J. Rogers, SE Ohio
Abstract
The fundamental constants of physics—c, G, h, and k_B—are traditionally regarded as intrinsic properties of the universe, discovered through experiment. This paper posits an alternative framework: these constants are not fundamental in themselves, but are emergent artifacts arising from the choice of an arbitrary, human-centric measurement system (SI units). We argue that the universe operates on a natural, orthogonal basis defined by the Planck scales, where the laws of physics reduce to simple identities. The relationship between our SI basis and the Planck basis is described by a diagonal Jacobian matrix, whose four diagonal elements represent the only true fundamental scaling factors. The familiar constants c, G, h, and k_B are shown to be specific algebraic constructions of these diagonal scaling factors. They function to correct for the inherent disproportions in our chosen units, creating apparent "off-diagonal" relationships between concepts (like mass and energy) that are unified in the natural basis. This reframes the constants from profound mysteries into predictable consequences of our metrology.
1. Introduction
The edifice of modern physics is built upon a foundation of fundamental constants. The speed of light (c), the Newtonian constant of gravitation (G), the Planck constant (h), and the Boltzmann constant (k_B) appear ubiquitously in our most foundational equations. They are typically presented as immutable, universal numbers whose precise values are determined by experiment and whose origins remain a deep mystery. The pursuit of a "theory of everything" is often framed as a quest to explain why these constants have the values they do.
However, the very existence of these constants, with their complex and seemingly unrelated units, hints at a deeper structural truth. The development of the Planck unit system (Planck, 1899) provided the first glimpse of this truth. By combining c, G, and h, it is possible to derive a "natural" set of units for mass, length, time, and temperature that are free from arbitrary human standards. The standard interpretation is that these constants are fundamental, and the Planck units are derived.
This paper inverts that logic. We propose that the Planck scales represent the true, fundamental basis of physical reality. Our arbitrary SI units represent a poorly scaled, non-orthogonal projection of that reality. Consequently, the fundamental constants are not discoveries, but inventions—mathematical conversion factors required to make physical laws work within our chosen, dissonant system. They are the components of a Jacobian matrix that translates between our system and nature's.
2. The Two Bases of Reality
To understand the role of the constants, we must first define the two coordinate systems in question.
2.1 The Planck Basis: An Orthogonal Reality
The Planck basis {m_P, l_P, t_P, T_P} represents the natural gridlines of the universe. In this system, physical laws achieve a state of ultimate simplicity and unity, where the numerical value of the constants becomes 1. This is not a normalization trick; it is a revelation of equivalence:
Mass = Energy: E = m
Energy = Frequency: E = f
Mass = Frequency: m = f
Momentum = Wavenumber: p = k (where k = 1/λ)
Temperature = Energy: T = E
In this basis, concepts we perceive as distinct—a quantity of mass, a rate of oscillation, an amount of energy, a measure of temperature, a unit of inverse length—are all revealed to be different facets of the same underlying physical reality, which can be described as a "process" or "frequency."
2.2 The SI Basis: A Human-Centric Abstraction
The SI basis {kg, m, s, K} is founded on historical accident and human convenience. The meter, kilogram, second, and Kelvin have no a priori relationship to one another. Their scales are wildly disproportionate to the universe's natural 1:1 equivalences. This choice of a "badly scaled" basis is the origin of the constants.
3. The Planck Scale as the Locus of Unification
The historical motivation for the Planck units was to find a system of measurement free from anthropocentric artifacts. However, the true significance of the Planck scale is not that it sets the constants c, G, and h to unity. That is a symptom, not the cause. The profound truth of the Planck scale is that it is the locus where the conceptual divisions between fundamental physical quantities dissolve, revealing a deep, structural unity. This is the unification physicists have sought, hidden in plain sight within the very definition of our constants.
3.1 The Great Equivalence Chain
When all physical quantities are expressed as dimensionless ratios relative to their corresponding Planck scale, a remarkable series of equivalences emerges. Let us denote a dimensionless quantity as Q̃ = Q / Q_P. In this natural unit system, the laws of physics conspire to produce a single chain of identity:
T̃ = f̃ = m̃ = Ẽ = p̃ = F̃ = 1/l̃
Let's break down this chain, which we shall call the Planck Equivalence Chain:
Temperature is Energy (T̃ = Ẽ): This equivalence comes from statistical mechanics, where E = k_B T. In Planck units, Ẽ = E/E_P and T̃ = T/T_P. Since k_B = E_P / T_P, the equation E/E_P = (E_P/T_P) * T / E_P simplifies to Ẽ = T̃. Temperature is revealed to be nothing more than the energy-per-degree-of-freedom of a system.
Energy is Mass (Ẽ = m̃): This is Einstein's celebrated E = mc². Since E_P = m_P c², dividing the equation by this identity yields E/E_P = (m/m_P) * (c²/c²), which is Ẽ = m̃. Mass is revealed to be the rest-energy content of a system.
Energy is Frequency (Ẽ = f̃): This is Planck's foundational quantum relation, E = hf. Recalling that h = E_P / f_P, dividing by the Planck relation gives E/E_P = (E_P/f_P) * f / E_P, which simplifies to Ẽ = f̃. Energy is revealed to be the fundamental frequency of a quantum system.
Energy is Inverse Length (Ẽ = 1/l̃): This relationship combines quantum mechanics (E=hf) and relativity (c=fλ). It can be expressed as E = hc/λ, where λ is wavelength. In Planck units, this becomes Ẽ = 1/λ̃. This powerfully connects energy to spatial dimension, implying that a high-energy event is synonymous with a process confined to a small length scale.
This chain unifies thermodynamics, special relativity, and quantum mechanics into a single coherent statement. It asserts that at the most fundamental level, there is only one type of physical quantity, which we might call "process" or "activity." A massive particle is a high-frequency process. A high-temperature system is a high-energy process. A short-lived interaction is a high-momentum process.
3.2 The SI Veil: How Constants Obscure Unity
The Planck Equivalence Chain is the simple reality. Our SI unit system obscures it. To express this single, unified statement in our fragmented human basis, we are forced to introduce the scaling factors that we call constants.
The single, unified statement:
T/T_P = f*t_P = m/m_P = E/E_P = p/p_P = F/F_P = l_P/l
Look what happens when we try to state even a simple part of this chain, like "mass is frequency," in SI units. We start with the dimensionless truth:
m/m_P = f*t_P
To solve for the mass m in kilograms, we must rearrange:
m = f * (m_P * t_P)
The term in parentheses, (m_P * t_P), is a fixed value derived from the Planck scales. When we calculate it, we find m_P * t_P = h/c². Thus, the simple identity m̃ = f̃ is warped by our measurement system into the complex form m = f(h/c²), which rearranges to mc² = hf.
The constants, therefore, are not fundamental properties. They are the shadows cast by unity onto the cave wall of our perception. Each constant is the specific "correction factor" required to translate one part of the Equivalence Chain into our arbitrary human language.
c is the factor needed to see length = time.
h is the factor needed to see energy = frequency.
k_B is the factor needed to see temperature = energy.
G is the factor needed to see mass = geometry.
The quest of physics has not been to find the values of these constants. The true quest has been to gather enough constants to reverse-engineer their structure, allowing us to erase them and finally read the simple sentence they were obscuring all along: that at the heart of reality, everything is one.
4. The Transformation Matrix: Constants as Jacobians
The relationship between these two bases can be described with the mathematical formalism of a Jacobian matrix.
4.1 The Diagonal Transformation of Basis Vectors
The transformation that maps the fundamental Planck basis vectors to our SI basis vectors is a simple diagonal scaling matrix. Our "kilogram" is nothing more than a scaled version of the Planck mass; our "meter" is a scaled version of the Planck length. There is no mixing of the fundamental concepts themselves.
[ kg ] [ α 0 0 0 ] [ masŝ_P ]
[ m ] = [ 0 β 0 0 ] [ lengtĥ_P ]
[ s ] [ 0 0 γ 0 ] [ timê_P ]
[ K ] [ 0 0 0 δ ] [ temp̂_P ]
The diagonal elements {α, β, γ, δ} are the four truly fundamental numbers of our physical world. They are the numerical values of the Planck units (derived from h) expressed in SI terms:
α = m_P = sqrt(hc/G) ≈ 5.456 x 10⁻⁸ kg
β = l_P = sqrt(hG/c³) ≈ 4.051 x 10⁻³⁵ m
γ = t_P = sqrt(hG/c⁵) ≈ 1.351 x 10⁻⁴³ s
δ = T_P = (1/k_B)sqrt(hc⁵/G) ≈ 3.551 x 10³² K
These four scaling factors are the complete description of how our measurement system relates to the universe's natural scale. All other constants are constructed from them.
4.2 Construction of the Apparent Constants
The famous constants c, G, h, and k_B emerge as specific combinations of these diagonal scaling factors. They are the precise conversion factors needed to enforce the unified laws of the Planck basis within our disjointed SI basis.
The Speed of Light, c: In the Planck basis, length = time. In our basis, β ≠ γ. The constant c is the ratio required to restore this equivalence:
c = β / γ = l_P / t_P ≈ (4.051 x 10⁻³⁵ m) / (1.351 x 10⁻⁴³ s) ≈ 3 x 10⁸ m/s
c is not a speed limit; it is the measure of the disproportion between our meter and our second.
The Planck Constant, h: In the Planck basis, energy = frequency (E = f = 1/t). Our units for energy (Joule = kg⋅m²/s²) and frequency (Hertz = 1/s) are not equivalent. h is the factor that bridges this gap:
h = (α ⋅ β²) / γ = (m_P ⋅ l_P²) / t_P ≈ 6.626 x 10⁻³⁴ J⋅s
h is not a fundamental quantum of action; it is the specific scaling factor required to translate "cycles per second" into "Joules."
The Gravitational Constant, G: In a unified theory, mass/energy is directly equivalent to spacetime geometry. In our system, we need a constant to relate kilograms to meters and seconds. G provides this complex conversion:
G = β³ / (γ² ⋅ α) = l_P³ / (t_P² ⋅ m_P) ≈ 6.674 x 10⁻¹¹ m³⋅kg⁻¹⋅s⁻²
The Boltzmann Constant, k_B: In the Planck basis, temperature = energy. k_B bridges the arbitrary human scale of temperature (Kelvin) with the physical scale of energy (Joules):
k_B = (α ⋅ β²) / (γ² ⋅ δ) = (m_P ⋅ l_P²) / (t_P² ⋅ T_P) ≈ 1.38 x 10⁻²³ J/K
5. Conclusion: A New Perspective on Physical Law
This framework reveals that the fundamental constants of nature are not features of the universe, but rather features of our description of it. The universe operates on a simple, unified scale. Our decision to use arbitrary, human-centric units forces the invention of constants to patch the seams of our disjointed perception.
This has profound implications:
Demystification: The constants are no longer mysterious numbers to be explained. They are predictable artifacts, constructed from the four primary scaling factors that define our system of units.
Unification: The "off-diagonal" behavior of constants in our equations (e.g., c mixing space and time in Lorentz transformations) is understood not as a fundamental mixing, but as a mathematical consequence of our non-unified basis.
Pedagogy: Physics education could be transformed. Instead of presenting constants as divine edicts, they can be introduced as logical necessities for translating between the simple, unified reality of the Planck scale and our everyday measurements.
The true constants of nature are not c, G, h, and k_B. The true constants are the four scaling factors—α, β, γ, δ—that measure how far our chosen rulers deviate from the universe's own. Physics is not the study of these constants; it is the process of seeing past them to the unified reality they inadvertently obscure.
Appendix A: The Category Error of the Reduced Planck Constant (ħ)
A core principle of this paper is the careful separation of fundamental reality from the artifacts of our measurement system. It is therefore essential to address the widespread use of the reduced Planck constant, ħ ("h-bar"), and to demonstrate why its use in a foundational analysis constitutes a significant category error.
The two components of ħ belong to entirely different conceptual domains:
The Constant 2π: This is a fundamental, dimensionless constant of geometry. It is the pure numerical ratio between a circle's circumference and its radius. It is a fact of mathematics, not an artifact of human metrology. It describes the intrinsic geometry of rotation and oscillation. It is, in a very real sense, a part of the universe's logical structure.
The Constant h: As demonstrated in the main body of this paper, the Planck constant h is not fundamental physics. It is a composite Jacobian scaling factor, h = (α ⋅ β²) / γ, constructed from the arbitrary scaling of our SI units for mass, length, and time. Its entire existence is an artifact of our decision to use the kilogram, meter, and second. It has no meaning in the unified Planck basis; it is a conversion factor, not a property of nature.
The reduced Planck constant is defined as ħ ≡ h / 2π.
Creating ħ involves dividing a metrological artifact (h) by a fundamental geometric constant (2π). This is the category error. This act creates a hybrid quantity that obscures the true nature of both of its components:
It taints the pure, geometric 2π by bundling it with the arbitrary scaling factors inherent in h.
It disguises the true nature of h as a mere scaling factor by giving it a false association with the fundamental geometry of wave mechanics and rotation.
The equation E = hf transparently states that energy (in our arbitrary Joules) is proportional to frequency (in our arbitrary Hertz), and h is the specific conversion factor our bad units require.
The equation E = ħω is conceptually muddy. It merges the unit conversion (h) and the geometric conversion from cycles to radians (2π) into a single symbol, ħ. This makes it notationally tidy but philosophically disastrous for anyone trying to understand the true structure of our physical laws. It encourages the very confusion this paper seeks to eliminate—the confusion between what is a property of the universe and what is a property of our rulers.
Therefore, for this analysis to remain logically consistent, it is imperative to use only h. The purpose of this framework is to isolate the scaling artifacts (α, β, γ, δ) and the composite Jacobians (h, c, G, k_B) that arise from them. To use ħ would be to deliberately re-contaminate the analysis by merging one of these artifacts with an unrelated, fundamental constant of geometry. ħ is not physics; it is a conflation of a scaling artifact with a geometric truth.
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