An Internal Inconsistency in the ħ-Based Planck Unit System Arising from the Assumption of Unified Scaling

J. Rogers, SE Ohio

Note to the Reader: A Clarification of the Core Argument

The central point of this paper can be understood through a simple observation about the ħ-based Planck unit system: while it allows us to write both "E = f" and "E = m" in Planck units, these two expressions represent fundamentally different dimensionless energy ratios that differ by a factor of 2π.

In the ħ-based system:

• When we express energy in terms of mass, we get E = m (in Planck units)
• When we express energy in terms of frequency, we get E = 2πf (in Planck units)

This discrepancy means that the system cannot simultaneously satisfy both the Principle of Unified Scaling (which requires E = m = f) and the Planck-Einstein relation (E = hf = 2πħf). The following paper demonstrates mathematically that this 2π factor is not merely a convention but represents an internal inconsistency that prevents the ħ-based Planck system from truly unifying physical concepts.

Abstract
The Planck unit system, based on the fundamental constants, is widely used in theoretical physics with the goal of simplifying equations and revealing a deeper unity among physical quantities. The standard formulation of these units uses the reduced Planck constant, ħ. This paper demonstrates that the ħ-based Planck system is internally inconsistent when combined with the foundational assumption that a "natural" unit system should exhibit unified scaling across all physical domains. We prove through direct derivation that demanding a consistent scaling relationship between mass (m) and frequency (f) within the ħ-based framework necessarily leads to the conclusion mc² = ħf. This result stands in direct contradiction to the experimentally inviolate Planck-Einstein relation, E = hf. We conclude that the standard ħ-based Planck unit system is mathematically incapable of representing a truly unified physical reality, as it creates a 2π discrepancy between the quantum nature of energy and its relativistic mass-equivalence.

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1. Introduction

Natural unit systems are a cornerstone of modern theoretical physics. By setting certain fundamental constants to the dimensionless value of 1, they aim to strip away human-centric measurement conventions to reveal the raw mathematical structure of physical laws. The most prominent of these is the Planck unit system, which in its standard formulation sets c = G = ħ = k_B = 1.

A core, often implicit, assumption behind this practice is that such a system will reveal a "unity" among physical concepts that are disparate in conventional SI units. This unity can be expressed mathematically as a unified scaling law, where quantities like mass, energy, frequency, and temperature all scale in a lockstep, proportional manner.

This paper tests that assumption. We will demonstrate that the standard choice of ħ (the reduced Planck constant, h/2π) as a foundational constant is not a matter of convention or convenience, but is instead a choice that introduces a fundamental and irreparable inconsistency into the system's ability to represent this unity.

2. The Principle of Unified Scaling

For a natural unit system to be truly "unified," it must allow for a consistent and proportional relationship between its base quantities. We can formalize this concept as the Principle of Unified Scaling. This principle states that for any physical system, the dimensionless ratio of a measured quantity to its corresponding Planck unit must be the same across all domains.

Let m_P, t_P, and E_P be the Planck mass, time, and energy, respectively. For a system with mass m, characteristic frequency f, and rest energy E, this principle is expressed as:

$$\frac{m}{m_P}=f\cdot t_P=\frac{E}{E_P}=\dots (1)$$

This equation is the mathematical statement of unity. This paper will perform a proof by contradiction by assuming this principle holds true for the standard ħ-based Planck system and showing that it leads to a physically incorrect result. We will denote the ħ-based Planck units with a subscript ħ (e.g., m_Pħ).

3. Derivation of the Mass-Frequency Relationship in the ħ-System

We begin by examining the relationship between mass and frequency implied by the Principle of Unified Scaling within the ħ-based system. From Equation (1), we select the mass-frequency link:

$$\frac{m}{m_{P\mathrm{\hslash }}}=f\cdot t_{P\mathrm{\hslash }}(2)$$

We can solve for mass, m, in terms of frequency, f:

$$m=f\cdot (m_{P\mathrm{\hslash }}\cdot t_{P\mathrm{\hslash }})(3)$$

To evaluate the term in parentheses, we must define the standard ħ-based Planck units in terms of the fundamental constants ħ, c, and G:

• Planck Mass: $ m_{P\hbar} = \sqrt{\frac{\hbar c}{G}} $

• Planck Time: $ t_{P\hbar} = \sqrt{\frac{\hbar G}{c^5}} $

Now, we compute their product:

$$m_{P\mathrm{\hslash }}\cdot t_{P\mathrm{\hslash }}=\sqrt{\frac{\mathrm{\hslash }c}{G}}\cdot \sqrt{\frac{\mathrm{\hslash }G}{c^5}}=\sqrt{\frac{{\mathrm{\hslash }}^{2}cG}{G{c}^5}}=\sqrt{\frac{{\mathrm{\hslash }}^2}{c^4}}$$

$$m_{P\mathrm{\hslash }}\cdot t_{P\mathrm{\hslash }}=\frac{\mathrm{\hslash }}{c^2}(4)$$

Substituting this result back into Equation (3), we find the relationship between mass and frequency that is mandated by the internal logic of the ħ-based unified scaling:

$$m=f\cdot \frac{\mathrm{\hslash }}{c^2}$$

Rearranging this equation yields a stunning result that takes the form of a mass-energy equivalence:

$$m{c}^2=\mathrm{\hslash }f(5)$$

4. The Inescapable Contradiction

The result derived in Equation (5) is not a statement of choice or convention; it is the necessary mathematical consequence of demanding unified scaling within the standard ħ-based Planck system. The term mc² is universally defined as the rest energy, E, of a particle of mass m. Therefore, the ħ-based unity framework predicts:

$$E=\mathrm{\hslash }f(\text{Prediction from ħ-System Unity})$$

However, the foundational and experimentally verified law connecting the energy of a quantum system to its fundamental frequency is the Planck-Einstein relation:

$$E=hf(\text{Observed Physical Law})$$

Since h = 2πħ, these two statements are mutually exclusive. The energy predicted by the ħ-based system's own assumption of unity is incorrect by a factor of 2π:

$$E_{\text{predicted by ħ-unity}}=\frac{1}{2\pi }{E}_{\text{actual}}$$

The system is therefore fundamentally inconsistent. It cannot simultaneously satisfy its own scaling laws and agree with the established laws of quantum mechanics. The initial assumption—that the ħ-based system can support the Principle of Unified Scaling—must be false.

5. Discussion and Conclusion

The inconsistency demonstrated is not a minor numerical issue; it is a catastrophic failure of the ħ-based Planck system to serve its primary purpose of unifying physical concepts. The choice to standardize on ħ was made for mathematical convenience in equations of motion and wave mechanics, which naturally involve radians (ω = 2πf), leading to the tidy expression E = ħω.

However, we have shown that this convenience comes at a devastating cost: it breaks the more fundamental unity between the inertial property of a system (mass) and its quantum oscillatory count (frequency). The ħ-system forces an artificial 2π factor between these domains, shattering the simple, direct scaling that a truly "natural" system ought to reveal.

In conclusion, the widespread use of ħ-based Planck units in the search for unified theories (such as String Theory and Quantum Gravity) is built upon a mathematically unsound foundation. Any theory that begins by setting ħ=1 has already, and perhaps unknowingly, accepted a framework that is inconsistent with the unified scaling of nature's laws. This paper formally demonstrates that the standard Planck unit system is disproven as a valid tool for this purpose, and its continued use in foundational research should be subject to immediate and critical re-evaluation. A consistent unified framework is only achievable if built upon the Planck constant, h.