Published Papers

The peer-reviewed papers collected here develop the line of work that runs through this site: a systematic reading of the universal constants and the equations of physics in terms of Planck-scale quantities and dimensionless ratios. Each paper builds on the previous ones, moving from the dimensional structure of Planck’s constant, to the decomposition of matter and radiation formulas, to the precision of constant measurements, to the full Planck-ratio reading of classical and quantum physics. Abstracts are reproduced verbatim; each entry includes a short note on what the paper contributes and where the ideas are developed further on this site.

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Please cite the journal versions of these papers. Where content from a paper appears on this site, the citation and DOI are provided. Figures reproduced from the peer-reviewed papers are attributed under the Creative Commons Attribution 4.0 International License where the publication permits; distribution of those figures must preserve the original attribution.

Peer-reviewed publications

Understanding the natural units and their hidden role in the laws of physics

Humpherys, D. (2024). Understanding the natural units and their hidden role in the laws of physics. European Journal of Physics, 45(5), 055802. https://doi.org/10.1088/1361-6404/ad61d3 · Open access (CC BY 4.0).

Abstract
The natural units of measure lauded by Max Planck more than 100 years ago are underutilized today. Many physical constants, including the Planck constant, the gravitational constant, the speed of light, vacuum permittivity, and vacuum permeability consist of natural units in their unit dimensions. The natural units are present in all formulas containing these constants. The defining characteristic of the natural units is an alignment of unit values at the Planck scale. This alignment gives a computational basis of proportionality from which the correlated properties and dynamics of elementary particles, including wavelength, period, mass, momentum, and energy, manifest in equal or inversely proportional ratios of the Planck scale. These correlations explain many of the defining equations of quantum mechanics, classical gravity, and electromagnetism.

What this paper contributes. A unified treatment of how the universal constants — including the electromagnetic constants — encode Planck-scale quantities, and the proof that physical formulas across quantum mechanics, gravity, and electromagnetism can be rewritten as Planck-scale quantities multiplied by dimensionless Planck-ratio factors. The paper establishes the framework that organizes the rest of the work on this site.

Read on the site: What are Planck units? · The Structure of Universal Formulas · Universal Constants

Measuring Planck’s constant with Compton scattering

Humpherys, D. (2023). Measuring Planck’s constant with Compton scattering. Applied Physics Research, 15(1), 24–30. https://doi.org/10.5539/apr.v15n1p24

Abstract
Measured values of the electron mass and Compton wavelength yield a value of Planck’s constant with a relative standard uncertainty of 3×10−10. This is only slightly larger than the 1.3×10−10 relative standard uncertainty in measurements performed using the Kibble balance. Compton scattering presents an alternative pathway for improving the value of Planck’s constant. Natural units of length, mass, and time offer viable solutions for improving the values of physical constants. While extensive values of the Planck units lie beyond the reach of present-day instrumentation, certain product and quotient pairs of Planck units such as the speed of light can be measured with relatively high precision. Better measurements of certain unit pairs will improve the value of the gravitational constant.

What this paper contributes. The result that product-and-quotient pairs of Planck units — though individually beyond present instrumentation — can be measured to high precision, with direct consequences for the precision of the gravitational constant. The triangle diagram relating measurable constant pairs is introduced here. Compton scattering is shown to yield Planck’s constant with uncertainty comparable to the Kibble balance.

Read on the site: What Planck units reveal about the precision of G (in preparation) · What is the Compton wavelength? (in preparation)

The implicit structure of Planck’s constant

Humpherys, D. (2022). The implicit structure of Planck’s constant. European Journal of Applied Physics, 4(6), 22–25. https://doi.org/10.24018/ejphysics.2022.4.6.227

Abstract
Max Planck derived natural units of length, mass, and time on the assumption that each of the universal constants embodies natural units in its unit dimensions. The four natural units and dimensions comprising Planck’s constant infuse more granular elements into the formulas enriching our understanding of the physical constants and the phenomena they represent. The natural units offer a consistent language for comparing classical and quantum mechanical formulas.

What this paper contributes. The dedicated treatment of Planck’s constant as a composite of Planck-scale quantities — the relationship $\hbar = l_{\mathrm{P}} m_{\mathrm{P}} c$ — and the observation that writing $\hbar$ in this form provides a common language for comparing classical and quantum formulas. Establishes the dimensional-decomposition approach that the later papers generalize.

Read on the site: What is Planck’s constant? · Planck’s constant reference

Natural Planck units and the structure of matter and radiation

Humpherys, D. (2021). Natural Planck units and the structure of matter and radiation. International Journal of Quantum Foundations, 3, 1–20. https://ijqf.org/archives/6382

Abstract
Planck’s constant and the gravitational constant embody natural units of length, mass, and time. When we replace the universal constants with natural Planck units, a hidden structure appears in the equations of physics comprising ratios of length, mass, and time to the Planck scale. It is the proportions of Planck units that define observable physical phenomena and not the composite values of the constants. Natural unit formulas offer more granular information about the structure of matter and radiation than equations written with $\hbar$ and $G$. These natural formulas reveal physical relationships explaining the correspondence between classical and quantum phenomena. Relationships between rest mass, velocity, and wavelength show how classical and quantum mechanical momentum and energy are related, suggesting that momentum is universally a function of wavelength and not velocity.

What this paper contributes. The unified treatment of matter and radiation momentum and energy in natural form — the result that a single formula $p = (l_{\mathrm{P}}/\bar{\lambda})\,m_{\mathrm{P}} c$ describes both, and that kinetic energy factors into a wavelength ratio times a velocity ratio. The argument that momentum is fundamentally a function of wavelength, not velocity, and the three conservation pairs (length–mass, length–momentum, time–energy) are introduced here.

Read on the site: What is momentum? · What is the physical meaning of E=mc2?

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