Physical Constants as Unit Harmonization: A Unified Origin for Scale and Symmetry

J. Rogers, SE Ohio

Introduction

The speed of light c, the gravitational constant G, and Planck's constant h are often described as the sacred pillars of modern physics—the fundamental constants that define the fabric of our universe. We ask why they have the values they do, imagining that these numbers hold the deepest secrets of cosmic design. This paper argues that this view is a profound misunderstanding.

We will demonstrate that these dimensional quantities are not fundamental properties of nature, but rather harmonization coefficients—artifacts generated by our system of measurement. They exist to correct for our historically arbitrary choice to define independent scales for length, time, and mass, when in reality these quantities are intrinsically linked. But this perspective does more than just clarify our notation; it reveals a foundational physical principle. We will show that this principle—the underlying unity of physical reality—provides a direct physical origin for the two great pillars of theory construction: scale invariance (dimensional analysis) and symmetry invariance (conservation laws).

The natural unit system in this paper is the original 1899 Planck units with h. This is because the simple proportion is E~f, not E~w.

Part I: Deconstructing Measurement

The Three Separate Concerns

Physical theories conflate three distinct concerns that should be separately defined:

1. Arbitrary measurement scales - Our conventional choices of meter, kilogram, second, etc. These have no physical content.

2. Invariant physical ratios - The actual relationships between physical quantities that don't depend on measurement choices. This is the real physics.

3. Dimensional constants - Numbers like c, G, h, k_B whose function is to harmonize our arbitrarily scaled measurement axes.

Dimensional vs. Dimensionless Quantities

We must distinguish two fundamentally different types of quantities:

• Dimensional constants like c, G, h, and k_B have units (m/s, J·s, etc.). These serve to harmonize our measurement systems and are the subject of our initial critique.

• Dimensionless ratios like the fine structure constant α ≈ 1/137 are pure numbers. These are coordinate-free, unit-independent quantities that represent the actual physics—the "fingerprints" of reality that remain regardless of how we choose to measure.

The Core Insight: The Equivalence Chain

We incorrectly assume our measurement axes are independent. But the underlying physics connects these quantities through fixed equivalences. The practice of using "natural units" reveals something profound: the quantities we measure as mass, length, time, and energy are not independent entities but are different facets of a single, unified physical reality.

In natural units, this is expressed by the Equivalence Chain:
T_nat = f_nat = m_nat = 1/length_nat = E_nat = p_nat = F_nat

This is the central physical postulate. The physics is this equivalence chain—the coordinate-free, identity relationships between quantities.

How Constants Emerge from Scaling Choices

To scale from natural units to our conventional SI units, we multiply each quantity by 1, expressed as its Planck unit ratio (e.g., E_P/E_P). This adds zero physical information.

Consider the energy-frequency equivalence: E_nat = f_nat.
Scaling to SI units gives:
E_si / E_P = f_si × t_P

Solving for E_si:
E_si = f_si × (E_P × t_P)

The product (E_P × t_P) is a composite scaling factor, which we call Planck's constant, h.

The physical law is the equivalence E_nat ≡ f_nat. The constant h is not part of the physical law; it is a conversion factor that emerges solely from our choice to measure energy in Joules and frequency in Hertz. Constants appear exactly where our independent axis scaling choices create an imbalance that doesn't self-cancel. For example, F = ma is free of constants because the scaling factors for force, mass, and acceleration happen to cancel perfectly.

Part II: Reconstructing Physical Law from a Unified Foundation

This framework does more than demystify constants; it provides a physical origin for the most powerful rules of theory construction.

Derivation I: The Physical Basis for Dimensional Analysis

In the standard paradigm, dimensional analysis is a powerful tool that lacks a clear physical justification. This framework provides it. Dimensional analysis works because it is the mathematical enforcement of the idea that physical laws must be independent of our arbitrary measurement conventions.

Dimensions are bookkeeping tags that track the independent scaling factors (l_P, t_P, m_P) we have introduced. An equation is dimensionally consistent if and only if these arbitrary scaling factors completely cancel out, leaving a pure, unit-independent statement about the underlying reality. An equation that is not dimensionally consistent is one whose truth value depends on whether one measures in meters or feet—a statement about convention, not physics.

The Buckingham Pi theorem is, therefore, not an abstract mathematical curiosity. It is the necessary procedural consequence of demanding that our equations describe the underlying, unified reality rather than our arbitrary, multi-scale measurement conventions.

Derivation II: The Physical Origin of Conservation Laws

Noether's theorem, which connects continuous symmetries to conserved quantities, is a cornerstone of physics. Our framework reveals its physical origin.

1. The Foundational Identities: We begin with the core postulate of physical identity from the Equivalence Chain:

• Energy is inverse time: E_nat ≡ 1/t_nat

• Momentum is inverse length: p_nat ≡ 1/l_nat

2. The Formal Machinery: The Principle of Stationary Action: Physical laws are derived from the principle that the Action, S, is stationary (δS = 0). In natural units, the Action (∫ E·dt or ∫ p·dx) is fundamentally dimensionless, as it represents (1/t_nat) × t_nat = 1. It is an invariant, pure number.

3. The Bridge: Isomorphism between Physical Identity and Formal Definition: Lagrangian mechanics provides a formal definition for the momentum p conjugate to a coordinate q: p ≡ δS/δq. The profound insight is that this is not an arbitrary definition; it is the formal mathematical expression of the physical identity p_nat ≡ 1/l_nat postulated in the Equivalence Chain. The physical quantity representing change along a coordinate is what the action principle identifies as the conjugate variable.

4. The Synthesis: Why Conservation Laws Exist: A continuous symmetry (e.g., spatial translation) is a statement that the underlying physical reality is homogeneous with respect to that coordinate q.

• Symmetry means reality is indifferent to the absolute value of q.

• The Equivalence Chain identifies momentum p as the physical embodiment of the frequency or "cost" associated with that coordinate q.

• Therefore, if reality is indifferent to q, the physical measure of change along q (its momentum) must itself be unchanging. It must be conserved.

A conservation law is the logical necessity that arises when a fundamental symmetry (homogeneity of reality) is viewed through the lens of a fundamental identity (the equivalence of momentum and spatial frequency). Noether's theorem is the mathematical engine that formalizes this physical principle.

Part III: Conclusion and Implications

Historical Context and the Software Analogy

Einstein understood this about mass-energy equivalence; he insisted the physics was E ∝ m, and that c² was merely the conversion factor. Newton designed his laws to be unit-invariant. Modern physics, however, has reified these conversion factors into "fundamental constants," mixing the business logic (invariant ratios) with the presentation layer (measurement conventions). Dimensional constants are "magic numbers" in the code of physics, obscuring the elegant architecture underneath.

Conclusion

The dimensional "constants" we treat as fundamental are conversion factors generated by our choice to scale measurement axes independently. The real physics consists of coordinate-free equivalences and dimensionless ratios.

This re-framing does more than clean up our equations. It reveals that the Equivalence Chain is a foundational postulate of reality from which the guiding principles of physics emerge.

• Scale Invariance (Dimensional Analysis) is the requirement that our equations be independent of our arbitrary scaling choices.

• Symmetry Invariance (Conservation Laws) is the requirement that our equations reflect the homogeneity of the underlying reality being scaled.

Both principles are consequences of demanding that our theories describe a single, unified physical reality rather than the arbitrary coordinate system we impose upon it. By separating the bookkeeping from the physics, we find a more parsimonious, powerful, and physically intuitive foundation for our understanding of the universe.