for addressing challenges like quantum gravity and the values of dimensionless constants, we must discard the flawed notion that reconciling c, h, and k is one of those challenges. Understanding these constants as unit-system-dependent scaling factors provides a clearer view, freeing the search for fundamental theories from this particular metrological confusion.
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