You may win a 5-liter barrel of Pilsner Urquell if you find a physically motivated mathematical proof of a fascinating identity.

*Pictures are sometimes hiding very different messages than what meets the eye.*

Monstrous moonshine started with the observation that the bizarre number \(196883+1\) appears at two seemingly totally unrelated pieces of mathematics. \(1,196883\) are the dimensions of the two smallest irreducible representations of the monster group, the most complete sporadic group in group theory, while \(196884\) is the second subleading coefficient in the expansion of the important function of a complex variable, the \(j\)-function\[

j(\tau) = \frac{1}{q} + 744 + 196884q+ \dots, \,\, q= e^{2\pi i \tau}.

\] Later, it was proven that the appearance of this number is no coincidence once the appropriate string-theory-like CFT with the monster group symmetry was constructed. The same two-dimensional CFT dealing with the Leech lattice, up to a possibly different treatment of the left- and right-movers and their correlation, is the dual boundary CFT of pure 3-dimensional gravity in \(AdS_3\), as Witten showed (see around 2007).

The holographic duality seems to work for \(k=1\) only, as Gaiotto showed in a newer negative result, but this highly curved \(k=1\) model of quantum gravity became one of my 5 most favorite models of quantum gravity on which I am testing ideas about the deepest wisdom about the spacetime and information.

What about the number of elements of the monster group?

The monster group is finite but, as the name indicates, rather large. Its order (number of elements) is

2It's almost \(N\approx 10^{54}\). Most of the primes up to \(71\) appear in the factorization of \(N\). Is there a dual calculation of this integer?^{46}· 3^{20}· 5^{9}· 7^{6}· 11^{2}· 13^{3}· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

≈ 8 · 10^{53}.

You may see that the number \(N\) is close to \(\exp(4\pi^3)\). In fact, I am offering you a more detailed expansion\[

\ln N = 4\pi^3 + \frac{1}{\pi^2} - \frac{1}{73\pi^7} - \frac{1}{5\cdot 73\cdot \pi^{12}} + \dots

\] You see that it's an expansion in \(\pi^{-5}\). Note that \(6\pi^5\) is the ratio of the proton and electron masses. ;-)

Does the expansion of \(N\) above make any sense? If the answer is Yes and if you want to find the physics-based proof that I may have or not have, here are some hints.

**Hints**

Find the "theory of everything" when it comes to the black hole microstates in every conceivable vacuum of string/M-theory and/or quantum gravity.

Realize that the black hole microstates are always linked with some symmetries – and with (complicated) monodromies. For generic black holes, they are monodromies of the field operators around the thermal circle near the horizon. This microstate-monodromy dictionary may be described in terms of the Raju-Papadodimas formalism but it's new for PR.

Fuzzball microstates of BPS black holes, whenever they exist, may be understood as "operators" that reshuffle the fields on the corresponding fuzzball background.

For the pure \(AdS_3\) gravity with the monster group holographic description, you may construct Einstein-Rosen bridges, as in ER-EPR, and study the monodromy around their throat. A well-defined monodromy around the bridge is an eigenstate that decomposes as a particular superposition of states of two connected black holes.

You will find Schur's lemma to be helpful when you relate the basis of the well-defined monodromy with the basis of the well-defined representations of the two individual connected black holes.

Find the relationship of the microstates of these black holes and Einstein-Rosen bridges with elements of the monster group.

Prove the monstrous beer expansion and win the beer.

Clarify the relationship between the monodromies and black hole microstates in Matrix theory, generic AdS/CFT vacua, and generic weakly-coupled, perturbative string theory backgrounds.

Solve all the remaining puzzles of quantum gravity that are close enough to the information paradox.

John Conway really wants to know one thing before he dies: why the monster group exists. Can you help him and others?

If you find the tasks above too difficult, you may at least make a bet whether I actually possess an elegant proof of all these claims about quantum gravity. (As far as I know at this moment, the incorporation of the proton-electron ratio was just an elaborate joke. So far?)

My mind-reading powers are failing me : is this post serious or not?

ReplyDeleteThis question is exactly the main question of the puzzle in this blog post and you need mathematical, physical, and creative powers rather than mind-reading powers to find the answer!

ReplyDeleteIf you actually had a proof which solves everything related to the black hole information problem, you'd be hard at work publishing an article which would win you the Milner prize. In that case, why would you invite others to scoop you by giving so many hints?

ReplyDeleteI barrack for you, Lumo, and for you brave battle with what looks like it could turn out to be the biggest proof ever to have a Pilsner beer as its preliminary prize - before the really big prize-monies start to flow-in.

ReplyDeleteHowever, if Grandmother Nature (or this Multiverse of ours ;->) nips your project in the bud you STILL deserve (do in many more minds and opinions than mine) fame and fortune for all the fun, fascination, and satisfying fight-backs (against so many sorts of foolishness) that you generate.

I'm calling your bluff.

ReplyDeleteFind the "theory of

everything" when it comes to the black holemicrostates in

every conceivablevacuum of string/M-theory and/orquantum gravity.

Really! String theorists still haven't classified every conceivable vacua of string/M-theory. Besides, once you go far enough from BPS states, string/M-theory becomes an incomprehensible mess.

In addition, since any arbitrary localized configuration of matter can be thrown completely into a sufficiently large black hole, if you know

everythingabout black hole microstates, then you also know everything about any arbitrary localized configuration of matter, including systems exhibiting quantum chaos and quantum computers. If you claim you know everything about quantum chaos and quantum computers, I don't believe you.One more opinion: Conway might be lucky to be told how - in mathematical relation to other fundamental physical and/or mathematical aspects of What Is going on - |M| exists, but never ever "why".

ReplyDeleteWhat are the coefficients of your expansion? I don't see where e.g. the 5*73 comes from.

ReplyDeleteSurface area of a sphere with radius pi, entropy of a black hole with radius pi? Just trying to figure out what 4 pi^3 could mean.

ReplyDeleteI would say that 73 follows the 71,last prime of that list, and the barrel of Pilsner beer follow the bottle of JackDaniels offered in year 74 :) http://archive.numdam.org/ARCHIVE/SDPP/SDPP_1974-1975__16_1/SDPP_1974-1975__16_1_A4_0/SDPP_1974-1975__16_1_A4_0.pdf

ReplyDelete--lovely video by Numberphile---Tim Burness is an ideal communicator---I am sure is a great prof. to have for courses.

ReplyDeletePeter---well, why does anything exist? That is an unanswerable teleological question.

ReplyDeleteSomeone asked Frank Wilczek "Why is there something rather than nothing" and was told because "nothing" is unstable...(origin of the comment is, I believe, Sidney Coleman).

In the Numberphile vid, Conway is asked why the Monster and his answer is "because that's the way it is"...

Hey, you just want us to figure this out for you :) I have a spare 15 minutes...

ReplyDeleteBTW I didn't realize that Pilseners were the size of ants--real Lilliputians, given the size of that 5 litre barrel.

There is a good "popular" book on the Monster by Mark Ronan called "Symmetry and the Monster"---but to understand it you need at least an undergrad math competence.

Conway (Simon Norton, and someone else) wrote an exhaustive Atlas of Groups.)

73 is the smallest prime integer greater than all the primes that appear in the decomposition of N.

ReplyDeleteA problem with your claims is that your comments did get approved.

ReplyDeleteI'll wait for Sheldon to solve it.

ReplyDeleteAway from BPS/supersymmetric states, physical quantities are harder to calculate - but that's a fact in *every* theory that some quantities are easier to calculate, others are harder. In some formulations, all the states including the non-BPS states are described by well-defined equations in principle. In other contexts, we may be missing some nonperturbative aspects of non-BPS states even in principle but the evidence is overwhelming that the theory exists and is perfectly defined throughout the landscape.

ReplyDeleteWhat one has to know to understand these things about microstates and information may be strictly smaller than what is needed to understand the structure of the string landscape but on the other hand, there is no proof that these aspects of string theory are strictly isolated from each other, either.

Quantum chaos and quantum computers is really just applied maths. I don't count any of these things to fundamental equations of physics. Some of these applications may be related to the fundamental equations, most others are not.

Dear JollyJoker, in Witten's AdS3/monster pure gravity, the entropy is

ReplyDeleteS = 4*pi*sqrt(k) (sqrt(L0)+sqrt(L0tilde))

By an expansion of sqrt(L0) around some fixed large values, realizing that the subleading terms go like pi^2, you get the additional factor of pi^2. But this is already extending the hint above.

I think that I understand Conway what he means, what kind of insights would count as "explaining why the monster group exists", I consider it among the top questions about the conceptual structure of mathematics as well, I am convinced that the explanation is deeply linked to theoretical physics, and I think we/I are actually close to knowing what the explanation is.

ReplyDeleteDear Peter, just to reveal the context. monstrous moonshine has been announced by promising some bottle of whiskey or something like that to someone.

ReplyDeleteSo the candidates were bottles of domestic rum and fernet - produced near Pilsen - but then I decided to beerize the enterprise because it's a more well-known product outside Czechia. ;-)

I get it. Well, if you can derive such a series from physics it would be quite amazing and I assume you would not just sit on it but upload it to the arxiv soon ...

ReplyDeleteAll of the Pilseners are microscopic, and many of them are employed as brewer's yeast cells.

ReplyDeleteThanks. I had a hunch "radius pi" would make no sense.

ReplyDeleteDear JollyJoker, a simpler way to make this factor natural is to demand the circumference (periodicity) 1 which means the radius equal to 1/(2*pi). The dual momentum is then a multiple of 2*pi and this momentum may be exponentiated to the second or higher power in L0 etc.

ReplyDeleteAs a complete noobling I am thinking the Amplituhedron must be connected to this structure?

ReplyDeleteThe monster group and the Higgs vacuum:

ReplyDeleteVh= 246.2196509 GeV

Electron= 5.109989276E-04 Gev=Me

Vh/Me = 481839.8584 ( pure number )

Electromagnetic Fine structure constant to zero momentum = (137.035999173)=alpha(0)^-1

Dim group E8 non zero roots= 240=112+128

( (196884+196883)+(alpha(0)^-1/Pi )-1 ) × ((alpha(0)^-1)/112) )- ((alpha(0)^-1)/112)-1 )= Vh/Me = 481839.8584

((196884+196883+(137.035999173/Pi -1))*(137.035999173/112) -( (137.035999173/112) -1 ) = Vh/Me

(196883/2^2 + 196883)=mh/Me =246103.75 ==> mh =125.7587 GeV

ReplyDeletemh = higgs bison

I suggest you bookmark this page and check for an answer every three decades or so ;)

ReplyDelete3 decades, hmm, I might have a bash myself then lol

ReplyDeleteIt looks like What Is can evolve very special thinking patterns of metabolizing matter that realize - without maths - that "What Is going on" is no_thing (or 'not thingy'), and - with maths - that the Monster Group can be degraded, as in made to appear meek and relatively minor, if shone a 'Lumoniferocious light' on with the logic of stringy duality.😊

ReplyDeleteWell, it's obviously the size of the program store used to simulate the universe we're in.... just don't know if its bits, nibbles or bytes! :-)

ReplyDeleteIt is very serious exciting fun ;-)

ReplyDeleteProfessor Motl,

ReplyDeleteCould you point me to a book or paper where the idea of calculating temperature from the connection between Tr exp(-H/T) and exp(iHt) is explained in detail, especially the reason why this works for analytically continued metrics. All the places where I've seen this derivation of temperature presents it in a very compact form, as though it is obvious, and I have been unable to fully grasp the logic behind it.

Vangel - It was predicted by many before the fact, using many different assumptions - assumptions that differed from yours ('blowback' from occupation). As I mentioned before, it is not the prediction I am arguing against but the assumptions and method by which it was generated.

ReplyDeleteThe 'logic' of your comments is fallacious, the psychology is non-existent, and the only empirical study you cite, that of Pape, looked only at the small fraction of democratic occupations in which suicide bombings occurred, failed to explain why, and neglected altogether suicide bombings in other contexts (the latest of which involved two young girls under the control of Boko Haram).

Extremists of all kinds, including jihadists, are well known to engage in behaviors they condemn in others. This is known as hypocrisy. I find it funny as well.

I am not surprised to hear that typical Muslims in the west want nothing to do with extremists. It confirms my own experience. But there remains a significant minority in the west, and majorities elsewhere, that you have failed to understand.

Finally, I note that your reply is not responsive to my comment and that I shall not respond to further repetitions of your few, tiresome talking points.

There are only nine additional regular Platonic solids in higher dimensions in addition to the five in 3D space, and they are simple affairs, so how can this massively multidimensional object be so symmetrical? Are they merely wrongly defining regularity so the Monster isn't really associated with a multidimensional polytope? Or doesn't geometry apply any more in group theory? What I find especially odd is how multiple dimensions sort of assumes orthogonal right angles between dimensions but that's just a Cartesian convention.

ReplyDeleteSo you verify Witten (i.e. symmetries of the black hole correspond to symmetries of the Monster) via EP=EPR?

ReplyDeleteThis is too big, you must publish;

What happens when we have instead a higher spin gravity/black hole in the bulk?

Finally do you know if there is a an up to date review/reference for the Monster/black hole and related issues?

Cheers ! Graphical analysis of number π (3.14...) I am working on analysis how π and e (natural algorithm 2,71 …) relates in famous formula e^(iπ) + 1 = 0 (peeyed)

ReplyDeleteThis is a good posting which shows Motl understands the beauty, power, inevitability and even humor of string theory. Would Professor Motl name for us 15 of the world's 30 best string theorists?

ReplyDelete;)

ReplyDeletewww.youtube.com/watch?v=DlhfpKPfBNE

8-)

ReplyDeleteSome of us are civilized others almost as barbarians mostly arabs and asians. According to my experience from Korean production factory with 1200 employee and from them 800 gypsies as project/quality manager, if learn habits and language of them, there is perfect resuts also with them. Anyway they respect me and then I respect them 8-)

The war party apologists predicted terrorist acts because they want to profit from further activities but note that these are the same people who have been making bad assumptions and lying all along. They are the same people who made up evidence to get us into the wars that they found so profitable in the first place.

ReplyDeleteI think that the logic that I have used is clearly correct. Invasions and occupations cause resentment and fuel terrorist activity for exactly the same reason that Plato noted. The fact that you choose to search for any explanation rather than the most obvious tells us a great deal about where you stand on the war party issue. Note that the marches against the activities show exactly what I stated. Most ordinary Muslims want nothing to do with terrorist activity even though that they explain why those idiots carried it out. Also note that such activity is not particularly unique to Muslims. Irish Catholics have acted in similar ways against English Protestants. Jewish extremists have used terror to silence critics. As for the freedom of speech argument let us note that the French government closed down the predecessor of the magazine for insulting Charles de Gaulle and that the American press engages in self censorship all the time.

LOL...Aren't 'WE' the same people who have a marching season that glorify imperialism and mocks Irish Catholics for battles that they lost hundreds of years ago? And don't those Irish Catholics engage in terrorist activity to strike back against the invaders? Arabs are not particularly all that different than the Anglo Saxons who invaded their lands. The English have slaughtered Africans, Irish, Scots and other people without much regret. Americans kept slaves and slaughtered the Plains Indians. They went into Vietnam and killed innocent women and children. Their president dropped an atomic bomb in Japan against the wishes of most of his staff and the military leaders who wanted to accept the Japanese surrender terms. The American leaders tortured innocent people and polls reveal that a majority of American citizens supported that torture.

ReplyDeleteWE are no better than THEM. In fact, seeing Arabs as a group rather than individuals is one of the reasons why we have all of these problems.

The last three elements of the monster group (47,59,71) multiply to 196883.

ReplyDeleteDear Luboš,

ReplyDeleteI see your contributor,

pEGO, must have very recently deregistered/deleted his original (Disqus?) ID and re-registered (or something along those lines) because the author of his 'fly-tipping' post below is now showing as 'Guest' (with no avatar), whereas his later posts now have his original ID and avatar. He's a sly cunt.Quantum chaos and quantum computers is really just applied maths.

ReplyDeleteI think you fail to appreciate the significance of complexity to fundamental physics.

If you can calculate the spectrum of a black hole exactly, then you can also calculate the spectrum of quantum computers, and systems with quantum chaos.

Hi, no, you are completely wrong. Fundamental physics - like quantum gravity - is full of various problems that may be solved exactly, integrable problems.

ReplyDeleteBut there are lots of other problems e.g. in classical and quantum chaos that are not integrable - in many cases, it's been demonstrated that they are not integrable. Their not being solvable is really a flagship defining property of classical and quantum chaos.

Not to mention that the "spectrum of a quantum computer" is a totally silly expression. When we consider quantum systems, the spectrum may only be one of an operator. Quantum computer is not an operator. At most, we may talk about its Hilbert space, and it's the trivial Hilbert space spanned by the 0010100111... basis vectors, 2^K of them. There's nothing mysterious about it.

This is like saying that being able to calculate the emission spectrum of a blackbody implies being able to solve all of classical physics. Determining the thermal physics of black holes from string microstates does not require you to actually track the exact microstate the black hole happens to be in. Solving the intractable problems you point to are not (or almost certainly should not be) necessary to solve the problem.

ReplyDeleteTake a look at the statistic referenced in this article:

ReplyDeletehttp://cnsnews.com/mrctv-blog/dan-joseph/27-french-youth-support-isis

In a few decades, the muslims will be majoritary in France, England, Scandinavian countries, etc. Then - at the ballot box or through violence - they will impose Sharia thoroughout western europe.

It's very unlikely the current western leaders will take the drastic measures that could stop this, at this late hour. Even now, they're actually denying this existential problem even exists.

Indeed, they - the political, economic and cultural elites - are the ones that commites the high treason - importing muslims for their votes given in exchange for welfarre, accompanied by the post-modernist/multi-culturalist/marxist/culpabilist/thought-crime ideological toxic mix served to the indigenous poeple - leading to the suicide of the western civilization.

I'm even starting to see the merits of Motl's vision for Russia. Sure, it's a dictatorship and not my first, second or even tenth choice.

But it is not infected by the western types of elites that de facto succeeded in destroying their own civilisation and is one of very few choices for a continuation of the western civilisation.

The arabs and blacks that will inherit western europe redundantly proved throughout history they're not up to the task of creating or maintaining an advanced civilisation.

None of these scenarios is "guaranteed" at this moment. And if the reaction to the Charlie Hebdo attack in France - and Europe - showed, it is that the Europeans are much more unambiguous about viewing the Islamic blackmailing of the public life as unacceptable.

ReplyDeleteTop American officials didn't attend because they're actually not certain whether they are closer to the Islamist killers or those who make fun of the religious symbols.

"the Europeans are much more unambiguous about viewing the Islamic blackmailing of the public life as unacceptable."

ReplyDeleteActually, the creep of accommodating more and more Sharia-inspired practices in public life has been quite relentless in a number of European countries in recent years, so it is way too early to tell if any serious opposition to allowing such creep to continue will result from these events.

A bit of perspective is required. France is a country that has a population more than 60 million. Of these about 5 million are of a Muslim background. Note that at least one of the terrorists was not exactly what one would call a devout Muslim. He had drank alcohol, smoked pot, engaged in premarital sex, and was quite Westernized.

ReplyDeleteThe fact is that the it is not just young French Muslims that have problems with the US and support anti-American activities. It is French youth that has problems with American activities and is anti-American.

Note that ISIS and al-Qaeda have trouble recruiting young Muslims because most Muslims are not interested in terrorism or even in politics. Since you cited an article that cited polls let me cite one of my own to support my statements.

The problem for a terrorist group like al-Qaeda is that its recruitment pool is Muslims, but most Muslims are not interested in terrorism. Most Muslims are not even interested in politics, much less political Islam. France is a country of 66 million, of which about 5 million is of Muslim heritage. But in polling, only a third, less than 2 million, say that they are interested in religion. French Muslims may be the most secular Muslim-heritage population in the world (ex-Soviet ethnic Muslims often also have low rates of belief and observance). Many Muslim immigrants in the post-war period to France came as laborers and were not literate people, and their grandchildren are rather distant from Middle Eastern fundamentalism, pursuing urban cosmopolitan culture such as rap and rai. In Paris, where Muslims tend to be better educated and more religious, the vast majority reject violence and say they are loyal to France.

Ref: http://www.juancole.com/2015/01/sharpening-contradictions-satirists.html

What I noted is the hypocrisy of the press. For some reason, as they were up in arms about censorship, the press forgot to mention that the magazine's predecessor publication was shut down by the government for insulting Charles de Gaulle and that at least one cartoonist was threatened by a Jewish extremist group. It also failed to mention that most French Muslims have little interest in politics and less interest in terrorism. Even worse is the failure to mention that the French government is not all that concerned with civil rights and with all its surveillance failed to stop people that it had been watching.

ReplyDelete"The fact is that the it is not just young French Muslims that have problems with the US and support anti-American activities."

ReplyDeleteWas it an American magazine that got attacked?

[(196883+196884+1)^4]/sqrt(4!+InIn(71))= Mpk(r)/Me

ReplyDeleteMpk(r)= reduced planck mass =[sqrt(h(bar)c/Gn)]/sqrt(8*Π)

Me = electron mass

The monster group is fundamental in the theory of unification of gravity and all others fields

Not at all. But most of these actions have little to do with the EU countries. They just happen to be convenient targets because they have supported American meddling in the Middle East recently. While their own meddling in the distant past has also stirred resentment the young people are not that familiar with past European nation crimes in Africa or the Middle East. The camps that are training these idiots are funded by people whose biggest beef is with American actions in Muslim nations. France is a failed nation that is on its way down and not that much of a concern for them other than what it does as a part of NATO. The magazine had little to do with anything. It was just a convenient soft target that would act as a catalyst that would drive the French to be oppressive against ordinary Muslims who want nothing to do with terrorism and are not that interested in politics or Islam. The terrorist leaders want discrimination that would help recruiting and fund raising drives. The best way to deny them what they want is to only go after individuals who are terrorists, not treat all Muslims alike.

ReplyDelete