A Formal Proof That "Reduced Planck Units" Are Mathematically Non-Existent

 J. Rogers, SE Ohio

Abstract

We provide a formal proof that "reduced Planck units" (derived from ℏ = h/2π rather than h) cannot exist as a distinct system of measurement. By examining the mathematical structure of dimensional transformations between natural and SI units, we demonstrate that any putative "reduction factor" cancels identically in all meaningful operations. This proves that Planck's original 1899 formulation using h represents the unique, mathematically consistent set of natural units, and that the later "reduced" convention represents a compounding of conceptual errors rather than a refinement.

1. Introduction

In 1899, Max Planck identified combinations of the fundamental constants c, G, and h that yield natural units of length, mass, and time:

l_P = √(hG/c³)
m_P = √(hc/G)
t_P = √(hG/c⁵)

Following the introduction of ℏ = h/2π in quantum mechanics, it became conventional to define "reduced Planck units" by substituting ℏ for h in these expressions. This paper demonstrates that such "reduced" units are mathematically non-existent—they vanish identically in the transformations where Planck units actually function.

2. The Role of Planck Units as Jacobians

Physical quantities in SI units relate to their natural (dimensionless) counterparts through Planck units acting as conversion factors:

Q_SI = Q_nat × Q_P

More precisely, since Q_nat is dimensionless, the transformation is:

Q_SI = Q_nat × (Q_P/Q_P)

where the ratio (Q_P/Q_P) serves as the Jacobian between natural ratios and SI measurements. This ratio equals 1 in any consistent unit system, making it an identity transformation.

3. The Cancellation Theorem

Theorem: Any multiplicative factor applied to Planck units cancels identically in dimensional transformations.

Proof:

Let Q_P represent a Planck unit, and suppose we define a "modified" Planck unit:

Q_P' = Q_P × k

where k is any dimensionless factor (such as √(1/2π) for reduced units).

In the transformation between natural and SI units:

Q_SI = Q_nat × (Q_P'/Q_P')
     = Q_nat × [(Q_P × k)/(Q_P × k)]
     = Q_nat × (Q_P/Q_P) × (k/k)
     = Q_nat × (Q_P/Q_P) × 1
     = Q_nat × (Q_P/Q_P)

The factor k cancels identically, leaving only the original Planck unit ratio.

Corollary: "Reduced Planck units" defined by E_P_reduced = E_P × √(1/2π) (and similarly for other quantities) reduce to:

E_P_reduced/E_P_reduced = E_P/E_P

Therefore, reduced Planck units are mathematically indistinguishable from non-reduced Planck units in all Jacobian operations. ∎

4. Implications

4.1 Non-Existence of Reduced Planck Units

Since Planck units only function meaningfully as ratios in dimensional transformations, and since the reduction factor cancels in all such ratios, "reduced Planck units" do not exist as a distinct mathematical entity. They are a notational artifact that disappears in calculation.

4.2 Preservation of Geometric Structure

Using h (non-reduced) preserves the actual geometric structure of physical laws. Consider Hawking temperature:

Using h (non-reduced):

T ~ 1/(16π²M)  →  T = c³h/(16π²GMk_B)

Using ℏ (reduced):

T ~ 1/(8πM)  →  T = c³ℏ/(8πGMk_B)

The non-reduced form preserves the true dimensionless geometric factor (16π²), while the reduced form obscures it by absorbing 2π into the unit definition. Since 1/2π is a geometric ratio that cannot be set to unity without creating inconsistencies elsewhere ("lumps under the carpet"), the non-reduced formulation is physically clearer.

4.3 Historical Resolution

The introduction of ℏ in the 1920s served a legitimate purpose in simplifying angular momentum expressions in quantum mechanics. However, the subsequent redefinition of Planck units using ℏ represented a category error: confusing a notational convenience in one domain (quantum angular momentum) with a fundamental redefinition of natural units.

Planck's original 1899 formulation using h remains the unique, mathematically consistent choice because:

1. It preserves real geometric factors
2. It avoids introducing canceling factors
3. It represents the minimal, non-redundant basis

5. Formal Statement

Proposition: There exists only one mathematically consistent system of Planck units, corresponding to Planck's original 1899 formulation using h.

Proof: Any modification of Planck units by a constant factor k yields a system that is:

• Mathematically equivalent (via the Cancellation Theorem)
• Physically obscure (hiding geometric structure)
• Conceptually redundant (introducing factors that immediately cancel)

Therefore, the choice k = 1 (non-reduced units using h) is the unique minimal, transparent representation. ∎

6. Conclusion

The mathematical structure of dimensional transformations proves that "reduced Planck units" cannot exist as claimed. The reduction factor √(1/2π) cancels identically in all meaningful operations, leaving only Planck's original 1899 non-reduced units.

This is not a matter of convention or preference—it is a formal mathematical result. The physics community's 125-year adoption of "reduced" Planck units represents a compounding of conceptual misunderstandings rather than progress. The original formulation was correct, and attempts to "improve" it through reduction have merely obscured the clean mathematical structure Planck discovered.

Natural units are the Jacobian transformations between dimensionless physical ratios and human measurement conventions. There is only one such transformation, and Planck found it in 1899.

---

References

Planck, M. (1899). "Über irreversible Strahlungsvorgänge." Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 5, 440-480.

---

Note: This paper presents a mathematical proof, not a physical theory. All results follow from standard dimensional analysis and the established role of Planck units in dimensional transformations. No new physics is proposed; we merely clarify the mathematics of existing practice.