Natural Units as Coordinate Alignment: A Structuralist Resolution to the "Miracle" of Dimensional Reduction

J. Rogers, SE Ohio

Abstract

In theoretical physics, the technique of "natural units" (simultaneously setting

        $c = h = G = k_B = 1$
      

) is ubiquitous, yet its ontological justification remains obscure. If physical dimensions (Length, Time, Mass) represent distinct, fundamental categories of reality—as asserted by the standard framework—then their simultaneous reduction to a single dimensionless integer violates the Leibnizian principle of the identity of indiscernibles. The standard view treats this reduction as a mere "computational convenience," leaving the coincidence of these scales unexplained. This paper argues that this instrumentalist interpretation is insufficient. Drawing on Ontic Structural Realism, we propose a formal resolution: the "measurement space" of physics is a fibered manifold where physical laws exist as invariant sections. In this model, fundamental constants act as Jacobian connection coefficients transforming between arbitrary local coordinate charts (SI) and the intrinsic geometry of the base manifold. "Natural units" are identified not as an arbitrary choice, but as the unique diagonalization of the coordinate transformation matrix. Crucially, this framework demands the normalization of the full Planck constant (

        $h=1$
      

) rather than the reduced constant (

        $\hbar=1$
      

), arguing that the latter conflates linear scaling with geometric rotation.

1. The Ontological Gap: The Problem of Coincidence

Standard pedagogy treats the constants of nature (

        $c, h, G, k_B$
      

) as fundamental parameters that define the scale of physical regimes. Yet, advanced work routinely sets these parameters to unity. This creates a tension:

1. The Distinctness Claim: Mass, Length, and Time are ontologically distinct categories.

2. The Identity Operation: Natural units assert

           $1 \text{ Mass} \sim 1 \text{ Length} \sim 1 \text{ Time}$
         

   .

If

        $c$
      

and

        $G$
      

are independent physical entities, their simultaneous unification is a "miracle"—a coincidence of scales that the standard framework utilizes but cannot explain. This effectively posits a universe where the most fundamental description requires the erasure of the very dimensional categories the theory is built upon. We term this the Problem of Dimensional Reduction: Why does nature permit a coordinate system in which distinct physical dimensions are numerically identical?

2. A Formal Model: The Measurement Bundle

To resolve this, we move from an entity-based ontology to a structuralist one, aligning with the work of Ladyman and Ross [1] and March-Russell [2]. We posit that physical quantities are not intrinsic properties of objects, but projections of dimensionless relations onto a "Measurement Bundle."

2.1 The Geometric Structure

Let

        $\mathcal{S}$
      

be the Substrate, a manifold of dimensionless physical events where laws exist as invariant proportionalities (e.g.,

        $E \propto f$
      

).
Let

        $\mathcal{F}$
      

be the Measurement Fiber, a vector space representing possible scale choices (e.g.,

        $\mathbb{R}^n$
      

spanning Mass, Length, Time).

A specific system of units (like SI) is a local trivialization of this bundle—a specific choice of basis vectors

        $\mathbf{e}_{SI} = \{kg, m, s, K\}$
      

.

2.2 Constants as the Jacobian Matrix

The fundamental constants are not scalar properties found in

        $\mathcal{S}$
      

. They are the components of the Jacobian matrix

        $\mathbf{J}$
      

that transforms the intrinsic, dimensionless basis of the Substrate

        $\mathbf{e}_{Nat}$
      

to the arbitrary basis of the observer

        $\mathbf{e}_{SI}$
      

.

Consider the linear relation

        $E_{SI} = h \cdot f_{SI}$
      

. In our formalism, this indicates that the Jacobian element mapping the intrinsic Time

        $^{-1}$
      

basis to the SI Energy basis carries a coefficient of

        $h$
      

:

        $$J_{f \to E} = h$$    

Similarly,

        $c$
      

is the Jacobian element mapping Time basis to Length basis (

        $J_{t \to \ell} = c$
      

), and

        $k_B$
      

maps Energy to Temperature. The matrix

        $\mathbf{J}$
      

effectively encodes the "metric" of the fiber, defining the scaling and orthogonality relations between our chosen units.

3. Natural Units as Matrix Diagonalization

In this formalism, the operation of "using natural units" ceases to be an arbitrary scaling choice and becomes a precise geometric operation: Diagonalization.

We seek a basis transformation such that

        $\mathbf{J}$
      

becomes the Identity matrix

        $\mathbf{I}$
      

. This condition:

        $$c = h = G = k_B = 1$$     

implies that we have aligned our measurement axes with the eigenvectors of the transformation matrix.

This forces the selection of the Planck basis:

        $$\ell_P = m_P = t_P = T_P = 1$$      

3.1 The Eigen-Scales of Measurement

The "Planck units" are therefore not merely "small" numbers. They are the eigen-scales of the measurement operator. They represent the unique point in the moduli space of unit systems where the projection of a unit interval onto any axis yields the same numerical value.

This provides the ontological justification the standard framework lacks: Natural units work because the underlying physics is defined on a dimensionless substrate, and the Planck basis is the unique isomorphism between that substrate and our description of it.

4. Implications and Predictions

This structuralist interpretation yields specific constraints that differ from conventional practice.

4.1 The

        $h=1$
      

vs.

        $\hbar=1$
      

Distinction

Standard QFT practice sets

        $\hbar = 1$
      

. This framework predicts this is a category error.

•         $h$
        

  is the linear scaling factor between frequency (cycles/time) and energy.

•         $\hbar = h/2\pi$
        

  introduces a geometric factor (

          $2\pi$
        

  ) related to rotation (radians/cycles).

To diagonalize the linear measurement matrix, one must normalize the linear scaling factor (

        $h$
      

), not the geometric one. Setting

        $\hbar=1$
      

essentially defines the natural unit of angle as the radian, whereas setting

        $h=1$
      

defines it as the cycle. Since the substrate relations (like

        $E \propto f$
      

) are linear proportionalities,

        $h=1$
      

is the topologically correct diagonalization.

4.2 Unification of Unit Systems via Sub-Bundle Diagonalization

Stoney units, Planck units, and Atomic units are often treated as competing conventions. In this framework, they are simply diagonalizations of different sub-matrices of

        $\mathbf{J}$
      

:

• Planck Units: Diagonalize the full matrix involving

          $G$
        

  (Gravity sector).

• Atomic Units: Diagonalize the sub-matrix involving

          $m_e, e, h$
        

  (QED sector).
  This explains their domain-specific utility: they are the natural orthonormal bases for different sub-bundles of the total theory.

4.3 Separation of Constants

This framework resolves the confusion between "dimensional" and "dimensionless" constants [3].

• Dimensional constants (

          $c, h, G$
        

  ) are artifacts of the coordinate chart

          $\mathbf{J}$
        

  . They are explained by this theory.

• Dimensionless constants (

          $\alpha, m_p/m_e$
        

  ) are intrinsic parameters of the invariant relations in

          $\mathcal{S}$
        

  . They remain unexplained here, but are cleanly isolated from coordinate artifacts.

5. Discussion: Structural Realism in Metrology

This framework extends Ontic Structural Realism to metrology. The relations (dimensionless ratios) are the ontic reality; the relata (dimensionful quantities) are artifacts.

The Buckingham

        $\pi$
      

theorem [4] can thus be reinterpreted: it is a method for finding the kernel of the Jacobian map

        $\mathbf{J}$
      

—identifying the invariant combinations that lie in the null space of the unit transformation.

6. Conclusion

The history of physics can be viewed as an asymptotic drift toward the natural coordinate basis, culminating in the 2019 SI redefinition [5] which fixed the constants, effectively defining units by intrinsic geometry.

This paper demonstrates that the "convenience" of natural units is actually the manifestation of a deep physical truth: the universe is fundamentally dimensionless. The "fundamental constants" are the artifacts of our deviation from this truth. By identifying constants as Jacobian connection coefficients and natural units as coordinate diagonalization, we resolve the explanatory gap, offering a rigorous ontological foundation for the operational success of modern physics.

---

References

1. Ladyman, J., & Ross, D. (2007). Every Thing Must Go: Metaphysics Naturalized. Oxford University Press.

2. March-Russell, C. R. C. (2006). "Dimensionless Constants and the Structure of Physical Law."

3. Wilczek, F. (2001). "Fundamental Constants." Physics Today.

4. Buckingham, E. (1914). "On Physically Similar Systems." Physical Review.

5. BIPM. (2019). The International System of Units (SI), 9th edition.

6. Rogers, J.  (2025).  The Structure of Physical Law as a Grothendieck Fibration