Dirac stated that "Any two of the very large dimensionless numbers occuring in Nature are connected by a simple mathematical relation, in which the coefficients are of the order of magnitude unity."

For this particular relation:

\begin{equation} \frac{\overbrace{T_{0}}^{\text{age of the Universe}}}{\underbrace{e^2/\left(m_{e}c^2\right)}_{\text{time it takes light to cross an atom}}}\approx \frac{\overbrace{e^2}^{\text{electromagnetic force between an electron and a proton}}}{\underbrace{Gm_{p}m_{e}}_{\text{gravitational force between an electron and a proton}}} \end{equation} It means that $G$ should vary with time, since the age of the Universe is obviously increasing with time.

My question is: Why such a belief is justified? I know this is in conflict with observations but I wanted to understand Dirac's mind. Why he considered this to be, in principle, something that could hold in the Universe.

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    $\begingroup$ Eddington (who was a calculating prodigy) had similar beliefs. They seem more like numerology than physics, to me. $\endgroup$ – PM 2Ring Sep 5 '19 at 6:27
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    $\begingroup$ Following @knzhou, Dirac's thought process may have been: the ratio of electric and gravitational force strengths is unnaturally large; ah, but it has the same order of magnitude as a certain combination of basic physical quantities; so perhaps the two expressions are equal for a reason. And one can devise theories (e.g. quintessence-type theories) in which this relationship holds; but they seem to be inconsistent with observation - atoms in the early universe and today, emit at the same frequencies, etc... But I just guess at Dirac's mind; to really know, please seek evidence like papers etc. $\endgroup$ – Mitchell Porter Sep 5 '19 at 11:35
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    $\begingroup$ "It means that G should vary with time, since the age of the Universe is obviously increasing with time.". Is that a problem? I've seen papers toy around with non-constant G that varies conjointly with c^2. $\endgroup$ – Maxter Sep 5 '19 at 17:32
  • $\begingroup$ If the universe is expanding, the time it takes light to cross an atom will increase along with $T_0$, and $G$ could still be a constant. $\endgroup$ – chepner Sep 5 '19 at 19:47
  • $\begingroup$ @chepner Not really, the atom doesn't really get bigger by an expanding universe. $\endgroup$ – kutschkem Sep 6 '19 at 11:25

Dirac thought large numbers shouldn't exist at all in physics, on purely aesthetic grounds. After all, "where would such a number come from?" Today this criterion is known as "Dirac naturalness".

You might have heard about naturalness in the press, but few practitioners actually use Dirac's original form, because we know it's not very reliable. For example, the phenomenon of dimensional transmutation can be used to automatically create enormous numbers from normal-sized inputs, and that is how the smallness of the proton mass is explained. The other reason is that Dirac naturalness is very difficult to test, because if a large number is "technically natural", then it can be embedded in a Dirac natural theory by new physics that appears at extremely high energies. (For context see here and here.) This means we usually can't easily "cash out" Dirac naturalness into hypotheses that we can test this millennium, besides the cosmological ones that have already been shown false. Any scientific hypothesis that can't be tested in a thousand years is, of course, completely uninteresting.

However, if you do take Dirac's point of view, saying that two large numbers are equal reduces the number of large numbers you have to explain, thereby making the theory "more natural". That's the motivation behind the hypothesis.

It's easy to criticize this aesthetic idea as being subjective, and hence unscientific, but that's not right. In school we are taught that science progresses by proposing hypotheses, testing them, and then refining them. There is little attention devoted to how hypotheses are created in the first place. This process is necessarily subjective.

You might say that instead of beauty, one should instead try the simplest hypothesis first, or the one that looks the most sensible. Both of these are, however, also completely subjective. Even simplicity is subjective because a hypothesis can be very complex or the simplest possible one depending on the formalism you choose to work in. And there is nothing inherently wrong with numerology; if it works, it's called discovery. Many of the greatest discoveries in physics, such as Newton's inverse square gravity and Maxwell's electromagnetic waves, were made in precisely this way.

We should think of Dirac's large numbers hypothesis as an interesting guess with a real scientific motivation, which just didn't pan out. 99% of all fundamental physics doesn't, that's the way it's always been, and that's how it must be, since we don't know ahead of time what the 1% will be.

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    $\begingroup$ "There is little attention devoted to how hypotheses are created in the first place. This process is necessarily subjective. " This is a very important point. That said, I also think it should be balanced with the point that one should not get too locked in to one particular way of making hypotheses. We need broader diversity of thought, not narrower. $\endgroup$ – The_Sympathizer Sep 6 '19 at 2:13

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