The Ghost in the Machine: How Measurement Artifacts Became Fundamental Constants, Obscuring the Path to Unification

J. Rogers, SE Ohio

Introduction

The speed of light c, the gravitational constant G, and Planck’s constant h are the sacred pillars of modern physics. We treat their values as cosmic DNA, holding the deepest secrets of the universe. We ask, "Why these numbers?" as if interrogating the mind of God. This paper argues that this is the wrong question, born from a profound misunderstanding.

These dimensional constants are not fundamental properties of nature. They are harmonization coefficients—ghosts in our theoretical machine, generated by the arbitrary way we measure the world. They exist only to correct for our historical decision to define independent scales for length, time, and mass, when reality is a single, unified entity.

By deconstructing this illusion, we will do more than just clean up our equations. We will demonstrate that the very unity obscured by these constants provides a direct physical origin for the two most powerful pillars of theory construction: scale invariance (enforced by dimensional analysis) and symmetry invariance (which gives us conservation laws). The constants are not the message; they are the static, and by tuning them out, we can hear the music of reality more clearly. Furthermore, this perspective reveals a crucial insight for the search for a "theory of everything": it should not be a theory of constants, but a theory of the underlying process that forces their commensurability.

Part I: Deconstructing the Constants

The Illusion of Independent Axes

Our physical theories conflate three distinct concepts:

1. Arbitrary Measurement Scales: The meter, kilogram, and second. These are human conventions with no intrinsic physical meaning.

2. Invariant Physical Relationships: The true, coordinate-free laws of nature. This is the real physics.

3. Dimensional Constants: Numbers like c, G, and h that harmonize our arbitrary scales with the invariant laws.

The core error is assuming our measurement axes are independent. We believe we can choose a unit of length and a unit of time separately. But the underlying physics connects them through fixed equivalences. The practice of using "natural units" reveals this profound truth: the quantities we measure are not fundamentally distinct, but are different, perfectly commensurate, manifestations of a single underlying physical reality.

This is best expressed through the Equivalence Chain, the central postulate of this framework:

T_nat = f_nat = m_nat = 1/l_nat = E_nat = p_nat = F_nat = ...

This chain is not a statement of identity. Energy is not literally frequency. Instead, the Equivalence Chain asserts that quantities like energy, momentum, and spatial frequency are distinct physical effects that are all co-scaled by a single, underlying degree of freedom of nature. They are different readouts from the same fundamental meter.

How Harmonization Coefficients Emerge

Constants are generated mechanically when we map this unified reality onto our fragmented measurement system. To convert from natural units to SI, we multiply by 1, expressed as a ratio of Planck units (e.g., E_P/E_P). This adds no physical information; it is purely a bookkeeping step. The novelty of this approach is demonstrating that the system of units itself creates the illusion of fundamental constants.

Let’s watch a constant materialize. The physics is the commensurability of energy and frequency. Now, we impose our measurement system:

1. Start with the physics: Energy and frequency are perfectly co-scaled.

2. Scale each side to SI: E_si / E_P = f_si × t_P

3. Solve for the SI energy: E_si = f_si × (E_P × t_P)

We define the composite scaling factor (E_P × t_P) as Planck’s constant, h.

Planck's constant is not a property of nature. It is a property of our measurement system, forced upon us when we try to simultaneously measure two perfectly commensurate effects using two independent, historically-contingent rulers.

Conversely, consider Newton’s second law, F = ma. Why is there no constant? Because when we trace the scaling factors, they happen to cancel out perfectly:

F_si / F_P = (m_si / m_P) × (a_si / a_P)

The relationship between the Planck units F_P, m_P, and a_P is such that they cancel, leaving F_si = m_si × a_si. No constant appears because our conventional definitions for force, mass, and acceleration are already harmonized. Constants appear precisely where our independent scaling choices create a dimensional imbalance.

• c harmonizes our independent time axis with our length axis.

• G harmonizes our mass-energy axis with our spacetime curvature axis.

• k_B harmonizes our energy axis with our temperature axis.

The real, invariant physics lies in dimensionless ratios like the fine structure constant (α ≈ 1/137) and the Equivalence Chain itself. The rest is just accounting.

Part II: Reconstructing Physics from Unity

This perspective does more than demystify constants; it provides a physical origin for the rules we use to build our theories.

Principle I: The Physical Origin of Dimensional Analysis

Dimensional analysis is often taught as a formal mathematical trick. This framework reveals its physical foundation. Dimensions are simply bookkeeping tags that track the arbitrary scaling factors (l_P, t_P, m_P) we 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 fails this test is not a statement about physics; its truth value depends on whether you use meters or feet. It is a statement about convention.

Therefore, the requirement of dimensional consistency is the mathematical enforcement of the demand that our laws describe nature, not our rulers. The Buckingham Pi theorem is not a mathematical curiosity; it is the procedural consequence of this fundamental principle. Dimensional analysis enforces measurement independence.

Principle II: The Physical Origin of Conservation Laws

Noether’s theorem mathematically connects continuous symmetries to conserved quantities. It is a cornerstone of physics, yet its physical origin remains abstract. This framework provides that origin by postulating that the quantities we measure are not fundamental, but are different, perfectly commensurate manifestations of a single underlying physical reality. This is where the novelty of this approach becomes apparent.

The derivation is as follows:

1. The Foundational Postulate: The Equivalence Chain (E_nat = f_nat = p_nat = 1/l_nat = ...) is not a statement of identity, but of perfect commensurability. It postulates that quantities like energy, momentum, and spatial frequency are distinct physical effects that are all co-scaled by a single, underlying degree of freedom of nature. They are different readouts from the same fundamental meter.

   • Momentum is the readout of this reality's expression over space.

   • Energy is the readout of this reality's expression over time.

2. The Bridge to Formalism: The Principle of Stationary Action is the engine of our theories. The Lagrangian formalism defines canonical momentum as p ≡ δS/δq. This framework asserts that this is not a mere mathematical convenience. It is the formal recognition that momentum is the physical effect generated by nature’s underlying degree of freedom as it varies with respect to the coordinate q. The formalism correctly identifies the conjugate variable because it is tracking the physical output of this deeper process.

3. The Synthesis and Derivation: A continuous symmetry (e.g., spatial translation) is a statement that the underlying physical reality is homogeneous—its fundamental state is indifferent to the absolute value of that coordinate q.

   • Symmetry means: The underlying reality does not depend on q.

   • The Postulate means: Momentum (p) is the measured physical effect of that reality's expression along q.

   • Therefore, the derivation is: If the underlying reality does not change with q, then its measured physical effect along q (the momentum) must be constant. It must be conserved.

This is a direct physical derivation of the principle of conservation, a novel claim distinct from a mere interpretation of Noether's theorem. It is a statement about its physical source. A conservation law is the necessary consequence of a symmetry of the underlying reality, which in turn forces the constancy of the corresponding measured physical effect.

This reframes conservation laws from being a property of our mathematical models to being a direct window into the structure of the unified reality that our models attempt to describe.

Part III: Implications for Unification

If the constants are artifacts, what does this imply for the ultimate goal of physics: a "theory of everything"?

The lesson is clear: The search for a theory of everything should not be a search for a theory of the constants. It should be a search for a theory of the underlying process or reality that forces their commensurability.

If the Equivalence Chain is correct, then the deepest laws of nature are not equations that relate individually measured quantities. They are the rules governing the single, unobservable degree of freedom whose "readouts" we perceive as mass, energy, time, and space.

The search for a unified theory is, therefore, a search for the fundamental "meter" of the universe and the rules by which it projects itself onto the conceptual axes we perceive.

Conclusion: Refactoring Physics

From a software engineering perspective, modern physics has mixed its concerns. The business logic (invariant relationships, co-scaling by underlying reality) is entangled with the presentation layer (our arbitrary SI units). The dimensional constants c, G, and h are magic numbers hardcoded throughout the system, obscuring the elegant, underlying architecture.

A refactored physics separates these concerns. It recognizes that the true laws of nature are the coordinate-free commensurabilities and the dimensionless ratios that emerge from them. This perspective reveals that our most powerful theoretical tools are not abstract rules, but direct consequences of a single, unified reality.

• 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 homogeneities of the underlying reality being scaled.

Both principles emerge from a single demand: that our theories describe the unified fabric of reality, not the patchwork quilt of coordinates we lay upon it. The "fundamental constants" are not profound truths. They are the seams in our quilt, and it is time we looked past them to the seamless reality they conceal. By recognizing the constants as ghosts in the machine, we can begin to build a physics that is truly unified, not merely harmonized.