Constants that Resolve to Unity

Note that all the non-grey items in the table below resolve to unity or are unity by definition, if one uses the Planck units.

The two grey items are included because they are linked to a constant that isn't formally recognised, as far as I know, which I have dubbed the "charge to structure ratio" for ease of reference.

Note that I also made up the terms “reduced permeability” and “raised permittivity” as I have not noticed any similar usage anywhere.  Perhaps the terms are already used, or something similar and, if so, again I’d appreciate knowing.

Normalising permeability and permittivity to unity is certainly not a new concept.  I use the term “resolve” because I am not really doing anything to these values above, other than expressing them in Planck units.  I consider the step of dividing or multiplying by 4π to be a “normalisation”.  This normalisation is applied to my selection of the Planck charge, such that q_Pl²=4πε₀ħc.  Note also that the term 4πε₀ appears in the expression for Coulomb's constant, k_e=1/4πε₀.  In other words, Coulomb's constant is merely the inverse of raised permittivity.  For that reason I have placed them together and generally consider them to be the same constant.

Wikipedia's page on Planck units, at time of writing, indicates that the selection of Planck charge as either q_Pl=√(4πε₀ħc) or q_Pl=√(ε₀ħc) is at the author's choice.  I disagree for two reasons.

First, Planck charge can be written in terms of the Boltzmann constant, noting again that k_e=1/4πε₀:

q_Pl=√(ħc/k_e)

Compare this with the equation for Planck mass:

m_Pl=√(ħc/G)

Second, it can be seen clearly that when q_Pl=√(4πε₀ħc) is selected, the expression for the fine structure constant becomes α=e²/q_Pl², indicating that the value is merely a representation of the ratio between the elementary charge and the Planck charge.  (See also Why I Like Planck, Constants that Resolve to Unity and Coupling Constants.)

 

Note that the precise value of the reduced permeability in SI units at 1.00x10^-7 is merely an artefact associated with the definition of the ampere (which flows through to the coulomb).  That definition included a 2, which is why the reduced permeability involves 4π, while the reduced Planck constant involves 2π.  If the ampere were “the constant current which, if maintained in two straight parallel conductors of infinite length of negligible circular cross section and placed one metre apart in a vacuum, would produce between these conductors a force equal to 1 × 10⁻⁷ newton per metre of length” (rather than the actual 2 × 10⁻⁷), then the normalisation adjustments would all involve 2π.