In natural units,the gravitational fine structure constant is equal to the square of the mass of a particle

[math]\alpha_G = m^2[/math]

The hope or immediate realization of this are attempts to find quantization of mass depending on factors of [math]n\hbar[/math].

The [math]Gm^2[/math] can be thought of as the gravitational charge of the system analogous to the electric charge [math]e^2[/math] (as stated before - but not necessarily equal in magnitude, they are only equal in the dimensional sense). Some mass formula have been suggested in literature (see What is special about the Planck mass? Sivaram). Before we look at the more advanced mass formula that will be suggested here, let’s take a look at what some have called ‘’the mysterious’’ Weinberg formula:

[math]m = (\frac{H_0\hbar^2}{Gc})^{\frac{1}{3}}[/math]

In which [math]H_0[/math] is the Hubble parameter. This will correspond to either one particle mass, or Weinberg had in mind a spectral property leading to different masses. In any case, this equation cannot predict all particle masses with this description alone. Alternatively, the Weinberg formula has been written as

[math]m^3 = \frac{H_0 \hbar^2}{Gc}[/math]

In this formulation, it has been suggested by Arun and Sivaram that it is ''unclear'' whether now he associates the equation to three particle masses [1] or as an attempt to find a fundamental unit of mass. Either way, we can see his ultimate aim would have been to describe fundamental particle masses from cosmological features, like the Hubble constant. Schwinger, Motz (et al). have demonstrated what it means to quantize a charge: Charge and mass are in fact so similar, you can indeed describe both in similar physics. Nature seems to posit some kind of universality with the presence of an electric charge being synonymous with mass. The only exception it appears in nature is a neutrino, which is expected to have no charge and a very small mass, but if it has a very small mass (you can also argue) it may possess a vanishing, but non-zero charge with a proportionality of [math]e^2 \propto Gm^2[/math]. To try and articulate how small this charge would be, it would have a mass and charge 1 millionth of that of an electron. Either way, the ultimate idea is that large numbers can determine the small in dynamic ways.

We can in fact restate Weinbergs formula for one that satisfies the gravitational interpretation of the charge. I provide this as a simple manipulation of his equation:

[math]Gm^2 = \frac{H_0 \hbar^2}{mc} = \frac{H_0 \hbar^2}{p}[/math]

In which we recognize a momentum term in the denominator.

Let’s have a quick look at a more advanced mass formula candidate

[math]m = nk[\frac{mc}{e}\frac{1}{2 \sqrt{T}}]^n M_e = nk [\frac{\sqrt{G}m}{2 e}]^n M_e[/math]

Immediately we can notice the use of the gravitational charge in the last term [math]sqrt{G}m[/math] - the only difference is that it has focused on the Planck mass definition of the charge. The Planck mass should not necessarily be considered fundamental, it seems like too much a basic unit of matter for any particles we have observed in the standard model. Though the middle term is good for string dynamics and superstring tension, the last term appears to be made of more fundamental assumptions which included the gravitational charge of the system. The adjustable parameters is what allows us to predict particle masses, for example n = 2, k = 3 gives the muon mass, n = 2, k = 4 gives the pion mass, n = 3, k = 4 the \Delta resonance, n = 3, k = 6 the D meson n = 3, k = 10 the \psi, n = 4, k = 4 the upsilon particle. According to Sivaram, several more particle masses can be obtained.

Anyway, the main point may have became a bit clearer: The more advanced suggested mass formula does indeed predict a wide range of particle masses, but more importantly, the Weinberg formula can be expressible also in a dimensionless form:

[math]\frac{m}{m_e} = nk [\frac{\sqrt{G}m}{2 e}]^n = nk [\sqrt{\frac{H_0 \hbar^2}{mc}}\frac{1}{2 e}]^n = nk [\sqrt{\frac{H_0}{mc}}\frac{\hbar}{2 e}]^n[/math]

So in this sense, we get to keep Weinberg’s attractive idea about determining fundamental parameters from cosmic parameters - dimensionless terms are well-known in physics to be the only true physical parameters of a theory. A good example would be Bernoulli's fluid equation for an example.

While this is all nice, it seems from Regge trajectories, the construction of certain particles strangely coincides with multiples of the mass squared. If we go back to the formula:

[math]m = nk [\frac{\sqrt{G}m}{2 e}]^n M_e[/math]

Then obtaining the mass squared is a simple procedure:

[math]m^2 = nk [\frac{Gm^2}{4 e^2}]^n M^2_e[/math]

As shown in the previous post, the part of the Langrangian which deviates from flat space is in units of [math]G = 1[/math] is:

[math]\mathcal{L}\psi = m^2\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi[/math]

And so perhaps an appropriate correction to formulate an equation to predict particle masses as a contribution also from curved space may involve the replacement of the appropriate mass squared term:

[math]\mathcal{L}\psi = n\mathbf{k} [\frac{Q^2_g}{4 e^2}]^n M^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n M^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi[/math]

As stated in one previous post on another thread, the role of [math]\frac{Gm^2}{e^2}[/math] has played a large role in the history of physics, including applications to astrophysics - in order to know these types of relationships, I reference wiki:

''

- (4.5) in Barrow and Tipler (1986) tacitly defines [math]\frac{\alpha}{\alpha_G}[/math] as [math]\frac{e^2}{Gm_pm_e}[/math]
* -even though they do not name the [math]\frac{\alpha}{\alpha_G}[/math] in this manner, *, it nevertheless plays a role in their broad-ranging discussion of astrophysics, cosmology, quantum physics, and the anthropic principle;

In our case we would be defining the inverse of the formula which described this as [math]\frac{\alpha_G}{\alpha}[/math]. In what way the wave function plays a role in the Langrangian density is uncertain to me if it is even required at all.

**References:**

https://pdfs.semanti...e0b4995919d.pdf

https://arxiv.org/pd...ics/0408056.pdf

https://arxiv.org/ft...7/0707.0058.pdf

C. Sivaram, Astrophys. Sp. Sci. 207, 317 (1993); 219, 135 (1994)

**Edited by Dubbelosix, 14 August 2019 - 04:21 AM.**