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I was wondering about the fundamental constants of Nature since several years, and still pondering on them. Of course, I have read a lot of papers on them, but never found any satisfying classification/description of their true "nature" (pun intendend). I'm especially interested in 6 constants: $\hbar$, $c$ (or more specifically $c^{-1}$), $k_{\mathrm{B}}$, $G$, $k \, e^2$ (more specifically $\alpha \equiv k \, e^2 / \hbar c$) and $\Lambda$ (the mysterious cosmological constant). I'm not counting the fundamental particles masses as "fundamental constants", since they could be derived/calculated in principle, from the particles interactions. I consider the masses as emergent parameters.

I noticed that the 6 constants above could be grouped into two classes, since they're appearing differently into physical theories and equations. Firstly, there are three unit conversion factors, which have values that are reflecting the human scale in our universe. Their physical dimension don't depend on the spacetime dimension $D$: \begin{align} \hbar &\sim 10^{-34} \, \mathrm{J \cdot s} \quad [\text{unit of action in spacetime}] \tag{1} \\[12pt] c^{-1} &\sim 10^{-9} \, \mathrm{s/m}, \quad [\text{unit of time in spacetime}] \tag{2} \\[12pt] k_{\mathrm{B}} &\sim 10^{-23} \, \mathrm{J/K}. \quad [\text{unit of entropy/information in spacetime}] \tag{3} \end{align} As a second class of constants, there are the coupling constants, which have dimensions that depend on the spacetime dimensions, except $\Lambda$ (see my comment below, about Poisson's equation). Of course, $D = 4$ in our universe: \begin{align} \frac{\hbar G}{c^3} &\sim \mathrm{L}^{D \,-\, 2}, \quad [\text{coupling of geometry and energy content in spacetime}] \tag{4} \\[12pt] \alpha \equiv \frac{k \, e^2}{\hbar c} &\sim \mathrm{L}^{D \,-\, 4}, \quad [\text{coupling of large scale electromagnetism and matter}] \tag{5} \\[12pt] \Lambda \equiv \frac{8 \pi G \rho_{\mathrm{vac}}}{c^4} &\sim \mathrm{L}^{- 2}. \quad [\text{coupling of large scale vacuum and geometry ??}] \tag{6} \end{align} I interpret $\Lambda$ (or $\rho_{\text{vac}}$? See my first comment below) as a measure of the non-trivial vacuum energy density. In principle, this vacuum energy could be calculated in QFT (using some hypothesis), but I think it can't. This energy depends on the number of fundamental fields in Nature (including the unknown Dark Matter) and may even depend on an unknown Quantum Gravity theory. I believe it's much better to interpret $\Lambda$ as a new independant fundamental constant that cannot be derived or calculated from any field theory without some arbitrary input/hypothesis.

Constants (1), (2), (3) are associated to the Human Scale, or define the only scale for which Life could be supported in our universe. It is natural to fix these three constants to define our measure units (especially mass, length and temperature units). Notice that these three constants don't depend on the spacetime dimension $D$! Constants (4), (5), (6) may define other scales (depending on dimensionality $D$) and are associated to some kind of macroscopic couplings.

So I have two questions:

I never saw that distinction of constants in the litterature. Is this "classification" new or have already been considered before? I would like to find some reference that could support it.

The dimensions of (4), (5) and maybe even (6) seem to suggest that these 3 constants could be "fields" by themselves, that could vary discretely from one universe to another one (in the multiverse, if this thing is real).

They also suggest that they may be dependant on the energy scale involved, i.e. be functionals of the energy included in spacetime: $G = f_1(E)$, $\alpha = f_2(E)$ (which is already known, in QFT) and maybe $\Lambda = f_3(E)$.

I suspect that the cosmological constant (as a measure of the vacuum energy) should be the most fundamental constant that represents the "content of spacetime" (not just the usual energy in the space section), and that $G$ and $\alpha$ could be dependant on $\Lambda$: \begin{align}\tag{7} G &= f_1(\Lambda), &\alpha &= f_2(\Lambda). \end{align} So is this idea have already been explored/studied before? What could support the view that $\Lambda$ is a true independant constant that may determine the value of $G$ and $\alpha$ for a given dimension $D$ and a given topology?

I know that all this stuff sounds very speculative (I'm expecting a lot of downvotes here), but there's already a lot of speculative theories in modern physics. The "classification" given above could have been already studied before. If not, it could shed some light on the fundamental constants of Nature.

Cham
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  • You may be interested in Dirac's Large Number Hypothesis: https://en.m.wikipedia.org/wiki/Dirac_large_numbers_hypothesis – Craig Dec 25 '18 at 19:11
  • I'm still wondering what should be the independant variable that must be considered, for the cosmological constant: $\Lambda$ or $\rho_{\text{vac}}$? If it's $\rho_{\text{vac}} \sim \mathrm{L}^{-D}$, then $\Lambda$ is derived from $\rho_{\text{vac}}$ ($\Lambda = 8 \pi G \rho_{\text{vac}}/c^4$) and there are three independant constants that determine a scale (if $D \ne 2, 4$): $G$, $\alpha$ and $\rho_{\text{vac}}$. – Cham Dec 27 '18 at 15:56
  • For anyone who thinks that it is meaningful to think about dimensionful "fundamental" constants as having values that can vary (as you imply towards the end of this post), this answer and its references are recommended reading. – ACuriousMind Dec 27 '18 at 16:07
  • Just to add that the constants $G$ and $\alpha$ are getting dimensions from their respective Poisson equation (or their relativistic generalisation) in $D - 1$ space dimensions:\begin{equation}\nabla^2 \Phi = \frac{\partial^2 \Phi}{\partial x_1^2} + \frac{\partial^2 \Phi}{\partial x_2^2} + \ldots + \frac{\partial^2 \Phi}{\partial x_{D - 1}^2} = 4 \pi G \rho,\end{equation} where mass density has dimensions $\rho \sim M/\mathrm{L}^{D - 1}$. Using Planck's constant $\hbar$ (so $M c / \hbar$ is $\mathrm{L^{-1}}$), energy density has dimension $1/\mathrm{L}^D$ in $D$ dimensional spacetime. – Cham Dec 27 '18 at 16:38
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    I think some of these points are up to debate. For example, you could say the cosmological constant measures the "coupling strength" of "vacuum and geometry", or you could say that it merely tells us how big the universe's length scales in Planck units, i.e. it's just another unit conservation. – knzhou Dec 27 '18 at 17:04
  • I want to emphasize that the units of $\hbar$, $c$ and $k_{\mathrm{B}}$ dont depend on the dimensionality of space. On the contrary, $G$, $\alpha$ and $\rho_{\text{vac}}$ have dimensionalities (and units) which depend on the space dimensionality. This is important for the classification that I'm discussing. – Cham Dec 27 '18 at 17:05
  • @knzhou, the problem with Planck's unit ($\ell_{\mathrm{P}} = \sqrt{\hbar G / c^3}$ when $D = 4$) is that it's dependant on the spacetime dimensionality. In a sense, it's not a "natural" unit without telling what is $D$. This is why I consider $G$ as a "coupling constant" which appears only in the dynamical equation (i.e Einstein or Poisson's equation). It doesn't have the same "status" as $\hbar$ or $c$, for example. – Cham Dec 27 '18 at 17:15
  • I like your question, even if it's probably unanswerable, imo, but I think the wording "in human terms" could be rephrased (after all, everything we observe is in human terms, one way or other), but more importantly, I think you should edit in your comments to your post (apologies if you have already) . Anyway, +1 for your thoughts on it. –  Dec 27 '18 at 17:46
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    @StudyStudy, thanks. Adding my comments to the question would make the Q much longer, and yet it's already pretty long. – Cham Dec 27 '18 at 17:48

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