The Upgraded Planck System of Units that Reaches from the Known Planck Scale All the Way Down to Subatomic Scales

1. Introduction and Motivation

1.1. Three Fundamental Systems of Units Under Consideration

In a recent paper [1], we used a cosmological system of units based on the speed of light c, Newton’s gravitational constant G, and MOND’s characteristic acceleration $a_0$[2,3,4]. Since G is a building block of this system, then $a_0$ can substitute for the universal MOND unit, the mysterious ${\mathcal{A}}_0\equiv a_{0}G$. Besides ${\mathcal{A}}_0$, pairs of the fundamental units produced two more de facto important dynamical units: the surface density ${\sigma }_0=a_0/G$, and the force $F_0=c^4/G$. Notice a deviation implicit in unit $F_0$: the inversion of G produces the term $c^{4}G$ which is only a composite unit of no particular interest with dimensions of [M]⁻¹[L]⁻¹[T]⁻⁶$=c^8/F_0$. This deviation singles out the unit of force $F_0=c^4/G$ as an important component of the cosmological system (along with the pair ${\mathcal{A}}_0$ and ${\sigma }_0$), but there is not much more one can do with it at this point, besides noting that the same unit of force appears in the famous Planck system of units as well, and that $F_0$ is a geometry-independent quantity, since both c and G do not carry an imprint of the geometry of our four-dimensional spacetime.

The cosmological system of units does not use Planck’s constant h[5,6], which turns out to be a derived unit of no particular interest, but current thinking forgives the oversight—after all, this is a cosmological system designed for measurements in universal scales. Nevertheless, we were drawn into comparisons with the Planck system which now uses Dirac’s $\cancel{h}=h/\left(2\pi \right)$ as a fundamental unit [7,8,9]; and, soon enough, we also added Hartree’s atomic system of units [10], which paradoxically does not use c (or G, for that matter) as a fundamental unit. The immediate implication is that the speed of light is not an upper limit in the atomic world, where the unit of speed is ${\mathsf{\alpha }}_{\cancel{h}}c\ll c$, where ${\mathsf{\alpha }}_{\cancel{h}}=1/137.036$ is the fine-structure constant. Under these circumstances, we ended up juggling three different fundamental systems of units, comparing and contrasting their building blocks and the assumptions that have gone into their architectures.

1.2. Dirac’s Problematic Constant ${\mathsf{\alpha }}_{\cancel{h}}$ and the Three Widely-Used Atomic Radii

The modern atomic and Planck systems use Dirac’s $\cancel{h}=h/\left(2\pi \right)$[7] instead of Planck’s original and purely physical constant h[5]. This modification is not trivial because the $2\pi$ carries the “unit” of radians which, although not a physical unit, alerts us to the presence of 2-D geometry. The descriptive word radian has been dropped by many authors and also by the SI system of units, leading to a widespread misunderstanding that $\cancel{h}$ simply absorbs a numerical factor of $2\pi$ with no further ramifications. The inconsistency was noted by Bunker et al. [11] who inserted the unit of radian in the definition of $\cancel{h}$ and the unit of cycle in the definition of h. The SI system must reinstate at least the radian “unit” as a descriptive word, because it alerts us to the presence of geometry (see below). The same holds for trigonometric functions whose arguments must always be in radians—although this is such a common knowledge that the radian is no longer mentioned. On the other hand, the radian is not dropped from the unit of angular velocity, which has always been radians/sec, where “radians” is a descriptive term and “sec” is the only physical unit.

Dirac believed that $\cancel{h}$ is the true universal constant, and we can only guess the reason why: the $2\pi$ in $\cancel{h}$ has introduced 2-D geometry into the constant, so, unlike Planck’s h, the constant $\cancel{h}$ is not purely physical, it is a composite constant. This fact was effectively proven by Leblanc et al. [12] who showed that the Compton radius $r_{\mathrm{c}}$ (where $r_c\propto \cancel{h}$) also includes a geometric component. The seminal results presented in Refs. [11,12] have important consequences in physics that become detectable when we write side-by-side the three famous electronic radii of the atomic world (where $m_{\mathrm{e}}$ is the electron mass and e is the fundamental positive charge):

$$\begin{array}{cc}\hfill \mathrm{Classical}\mathrm{radius}:& r_{\mathrm{e}}=e^2/\left({ϵ}_0{m}_{\mathrm{e}}{c}^2\right)=r_{\mathrm{c}}{\mathsf{\alpha }}_{\cancel{h}}\hfill \\ \hfill \mathrm{Compton}\mathrm{radius}:& r_{\mathrm{c}}=\cancel{h}/\left(m_{\mathrm{e}}c\right)\hfill \\ \hfill \mathrm{Bohr}\mathrm{radius}:& r_{\mathrm{b}}={ϵ}_0{\cancel{h}}^2/\left(m_{\mathrm{e}}{c}^2\right)=r_{\mathrm{c}}{\mathsf{\alpha }}_{\cancel{h}}\hfill \end{array}$$

(1)

Here, ${\mathsf{\alpha }}_{\cancel{h}}$ is the fine-structure constant and ${\cancel{ϵ}}_0$ is the reduced vacuum permittivity defined by ${\cancel{ϵ}}_0\equiv 4\pi {ϵ}_0$, an equation that shows how the stereometry of space modifies the physical unit ${ϵ}_0$ of the vacuum. Therefore, we have an SI unit problem here too, just as Bunker et al. [11] discovered for $\cancel{h}$. The vacuum is a three-dimensional space, hence the stereometric term of $4\pi$; thus, the units of ${\cancel{ϵ}}_0$ must also include the descriptive word steradians.1 Now, these geometric considerations show why three different radii do exist in atomic physics: although they have the same physical dimension of length [L], they capture entirely different geometries; the electrons in the atoms venture in 3-D space (their orbitals are 3-D structures, hence, the $1/\left(4\pi \right)$ in $r_{\mathrm{e}}$); the emitted photons only “see” two dimensions (see footnote 1); and the electrons in the Bohr model of the atom are quantized and they see only discrete sectors embedded in 3-D space (hence, the $1/\pi$ factor in $r_{\mathrm{b}}$). The factor of 1/4 “missing” from the $1/\pi$ is however applied to the energy levels, because this factor is included in the Rydberg energy $E_{\mathrm{R}}$ (see below).

Deriving the geometric pattern of the quantized radii $r_n$ of the Bohr model is a little harder, yet within our grasp.2 In any case, the factor of 1/4 is necessarily missing from $r_{\mathrm{b}}$, so that the quantized angular momentum ${\mathcal{L}}_n$ is truly a 2-D quantity (${\mathcal{L}}_n\propto \cancel{h}\propto 1/\left(2\pi \right)$), and the associated Rydberg energy $E_{\mathrm{R}}$ is independent of geometry (although the abolished geometry contributes a unitless constant of 1/4, i.e., $E_{\mathrm{R}}\propto 1/{{\cancel{ϵ}}_0\cancel{h})}^2\propto 1/{[4\pi /\left(2\pi \right)]}^2=1/4$, the same factor as that “missing” from $r_{\mathrm{b}}$).

Lastly, the Bohr radius is the fundamental unit of length in the atomic system [10], but we argue that the Compton radius is actually the most important unit because its definition in equation (1) does not contain the fine-structure constant ${\mathsf{\alpha }}_{\cancel{h}}$. Furthermore, there is more circumstantial evidence that $r_{\mathrm{c}}$ is important among the three radii shown in equation (1): $r_{\mathrm{c}}$ is the geometric mean of $r_{\mathrm{e}}$ and $r_{\mathrm{b}}$ (i.e., $r_{\mathrm{c}}=\sqrt{r_{\mathrm{e}}{r}_{\mathrm{b}}}$), and this implies that the Compton radius $r_{\mathrm{c}}$ is also the geometric mean of all three length scales combined together, viz.

$$r_{\mathrm{c}}=\sqrt[3]{r_{\mathrm{e}}{r}_{\mathrm{c}}{r}_{\mathrm{b}}}.$$

(2)

Thus, $r_{\mathrm{c}}$ is singled out among the three electronic radii for further duty in all systems of units (but probably without the $2\pi$ term, in order to remove the artificially inserted 2-D geometry). Furthermore, it may not be as obvious yet, but geometric averaging plays a huge role in nature as was first discovered in Ref. [1]. The above geometric means (hereafter, denoted as G-Ms, to avoid confusion with the famous heliocentric constant “$GM$”) are only a prelude to their ubiquitous appearances in many G-M combinations of natural constants and physical quantities as well.

1.3. Dimensionless Constants

The general notion about constructing a system of units is that one is free to choose any units to be the building blocks. Dimensionless constants do not have units to offer, thus they are not chosen as building blocks. They remain as passive invariants in any adopted system of units, and they serve mostly as cross-checks of the various calculations performed between dimensional quantities. The current thinking is summarized in the following excerpt from Zeidler [14]:

“A special role is played by those physical quantities which are dimensionless in the SI system. We expect that such quantities are related to important physical effects. The experience of physicists confirms this.”

So, we suspect that such constants are important in physics, but we do not really know what to do with them beyond their ascertained invariance, simply because they lack units.

The general notion about constructing a physical system of units is wrong on two counts: (1) Although unitless constants do not have units to offer, they must be actively included in systems of units, because they introduce the natural forces that cause the important physical effects mentioned by Zeidler [14]. (2) We are not free to choose any dimensional units as building blocks; we must choose wisely the units that measure the fundamental forces and, in addition, those units dictated by the vacuum itself. In particular, choosing a favorite particle to supply its properties for building blocks could be a bad idea,3 and the reason is that such favoritism would violate a principle of fairness in this world. As will be seen below, nature does not at all favor or neglect any particle or force field, not even the “very small” ones against the “very large” ones, and vice versa.

One or both of the above defects have crept into our systems of units, where they selectively crippled or eliminated entirely some fundamental forces of nature. A natural force is crippled when its dimensional or its dimensionless constant is not included as a building block of a system of units; and a force is eliminated entirely when both of its defining constants are not included in a system of units.

1.4. Outline

In this work, we construct a self-consistent system of units that does not suffer from the above defects and that includes gravity, electromagnetism, and the weak interaction (§  2). The coupling constant of the strong interaction is not included yet, because massive particles at the TeV energy scale have not been discovered. We are going to test this system’s performance in atomic and subatomic scales, and in the Planck and the macroscopic scales as well (§ 3).

It turns out that the Planck system is easier to upgrade, because it already includes the appropriate dimensional constants $\{c,Z_0,G,\cancel{h}\}$, although the impedance of free space $Z_0$ has so far been sidelined. So, what we need to do for the upgrade is to activate the unitless coupling constants ${\mathsf{\alpha }}_G$ (gravitational coupling constant) and ${\mathsf{\alpha }}_{\cancel{h}}$ (fine-structure constant); and to repair the damage that $\cancel{h}$ has caused by inadvertently introducing geometry in them (see footnote 9 for details), besides the well-intended quantum forces.

In § 2, we describe the building blocks of the upgraded Planck system of units. In § 3, we collect the new results concerning masses, charges, and lengths in the new system. In § 4, we discuss the results, and, in § 5, we summarize potential issues still lingering in this system of units, as well as some future research prospects.

Finally, in § 6, we list the most important highlights of our investigation, including the results obtained in the Appendices. In Appendix A, we derive from first principles the long-sought physical significance of Koide’s lepton constant $K=2/3$[15] of atomic physics; the Higgs couplings to the vector bosons [16] and the bottom quark; and the Weinberg angle [16] in terms of K. In Appendix B, we discuss the universality of the Tully-Fisher/Faber-Jackson relation [17,18] discovered in spiral and elliptical galaxies, respectively (see also Ref. [1]). This fundamental relation that relates the fourth power of a kinetic scalar to a surface density signifies a new universal law of nature that has manifestations in several other parts of physical science besides astrophysics.

2. The Building Blocks of the Upgraded Planck System

The upgraded Planck system (UPS) includes the following building blocks at the very least:

$$\mathrm{UPS}:=\{c,Z_0,G,{\mathsf{\alpha }}_G,\cancel{h},{\mathsf{\alpha }}_{\cancel{h}}\},$$

(3)

where $\{c,Z_0,G,\cancel{h}\}$ are dimensional units and $\{{\mathsf{\alpha }}_G,{\mathsf{\alpha }}_{\cancel{h}}\}$ are dimensionless units. We use a slash to indicate the presence of geometric units, which are undesirable. This is a problem we have to contend with throughout this work. In § 2.2, we will be ready to replace the units $\{\cancel{h},{\mathsf{\alpha }}_{\cancel{h}}\}$ with Planck’s units $\{h,{\mathsf{\alpha }}_h\}$ in the UPS in order to eliminate the 2-D geometry introduced to Dirac’s constants by the $2\pi$ term (see also footnote 27 below).

2.1. Dimensional Units

For future reference, we need to recall and emphasize a gem of natural units, the fundamental dimensional relation between gravity (supplying G) and electromagnetic (EM) forces (supplying also the vacuum’s constant ${\left(4\pi {ϵ}_0\right)}^{-1}$). This is obtained by equating the dimensions of the forces in Newton’s gravitational law and Coulomb’s law for two electrons. We find, in dimensional form, that

$$G{m}_{\mathrm{e}}^2\sim {\left(4\pi {ϵ}_0\right)}^{-1}{e}^2,$$

(4)

where $m_{\mathrm{e}}$ is the mass of the electron and e is the elementary positive charge. Here, we write down explicitly the vacuum permittivity ${\cancel{ϵ}}_0$ as $4\pi {ϵ}_0$ to ensure that its geometric content (the $4\pi$ factor in the EM term) is clearly noticeable.

Each side of equation (4) becomes unitless when divided by $\cancel{h}c$, as is done separately in the definitions of the two fundamental coupling constants. Unfortunately, the $\cancel{h}$ introduces then additional geometry into the gravitational part and eliminates geometry from the EM part, clearly altering the original geometrical characteristics of the two coupling constants (see § 2.1.2 below). The unit of [rad]⁻¹ has been dropped from $\cancel{h}$ by international agreement, so this intrusion of geometry is no longer visible [11]; going as far back as Schrödinger [19], our community is under the mistaken impression that it is only a pure numerical factor of $2\pi$ which has been absorbed in the definition of $\cancel{h}$. This is a terrible mistake that has set us back for the past 100 years or so.4

2.1.1. Constants Imposed by the Vacuum

The speed of light is not related to a natural force. It is passively produced by the vacuum in order to dictate the speed of EM waves, and it also sets an upper limit to the motion of material objects possessing mass. The vacuum does that incidentally, by providing the smallest possible natural resistance to any kind of motion. The magnitude of c is set by the G-M of two inverse properties of the vacuum [13], viz.

$$c=\sqrt{{{\cancel{ϵ}}_0}^{-1}{{\cancel{\mu }}_0}^{-1}}.$$

(5)

The SI value of c is $c=2.9979\times {10}^8\mathrm{m}{\mathrm{s}}^{-1}$, and its dimensions are [length][time]⁻¹ [23].

In equation (5), the reduced values of vacuum permittivity and vacuum permeability combine in a way that removes geometric constraints from this speed (see also footnote 1); the maximum permitted velocity of a combined EM wave or a massive object must be the same in any direction. In contrast, it is understood that a static electric field in vacuum must adjust to the geometric constraint imposed by ${{\cancel{ϵ}}_0}^{-1}$, and this is why the vacuum’s inverse permittivity appears in Coulomb’s law. In fact, ${{\cancel{ϵ}}_0}^{-1}$ is the slope between the electric field $\mathcal{E}$ and the charge surface density $e/r^2$[1], where r represents distance (see also Appendix B for the role that various surface densities play in disjoint parts of physics).

Since equation (5) can be written in the equivalent form

$$c=\sqrt{{ϵ}_0^{-1}{\mu }_0^{-1}},$$

(6)

we can surmise that the physical quantities ${ϵ}_0$ and ${\mu }_0$ are geometry-free. Indeed, after some manipulations involving also the geometry-free fine-structure constant ${\mathsf{\alpha }}_{\cancel{h}}$ (as this was inadvertently defined long ago using Dirac’s $\cancel{h}$), we find that

$${ϵ}_0^{-1}=c\left(\frac{2h{\mathsf{\alpha }}_{\cancel{h}}}{e^2}\right),$$

(7)

and

$${\mu }_0^{-1}=c\left(\frac{e^2}{2h{\mathsf{\alpha }}_{\cancel{h}}}\right).$$

(8)

There is no geometric influence on the right-hand sides of these equations. The quantity that is inverted from one equation to the other, $h/e^2={\mu }_{0}c/\left(2{\mathsf{\alpha }}_{\cancel{h}}\right)$, is proportional to the impedance of free space $Z_0=\sqrt{{\mu }_0/{ϵ}_0}=376.730\Omega$, which is the G-M of ${\mu }_0$ and ${ϵ}_0^{-1}$; thus, $h/e^2$ has dimension of [ohmic resistance] (see Ref. [24] and footnote 23 below). Thus, ${ϵ}_0^{-1}$ and ${\mu }_0^{-1}$ can effectively be expressed as G-Ms involving the squares of c and $Z_0$ (i.e., $\sqrt{c^2{Z}_0^2}$ and $\sqrt{c^2(1/Z_0^2)}$, respectively); the first G-M involves a direct multiplication of the two constants involved, whereas the second G-M uses the Lie-type inversion of one of the two constants [13]. We will pick up again this important inference in § 4.2 below.

2.1.2. Dirac’s Constant $\cancel{h}=h/\left(2\pi \right)$

Dirac’s constant is the slope between the energy E carried by a single photon and its angular frequency $\omega$, viz.

$$E=\cancel{h}\omega .$$

(9)

Its SI value is $\cancel{h}=1.0546\times {10}^{-34}\mathrm{J}\mathrm{s}{\mathrm{rad}}^{-1}$[11,23], and its dimensions are ${\left[\mathrm{action}\right]\left[\mathrm{rad}\right]}^{-1}$ or, equivalently, [moment of inertia] [second]⁻¹ [rad]⁻¹. Here, the physical unit is J s, while the descriptive unit [rad]⁻¹ alerts us to the presence of 2-D geometry.

This new awareness, that inertia is built into $\cancel{h}$ (and Planck’s h), may be the spark we need to theorize that the weak equivalence principle [25] is embedded into the microcosm as well, where gravity is not important. Action integrals [26], in particular, may be viewed as carrying the physical units of [moment of inertia] [second]⁻¹, thus each action is a measure of the rate of change of moment of inertia at all scales of the universe, large and small (see also § 3.5 below).

In the spirit of equations (7) and (8), Planck’s reduced constant may also be split into a product of two G-Ms, viz.

$$\cancel{h}=\sqrt{\cancel{h}\left({\cancel{ϵ}}_{0}c\right)}\sqrt{\cancel{h}\left(\frac{1}{{\cancel{ϵ}}_{0}c}\right)}=e\left(\frac{\cancel{h}}{e}\right);$$

(10)

the first G-M on the right-hand side is geometry-independent; the next G-M is influenced by 2-D geometry since it is directly proportional to

$$\sqrt{\cancel{h}({\cancel{ϵ}}_0-1)}\propto \sqrt{{\left(4{\pi }^2\right)}^{-1}{\left(2{ϵ}_0\right)}^{-1}}\propto {\left(2\pi \right)}^{-1}.$$

This G-M that reduces to $(\cancel{h}/e)$ has dimensions of [magnetic flux] = [magnetic field][area].5 It is understood from the G-M decomposition (10) that the vacuum quantity ${\cancel{ϵ}}_{0}c=4\pi /Z_0$ can couple to $\cancel{h}$, and thus influence quantum phenomena; and it does so in the definition of the fine-structure constant (§ 2.2.1).

2.1.3. Newton’s Gravitational Constant G

Newton’s gravitational constant G is the slope between the gravitational field $a\left(r\right)$ (i.e., acceleration) and the surface mass density $\sigma \left(r\right)\equiv M\left(r\right)/r^2$[1] on the surface of a sphere of radius r enclosing a total mass of $M\left(r\right)$, viz.

$$a\left(r\right)=G\sigma \left(r\right).$$

(11)

Its SI value is $G=6.67430\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$[23], with dimensions of [acceleration] [surface density]⁻¹ [1].

In the spirit of equation (10), Newton’s gravitational constant can also be split into a product of two G-Ms, viz.

$$G=\sqrt{G{M}^2}\sqrt{G\left(\frac{1}{M^2}\right)},$$

(12)

which shows the potential of mass M to couple to G, and thus influence gravitation. The $M^2$ term does that in the definition of the gravitational coupling constant (§ 2.2.2). According to equation (4), the first G-M reduces to $e/\sqrt{{\cancel{ϵ}}_0}=\sqrt{\cancel{h}c}$ and the second G-M reduces to $G(\sqrt{{\cancel{ϵ}}_0}/e)=G/\sqrt{\cancel{h}c}$.

2.2. Dimensionless Units

We now come to the operations and the properties of the unitless coupling constants $\{{\mathsf{\alpha }}_G,{\mathsf{\alpha }}_{\cancel{h}}\}$. We show how these units rectify the Planck system of units and make it functional over all scales of the universe, including atomic and subatomic scales as well.

2.2.1. Fine-Structure Constant ${\mathsf{\alpha }}_{\cancel{h}}$

The fine-structure constant has been defined as

$${\mathsf{\alpha }}_{\cancel{h}}=\frac{e^2}{{\cancel{ϵ}}_0\cancel{h}c}.$$

(13)

Its value has been measured [27] to be very close to ${\left(137.036\right)}^{-1}$ (or ${\mathsf{\alpha }}_h={\left(861.022576\right)}^{-1}$ for the wiser choice $h\to \cancel{h}$ in the definition). Other than that, ${\mathsf{\alpha }}_{\cancel{h}}$ brings no geometry and no units into the system of units. In particular, the geometry embedded in the electric field (and carried on by ${\cancel{ϵ}}_0$) has been inadvertently eliminated by the insertion of $\cancel{h}$ in the modern definition (13).

Nevertheless, definition (13) provides a powerful tool (§ 2.2.3), which we have not taken advantage of in the past: being a measurable constant, ${\mathsf{\alpha }}_{\cancel{h}}$ may serve as the reference UPS unit against which we can quantify all the other unitless coupling constants. For instance, the gravitational coupling constant ${\mathsf{\alpha }}_G$, which we describe next, acquires a quantitative meaning by comparison to ${\mathsf{\alpha }}_{\cancel{h}}$ in the ratio (${\mathsf{\alpha }}_G/{\mathsf{\alpha }}_{\cancel{h}}$); most importantly, $\cancel{h}$ and its artificial dependence on geometry drop out of this comparative ratio, which is another strong hint that $\cancel{h}$ should not have been used in equation (13) above.

2.2.2. Gravitational Coupling Constant ${\mathsf{\alpha }}_G$

Using G and the electron mass $m_{\mathrm{e}}$, the gravitational coupling constant has been defined as

$${\mathsf{\alpha }}_G=\frac{G{m}_{\mathrm{e}}^2}{\cancel{h}c}.$$

(14)

Its value is $1.7518\times {10}^{-45}$ (or ${\mathsf{\alpha }}_G=2.7881\times {10}^{-46}$, the geometry-free value obtained for $h\to \cancel{h}$ in the definition), as determined by calculation.

Comparing the definitions (14) and (13), we see that ${\mathsf{\alpha }}_G$ is, unfortunately, geometry-dependent. This problem did not exist during Max Planck’s heydays, when h was in use and $\cancel{h}$ did not exist. In general, the problem with the modern definitions of constants and variables is that $\cancel{h}$ necessarily introduces 2-D geometry and a [rad] descriptive unit, in addition to the intended physical constant h. We must pronounce this Dirac’s miscue [7,8].

The geometry dependence so artificially inserted in ${\mathsf{\alpha }}_G$ will be taken out entirely in the calculations that follow. We must emphasize up front that reinstating the physical nature of ${\mathsf{\alpha }}_G$ (and ${\mathsf{\alpha }}_{\cancel{h}}$) is necessary for the successful repair of the modern Planck system, and it leads to the determination of natural scales of mass, length, and charge for the chosen mass-to-charge ratio $(m_{\mathrm{e}}/e)$ of the electron (§ 2.2.3), or any other chosen particle for that matter (see footnote 10 for details).

2.2.3. Relative Strength of Gravitational Coupling ${\mathsf{\beta }}_G$

Leaving aside the descriptive [rad] unit in the above coupling constants, we come now to the only known method of actively using such dimensionless (pure) numbers. Being pure numbers, these constants have absolutely no meaning or practical use, but they become useful in ratios, in which their strengths are compared against other dimensionless constants; in such comparisons, these ratios acquire quantitative meaning, and then their relative strengths are, for all practical purposes, measurements of the same stature and importance as dimensional quantities (which, incidentally, are also measured by comparisons to international standards). One unitless coupling constant should however be included in the system in absolute terms in order to provide the reference value for the comparisons.6

For the UPS, we choose ${\mathsf{\alpha }}_h$ (let $h\to \cancel{h}$ in equation (13); Ref. [28]) for this duty because it has been measured by experiment [27], and its physical meaning has now become clear (see § 3.1 and Table A1 below): the factor $\sqrt{{\mathsf{\alpha }}_h}\simeq 1/30$ is a fundamental scale used by the Higgs field to couple to the bottom quark, and then on to all lower particle masses.

In this study, we assume that the coupling constants do not vary at the low energies ($<246.22$ GeV; Ref. [29]) of the subatomic particles. Then, we calculate the UPS unit

$${\mathsf{\beta }}_G\equiv \frac{{\mathsf{\alpha }}_G}{{\mathsf{\alpha }}_{\cancel{h}}}=\frac{{\cancel{ϵ}}_{0}G{m}_{\mathrm{e}}^2}{e^2}=2.4006\times {10}^{-43},$$

(15)

a pure comparative number which is independent of $\cancel{h}$ and h (and c, for that matter—as would be expected, the vacuum does not at all contribute to such a ratio of forces). This “measurement” of ${\mathsf{\beta }}_G$ represents the strength of gravitational coupling relative to that of the EM coupling obtained from electrons (see footnote 10 for considering protons instead, and constructing another UPS with different scales, but with the same elementary particles).

Compared to the relation between units shown in equation (4), ${\mathsf{\beta }}_G$ carries a lot more weight because it can be used in quantitative calculations (although it was equation (4) that gave us a reason to define ${\mathsf{\beta }}_G$). The numerical value obtained in equation (15) does not tell us that gravity is weak and the EM force is strong;7 it only tells us about the relative couplings of these forces in the particular system of units that includes ${\mathsf{\beta }}_G$ as a building block. Gravity is attractive and has always had a chance to grow past the other forces in extraordinarily massive8 settings ($M\gg m_{\mathrm{e}}$)—something that is actively occurring in many places in the present universe. In contrast, the Coulomb force cannot do the same, because its attraction brings together charges of opposite signs that cancel each other out [30].

Also, equation (15) does not tell us that, in the distant past, gravity could have been much stronger in the atomic world, and it only got weaker going forward in time [8,9] because of the expansion of the universe. The gravitational force has always been weak in the atomic world because the characteristic atomic masses are too small (much smaller than the Planck mass $M_{\mathrm{p}}=3\times {10}^{19}$ GeV/$c^2$). So, if it should come to a bout of conjectures, probably the safest choice is the assumption that these constants do not vary in time either, just as they do not vary in space, and some physical reason must be found to the contrary rather than relying on pure speculation.

2.3. Determining a New Atomic Mass Scale

Definitions (13) and (14) have both incorporated $\cancel{h}$, thus the coupling constants have been defined in the microcosm. Here, we use the above results to establish a new atomic mass scale, after correcting for the unintended insertion of planar geometry into the coupling constants.

Looking at equation (14), we see two problems that need to be addressed: (a) Despite the apparent lack of units (not entirely true, since $\cancel{h}$ also carries descriptive radians), ${\mathsf{\alpha }}_G$ is not influenced by EM coupling (there is no e in the definition, only mass $m_{\mathrm{e}}$, and the two long-range forces are not linked to one another, although they are in the real world). (b) The unfortunate use of $\cancel{h}$ has had the unforeseen consequence of arbitrarily adding spurious geometry into the dimensionless mix.9

We can solve both problems by adopting equation (15) to help us define a new atomic mass scale $M_{\mathrm{A}}$ in the UPS. The relative ratio ${\mathsf{\beta }}_G$ carries both forces, and the composite unit $\cancel{h}$, which was not appropriate in the first place, has been eliminated (correcting thus a century-old oversight). One unavoidable conclusion is that the geometry of the vacuum (the ${\cancel{ϵ}}_0$) is still present in ${\mathsf{\beta }}_G$. This comes from the geometric dependence of the electric field, which will now overtly influence the new mass scale $M_{\mathrm{A}}$ (see also § 5 below). This appears to be a realistic relationship since mass also carries charge and the carried electric field is influenced by the vacuum’s ${\cancel{ϵ}}_0$.

Based on these considerations, we return to equation (14), and we rewrite this definition by making the following substitutions: ${\mathsf{\beta }}_G\to {\mathsf{\alpha }}_G$, $h\to \cancel{h}$, and $M_{\mathrm{A}}\to m_{\mathrm{e}}$. We find a new equation, viz.

$${\mathsf{\beta }}_G=G{M}_{\mathrm{A}}^2/\left(hc\right),$$

(16)

in which both sides are dimensionless; and substituting for ${\mathsf{\beta }}_G$ from equation (15), we obtain the new atomic mass scale

$$M_{\mathrm{A}}=\sqrt{{\cancel{ϵ}}_{0}hc}\left(\frac{m_{\mathrm{e}}}{e}\right)=2.6730\times {10}^{-29}\mathrm{kg}.$$

(17)

We repeat here the ingredients that form the physical basis for this mass: (i) the unitless ratio ${\mathsf{\beta }}_G$ in equation (15) has no dependence on $\cancel{h}$, or h, or c; (ii) the substitution $h\to \cancel{h}$ produces a truly unitless equation (16); there are no loose [rad] units in this equation, covertly suppressed by SI conventional practices (although here the “cycle” unit has indeed been suppressed in h, since it does not signify insertion of geometry; see also [11]); and (iii) the ratio of electric charges $e/\sqrt{{\cancel{ϵ}}_{0}hc}=1/30$ is the same deflation factor mentioned above (e.g., in footnote 8) and in § 3.1 below.

It is quite interesting that only the ratio $(m_{\mathrm{e}}/e)$ of the characteristic parameters of the electron ends up being a building block of the new mass scale $M_{\mathrm{A}}$. The reciprocal ratio, i.e., $e/m_{\mathrm{e}}=1.7588\times {10}^{11}$ C kg⁻¹, was first measured by J. J. Thomson [31], years before the electronic charge itself was finally measured by experiment (see also § 4.2).

The presence of ${\cancel{ϵ}}_0$ (coupled to h, as shown in the Lie-type G-Ms given in § 2.1.2) in the new mass scale $M_{\mathrm{A}}$ is necessary (the vacuum’s ${\cancel{ϵ}}_0$ is a building block of the electrostatic field): after some algebraic manipulations, we recast equation (17) (or equation (16)) to the equivalent form

$$M_{\mathrm{A}}=\sqrt{\left(\frac{hc}{G}\right)\left(\frac{E_{\mathrm{grav}}}{E_{\mathrm{elec}}}\right)},$$

(18)

where the ratio of energies,

$$\frac{E_{\mathrm{grav}}}{E_{\mathrm{elec}}}={\mathsf{\beta }}_G,$$

was determined from the corresponding forces acting between two interacting electrons.10 In dimensional analysis, this ratio is 1 (see equation (4) above), but here, ${\mathsf{\beta }}_G$ plays an important quantitative role: the unitless factor

$$\sqrt{{\mathsf{\beta }}_G}=4.900\times {10}^{-22}$$

(19)

scales the original Planck mass [5] $(M_{\mathrm{p}}=\sqrt{hc/G}=5.4555\times {10}^{-8}\mathrm{kg})$ down to the atomic world. This scaling is a significant result of our work, as it connects the original Planck mass scale with the $M_{\mathrm{A}}$ scale of the atomic world,11 viz.

$$M_{\mathrm{A}}=M_{\mathrm{p}}\sqrt{{\mathsf{\beta }}_G}.$$

(20)

We note that $M_{\mathrm{A}}$ and $M_{\mathrm{p}}$ are mass scales related by this equation, and they do not (cannot) correspond to any real particle or object in nature (see also § 4.3).

Notice the complete absence of $\cancel{h}$ from equations (15)-(18). The only geometric dependence entering these equations is that which is imposed by the vacuum on to the electrostatic field (hence, $M_{\mathrm{A}}\propto \sqrt{{\cancel{ϵ}}_0}\propto 2\sqrt{\pi }$). The $\sqrt{\pi }$ does not carry any angular units since $E_{\mathrm{elec}}\sim e^2/{\cancel{ϵ}}_0\sim \left[\mathrm{Joule}\right]$ in equation (18)—just like the $1/\pi$ in the Bohr radius and the factor of 1/4 in the Rydberg energy (see the analysis following equation (1) in § 1.2). Therefore, besides introducing the speed of light, the vacuum manages to imprint $M_{\mathrm{A}}$ only with a unitless, purely numerical constant12 (see also footnote 5).

3. Results within the UPS Realm

3.1. Subatomic Masses

This new UPS mass scale (17) corresponds to the value

$$M_{\mathrm{A}}=15.0\mathrm{MeV}/c^2,$$

(21)

thus it lands near the subatomic world of the low-mass up and down quarks, with corresponding masses $m_{\mathrm{u}}=2.16$ MeV/$c^2$ and $m_{\mathrm{d}}=4.67$ MeV/$c^2$ [32]; and it is smaller than the G-M defined for the hydrogen atom

$$\sqrt{m_{\mathrm{e}}{m}_{\mathrm{p}}}=21.9\mathrm{MeV}/c^2,$$

(22)

where $m_{\mathrm{p}}$ is the proton mass.

The new mass scale $M_{\mathrm{A}}$ appears to be important for the standard model of particle physics, and it should be investigated further theoretically (there is no elementary particle corresponding to this energy). So far, we have derived the following useful relations (sufficient to lead us to a clear physical interpretation of Koide’s enigmatic constant and other constants in the Standard Model; for details, see Appendix A):

(1)

The mismatch between $M_{\mathrm{A}}$ and $\sqrt{m_{\mathrm{e}}{m}_{\mathrm{p}}}$ may be related to Koide’s K-constant $K=2/3$ [15], viz.

$$M_{\mathrm{A}}/\sqrt{m_{\mathrm{e}}{m}_{\mathrm{p}}}=0.6850,$$

(23)

connecting thus the masses of leptons to the atomic constants $M_{\mathrm{A}}$ and $m_{\mathrm{p}}$.13

(2)

Using the above values of first-generation quark masses and the mass of the strange quark, $m_{\mathrm{s}}=93.4$ MeV/$c^2$ [32], we find that

$$\sqrt{m_{\mathrm{u}}{m}_{\mathrm{s}}}/M_{\mathrm{A}}\simeq 0.95,$$

(24)

and

$$\sqrt{m_{\mathrm{d}}{m}_{\mathrm{s}}}/\sqrt{m_{\mathrm{e}}{m}_{\mathrm{p}}}\simeq 0.95,$$

(25)

showing only a 5% deviation of both quark G-Ms from the two atomic mass constants. The results indicate that the mass of the second-generation strange quark is connected to both $M_{\mathrm{A}}$ and the masses of the first-generation quarks. Thus, a connection should exist for the charm quark too,14 and so on for the third generation of quarks as well.

(3)

It certainly appears that there exists a ladder-type mechanism that uses G-Ms (and some scaling coefficients) to relate various particle masses (see also Table A1 in Appendix A below). Some examples (and their corresponding deviations from experiment) are:

$$m_{\mathrm{s}}=\sqrt{m_{\mathrm{d}}{m}_{\tau }}(2.5\%),$$

(26)

where $m_{\tau }=1.777$ GeV/$c^2$ is the tauon mass;

$$m_{\mathrm{s}}=\sqrt{m_{\mathrm{u}}{m}_{\mathrm{b}}}(1.7\%);$$

(27)

$$m_{\mathrm{c}}=\sqrt{m_{\mathrm{p}}{m}_{\tau }}(1.7\%);$$

(28)

$$m_{\mathrm{c}}=\sqrt{2m_{\mathrm{d}}{m}_{\mathrm{t}}}(0.054\%),$$

(29)

where $m_{\mathrm{t}}=172.5$ GeV/$c^2$ is the top quark mass;

$$m_{\mathrm{u}}=\sqrt{2m_{\mathrm{d}}{m}_{\mathrm{e}}}(1.1\%);$$

(30)

$$m_{\mathrm{p}}=\sqrt{2m_{\mathsf{\mu }}{m}_{\mathrm{b}}}(0.17\%),$$

(31)

where $m_{\mathsf{\mu }}=105.66$ MeV/$c^2$ is the muon mass;

$$m_{\mathrm{b}}=\sqrt{m_{\mathsf{\mu }}{m}_{\mathrm{t}}}(2.1\%);$$

(32)

and

$$M_{\mathrm{A}}=\sqrt{m_{\mathsf{\mu }}{m}_{\mathrm{u}}}(0.71\%),$$

(33)

$$M_{\mathrm{A}}=\sqrt{\sqrt{m_{\mathrm{e}}{m}_{\mathsf{\mu }}}\sqrt{m_{\mathrm{e}}{m}_{\tau }}}(0.80\%).$$

(34)

(4)

The Higgs boson ($m_{\mathrm{H}}=125.25$ GeV/$c^2$) is certainly special, although unavoidably a part of the mass ladder. This is the only particle that is not involved in simple G-Ms with the low-mass particles. Two of its complex relations are the following:

$$m_{\mathrm{b}}=\sqrt{m_{\mathrm{s}}(m_{\mathrm{H}}/K)}(0.21\%),$$

(35)

where $K=2/3$ [15]; and

$$\frac{m_{\mathrm{H}}}{m_{\mathrm{b}}}=30.0\simeq \frac{M_{\mathrm{A}}}{m_{\mathrm{e}}}(2.0\%).$$

(36)

This relation shows how the Higgs boson manages to assign mass to the much lower-mass bottom quark by using a novel mechanism, not related to a G-M or Koide’s scale factor (see below).

(5)

The vacuum expectation value (VEV) of the Higgs field is $v=246.22$ GeV/$c^2$[29]. To within a deviation of 1.8%, we find for the compact15 triplet H-t-v that

$$m_{\mathrm{t}}=\sqrt{m_{\mathrm{H}}v},$$

(37)

which shows exactly where the most massive quark is located at the top of the mass ladder. Furthermore, the Higgs mass is the G-M of the top quark mass and the mass of the Z⁰ boson $m_{{\mathrm{Z}}^0}=91.1876$ GeV/$c^2$ (a deviation of only 0.13%), viz.

$$m_{\mathrm{H}}=\sqrt{m_{\mathrm{t}}{m}_{{\mathrm{Z}}^0}}.$$

(38)

Obviously, the top quark receives its mass from the Higgs mechanism, and then it participates in the G-Ms that define the masses of the other particles (see Table A1 in Appendix A). The high-mass geometric sequence Z⁰-H-t-v appears to be very compact indeed (footnote 15), and its common ratio is about 1.38.16 We note that W^± (mass $m_{W^{\pm }}=80.377$ GeV/$c^2$) is not a member of this sequence since $m_{{\mathrm{Z}}^0}/m_{{\mathrm{W}}^{\pm }}=K^{-1/4}\simeq 1.11$.17 This relation provides another definition of Koide’s K in terms of the decay products of the Higgs boson (deviation 2.5%).

(6)

On the other hand, the G-M of $m_{\mathrm{H}}$ and $m_{{\mathrm{W}}^{\pm }}$ is 10% larger than $m_{{\mathrm{Z}}^0}$; but using empirically Koide’s constant, we find that

$$m_{{\mathrm{Z}}^0}=\sqrt{\left(K^{1/2}{m}_{\mathrm{H}}\right)m_{{\mathrm{W}}^{\pm }}},$$

(39)

an important relation with a deviation of the G-M from the measured $m_{{\mathrm{Z}}^0}$ value of only 0.57%. Furthermore, the relation $m_{{\mathrm{W}}^{\pm }}=\sqrt{K{m}_{\mathrm{H}}{m}_{{\mathrm{W}}^{\pm }}}$ also appears to hold (1.9% deviation), which then implies that

$$m_{{\mathrm{W}}^{\pm }}=K{m}_{\mathrm{H}}.$$

(40)

This relation helps us understand the important role of the exact constant $K=2/3$[15]: K is a numerical scale factor that relates some close pairs of particle masses. Here, the Higgs field connects to Z⁰ by an inverse-mapping G-M,18 viz.

$$m_{{\mathrm{Z}}^0}=\sqrt{m_{\mathrm{H}}^3\left(1/v\right)},$$

and to W^± by the simple scale factor K, as seen in equation (40). In hindsight, the Higgs mechanism could not assign two different (but comparable) masses to Z⁰ and W^±, both by using G-M averages, so it used two different couplings involving K and $1/v$, respectively.

(7)

Returning now to equation (35), we see the Higgs mass is scaled by $1/K$ to participate in a G-M with $m_{\mathrm{s}}$ and $m_{\mathrm{b}}$. Although we have only a limited view of the dynamics of the Higgs mechanism in the above equations, it is apparent that this mechanism uses a set of scaling rules in the various coupling factors that appear in the Lagrangians. The origin of these scaling rules is unknown to us at this moment, but we feel confident that we have made a step in the right direction with this analysis (see Appendix A.3 for more detais).

(8)

The next and considerably more difficult step concerns the assignment of mass to the bottom quark, whose mass is much lower than the Higgs mass and the masses of its decay products. We were surprised to find that yet another method is used by the Higgs mechanism for this coupling (no G-M can reach down to $m_{\mathrm{b}}$ because of the barrier set by the Higgs VEV): The only way that we could find for this coupling was the deflation factor of 1/30 mentioned above.

Notice the unitless factor of 30.0 in equation (36). This equation suggests that the mass scale $M_{\mathrm{A}}$ and the electron mass $m_{\mathrm{e}}$ are related to the mass ratio $m_{\mathrm{H}}/m_{\mathrm{b}}$. But $m_{\mathrm{e}}$ is not a mass scale and $M_{\mathrm{A}}$ is not a particle mass, so the proportion in equation (36) involving the ratio $M_{\mathrm{A}}/m_{\mathrm{e}}$ would be at least obscure if it were not for similar mass and charge ratios discussed above in footnote 8 and in item (iii) following equation (17). Using equation (58) derived below and the equations of § 2.3, we can rewrite proportion (36) in a palatable (physical) form, viz.

$$\frac{m_{\mathrm{b}}}{m_{\mathrm{H}}}=\frac{1}{30}\simeq \sqrt{{\mathsf{\alpha }}_h}(2.2\%),$$

(41)

where ${\mathsf{\alpha }}_h={\left(861.022576\right)}^{-1}$ is given by equation (13) after the corrective substitution $h\to \cancel{h}$ that restores Planck’s constant h [5,6] in the definition of the fine-structure constant.

Thus, the mass of the bottom quark $m_{\mathrm{b}}$, which is 30 times lower than $m_{\mathrm{H}}$, is determined self-consistently from this scaling equation by effectively using the ratio of scales $M_{\mathrm{p}}/M_{\mathrm{A}}$ in intermediate steps and the Planckian fine-structure constant19${\mathsf{\alpha }}_h=e^2/\left({\cancel{ϵ}}_{0}hc\right)$ in the final step. This is the third method employed by the Higgs boson to couple to other particles. In particular, it uses this 1/30 scaling to get down to the bottom quark and, then, into the regime of the lower particle masses (Table A1 below). If the $m_{\mathrm{b}}$ coupling also involves the W⁻ boson (which carries Koide’s scale factor K) to deliver charge to the bottom quark, then equations (40) and (41) combine to show that $m_{\mathrm{b}}=\left(m_{{\mathrm{W}}^{-}}\right)(\sqrt{{\mathsf{\alpha }}_h}/K)\simeq 0.05\left(m_{{\mathrm{W}}^{-}}\right)$. The physical significance of Koide’s scale for the high-mass quarks (c, b, t) and the leptons is discussed in detail in Appendix A.2.

3.2. The Planck Charge

The Planck charge $q_{\mathrm{p}}$ is a prime example of the state of confusion in the field: not understanding the meddling of geometry in the modern Planck units, people adopted different definitions of $q_{\mathrm{p}}$ by arbitrarily choosing between ${\cancel{ϵ}}_0$ and ${ϵ}_0$ and between $\cancel{h}$ and h. In the end, this unit, along with the Planck units of magnetic flux ${[\cancel{h}/{\cancel{ϵ}}_{0}c)]}^{1/2}$ and ohmic resistance ${\left({\cancel{ϵ}}_{0}c\right)}^{-1}$, fell out of favor.20

Now, we know better. The definition of the Planck charge $q_{\mathrm{p}}$ must be geometry-free, viz.

$$q_{\mathrm{p}}\equiv \sqrt{{\cancel{ϵ}}_0\cancel{h}c}=\sqrt{2{ϵ}_{0}hc}.$$

(42)

Absence of geometry is required, first because this is a unit of charge, and second because $q_{\mathrm{p}}$ provides an alternative definition of the fine-structure constant (which is geometry-independent in its current definition (13)), viz.

$${\mathsf{\alpha }}_{\cancel{h}}={\left(\frac{e}{q_{\mathrm{p}}}\right)}^2.$$

(43)

We find that $q_{\mathrm{p}}=1.8755\times {10}^{-18}\mathrm{C}=11.7062e$ (where $11.7062=\sqrt{137.036}$). Once again, nature shows us here her principle of fairness (or impartiality). As in the case of the electron mass $m_{\mathrm{e}}$, the elementary charge e here is not related to the fundamental unit of charge $q_{\mathrm{p}}$ by a rational numerical factor; instead, $q_{\mathrm{p}}$ is chosen as the UPS scale of charge that does not correspond to the charge multiple of any specific particle or field.

3.3. A New Atomic Length Scale

Equation (1) can help us determine a new length scale for the UPS, a scale that certainly does not correspond to any of the three atomic radii in equation (1): based on nature’s apparent principle of fairness, we understand that none of the known electronic radii can be the fundamental unit of length. But we know that scale values generally fall between particle values and vice versa. To proceed then, we use the G-M of $r_{\mathrm{e}}$ and $r_{\mathrm{c}}$ to determine a new atomic length scale $L_{\mathrm{A}}$.21

The G-M of $r_{\mathrm{e}}$ and $r_{\mathrm{c}}$ gives

$$L_{\mathrm{A}}=r_{\mathrm{c}}\sqrt{{\mathsf{\alpha }}_{\cancel{h}}}=\sqrt{\frac{\cancel{h}}{{\cancel{ϵ}}_0{c}^3}}\left(\frac{e}{m_{\mathrm{e}}}\right),$$

(44)

and $L_{\mathrm{A}}=3.2987\times {10}^{-14}\mathrm{m}=r_{\mathrm{c}}/11.7062$. The numerical value 11.7062 is the same with that found for the ratio $q_{\mathrm{p}}/e$ (equation (43)) because

$${\mathsf{\alpha }}_{\cancel{h}}={\left(L_{\mathrm{A}}/r_{\mathrm{c}}\right)}^2,$$

(45)

and then, the following proportion (cross-multiplied) holds exactly:

$$L_{\mathrm{A}}{q}_{\mathrm{p}}=r_{\mathrm{c}}e.$$

(46)

This relation implies that the G-M of the new scales $L_{\mathrm{A}}$ and $q_{\mathrm{p}}$ is equal to the G-M of the traditional and widely-used electronic constants $r_{\mathrm{c}}$ and e; and it brings to light a previously unused combination of units with dimensions of [length][charge]. These dimensions are equivalent to

$$\frac{\left[\mathrm{momentum}\mathrm{flux}\right]}{\left[\mathrm{magnetic}\mathrm{flux}\right]}=\frac{\left[\mathrm{momentum}\right]}{\left[\mathrm{magnetic}\mathrm{field}\right]}=\frac{\left[\mathrm{energy}\right]}{\left[\mathrm{electric}\mathrm{field}\right]};$$

these interesting units compare mass flows (“matter waves”) to EM waves (“energy flows”), and energy/momentum to EM field components. These quotients also indicate a close correspondence between the relativistic energy-momentum (E-p) equation

$$E=cp,$$

(47)

and Maxwell’s EM amplitudes (${\mathcal{E}}_0$, ${\mathcal{B}}_0$; [36]) relation

$${\mathcal{E}}_0=c{\mathcal{B}}_0.$$

(48)

The above dimensional ratios of units are obtained easily by dividing these two equations. We see then that ${\mathcal{B}}_0$ (current flow) is to EM waves what momentum p (mass flow) is to dynamics, and similarly for amplitude ${\mathcal{E}}_0$ and energy E.

Length $L_{\mathrm{A}}$ is much larger than the modern Planck length $L_{\mathrm{p}}=\sqrt{\cancel{h}G/c^3}=1.6163\times {10}^{-35}$ m. (The modern definition of $L_{\mathrm{p}}$ must be used here, because $r_{\mathrm{c}}$ in equation (44) brought its 2-D geometry into $L_{\mathrm{A}}$, and ${\mathsf{\alpha }}_{\cancel{h}}$ is accidentally geometry-free.) In this case, $L_{\mathrm{A}}$ must be scaled down to produce $L_{\mathrm{p}}$; thus, $L_{\mathrm{p}}=L_{\mathrm{A}}\sqrt{{\mathsf{\beta }}_G}$. This scaling-down of $L_{\mathrm{A}}$ should be contrasted to the scaling-up of $M_{\mathrm{A}}$ to produce the original Planck mass $M_{\mathrm{p}}$ (i.e., $M_{\mathrm{p}}=M_{\mathrm{A}}/\sqrt{{\mathsf{\beta }}_G}$; see § 2.3).

3.4. Cosmological Scales and Some Ambivalent Superatomic Particles

In §§ 2.3 and 3.3 above, we rescaled the fundamental scales of the UPS to obtain the corresponding Planck scales. These “A” and “p” values do not describe any specific particle or object in the universe; that would not be a fair choice by nature. But we can extend both scales into the macrocosm by running the G-Ms toward larger masses and lengths.

(a) Cosmological Mass Scales.—We evaluate a geometric progression that starts with scales $M_{\mathrm{A}}$ and $M_{\mathrm{p}}$, and moves on to larger mass scales:

$$\{M_{\mathrm{B}},M_{\mathrm{C}},M_{\mathrm{D}}\}=\{1.113\times {10}^{14},2.271\times {10}^{35},4.633\times {10}^{56}\}\mathrm{kg}.$$

(49)

Mass scale $M_{\mathrm{D}}$ is 2-3 orders of magnitude larger than the current estimates of the mass of the universe [1], so we can halt the sequence at $M_{\mathrm{D}}$. The common ratio of the geometric progression is $M_{\mathrm{p}}/M_{\mathrm{A}}=1/\sqrt{{\mathsf{\beta }}_G}=2.041\times {10}^{21}$. The G-M of $M_{\mathrm{B}}$ and $M_{\mathrm{C}}$ is equal to 0.84 earth masses; and the G-M of $M_{\mathrm{C}}$ and $M_{\mathrm{D}}$ is $5\times {10}^{15}$ solar masses and identifies universal structures much larger than individual galaxies (e.g., galaxy clusters).

(b) Cosmological Length Scales.—We evaluate a geometric progression that starts with scales $L_{\mathrm{p}}$ and $L_{\mathrm{A}}$, and moves on to longer length scales:

$$\{L_{\mathrm{B}},L_{\mathrm{C}}\}=\{6.730\times {10}^7,1.373\times {10}^{29}\}\mathrm{m}.$$

(50)

Length scale $L_{\mathrm{C}}$ is 2-3 orders of magnitude larger than the current estimates of the size of the universe [1], so we can halt the sequence at $L_{\mathrm{C}}$. The common ratio of this geometric progression is $L_{\mathrm{A}}/L_{\mathrm{p}}=1/\sqrt{{\mathsf{\beta }}_G}=2.041\times {10}^{21}$, the same as the common mass ratio given in item (a) above. The G-M of $L_{\mathrm{B}}$ and $L_{\mathrm{C}}$ is equal to 98.5 parsecs, a value typical of giant molecular cloud complexes in spiral galaxies.

(c) Cosmic Microwave Background (CMB).—We convert the temperature of the CMB, $T_{\mathrm{CMB}}=2.7255$ K, to an equivalent mass $m_{\mathrm{CMB}}=3.52\times {10}^{-10}$ MeV/$c^2$. Since $m_{\mathrm{CMB}}\ll M_{\mathrm{A}}$, we need to extend the geometric progression of mass scales to much lower masses as well. At the low-mass end of the geometric sequence $\{M_0,M_{\mathrm{A}},M_{\mathrm{p}}\}$, the tiny mass scale $M_0$ is found to be $M_0=7.35\times {10}^{-21}$ MeV/$c^2$. Then, the G-M relation

$$m_{\mathrm{CMB}}=\sqrt{M_0{M}_{\mathrm{A}}},$$

(51)

holds to within a 5.7% deviation between the two sides. This deviation is relatively small, given the enormous difference in scales (by 21.3 orders of magnitude) involved on the right-hand side of equation (51).

(d) A Superatomic Particle Near the Planck Mass?—The equivalent mass of the CMB photons is so low, that extending the geometric sequence of $\{m_{\mathrm{CMB}},m_{\mathrm{H}}\}$ to higher masses, we obtain a potential particle mass of $M_{\mathrm{S}}=m_{\mathrm{H}}^2/m_{\mathrm{CMB}}=4.453\times {10}^{16}$ GeV/$c^2\simeq 1.455\times {10}^{-3}{M}_{\mathrm{p}}$, which is at the scales where the strong force supposedly joins in with the other forces [37]. Since the Higgs mass is $m_{\mathrm{H}}=125.25$ GeV/$c^2$, then the energy ratio $\sqrt{{\mathsf{\beta }}_W}$ (analogous to $\sqrt{{\mathsf{\beta }}_G}$ in § 2.3) that scales the strong interaction down to the weak interaction is

$$\sqrt{{\mathsf{\beta }}_W}=\sqrt{\frac{E_{\mathrm{W}}}{E_{\mathrm{S}}}}=\sqrt{\frac{m_{\mathrm{H}}}{M_{\mathrm{S}}}}=5.30\times {10}^{-8}.$$

(52)

This value is smaller by a factor of 20 compared to the usually quoted coupling constant ratio of the weak to the strong interaction. One reason is that the quoted estimates of this ratio in particle physics depend on microphysics [37]; these values are not really constants, since they show a secular dependence on particle energy [38,39]. In any case, it is doubtful that the Higgs field can assign masses above its VEV of 246.22 GeV/$c^2$[40]; a phase transition to the mass $m={10}^{18}$ GeV/$c^2$ quoted in footnote 8 may be necessary.

(e) Sub-TeV Particles?—In the atomic world, the Higgs VEV appears to be a barrier against growing more massive nuclei and particles. Nevertheless, researchers are searching the TeV scales in hopes of discovering such particles [41]. If there is a way to jump across the Higgs VEV (which we do not currently see; see also [40]), then the next few particle slots generated by the high-mass geometric progression Z⁰-H-t-$v\cdots$ will have energies of 0.351, 0.502, 0.716, and 1.022 TeV.

3.5. Units Jumping to the Forefront

Consider the units of mass M and area $R^2$; think of them, in abstract terms, as gravity and 2-D geometry. It has long been known that one can form two new units from them, the surface density $\sigma =M/R^2$ and the moment of inertia $I=M{R}^2$. These have been used in physics applications, but they are not very popular, certainly not as popular as the unit of volume density we get from M and $R^3$. It has become clear to us that $\sigma$ and I have not been given due attention in the past, but now they seem to come to the forefront in our analyses of various astrophysical problems.

Surface density $\sigma$ plays an important role in our recent work on varying-G gravity [1], and provided the cosmological system of units with an important relation used in unit reductions and in the sourcing of the gravitational field: $\sigma \sim a/G$, where a is acceleration (or, equivalently, the gravitational field), so that

$$\left[\mathrm{surface}\mathrm{density}\right]=\frac{\left[\mathrm{gravitational}\mathrm{field}\right]}{\left[\mathrm{gravitational}\mathrm{constant}\right]},$$

indicating that the surface density $\sigma \left(r\right)=M\left(r\right)/r^2$ is a dynamical quantity of considerable influence. Moment of inertia I has already played a role in our calculations of units, as we hinted in § 2.1.2. Here, we discuss its potential significance for inertial masses and the weak equivalence principle [25]. We use dimensional analysis throughout this subsection, so all symbols (having their usual meanings) represent units.

Bunker et al. [11] righted a wrong in physics when they pointed out that, in units, $\cancel{h}=E/\omega$. (In this notation, [rad] is a descriptive unit carried by $\omega \sim$ [rad][second]⁻¹.) The current SI system suffers from this inconsistency, and it should be repaired as soon as possible. The problem that concerns us here is the following: in the original Planck system, one of the fundamental units was h, and action integrals had dimensions of $h=ET=M{R}^2{T}^{-1}=\mathcal{L}$, where $\mathcal{L}$ denotes angular momentum. This description has been in the books since Dirac and Landau’s times circa 1930 [8,9,26].

In their dimensional analysis of Newton’s G, Landau & Lifshitz [26] used only basic cgs-system units, and they did not mention any interpretation of the units of action. Other authors have mentioned that the action integrals $\mathcal{S}$ appear to have the same unit as angular momentum $\mathcal{L}$, so $\mathcal{S}=\mathcal{L}=ET$. This is an interesting conclusion, but not a fundamental one, since there exist objects and fields in nature that neither rotate nor revolve (e.g., free spin-0 particles in vacuum and tenuous gases in galaxy clusters).

We have a different interpretation of the units of action $\mathcal{S}$, which appears to be fundamental, viz.

$$\mathcal{S}=M{R}^2/T=I/T;$$

that is, $\mathcal{S}$ describes the rate of change of the moment of inertia $I/T$, where $I=M{R}^2$, and M is bona fide inertial mass. We believe that this interpretation has important ramifications going forward, because it introduces the concept of inertia into the units of all action integrals, including those of non-gravitating (atomic and nuclear) fields. But if the massless fields know about inertia in their variations, then the weak equivalence principle (and its inertial mass) may have already permeated all other physical processes besides gravity. In other words, macroscopic mass is, in a major way, gravity; but its inertia appears to be a universal property,22 communicated by the vacuum itself (i.e., by the “resisting” properties c and $Z_0$) in the same way that geometry-dependent vacuum properties (${\cancel{ϵ}}_0$ and ${\cancel{\mu }}_0$) are also installed in the universe.

To conclude, we believe that force actions cannot be unified before the influence of the vacuum on every single field is delineated and becomes fully understood. This is not going to be an easy task; as we mentioned above, the vacuum interferes in the universe by three distinct ways: dimensional geometry-dependent constants (${\cancel{ϵ}}_0$, ${\cancel{\mu }}_0$), dimensional geometry-free constants (c, $Z_0$), and purely numerical imprinted values. (We note that it is this vacuum “activities list” that prompted us to include $Z_0$ as another building block of the UPS, as shown in equation (3).)

4. Discussion

4.1. Pairs of Fundamental Dimensional Units

Equation (45) shows that two lengths are needed to produce the fine-structure constant ${\mathsf{\alpha }}_{*}*h**$ in any system of units: the fundamental scale $L_{\mathrm{A}}$ and a Compton-type scale such as $r_{\mathrm{c}}$. This subsidiary scale cannot be defined by using the fundamental mass scale (then, one gets ${\mathsf{\alpha }}_{\cancel{h}}=1$). Therefore, equation (45) defines $r_{\mathrm{c}}$ independently of mass $M_{\mathrm{A}}$. In our case, this definition is obtained easier from equation (46): $r_{\mathrm{c}}=L_{\mathrm{A}}(q_{\mathrm{p}}/e)$. Using the definition of the fine-structure constant is an integral part of the above derivation of $r_{\mathrm{c}}$; and this example justifies our statement that all systems besides the UPS are incomplete, missing at least the unitless coupling constants, and thus, they are incapable of describing all scales and forces in the universe.

Next, we consider Planck’s original set of dimensional units $\{c,G,h\}$, with h in place of $\cancel{h}$ to avoid misunderstandings from the introduction of geometry into the units. The speed of light barrier is applicable to all systems of units, but h is not fundamental in the cosmological system and G is not fundamental in the atomic system for “obvious” (now known to be obviously wrong) reasons: negligibly weak influences should not be building blocks at the core of a system. We realize now that all three constants are necessary building blocks, and that the vacuum-force pairs $\{c,G\}$ and $\{c,h\}$ serve two different (but complementary) functions within the UPS:

(a) The pair of constants $\{c,G\}$ with its universal unit of force23$F_0=c^4/G$ and the corresponding unit imprint of the famous Tully-Fisher/Faber-Jackson relation [17,18]$c^4=GM{a}_0$ (where $F_0=M{a}_0$; [2,3,4]) was analyzed in our companion paper [1] in the cosmological system of units. (We discuss the universality of this relation in Appendix B.) Combined with Newton’s G, powers of c define units whose purpose is to monitor the effectiveness of forces F in producing motion (speed V). Some of these units are very well-known: $c^2/G\sim F/V^2=M/R_{\mathrm{S}}$, $c^3/G\sim F/V=Z_{\mathrm{m}}$, $c^4/G\sim F$, and $c^5/G\sim FV=P$. Here, M is mass, $R_{\mathrm{S}}$ is (Schwarzschild) radius, $Z_{\mathrm{m}}$ is mechanical impedance, and P is power.

(b) With the notable exceptions of $\sqrt{h\left({ϵ}_{0}c\right)}\sim q$ (charge) and its Lie-type inversion $\sqrt{h/\left({ϵ}_{0}c\right)}\sim {\Phi }_{\mathrm{B}}$ (magnetic flux) (§ 2.1.2), the pair of constants $\{c,h\}$ can only generate composite units, which cannot be viewed as fundamental units in the physical world; although these units do afford some interesting symmetries. For instance, examine the sequence of units $hc\to \left[E\right]\left[L\right]$, $h\to \left[E\right]\left[T\right]$, $h/c\to \left[M\right]\left[L\right]$, and $h/c^2\to \left[M\right]\left[T\right]$, before the next powers of c generate some lower-level subsidiary units, e.g., $h/c^3\to \left[M\right]{\left[a\right]}^{-1}$. Combining powers of c with Planck’s h, these units are designed to monitor the action integral $\mathcal{S}$ (i.e., integrated energy in time) during motion, although they are not as well-known: $h/c^3\sim \mathcal{S}/V^3$,  $h/c^2\sim \mathcal{S}/V^2$,  $h/c\sim \mathcal{S}/V$,  $h\sim \mathcal{S}$, and  $hc\sim \mathcal{S}V$. Since action $\mathcal{S}$ determines both speed V and acceleration a, this sequence of units can also be interpreted as: $h/c^3\sim (E/V^2)/a=M/a$,  $h/c^2\sim (E/V)/a=p/a$,  $h/c\sim E/a$,  $h\sim \left(EV\right)/a$, and  $hc\sim \left(E{V}^2\right)/a$, where E represents energy and p represents momentum.

The above symmetries are naturally propagated also to derivative units. As a typical case, we discuss the sequence of composite units $M/T^n$ (for integer n) generated by the widely-used pair of units of mass and time $\{M,T\}$, because this sequence holds some surprises. These units apparently measure resistive properties in the material world:

$$\begin{array}{cccc}M/T\hfill & \sim F/(L/T)\hfill & =Z_{\mathrm{m}}\hfill & \left[\mathrm{mechanical}\mathrm{impedance}\right]\hfill \\ M/T^2\hfill & \sim F/L\hfill & =S_{\mathrm{m}}\hfill & \left[\mathrm{mechanical}\mathrm{stiffness}\right]\hfill \\ M/T^3\hfill & \sim F/\left(LT\right)\hfill & ={\sigma }_P\hfill & \left[\mathrm{power}\right]{\left[\mathrm{area}\right]}^{-1}\hfill \end{array},$$

(53)

where L represents length and the index P represents power. It is surprising that the unit $M/T$ (of the ubiquitous $\dot{M}$ of accretion physics) turns out to be a resistive property of inflowing matter. It is also quite surprising that the “power surface density” ${\sigma }_P$ is a member of this sequence of units that describe the various types of mechanical resistance. In Appendix B, we find that power surface density is a universal dynamical quantity, although it appears prominently only in the Stefan-Boltzmann law [42,43]. Its resistive character becomes apparent when we rewrite it in terms of force F and moment of inertia I, viz.

$${\sigma }_P=F^2/(I/T),$$

(54)

where $(I/T)$ represents resistance due to the rate of change of the moment of inertia. In this equation, we recognize the importance of the force squared $F^2$ in ${\sigma }_P\sim M/T^3$. Coming full circle to expressing the resistances in terms of $F^2$, we find for the impedance and the stiffness that $Z_{\mathrm{m}}=F^2/P$ and $S_{\mathrm{m}}=F^2/E$, respectively, where E represents energy. Therefore, the magnitude of $F^2$ appears to be regulated by power in impedance, by energy in stiffness, and by inertial changes in power surface density.

Furthermore, the inertial magnitude itself appears in the next term of the sequence (53), i.e., $M/T^4\sim F^2/I$, and the integrated quantity $\left(IT\right)$ appears next in $M/T^5\sim F^2/\left(IT\right)$. Obviously then, the units of the sequence $M/T^n$ describe resistive properties in which $F^2$ is regulated by the temporal variations of inertia.

4.2. The Varied Contributions of the Vacuum

The free space known as the vacuum is described by four interdependent constants (${ϵ}_0$, ${\mu }_0$,  $c=1/\sqrt{{ϵ}_0{\mu }_0}$,  $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$). When the vacuum wishes to also promote geometry in some parts of the natural world, then it introduces either ${\cancel{ϵ}}_0\equiv 4\pi {ϵ}_0$ or ${\cancel{\mu }}_0\equiv {\mu }_0/\left(4\pi \right)$ or both, provided they are not introduced in a product (there is no geometry in ${\cancel{ϵ}}_0{\cancel{\mu }}_0=1/c^2$).

From the nongeometric vacuum quantities ${ϵ}_0$ and ${\mu }_0$, only two additional purely physical quantities can be constructed by simple G-Ms, the speed of light c and the impedance of free space $Z_0$ (§ 2.1.1). They both represent upper limits in nature, the only known upper limits communicated by the vacuum to all scales and in all directions within the universe. Their origin is the least (but nonzero) resistance that the vacuum mounts passively against all motions in the material world (see also § 5 below).

Next, we wish to track down the geometry that is imposed selectively by the vacuum, so we rewrite the fundamental G-Ms discussed in § 2.1.1 as follows:

$$\sqrt{{{\cancel{ϵ}}_0}^{-1}{{\cancel{\mu }}_0}^{-1}}=c,$$

(55)

and

$$\sqrt{{{\cancel{ϵ}}_0}^{-1}\left(\frac{1}{{{\cancel{\mu }}_0}^{-1}}\right)}=\frac{Z_0}{4\pi }.$$

(56)

The G-M (55) is clearly geometry-free, whereas the G-M (56) attaches the $4\pi$ of 3-D space to the geometry-free impedance of free space $Z_0$.24 This is an important conclusion: when ${{\cancel{ϵ}}_0}^{-1}$ or ${{\cancel{\mu }}_0}^{-1}$ appears in equations, or they both appear in a combination other than their product (55), then they carry 3-D geometry with them. These composite vacuum constants show us how free space manages to interfere in the construction and evolution of additional (ready-to-interact with one other) physical entities, such as mass and electric charge, that characterize the underlying force fields.

We emphasize here that mass and charge are not actually fundamental quantities, as is widely believed; they can only be derived and clearly understood, if the contributions of the vacuum and the unitless coupling constants are also taken into account. We demonstrate this point here, with exact calculations:

(a) Consider first equation (13). Solving for the charge e, we obtain a scaled-down G-M relation of the form

$$e={{\alpha }_{\cancel{h}}}^{1/2}\sqrt{h\left(2{ϵ}_{0}c\right)}={{\alpha }_{\cancel{h}}}^{1/2}{q}_{\mathrm{p}}.$$

(57)

So, Planck’s physical constant h and the vacuum’s combination of $\left(2{ϵ}_{0}c\right)$ determine e as a geometry-free, G-M quantity. From this point of view, we can also see how dimensionless constants resize properties of the material world: the G-M is scaled down by the geometry-free factor of ${{\alpha }_{\cancel{h}}}^{1/2}\simeq 1/\sqrt{137}\simeq 1/11.7062$ (see also § 3.2).

(b) Consider next equation (14). Solving for the mass $m_{\mathrm{e}}$, we obtain a G-M relation of the form

$$m_{\mathrm{e}}={\left(\frac{{\mathsf{\alpha }}_G}{2\pi }\right)}^{1/2}\sqrt{\left(\frac{h}{G}\right)c}={\left(\frac{{\mathsf{\alpha }}_G}{2\pi }\right)}^{1/2}{M}_{\mathrm{p}}.$$

(58)

In this case, $m_{\mathrm{e}}$ is determined by the G-M of the composite physical constant $h/G$ and the vacuum’s c. (G participates because a mass is determined here.) The G-M is scaled down by a factor of ${[{\mathsf{\alpha }}_G/\left(2\pi \right)]}^{1/2}=1.670\times {10}^{-23}$ relative to $M_{\mathrm{p}}$. Because of the inclusion of $2\pi$, this factor is geometry-free, and so is $m_{\mathrm{e}}$ (since we defined our $M_{\mathrm{p}}$ as a geometry-free constant).

(c) By dividing equations (57) and (58), and neglecting for the moment the dimensionless, geometry-free factor ${(4\pi /{\mathsf{\beta }}_G)}^{1/2}=7.235\times {10}^{21}$, we obtain a geometry-independent G-M for the electron’s charge-to-mass ratio, viz.

$$\frac{e}{m_{\mathrm{e}}}\propto \sqrt{{ϵ}_{0}G}.$$

(59)

Thus, the ratio $e/m_{\mathrm{e}}$ is determined mainly by the G-M of the nongeometric constants ${ϵ}_0$ and G (vacuum and gravity, respectively); and the neglected scale factor carries the relative strength of the two unitless coupling constants ($\sqrt{4\pi /{\mathsf{\beta }}_G}\propto \sqrt{{\mathsf{\alpha }}_{\cancel{h}}/{\mathsf{\alpha }}_G}$) with the geometry due to the electrostatic field taken out of the ratio ${\mathsf{\beta }}_G$.

4.3. Sparse Geometric Averaging in Nature

We think we understand why virtually all pairs of constants and units ($U_1,U_2$) combine in G-Ms25 involving the direct form $U_1{U}_2$ or the inversion form $U_1{U}_2^{-1}$ (or $U_1^{-1}{U}_2$). Physically, two basic G-M quantities can be derived from each pair of units. Mathematically, these two operations result in mappings that are aways “smooth” since they involve constants; thus, the units of a system of units form a Lie group [13], and the associated Lie algebra can be carried out with ease.

One remaining question is why there are also square roots on top of these unit combinations, establishing thus G-Ms. We fall back to what is already known about G-Ms: compared to the commonly used arithmetic means, G-Ms place significantly more weight to the smaller of the two values. Thus, the most obvious property of the geometric averages $\sqrt{U_1{U}_2^{\pm 1}}$ is that they assist the smaller physical constants to leave their indelible marks in their combinations with larger constants. In a sense, by not letting small constants become negligible (or dominant) when they are combined with large constants,26 nature seems to follow a principle of fairness or impartiality at all scales of the universe. The degree of support for the small constants can be quite dramatic for much differing constants, as equations (15) and (19) vividly demonstrate: the G-M $\sqrt{{\mathsf{\beta }}_G}$ gains 21.3 orders of magnitude relative to the ratio ${\mathsf{\beta }}_G$, and it connects the Planck scale with the atomic world.

Consider now the subatomic particles discussed in § 3.1. Nature did not make a particle in each individual G-M slot. The mass ladder is mostly empty, and just a few actual particles have materialized in the subatomic scales of the universe [37]. So, there are additional selection criteria (scaling laws) on top of the G-Ms that regulate the creation of particles. Besides the factors of 2 and $\sqrt{2}$ in the equations of § 3.1, we have seen that the Higgs boson does not rely on pure G-Ms to reach down to lower masses; it uses, in addition, two different scale factors, Koide’s $K=2/3$ and $\sqrt{{\mathsf{\alpha }}_h}\simeq 1/30$ (equation (41)), to bypass many available particle slots (see also Table A1 below). In particular, the dramatic drop by 121 GeV/$c^2$ from the Higgs mass to the mass of the bottom quark can only be described as a deflation of particle mass that bypasses 10 G-M particle slots intervening between $m_{{\mathrm{Z}}^0}$ and $m_{\mathrm{b}}$.

5. Lingering Issues, Future Prospects, and a Brief Summary

The UPS was summarized in equation (3). The system is not flawless yet, and several issues must be investigated and resolved in the future (see, e.g., footnote 27 below). These issues can be traced to Dirac’s introduction of $\cancel{h}=h/\left(2\pi \right)$ in place of Planck’s h.

It is certainly true that in quantum mechanics, Dirac’s composite constant $h/\left(2\pi \right)$ always appears in form, and this also prompted Schrödinger [11,19] to absorb the $2\pi$ into a convenient new constant K. This tactic tells us that Schrödinger was not aware that he was including geometry into his constant K. Dirac [7,8,9], on the other hand, believed that $\cancel{h}=\mathtt{K}$ is the true constant (not h), so we can guess that he sensed that the two constants are fundamentally different in their makeup (see § 1.2 for more details).

Dirac’s reform has modified quite substantially the systems of units that have adopted $\cancel{h}$, but this modification came with a heavy price. Planck’s purely physical constant h cannot be dropped so nimbly, because then, we introduce errors in the definitions of the coupling constants. Dimensionless coupling constants should not include geometric dependencies other than ${\cancel{ϵ}}_0$ or ${\cancel{\mu }}_0$ (and these enter only via EM terms); geometry would give the constants an additional descriptive unit of [rad] and it would alter their nature. On the dimensional side of vacuum units, c and $Z_0$ (§ 4.2) are also geometry-free constants for a good reason: they represent upper limits set by the vacuum to be applicable in any direction of space irrespective of the dimensionality of space.

We note another issue concerning $\cancel{h}$: In the dimensional part of the UPS, the constant $\cancel{h}$ is the only fundamental dimensional unit that introduces geometry in the physical units. This is unusual and a singular property. Although we were inclined to adopt Planck’s h in place of $\cancel{h}$, we did not do so, because we do not know how to choose between the two constants. It seems from the calculations above that the use of h in the definitions of scales (Planck units, coupling constants) is mandatory, but then $\cancel{h}$ may be more appropriate to be retained for particles and fields, as Dirac [7,8,9] also thought. Perhaps, both constants should be retained in a modified UPS, along with ${\mathsf{\alpha }}_h$ and ${\mathsf{\beta }}_G$ (see the UPS as described in footnote 27).

Examining now the definitions of the dimensionless units that we summarized in § 2.2 (equations (13) and (14)), we see that ${\mathsf{\alpha }}_{\cancel{h}}$ is indeed geometry-free (${\cancel{ϵ}}_0\cancel{h}=2{ϵ}_{0}h$), but ${\mathsf{\alpha }}_G$ is not (${\mathsf{\alpha }}_G\propto 1/\cancel{h}\propto 2\pi$). We think this is an enormous oversight flying undercover at least since Dirac [8] introduced his “large numbers hypothesis;” and it has prevented physicists from defining an atomic mass scale in the modern Planck system, creating thus an insurmountable obstacle to force unification. The state of confusion can best be seen in the widespread misconception “that G carries units into the action of general relativity, thus gravity is not like the other forces of nature,” taught to thousands upon thousands of physics students for nearly a century. We now understand that gravity is just like the other forces, and it enters the “ring” with one dimensionless (${\mathsf{\alpha }}_G$) and one dimensional (G) constant, just as the EM forces do too.

Owing to the omnidirectional nature of the gravitational force, both of its constants should be geometry-free. For this reason, we tried to bypass the problem with the definition (14) of ${\mathsf{\alpha }}_G$ (it effectively carries a descriptive unit of [rad], thus it cannot be utilized) and to define new consistent atomic units within the UPS. First, we created a dimensionless ratio ${\mathsf{\beta }}_G={\mathsf{\alpha }}_G/{\mathsf{\alpha }}_{\cancel{h}}$ of the coupling constants that describes their relative strength; the $\cancel{h}$ does not partake in this ratio, and the only geometric influence left comes from the EM field. But this does not affect the makeup of the relative strength ${\mathsf{\beta }}_G$, since ${\mathsf{\beta }}_G$ is expressed as a ratio of energies.27

Next, we created a dimensionless geometry-free combination of fundamental units to attach to ${\mathsf{\beta }}_G$, viz.

$${\mathsf{\beta }}_G=G{M}_{\mathrm{A}}^2/\left(hc\right)={\left(M_{\mathrm{A}}/M_{\mathrm{p}}\right)}^2,$$

where $M_{\mathrm{p}}$ is the mass scale of the original [5] system of units. Finally, the new atomic mass scale $M_{\mathrm{A}}$ was derived from the known values of ${\mathsf{\beta }}_G$ and $M_{\mathrm{p}}$, viz.

$$M_{\mathrm{A}}=M_{\mathrm{p}}\sqrt{{\mathsf{\beta }}_G}.$$

The interpretation of this relation is straightforward: the ratio of the two widely different mass scales $M_{\mathrm{A}}/M_{\mathrm{p}}=4.9\times {10}^{-22}$ is precisely equal to the square root of the relative ratio of the two coupling constants ${\mathsf{\beta }}_G={\mathsf{\alpha }}_G/{\mathsf{\alpha }}_{\cancel{h}}=2.4\times {10}^{-43}$.

In § 3, we tested the influence of this mass scale in the atomic and subatomic world, and the results appear to be strong. The mass constant $M_{\mathrm{A}}$ has no trouble meddling in G-Ms (§ 3.1) along with particle (sub)atomic masses that have been measured by experiment [23,32] (but see also footnote 10 for UPS′, an alternative system of units based on proton’s parameters). In the process, we also clarified the confusion surrounding the so-called Planck charge (§ 3.2), and we also derived a new length scale that had no trouble meshing in G-M calculations with the already-known atomic radii (§ 3.3 and § 1.2).

In §§ 3.1 and 3.4, we calculated both mass scales and actual particle masses at practically all scales of the universe. The Higgs mechanism uses a multitude of scalings and couplings to distribute masses to (sub)atomic particles. This diversity of methods is, in part, responsible for hindering progress in the effort to unify the four fundamental forces of nature. The other part concerns the role of the vacuum (§§ 2.1.1 and 4.2). The behavior of the vacuum is not at all what our books describe [e.g., [20,33,37]. As far as we can see, the vacuum is not subject to forcing of any kind, and it seems to be unresponsive to quantum fluctuations, which occur exclusively in fields. By and large, the vacuum appears to be a passive independent entity, with no physical properties of its own, that imposes implicitly certain rules (by resisting the least) to the material world, that all inhabitants must necessarily observe and obey (to within the bounds of the uncertainty principle of course; see also Appendix B.2). Under these circumstances, there is no back reaction from the material world on to the vacuum itself. In hindsight, this conclusion makes sense—how can anything tangible manage to tangle up that which is the epitome of nothingness?

6. Highlights

6.1. Conclusions

(1)

Current systems of units are incomplete and incapable of describing all aspects of this universe. They do not include some of the fundamental dimensional constants, the dimensionless coupling constants, and all the restrictions installed by the vacuum itself onto the material world.

(2)

Each force of nature must be represented in a system of units with a dimensional and a dimensionless coupling constant. If Planck’s h is dropped, then the system cannot measure quantities related to quantum phenomena. If Newton’s G is dropped, then the system does not include gravity. The vacuum also comes in with any two of its four interdependent constants $\{c,Z_0,{ϵ}_0,{\mu }_0\}$.

(3a)

The fine-structure constant ${\mathsf{\alpha }}_h={\left(861.022576\right)}^{-1}$, not multiplied by $2\pi$, is the only coupling constant that must be included in absolute terms. It has been measured by experiment, and it provides the scale factor $\sqrt{{\mathsf{\alpha }}_h}\simeq 1/30$ used by the Higgs mechanism to deflate and couple to the bottom quark; and then, to reach down to all the other lower-mass particles (Table A1 below). Furthermore, the Higgs mass is apparently related to the masses of vector bosons, quarks, and leptons by G-M averaging and Koide’s scale of 2/3 in various incarnations. The above scales should be present in the coupling constants of the various fields (see Appendix A.3).

(3b)

All other unitless constants must be included in relative terms because only ratios of coupling constants have physical meaning—they provide relative strengths, just like the dimensional quantities.

(3c)

The modern definitions of the unitless coupling constants are incorrect because $\cancel{h}$ was used instead of Planck’s physical constant h. Dirac’s $\cancel{h}$ is a composite constant that also carries planar 2-D geometry and a descriptive unit of [rad]⁻¹; the $2\pi$ term in $\cancel{h}$ has inadvertently reversed the influence of geometry on to the coupling constants.

(4)

The vacuum is a passive entity and not subject to forcing of any kind by the material world. By providing the least (but nonzero) resistance to all motions that occur in its domain, the vacuum installs upper limits to the material world (c and $Z_0$ in nearly perfect dielectrics), which must then be included in systems of units as well. These two geometry-free constants also bring $4\pi {ϵ}_0$ and ${\mu }_0/\left(4\pi \right)$ with them, in which the influence of 3-D geometry (the $4\pi$ term) is apparent. (Here, the vacuum’s ${ϵ}_0$ and ${\mu }_0$ are both lower limits.) In unit combinations, such as $4\pi {ϵ}_0\cancel{h}$ and ${\mu }_0/\left(4\pi \cancel{h}\right)$, geometry inadvertently cancels out leaving behind unitless numerical imprints in the equations (see the three atomic radii in §1.2).

(5)

The rate of change of the moment of inertia is the universal unit of all action integrals, so it appears that inertia and the weak equivalence principle have been built into all scales of the universe, large and small, gravitating or not.

(6a)

There exists a new atomic mass scale $M_{\mathrm{A}}=15.0$ MeV/$c^2$ that we found by deflating the original Planck mass $M_{\mathrm{p}}$ by $\sqrt{{\mathsf{\beta }}_G}=4.900\times {10}^{-22}$, where ${\mathsf{\beta }}_G$ is the relative ratio of the coupling constants of gravity and fine structure. Of course, in our expanding universe, the event took place in reverse ($M_{\mathrm{A}}/\sqrt{{\mathsf{\beta }}_G}\to M_{\mathrm{p}}$). This inflation of scale accounted for 21.3 orders of magnitude in mass, and explains how the Planck scale is connected to the atomic world. (At the same time, the atomic scale of length was deflated by precisely the same amount to produce the tiny Planck length.)

(6b)

No (sub)atomic particle is found to occupy a scale value, and the measured masses in the atomic world are connected mostly by G-M averaging. By using G-M averaging, nature (a) remains impartial to designating any one of the particles as being more significant than the other particles; and (b) assigns more weight to the smaller participant in the G-M, thereby assisting smaller forces to leave their marks on the universe.

(6c)

We can relate characteristic atomic constants (charge e, mass $m_{\mathrm{e}}$, G-M $\sqrt{m_{\mathrm{e}}{m}_{\mathrm{p}}}$, Compton radius $r_{\mathrm{c}}$) to scale values ($q_{\mathrm{p}},M_{\mathrm{p}},M_{\mathrm{A}},L_{\mathrm{A}}$, respectively), but this is not how these physical entities were created; they were created by the Higgs scalings (1/30 and 2/3) and by G-M averaging of other nearby physical entities.

(7)

Leptons, quarks, and bosons get their masses from the Higgs field. The boson-quark mass ladder is shown in Table A1 below. How the Higgs field acquires its mass $m_{\mathrm{H}}$ and its vacuum expectation value v remains a mystery; the only hints in the known masses [32] are that $m_{\mathrm{H}}\simeq v/2$ (to within a deviation of 1.7%) and the G-M $m_{\mathrm{t}}=\sqrt{m_{\mathrm{H}}v}$ (equation (37), deviation 1.8%).

(8)

Koide’s lepton constant $K=2/3$ is one of the scaling constants used by the Higgs field in its couplings to other particles. We derived it from first principles in Appendix A, and it is also applicable to the high-mass quark triplet c-b-t. We also derived two additional Koide-type constants, $J=4/7$ (from the low-mass quark triplet u-d-s) and $B=0.336$ (from the Higgs bosons W^±-Z⁰-H). Constant B is barely 0.8% larger than the absolute minimum value of 1/3 that occurs for three equal masses.

(9)

In Appendix B, we pointed out four instances of a universal law that has the general form

$$\left(\mathrm{a}\mathrm{surface}\mathrm{density}\right)\propto {\left(\mathrm{a}\mathrm{kinetic}\mathrm{scalar}\mathrm{quantity}\right)}^4,$$

in which the power of 4 is the sum of the 3 spatial degrees of freedom and 1 additional degree of freedom for the scale of the underlying scalar quantity. The three types of surface density involved describe force F, power P, and moment of inertia I, all divided by surface area A. Pressure $F/A$ appears in the Higgs field and the Casimir effect; intensity $P/A$ appears in the Stefan-Boltzmann law; and mass $I/A$ appears in the Tully-Fisher/Faber-Jackson relation in spiral/elliptical galaxies. It certainly appears that the dynamics of the present universe is driven by the surface densities of various fundamental quantities (see also Appendix B.2).

6.2. Critical Questions and Answers

(Q1)

How does Planck mass relate to the atomic world?

—The atomic mass scale $M_{\mathrm{A}}=15$ MeV/$c^2$ inflates precisely to the Planck mass

$M_{\mathrm{A}}/\sqrt{{\mathsf{\beta }}_G}\to M_{\mathrm{p}}$, where the ratio $\sqrt{{\mathsf{\beta }}_G}=4.900\times {10}^{-22}$, a comparative dimensionless quantity (i.e., a ratio between two dimensionless constants).

(Q2)

What is the physical meaning of number 137?

—Number $137=861/\left(2\pi \right)$, where the $2\pi$ is a geometric term carrying the descriptive unit of radian [11]; so, 137 is a composite constant, and this is the reason that we did not figure out its physical significance in the past 100 years. The actual physical constant is 861, and the scale factor $\sqrt{861}\simeq 30$ is used by the Higgs boson to assign masses dynamically to much lighter particles, starting with the bottom quark and moving on down the mass ladder (Table A1). So, the deflation scale 1/30 should appear in the Higgs couplings of the lower-mass particles.

(Q3)

What is the physical meaning of Koide’s constant?

—Koide’s $K=2/3$ is another scale factor used in the Higgs couplings to assign masses to lighter vector bosons (the particles W^± and Z⁰). Koide’s formula holds exactly for the leptons e-$\mathsf{\mu }$-$\tau$ and for the high-mass quarks c-b-t (proofs are given in Appendix A.2).

(Q4)

How does the top quark get its mass?

—The top quark mass is the geometric mean of the Higgs mass and the Higgs vacuum expectation value $v=246.22$ GeV/$c^2$, so that $m_{\mathrm{t}}=\sqrt{m_{\mathrm{H}}v}$ to a precision of 1.8% relative to the measured experimental values $m_{\mathrm{H}}$ and v [32].

(Q5)

How do Higgs vector bosons get their masses?

—By two different mechanisms (couplings): In the ordered compact high-mass triplet, Z⁰-H-t, the Higgs mass is the geometric mean of the Z⁰ mass and the top quark mass, viz. $m_{{\mathrm{Z}}^0}=m_{\mathrm{H}}^2/m_{\mathrm{t}}$. In contrast, the W^± coupling involves Koide’s scale since ${\mathrm{W}}^{\pm }=K{m}_{\mathrm{H}}$, where $K=2/3$.

(Q6)

How does the bottom quark get its mass?

—By a third coupling mechanism: The Higgs mass is scaled down by the factor of 30 derived in (Q2) above, so that $m_{\mathrm{b}}=m_{\mathrm{H}}/30$. We have tried empirically several other scalings and G-Ms, but none of these patterns matched the experimental value of $m_{\mathrm{b}}$, which is much lower than $m_{\mathrm{H}}$ (the mass-energy gap is 121 GeV; Table A1). The assignment of mass to the bottom quark is becoming a major issue to resolve in the future by any theory that purports to explain mass assignments to low-mass quarks.

(Q7)

Is $\cancel{h}$, rather than h, the true universal constant?

—They both are, but Planck’s h is a pure physical constant, whereas Dirac’s $\cancel{h}=h/\left(2\pi \right)$ is composite and includes also the 2-D geometric term $2\pi$ that carries the descriptive unit of [rad]. Because of this geometric content, a miscue was committed in the post-Planckian era when $\cancel{h}$ was adopted for the modern definitions of the fine-structure constant and the gravitational coupling constant in which the role of geometry reversed its character.

Appendix A.1. Physical Interpretation

The physical interpretation of nature’s choice of $f=4$ is deduced from equation (A2), rewritten in the accessible form

$$\frac{1}{3}\left(m_{\mathrm{e}}+m_{\mathsf{\mu }}+m_{\tau }\right)=\frac{4}{3}\left(\sqrt{m_{\mathrm{e}}{m}_{\mathsf{\mu }}}+\sqrt{m_{\mathrm{e}}{m}_{\tau }}+\sqrt{m_{\mathsf{\mu }}{m}_{\tau }}\right).$$

(A8)

The factor of 1/3 indicates that the left-hand side is the arithmetic mean of the lepton masses. The factor of 4/3 on the right-hand side (also the G-M $\sqrt{K_1{K}_2}$ to within 1%) is $C_F$, the quadratic Casimir charge of the SU(3) fundamental representation [33]. It seems then that the assignment of masses to the leptons is constrained also by the delivery of charge, and this is why

$$f=3C_F,$$

(A9)

appears in the right-hand side of equation (A2). This last equation is a special case of the general formula ([33], section 4.5) of quantum chromodynamics, viz.

$$\frac{N_A}{2}=N{C}_F,$$

(A10)

as applied to SU(3), where $N=3$ dimensions, the indices $1,2,\cdots ,N_A$ label the $N_A\times N_A$ color generators in the “octet” quark-antiquark state, $N_A=N^2-1=8$, and $f=N_A/2$; thus, we find that $f=4$ and $C_F=4/3$ in SU(3). Finally, Koide’s constant turns out to depend only on the SU(3) octet number $N_A$, viz.

$$K=\frac{N_A}{N_A+4},$$

(A11)

or, equivalently, on the dimensionality $N=3$ of space. This result effectively does not support higher dimensional theories of space, such as strings and their variants. For as long as $K=2/3$, space is strictly three-dimensional, and there are no additional hidden dimensions (such as described in Ref. [48] and many other similar textbooks).

Appendix A.2. Additional Koide-type Constants

The G-M relations of § 3.1 can help us make physical sense of various other triple combinations of masses involving quarks or bosons that have been previously derived by numerology. Here, we summarize the calculations for three triplets that naturally come to mind, the quarks c-b-t and u-d-s, and the bosons W^±-Z⁰-H:

Table A1. Boson-Quark mass ladder in terms of the Higgs mass $m_{\mathrm{H}}=125.25$ GeV/$c^2$[32]. Two scales are used, Koide’s $K=2/3$ and the 1/30 deflation of $m_{\mathrm{H}}$ down to the bottom quark mass $m_{\mathrm{b}}$. Lepton masses and proton mass are also shown in terms of $m_{\mathrm{H}}$ for a comparison of scales.

Particle	Mass-Energy	Mass Relation(a)	Deviation(b)
	(MeV)		(%)
Vector Bosons
Z0	$9.1118\times {10}^4$	$m_{{\mathrm{Z}}^0}=K^{3/4}{m}_{\mathrm{H}}$	+1.3
W±	$8.0377\times {10}^4$	$m_{{\mathrm{W}}^{\pm }}=K{m}_{\mathrm{H}}$	+3.9
Quarks
top	$1.725\times {10}^5$	$m_{\mathrm{t}}=K^{-3/4}{m}_{\mathrm{H}}$	$-1.6$
bottom	$4.180\times {10}^3$	$m_{\mathrm{b}}=m_{\mathrm{H}}/30$	$-0.12$
charm	$1.270\times {10}^3$	$m_{\mathrm{c}}=2\sqrt{K/{30}^3}{m}_{\mathrm{H}}$	$-2.0$
strange	93.4	$m_{\mathrm{s}}=\left(K/{30}^2\right)m_{\mathrm{H}}$	$-0.67$
down	4.67	$m_{\mathrm{d}}=2\left(K^{7/4}/{30}^3\right)m_{\mathrm{H}}$	$-2.3$
up	2.16	$m_{\mathrm{u}}=\left(K^2/{30}^3\right)m_{\mathrm{H}}$	$-4.5$
Leptons & Proton
electron	0.511	$m_{\mathrm{e}}=\sqrt{\frac{K}{2\times {30}^7}}{m}_{\mathrm{H}}$	$-4.3$
muon	105.66	$m_{\mathsf{\mu }}={22}^2\frac{K^{5/2}}{\sqrt{2\times {30}^7}}{m}_{\mathrm{H}}$	$-0.45$
tauon	$1.777\times {10}^3$	$m_{\tau }={40}^2\frac{K^{-3/2}}{\sqrt{2\times {30}^7}}{m}_{\mathrm{H}}$	$-0.93$
&
proton	938.272	$m_{\mathrm{p}}=22{\left(\frac{2K^5}{{30}^9}\right)}^{1/4}{m}_{\mathrm{H}}$	$-0.12$

Notes: ^(a)The top three masses do not depend on the deflation scale 1/30, whereas the mass of the bottom quark $m_{\mathrm{b}}$ is the only one that does not depend on Koide’s scale $K=2/3$. ^(b)Deviation = [(right-to-left side) $-1$]$\times 100\%$;  left side is taken from Ref. [32].

Table A1. Boson-Quark mass ladder in terms of the Higgs mass $m_{\mathrm{H}}=125.25$ GeV/$c^2$[32]. Two scales are used, Koide’s $K=2/3$ and the 1/30 deflation of $m_{\mathrm{H}}$ down to the bottom quark mass $m_{\mathrm{b}}$. Lepton masses and proton mass are also shown in terms of $m_{\mathrm{H}}$ for a comparison of scales.

Particle	Mass-Energy	Mass Relation(a)	Deviation(b)
	(MeV)		(%)
Vector Bosons
Z0	$9.1118\times {10}^4$	$m_{{\mathrm{Z}}^0}=K^{3/4}{m}_{\mathrm{H}}$	+1.3
W±	$8.0377\times {10}^4$	$m_{{\mathrm{W}}^{\pm }}=K{m}_{\mathrm{H}}$	+3.9
Quarks
top	$1.725\times {10}^5$	$m_{\mathrm{t}}=K^{-3/4}{m}_{\mathrm{H}}$	$-1.6$
bottom	$4.180\times {10}^3$	$m_{\mathrm{b}}=m_{\mathrm{H}}/30$	$-0.12$
charm	$1.270\times {10}^3$	$m_{\mathrm{c}}=2\sqrt{K/{30}^3}{m}_{\mathrm{H}}$	$-2.0$
strange	93.4	$m_{\mathrm{s}}=\left(K/{30}^2\right)m_{\mathrm{H}}$	$-0.67$
down	4.67	$m_{\mathrm{d}}=2\left(K^{7/4}/{30}^3\right)m_{\mathrm{H}}$	$-2.3$
up	2.16	$m_{\mathrm{u}}=\left(K^2/{30}^3\right)m_{\mathrm{H}}$	$-4.5$
Leptons & Proton
electron	0.511	$m_{\mathrm{e}}=\sqrt{\frac{K}{2\times {30}^7}}{m}_{\mathrm{H}}$	$-4.3$
muon	105.66	$m_{\mathsf{\mu }}={22}^2\frac{K^{5/2}}{\sqrt{2\times {30}^7}}{m}_{\mathrm{H}}$	$-0.45$
tauon	$1.777\times {10}^3$	$m_{\tau }={40}^2\frac{K^{-3/2}}{\sqrt{2\times {30}^7}}{m}_{\mathrm{H}}$	$-0.93$
&
proton	938.272	$m_{\mathrm{p}}=22{\left(\frac{2K^5}{{30}^9}\right)}^{1/4}{m}_{\mathrm{H}}$	$-0.12$

Notes: ^(a)The top three masses do not depend on the deflation scale 1/30, whereas the mass of the bottom quark $m_{\mathrm{b}}$ is the only one that does not depend on Koide’s scale $K=2/3$. ^(b)Deviation = [(right-to-left side) $-1$]$\times 100\%$;  left side is taken from Ref. [32].

High-mass quarks c-b-t.—Based on experimental masses, equation (A1) with c-b-t values in place of e-$\mathsf{\mu }$-$\tau$ values produces a constant of 0.669 on the right-hand side, only 0.35% higher than $K=2/3$. This is a solid physical result. From the equations of § 3.1, we find that $m_{\mathrm{t}}=30m_{\mathrm{b}}{K}^{-3/4}$, $m_{\mathrm{c}}=2m_{\mathrm{b}}\sqrt{K/30}$, and a corresponding constant of 0.668 with a deviation of 0.20% from $K=2/3$.

Low-mass quarks u-d-s.—Koide’s K is not produced by the masses of the u-d-s triplet. Based on their experimental masses, equation (A1) with u-d-s in place of e-$\mathsf{\mu }$-$\tau$ produces the constant $J=0.567$ on the right-hand side, probably a value of no interest to numerology. We, on the other hand, have derived this constant analytically by utilizing the G-M relations of § 3.1 and by expressing the u-d-s quark masses in terms of the Higgs scales $m_{\mathrm{b}}/m_{\mathrm{H}}=1/30$ (deflation) and $K=2/3$ (Koide), used in assignments of masses lower than $m_{\mathrm{H}}$. It turns out that the entire boson-quark mass ladder has to be calculated in the process. The results of our calculations are listed in Table A1. Using the values obtained for the u-d-s masses at the bottom of the quark mass ladder, we find that

$$(m_{\mathrm{u}}+m_{\mathrm{d}}+m_{\mathrm{s}})/{\left(\sqrt{m_{\mathrm{u}}}+\sqrt{m_{\mathrm{d}}}+\sqrt{m_{\mathrm{s}}}\right)}^2=0.570,$$

(A12)

a constant that deviates only by 0.53% from the experimental value of $J=0.567$. In this case, we find that $0.570\to 4/7$, $f=2C_F=N_A/N=8/3$, and $J=N_A/(N_A+2N)=4/7$ in SU(3) (in place of equations (A9)-(A11)).

Higgs bosons W^±-Z⁰-H.—Based on experimental masses, equation (A1) with W^±-Z⁰-H masses in place of e-$\mathsf{\mu }$-$\tau$ masses produces a constant of $B=0.336$ on the right-hand side, only 0.80% higher than the lowest attainable value of $1/3$ obtained for three equal masses. From equations (39) and (40), we find that $m_{{\mathrm{Z}}^0}=K^{3/4}{m}_{\mathrm{H}}$ and $m_{{\mathrm{W}}^{\pm }}=K{m}_{\mathrm{H}}$ (see also Table A1), and then, equation (A1) for the W^±-Z⁰-H triplet is transformed to

$$1+K^{3/4}+K=0.336{\left(1+K^{3/8}+K^{1/2}\right)}^2,$$

(A13)

with an accepted root at $K_1=0.662$ (deviation 0.70% from $K=2/3$) and a rejected root at $K_2=1.541$. We conclude that $K_1$ is Koide’s constant, in which case, $f=1.012$ on the right-hand side of equation (A2) and $B=f/(f+2)$ (let $B\to K$ in equation (A7)). The quark-antiquark color octet number $N_A$ and the quadratic Casimir charge $C_F$ are not involved in these calculations (equations (A9)-(A11) are not applicable to vector bosons).

Appendix A.3. Determination of Various Coupling Factors and Constants of the Standard Model

Although approximate, the empirical equations for the mass ladder listed in Table A1 constitute a scaling model, and we can use them to calculate theoretical values for various constants of the Standard Model, thereby eliminating many of the free gauge couplings and particle masses from the model (currently being considered out of reach and expected to be measured by experiment). We briefly describe the determinations of several such constants below.

Appendix A.3.1. The Weinberg Angle

The weak mixing angle or Weinberg angle ${\theta }_{\mathrm{w}}$ is usually defined by [16,20]

$$cos{\theta }_{\mathrm{w}}\equiv \frac{m_{{\mathrm{W}}^{\pm }}}{m_{{\mathrm{Z}}^0}}.$$

(A14)

Its experimental value is 0.492 radians. Substituting the boson masses of the mass ladder (Table A1), we find that

$$cos{\theta }_{\mathrm{w}}=K^{1/4},$$

(A15)

that gives the theoretical value 0.443 radians (effectively 2.8 degrees smaller). CODATA [23] tabulates the value of ${sin}^2{\theta }_{\mathrm{w}}=0.223$, whereas our ${\theta }_{\mathrm{w}}=0.443$ gives ${sin}^2{\theta }_{\mathrm{w}}=0.184$. The Weinberg angle appears in several of the equations that follow.

Appendix A.3.2. Vector-Boson g-factors and Electric Charge

We determine first an equation for the Higgs VEV: using equation (37) and the top-quark relation in the mass ladder (Table A1), we find that

$$v=K^{-3/2}{m}_{\mathrm{H}}.$$

(A16)

This relation gives the ratio $m_{\mathrm{H}}/v=0.54$, whereas its experimental value is 0.51 [23].

The masses of the vector bosons are given by [16,20]

$$m_{{\mathrm{W}}^{\pm }}=g(v/2),$$

(A17)

and

$$m_{{\mathrm{Z}}^0}=\sqrt{g^2+g^{\prime 2}}(v/2),$$

(A18)

where g is the SU(2) weak isospin coupling and $g^{\prime }$ is the U(1) weak hypercharge coupling.

Substituting the boson masses of the mass ladder and the VEV of equation (A16), we find that

$$g=2K^{5/2},$$

(A19)

and

$$g^{\prime }=2K^{9/4}\sqrt{1-\sqrt{K}}.$$

(A20)

These relations give $g=0.726$ and $g^{\prime }=0.344$. The experimental values are 0.653 and 0.350, respectively. The calculated g-factor shows a larger deviation because of the relatively large (+3.9%) deviation of the calculated ratio $m_{{\mathrm{W}}^{\pm }}/m_{\mathrm{H}}$ in Table A1.

The electric charge in particle physics is defined in natural units as29

$$e\equiv \sqrt{4\pi {\mathsf{\alpha }}_{\cancel{h}}}=0.303.$$

(A21)

This is the constant of the EM interaction obtained in the Standard Model from the equation [16,20]

$$e=gsin{\theta }_{\mathrm{w}}=g^{\prime }cos{\theta }_{\mathrm{w}}=0.308.$$

(A22)

Using the equations of the mass ladder, we find that

$$e=2K^{5/2}\sqrt{1-\sqrt{K}}=0.311.$$

(A23)

Just as the constant of EM interaction e is defined in terms of the fine-structure constant ${\mathsf{\alpha }}_{\cancel{h}}$ in equation (A21), so is the the weak isospin g-factor in terms of the intrinsic strength of the weak interaction ${\mathsf{\alpha }}_{\mathrm{w}}$, viz.

$$g\equiv \sqrt{4\pi {\mathsf{\alpha }}_{\mathrm{w}}}=0.653.$$

(A24)

From the second equality, we get

$${\mathsf{\alpha }}_{\mathrm{w}}\simeq \frac{1}{30},$$

(A25)

which is effectively the deflation scale that we also found for $m_{★}/M_{\mathrm{p}}$ in footnote 8 and for $m_{\mathrm{e}}/M_{\mathrm{A}}$ in equation (36) above. Therefore, the deflation scale appears to be the strength of the weak interaction. Furthermore, these coincidences involving the scale 1/30 suggest that a particle of mass $m_{★}=M_{\mathrm{p}}/30\simeq 1.8\times {10}^{-9}$ kg could indeed exist at the unification scale of ${10}^{18}$ GeV (30 times below the Planck scale), just as the electron does at 30 times below the subatomic scale of 15 MeV (equation (21)).

Appendix A.3.3. Electron Yukawa Coupling Factor

The Higgs Yukawa coupling to the electron $y_{\mathrm{e}}$ [20,49] determines the mass of the electron, viz.

$$m_{\mathrm{e}}=y_{\mathrm{e}}\left(v/\sqrt{2}\right).$$

(A26)

Its experimental value is $y_{\mathrm{e}}=2.935\times {10}^{-6}$. Using the mass relation for the electron (Table A1) and equation (A16), we find that

$$y_{\mathrm{e}}=\frac{K^2}{{30}^{7/2}}=3.005\times {10}^{-6}.$$

(A27)

Appendix A.3.4. Quark g-factors

In the Glashow-Weinberg-Salam theory of electroweak interactions [20], the lowest-mass quarks (u and d) have masses

$$m_{\mathrm{u}}=g_{\mathrm{u}}\left(v/\sqrt{2}\right),$$

(A28)

and

$$m_{\mathrm{d}}=g_{\mathrm{d}}\left(v/\sqrt{2}\right),$$

(A29)

where $g_{\mathrm{u}}$ and $g_{\mathrm{d}}$ are the coupling constants of the corresponding Higgs interactions with experimental values $g_{\mathrm{u}}=1.241\times {10}^{-5}$ and $g_{\mathrm{d}}=2.682\times {10}^{-5}$. Using the corresponding mass relations (Table A1) and equation (A16), we find that

$$g_{\mathrm{u}}=\frac{\sqrt{2}{K}^{7/2}}{{30}^3}=1.267\times {10}^{-5},$$

(A30)

and

$$g_{\mathrm{d}}=\frac{2\sqrt{2}{K}^{13/4}}{{30}^3}=2.805\times {10}^{-5}.$$

(A31)

Finally, we use the coupling equation for the mass of the bottom quark, viz.

$$m_{\mathrm{b}}=g_{\mathrm{b}}\left(v/\sqrt{2}\right),$$

(A32)

where $g_{\mathrm{b}}$ is the coupling constant of the Higgs interaction with experimental value $g_{\mathrm{b}}=2.401\times {10}^{-2}$. Using the mass relation for the bottom quark (Table A1) and equation (A16), we find that

$$g_{\mathrm{b}}=\frac{\sqrt{2}{K}^{3/2}}{30}=2.566\times {10}^{-2}.$$

(A33)

Appendix B.1. Dimensional Analysis of Surface Densities

Dimensional analysis can help us understand the meaning of these surface densities, but not by reducing their definitions to the fundamental units of the UPS. We have to search a little harder to find any common properties between these quantities. We begin with the power surface density (wave intensity) ${\sigma }_P$ that assumes the simplest form among the surface densities:

$${\sigma }_P=\frac{F^2}{(I/T)}.$$

(A38)

In EM interactions, the rate of change of moment of inertia can be replaced by

$$I/T=q^2{Z}_0,$$

(A39)

where q is charge and $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$ is vacuum impedance; we find that

$${\sigma }_P=Z_0^{-1}{\mathcal{E}}^2,$$

(A40)

where the electric field is given by $\mathcal{E}=F/q$.

For gravitational power, the force in equation (A38) is also modified by the vacuum, but only by coupling to Newton’s G. Rewriting equation (A38) in terms of the gravitational field (acceleration) a, we find that

$${\sigma }_P=\left(\frac{c}{G}\right)a^2.$$

(A41)

Force is $F=P/c$ in terms of power P, and the force surface density ${\sigma }_F$ takes the corresponding forms

$${\sigma }_F=\frac{{\sigma }_P}{c}={ϵ}_0{\mathcal{E}}^2=G^{-1}{a}^2,$$

(A42)

where the vacuum’s c drops out from gravity’s ${\sigma }_F$. This is a fundamental difference as compared to the EM field’s ${\sigma }_F$, in which the vacuum (the ${ϵ}_0$ here) is permanently attached.

Finally, as was probably expected, the moment-of-inertia surface density ${\sigma }_I=M$ does not quite conform to the above picture. We find that

$${\sigma }_I=\frac{F^2}{(I/T^4)}=\frac{{\mathcal{J}}^2}{(I/T^2)},$$

(A43)

where $\mathcal{J}$ is impulse and $I/T^2$ is energy. Mass is already built with inertia, and it is not surprising that it does not scale as ${(I/T)}^{-1}$, as the other densities do. To find out how this force squared and impulse squared are regulated, we rewrite the terms in the denominators. It turns out that

$$I/T^3=P,$$

(A44)

so that equation (A43) then takes the form

$${\sigma }_I=\frac{F^2}{(P/T)}=\frac{{\mathcal{J}}^2}{PT},$$

(A45)

where the integrated quantity $PT$ represents energy E. Thus, $F^2$ is regulated by the rate of change of power $P/T$, and ${\mathcal{J}}^2$ is regulated by energy $PT$, both of which are restricted by the speed of light.

Appendix B.2. Physical Properties of Surface Densities

We conclude with a summary of properties of the above three surface densities:

(a)

The densities ${\sigma }_P$ and ${\sigma }_F$ (wave intensity and pressure, respectively) are both modified by the rate of change of inertia $(I/T)$ at all scales (equations (A38) and (A42)).

(b)

Density ${\sigma }_I$ (i.e., mass) is not modified by inertia, it is inertia; instead, we can say that mass is force squared $F^2$ regulated by the rate of change of power $(P/T)$, or impulse squared ${\mathcal{J}}^2$ regulated by energy E (equation (A45)), where E should be viewed here as the rate of change of the action integral, i.e., $(\mathcal{S}/T)$.

(c)

Vacuum constants are explicitly present in ${\sigma }_P$ (equations (A40) and (A41), where ${\sigma }_P$ is written in terms of the force fields squared ${\mathcal{E}}^2$ and $a^2$, respectively).

(d)

The vacuum remains present in the ${\sigma }_F$ of the EM field, but it drops out from the ${\sigma }_F$ of the gravitational field (both behaviors are shown in equation (A42)).

(e)

The force surface density ${\sigma }_F$ (equation (A42)) represents the conventional energy density of the force fields, whereas ${\sigma }_P$ (equation (A38)) shows that inertia is present during the action of all forces.

(f)

Both sides of equation (A39) have dimensions of Planck’s constant h, thus $I/T\sim q^2{Z}_0=\left[\mathrm{action}\right]$. Higher powers of T in $I/T^n$ ($n=2,3$) are also physically quite important: $I/T^2=\left[\mathrm{energy}\right]$ and $I/T^3=\left[\mathrm{power}\right]$. Equation (A44) for $I/T^3$ then implies that power stems from the third time derivative of the moment of inertia, a property that is fundamental for the emission of gravitational waves. The same relation, applied to EM waves, produces the ohmic power with dimensions of [electric current]²[ohmic resistance].

(g)

Equations (A34)-(A37) all have the characteristic form

$$\left[\mathrm{a}\mathrm{surface}\mathrm{density}\right]\propto {\left[\mathrm{a}\mathrm{kinetic}\mathrm{scalar}\mathrm{quantity}\right]}^4,$$

in which the power of 4 represents $N+1$ degrees of freedom, with $N=3$ for the spatial dimensions, plus 1 degree of freedom for the scale of the underlying scalar quantity.

(h)

The intensity ${\sigma }_P$ and the pressure ${\sigma }_F$ are proportional to kinetic terms that express ${\left[\mathrm{specific}\mathrm{energy}\right]}^4$. The Tully-Fisher/Faber-Jackson relation $M\sim V^4\sim {\left[\mathrm{specific}\mathrm{energy}\right]}^2$[17,18] then indicates that, in units where $c=G=1$ ([20], p. xix), it is the square of the mass

$$M^2={\left({\sigma }_I\right)}^2\sim {\left[\mathrm{specific}\mathrm{energy}\right]}^4,$$

that falls in the same category. Thus, we believe that the $M^2$ term here is meant to signify that the inertial and the gravitational mass are included on an equal footing (just as in equation (12) discussed in § 2.1.3).