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Paul Dirac on dimensionless constants and the "large number hypothesis"

Paul Dirac (see TRF biography) died exactly 30 years ago, on October 20th, 1984.

He was an eminent physicist, a co-father of quantum mechanics, the author of the Dirac equation, and a man who was convinced that the

Physical law should have mathematical beauty.
He wrote down this important sentence in capital letters on a blackboard during his 1955 lecture in Moscow. ;-)

Later, the English physicist would move to Florida where he was sort of happy – the place reminded him of England.

The video (well, picture+audio) at the top boasts Paul Dirac who describes how special physical constants are if they are dimensionless. The numerical values of those don't depend on the units – so even other (extraterrestrial) civilizations will unavoidably agree about these numbers. Due to this objectivity, there should be a calculation that clarifies why the numbers are what they are.

It's been an extremely important motivator for my efforts to get "deeper" to the laws of Nature, too. At some moment, I understood why string theory does guarantee that there are no continuously adjustable non-dynamical parameters in Nature. It means that in principle, every such dimensionless constant we observe in Nature is calculable with an arbitrary accuracy – as long as you learn a finite amount of discrete information about the right compactification of string theory.

Most laymen don't appreciate how incredible progress this provable characteristic of string theory represents. For the first time in the history of science, string theory gives us a framework where there doesn't exist a single "lever" that could be adjusted or fudged. Even if the number of relevant string theory vacua were really \(10^{500}\), one would still need just 500 digits of information (it's not that far from things I was able to memorize!) to understand really everything, with an arbitrary precision.

Independently of string theory, particle physics also reparameterized the examples of the dimensionless numbers that Dirac used in his monologue. The proton-electron mass ratio, about \(6\pi^5\sim 1836.15\), should be explainable from the basic laws. But it is not really the "simplest" thing. The electron is a pretty simple elementary particle but we know that the proton isn't too simple. It is a somewhat messy (even though the lightest [nearly?] stable) bound state of three valence quarks and lots of extra gluons and quark-antiquark pairs etc. In lattice QCD, a computational approach to the theory of the strong force, it's damn difficult to calculate the mass of the proton.

Moreover, the result depends on many other "more fundamental" constants such as the bare mass quarks, the QCD coupling (which runs as a function of energy, so we may replace this information by the QCD confinement scale), and other things. Why the proton mass is a rather messy quantity is easy to see: the proton itself became messy in the late 1960s and early 1970s.

Dirac's other example is the fine structure constant,\[

\alpha=\frac{e^2}{4\pi \epsilon_0\hbar c} \approx \frac{1}{137.036}.

\] That's arguably the most popular number among the physics-oriented numerologists. The idea has been that this is so simple to be defined that the calculation of the number in the final theory should be rather "straightforward". However, the electroweak theory showed us one reason why it isn't. The electrostatic force itself isn't quite "elementary". The \(U(1)_{em}\) gauge group behind it is really just a combination (diagonal group) of a more fundamental "hypercharge" \(U(1)_Y\) group and the \(Spin (2)\) subgroup of the \(z\)-axis rotations in the \(Spin(3)\sim SU(2)_W\) non-Abelian group producing the W-bosons (and their neutral friend, a combination of the photon and the Z-boson).

Because of this "mixed nature" of electromagnetism, the electromagnetic coupling \(e\) is really a function of "more fundamental" couplings \(g_1,g_2\) of the \(U(1)_Y,SU(2)_W\) groups, respectively:\[

e = \frac{g_1 g_2}{\sqrt{g_1^2+g_2^2}}

\] Now, another complication is that even these "more fundamental" couplings \(g_1,g_2\) are "problematic" because they are (logarithmically) running – they have an extra slow, quantum-mechanics-induced dependence on the energy scale and their values at "very low energies" \(g_{1},g_2(E\to 0)\) are actually relevant for the fine-structure constant \(1/137.036\dots\). However, physicists have understood that the values of the coupling constants that have a reason to result from a "simpler" calculation are, on the contrary, the values at very high energies,\[

g_1,g_2(E\to \infty)

\] or at the GUT or Planck scale, and so on. That's where you might reduce \(g_1,g_2\) to the knowledge of some other constants in a grand unified theory, or calculate them from "nothing or pure rigid geometry" in string theory. Once you do that, you may extrapolate them to \(E\to 0\) via the renormalization group equations and finally you compute the fine-structure constant using the formulae written above.

My main general point is that the dream of calculating the constants "out of nothing" is still here – and we already "surrendered a bit" so that (almost?) no one thinks it's too likely that someone will actually present the complete calculation by Christmas 2014 – but the precise idea about "which quantities should be given a truly elegant calculation of the value" is developing along with physics. The electroweak theory, the renormalization group, and (with some uncertainties surviving) grand unification and string theory imply that the previously "simple, elementary" dimensionless constants are actually a bit contrived and other constants are more fundamental even though it is not so straightforward to extract them from simple observations.

The second part of Dirac's video is about his large number hypothesis. Some large dimensionless constants such as the electron mass over the Planck scale may be extreme – much smaller than one or (which is the same because one may invert numbers) much greater than one – may be so extreme because they are functions of the "elementary" large number, the age of the Universe in the Planck units.

That's a wonderful idea to solve all these problems. Fortunately or unfortunately (it depends on your perspective), it predicts that the values of the dimensionless constants should actually change with time because the age of the Universe is surely changing with time, by its definition. Because this evolution of the constants seems to contradict the observations, the whole hypothesis by Dirac seems to be ruled out.

Note that the experiments don't exclude merely one particular, special version of the Dirac's "large number hypothesis" applied to one number. Because the constants were demonstrably the same many billions years ago and some evolution would have had a much bigger impact on all the values, we may really say that the whole philosophy or paradigm that Dirac offered in the talk has been killed by the experimental evidence.

I am emphasizing this point because most laymen tend to think that a theory may be ruled out only if one exactly says what the theory is, what are the exact values of the parameters, and what are the precise sets of problems where it should apply – and if one calculates the exact particular predictions for these quantities. But as this example (and many other examples) shows, this just isn't the case. Theories and frameworks, even incomplete and somewhat vaguely defined ones, predict rather specific patterns, regularities, or at least inequalities even before they are fully well-defined. And in most cases, they just contradict the evidence, so even the very general ideas and even "philosophies" may be falsified by the evidence long before someone manages to make the theory "truly well-defined and particular".

It's extremely unusual for a sufficiently robust, natural, and ambitious theory – one that is clearly capable of reproducing the quantitative success of other successful theories – to withstand the many possibilities by which it could have been falsified. String theory's competitors proposed as unified theories of all interactions die within minutes – and that's why the 45-year-long survival of string theory is incredibly strong circumstantial evidence that string theory is an extremely important set of ideas to be considered, even in the absence of a rigorous or irreversible proof that it applies to Nature.

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snail feedback (25) :

reader kashyap vasavada said...

Very interesting blog. Are string theorists hopeful for having a fundamental constant in ST? If I understand the string tension constant would depend on the model of compactification. Is it?

reader Luboš Motl said...

Sorry, I don't understand anything you wrote.

As I explained again and is well-known, string theory predicts the precise value of fundamental dimensionless constants - while it needs a discrete input.

This is not about hopes; this ability is a mathematically demonstrated fact, much like 2+2=4.

On the other hand, the string tension is *not* a *dimensionless* quantity, so its numerical value can't be calculated from the first principle. In perturbative string theory, we often use units in which string tension is

T = 1 / pi.

For certain reasons, it's so. The corresponding inverse (up to numerical constant 2.pi) Regge slope is alpha'=1/2, although people often use alpha'=2 if they analyze "only open strings".

reader kashyap vasavada said...

OK. I understand. Thanks. Of course, I did not mean that other contributions of ST are not useful if ST does or does not have a fundamental constant.

reader Luboš Motl said...

Sorry, what do you mean by "a" fundamental constant and what does it mean to "have it"? Your sentences seem to make no sense and you continue repeating them.

reader Gene Day said...

This blog is very helpful for me, Lubos. Thanks.

reader kashyap vasavada said...

Sorry. I did not word the question properly. But now it is clear what you mean. Thanks.

reader john said...

Dear Lubos, I am not sure what you did say. Are you saying that the string theory has only a finite number of discrete solutions. I mean, there are 10^500 different universes and nothing more ?

reader Luboš Motl said...

Hi, strictly speaking no, in practice probably yes.

The number of all vacua is infinite. For example, the AdS5 x S5 vacua with different radii - i.e. different values of N in SU(N) - are enough to produce a countably infinite family. There are others.

(There are also uncountable families - and these SU(N) AdS/CFT vacua are really uncountable sets - because they have "moduli", adjustable parameters that are however dynamical and cause additional forces of Nature. But in our world, such forces almost certainly don't exist which is equivalent to saying that vacua with the moduli - and therefore the uncountable families - are ruled out. We only look at the "stabilized" vacua of string theory and their number is countable.)

But if one imposes a bound on the effective size of the extra dimensions, to agree with the observed fact that only 3+1 dimensions are large enough to be seen, there exists circumstantial evidence that the number of such vacua is strictly finite and 10^500 is the most famous estimate of their number (the "exact" value is in no way known but the paper that ended up with this number is in no way an unjustified guesswork, either).

reader Luboš Motl said...

A fun story, Magyar Ember. The legend about Horthy as the saint defender of minorities etc. is nice, at least as an amusement. OK.

Concerning the economic policies, well, I would probably still call it Goulash socialism, and I would in fact have some trouble to distinguish it from the Goulash socialism of Janosz Kadar.

It's clear that the economic history - even during communism - of Czechoslovakia and Hungary was extremely different. It's sort of remarkable. Concerning the banking habits, one would assume that Hungary would have similar attitudes as Czechia and Austria - which are similarly conservative etc.

But Hungary jumped on the train of high inflation and excessive debt etc. That eroded any confidence in its own currency, so loans were primarily in foreign currencies, and so on, and so on - these are completely alien ideas from our viewpoint. The debt was zero throughout communism and the Czechs always trusted our currency and made all loans in it.

The unifying theme is that you look for and find many excuses for why the economy shouldn't be liberalized. These guys robbed us, and then those guys, and so on. At the end, the logic makes little sense. One either believes that capitalism creates values in a more efficient way, or he doesn't. Or something in between. The history forms one opinion but if the trust in capitalism isn't there, the party isn't quite right-wing economically.

I would disagree with your comments that the energy industry is strategically and politically important, and so on. The energy industry is an industry like every other - even though I do remember that shortly after 1989, this point would be questioned by many people here, too.

It's also untrue that "some industry's being a monopoly" is a justification for nationalizing it. Some competition may always arise if the private company, whether a monopoly or not, really works without subsidies. These days, we have a huge competition in the energy distribution market, telephone providers and even railways etc. although many of those things would be claimed to be naturally "demanding a state" some decade(s) ago. But those were myths.

For those and related reasons, I think that the term "right-wing" is as gross oversimplification of the Fidesz economic policies as Liam's "statism". But just to be sure, Fidesz is just some permuted superposition of left- and right-wing parties we know from other countries, too, so it can't be illegitimate in any way.

reader Magyar Ember said...

"A fun story, Magyar Ember. The legend about Horthy as the saint defender
of minorities etc. is nice, at least as an amusement. OK." really? I think you would have deleted this comment for its quality if it wasn't yours...

Horty participated in the Nuremberg trials as a witness not charged nor convicted of any war crimes.

regarding your economic arguments, which are more subtle, but not much more, you are trying to take it to the extreme as if it were about economy being liberalized or not. Its not what I said. all I said is that a country is not by any means a self regulating system, a country should be governed, and in case of a near bankruptcy, unorthodox measures may be the only way.

Regarding energy. I would find it paradoxical if you would argue that energy is not of strategic and political importance when every single energy producer country uses their position as a political weapon. ie.: Russia, USA, Saudi Arabia etc (and be sure to note that my argument was not about some industry, it was specifically energy.) not to mention the climate hysteria which is an international effort to make the energy sector even more unseparable from politics.

reader Magyar Ember said...

How Orbán sees the future of Hungary? Here's a short video of him telling exactly that in English. You decide:

reader QsaTheory said...

For entertainment purpose only

(proton-electron mass ratio=1836.152672)

1/ \textbackslash{}alpha =3\^{}3/2*(M\_p/M\_e) + 1+2/(3\^{}2*3\^{}2)=137.036000

reader andyd said...

That's hilarious!

reader QsaTheory said...

Wait for the second act, just kidding!

reader lukelea said...

Though I am unqualified to say such a thing, Dirac's "large number hypothesis" always struck me as sort of loopy, not worthy of him.

reader QsaTheory said...

If you check the wiki page you can see that people still ponder his keen observation.

reader Tony said...

Living in Florida myself, having been in UK a few times in the past, with a few family members married to Brits (so I am not totally out of touch) I really have a hard time imagining what in FL reminded him of England? !

reader Luboš Motl said...

Dear Magyar Ember, he may have been less guilty than the top 5 fathers of the Holocaust and other big shots in Germany. But if you need to be reminded which laws on Jews Horthy personally passed in Hungary in 1938-1941, see

Your praise that almost makes him look like a democrat is totally ludicrous.

I disagree that the society in a nation isn't a self-regulating mechanism that "needs to be governed" and I also disagree that the "unorthodox measures" should be used to mitigate bankruptcies. Bankruptcies are right, creative, fair, productive, and often inevitable - both bankruptcies of smaller and bigger companies as well as countries that run amok.

Russia and others are using fossil fuels (and other things!) to make a political point but America and others are doing the same thing with hi-tech electronics technologies and everyone else is doing a point with whatever it has or produces. None of these things implies that any of these industries should be managed by governments.

Climate hysteria is trying to make pretty much *everything* controlled by government, not just the "energy sector". They really want to regulate everything where *energy* flows, which really means everything, not just the energy sector. On Thursday, I am giving a climate talk on a conference on "House and Energy" for architects where the other "outside preachers" are of course alarmists who are telling them that the climate hysteria is important for what *they* are doing. In other countries, there are tons of such events in all sectors of the industries and in most of them, they only invite folks who have no clue what they are talking about.

reader n00b said...

JOhn Horgan interviews the E8 Surfer Dude

reader Peter F. said...

May be there is a relevant but yet not recognized aspect of the (or this multiverse-made) universe that does NOT change (as the for physicists and most of the rest of us conveniently calculated with chronology or 'timely dimension'). Hmm %~?
So Dirac might somehow have seen further ahead of fundamental physics than its reach even as of today.(???)

reader Giotis said...

Absence of dimensionless freely adjustable parameters is a prerequisite for a theory of everything; it means that you don’t need another theory on a more fundamental level to explain their values. It’s the end of the road.

In String theory there are no such parameters and it fits the bill once again. Everything is calculable within the theory by the theory.

reader br said...

Could you explain this some more? My impression (going by what Lubos was saying), is that if there are 10^500 observably different string theory solutions, this would require 500 digits of information to describe. But surely one has to select the right set? Of these 10^500 solutions, there is nothing to choose between them (they are all self-consistent, so we can't tell the right one), but one has to choose the right set to match observation?

reader Giotis said...

But this 10^500 is applicable for the low energy effective 4d world. The full theory at high energies (string scale) has no free moduli.

reader br said...

In that interview, Lisi says:
"When I do this for E8, there’s a symmetry called “triality” linking three different sheets of spacetime; with respect to each different sheet, each of the three different generations of fermions comes out right."
which he seems to claim would fix his admittedly not-yet-working theory, and the job would be done. Does this sound plausible? (I guess the answer will be 'no', but I would appreciate a more informed view).

reader br said...

OK, I didn't appreciate that - but does that help? Isn't there still a considerable freedom that can only be pinned down by matching with observation? It's just that your phrase 'Everything is calculable within the theory by the theory' doesn't sound right, though I'm sure you know more about it than I do.