The Reference Frame

Does global warming cause tornadoes?

2013-05-23T11:34:00.000+02:00

It was sort of inevitable that the deadly tornadoes in Oklahoma would ultimately be blamed on global warming and CO2 by someone. While most people – including those alarmed by "climate change" – reject this attribution, you can find pretty powerful people who promote this incredible link.

Senator Barbara Boxer was perhaps the most powerful person who enthusiastically supported the idea that the tornado outbreak was a message from Nature telling us to introduce new carbon taxes. She really sounds religious.

You may find lots of stories in the media that discuss a possible connection between tornadoes and the enhanced greenhouse effect. Thankfully, almost all of them (e.g. NY Daily News, Washington Examiner) say that there's no connection. But Barbara Boxer knows that such a connection would strengthen the case for the new taxes – so it must be a part of the consensus, right?

Without actually thinking about the science or asking researchers, leftwingers generally assume that whatever is convenient for their "cause" must be a part of the "scientific consensus".

But let us look at another magazine, National Geographic. It concludes by saying that a hypothetical influence of "global warming" on the frequency and strength of tornadoes could go in both ways and there is no evidence of a trend in either direction. They quote Roger Pielke Jr, among others. Still, the title asks whether there is a connection and it offers some ideas that could support such a link.

Which ideas?

Before they admit that it's always tricky to link a particular weather event to climate change, they offer these two paragraphs:

It sounds intuitive: Of course global warming should lead to more—and more powerful—tornadoes.

We're adding energy to the atmosphere by trapping heat with greenhouse gases, and tornadoes are the very picture of terrifying atmospheric energy.

What I find problematic is that it is not explained why these would-be arguments in favor of the connection are wrong. So many readers may just think that they're actual valid arguments in favor of the connection.

The quote above says that tornadoes are "the very picture of terrifying atmospheric energy" and we are adding it, so we are probably strengthening tornadoes. Is that a valid reasoning?

All forms of energy are convertible to each other – only the total energy is conserved. This is the statement known as the first law of thermodynamics. So adding energy in one form may increase the energy in other forms, too. However, there is also the second law of thermodynamics that says that you can't construct a device that does mechanical work by extracting the thermal energy (heat) from a colder object (of from an object indefinitely). In other words, the perpetual motion machine of the second kind is impossible.

(The perpetual motion of the first kind is a gadget that would create energy out of nothing and it is also impossible – because of the conservation of energy or, equivalently, the first law of thermodynamics.)

But the creation of a tornado out of the "global warming" would be exactly such a perpetual motion machine of the second kind! The reason is that whether we like it or not, a tornado is a place with concentrated "useful" mechanical energy – stored in the macroscopic ensembles of molecules – while a warmer atmosphere is the "useless" thermal energy i.e. energy stored in the chaotic motion of individual atoms. And one simply can't create the former (useful, concentrated energy) out of the latter (useless, chaotic energy).

Once the energy is converted to heat, i.e. universal chaos, it can't be converted to the "useful" mechanical forms anymore! Only if the temperature is non-uniform, the temperature differences may be exploited to do some mechanical work, like in "heat engines". But global warming creates no new temperature differences so it creates no new opportunity for such "natural heat engines" to produce new mechanical work!

Why is it so? Because the global warming is global, stupid. The widely discussed effect is caused by the increased CO2 concentration which adds something to the expected temperature at any place. But because the carbon dioxide molecules are almost instantly (within weeks at most) spread uniformly over the whole Earth's atmosphere, the greenhouse effect is equally strong everywhere. Well, there is some dependence on the latitude etc., and especially the feedbacks (e.g. the ice-albedo feedback) depend on the latitude, but this dependence is extremely slow. To extract a useful mechanical energy e.g. to create a tornado, the greenhouse effect would have to seriously change from one mile to another and it is clearly something that the uniform, "global" rise of the CO2 concentration isn't capable of doing.

In other words, the enhanced greenhouse effect doesn't change the magnitude of local non-uniformities, so it can't contribute to the tornadoes, either. The same comment is true not only for tornadoes but for any "special", localized weather event. These events don't really care about the overall temperature shifts and the overall temperature shifts (probably very modest ones) are the only thing that extra greenhouse gases may cause.

So at least in the leading approximation, CO2 isn't capable of changing the frequency of tornadoes – in either way. But you could hypothesize that there is some subtler, higher-order effect that causes such an influence, anyway. For example, the thickness of the troposphere (the circulating part of the atmosphere between the surface and the tropopause, about 10-20 km above the surface, where all the "complicated weather" takes place) depends on the overall strength of the greenhouse effect on the Earth.

It's plausible that as the troposphere gets thicker, there is more room for circulation and various manifestations of the circulation may strengthen. That's great but we should be interested not only about the Yes/No answer to the question "whether such an influence may exist" – in principle, almost everything influences everything else – but also about the estimated strength of such an influence. These estimates will be order-of-magnitude estimates of a sort; we won't discuss whether the effects boil down to wind shear, jet streams, funnel clouds, supercells, or other fancy concepts in atmospheric physics.

We should ask: by how many percent the frequency of tornadoes could rise or decrease (we really don't know the sign) if CO2 raised the global temperature by 1 °C?

That's a meaningful question, suggesting that we want to understand these things beyond demagogic slogans. So let's try to think a little bit. I proposed a mechanism suggesting that the strength or frequency of tornadoes could scale with (a power of) the thickness of the troposphere or, almost equivalently, with the total temperature increase caused by the greenhouse effect on Earth.

So how many percent may this unknown influence add? The key realization is that the greenhouse effect caused by the man-made CO2 is a small fraction of the overall greenhouse effect on Earth. The greenhouse effect on Earth is dominated by water vapor which adds over 30 °C to the temperature of pretty much every place on our blue, not green planet. Even if the whole 20th century warming was due to CO2 (which seems unlikely to me), we have only added about 2 percent to the overall warming caused by the greenhouse gases. We may add another percent, perhaps, but we're still talking about a few percent.

The thickness of the troposphere may have risen as a power law which also means by "several percent" and the same thing holds for all quantities describing the wind speeds, the total number of vortices, and so on, and so on. I actually think that this is an overestimate and because the troposphere is getting thicker, the atmospheric phenomena and energy are actually diluted into a larger volume so less is left for the near-surface weather. But let's be agnostic about the sign, exponents, and coefficients.

It's still true that these unknown influences only add or subtract several percent to or from the strength and frequency of the tornadoes we ultimately observe. And this is such a tiny change that it's ultimately undetectable. Why is it undetectable?

Because the number of tornadoes is notoriously variable. The interannual noise is so large that it totally prevents us from seeing a hypothetical signal that would only scale as a few percent. For example, look at this list of tornadoes in the U.S. spawn by tropical cyclones. The records (with lots of holes) begin around 1811. You see a few tornadoes a year, sometimes a dozen or two. There are exceptions like 115+ tornadoes in 1967 and 103+117 in 2004. It's pretty clear that in the past, the actual numbers were larger but people couldn't see everything due to the limitations of their observational technology.

At any rate, if the frequency of tornadoes grew by 2%, it would mean that there should have been just 113+ tornadoes and not 115+ tornadoes in 1967. Similarly for other numbers. The years for which the number of tornadoes are of order one are shifted by a tiny fraction of a single tornado. Try to statistically evaluate these chaotic tables in any way. It is absolutely clear that you couldn't possibly see a trend, whether it is a positive one or a negative one.

I believe it is a sufficient reason for us not to talk about such influences. Given the fact that we can't observe such an influence empirically, every proposed claim about such an influence has to suspected to be an artifact of errors, neglected terms and effects, and other things. We just don't know how large the influence is and what its sign is. Within the error margins, we observe the influence to be zero. So we should always act as if we were assuming that the influence is zero. Anything else amounts to bias – and violations of the presumption of innocence and other things.

Incidentally, I wrote about "noise" in the number of tornadoes on a given year. But the word "noise" is actually too disrespectful because what we really meant is "everything that has nothing to do with CO2". However, there could be – and there almost certainly are – many much stronger and signal-like influences on the frequency of tornadoes than the CO2 concentration. For this reason, it's already tendentious to pretend that the data are composed of a "signal" and of "noise", especially if we want to implicitly claim that CO2 is the only "signal" although it's almost certainly not the case.

Please, Ms Barbara Boxer and others, stop talking about these medieval hypothetical links that sound almost identical as the accusations against the witches in Salem, Massachusetts. Science supports none of your fantasies and every attempt by a person to rationalize such fantasies shows that the person lacks scientific integrity – and sometimes plain human honesty, too.

And that's the memo.

Augustin-Louis Cauchy: an anniversary

2013-05-23T10:54:00.000+02:00

By the number of mathematical papers he wrote, Augustin-Louis Cauchy was second just to Leonhard Euler. As many college freshmen may testify, more theorems and concepts in mathematics were named after Cauchy than anyone else. And a conservative theoretical physicist shouldn't omit a CV of Cauchy because Cauchy was... well... very conservative!

He died on May 23rd, 1857, i.e. exactly 156 years ago. But before he managed to do that, he had to do many other things. For example, he had to be born – in August 1789, just a month after Bastille was stormed by a crowd on the street, a mess we often call the beginning of the French Revolution.

Louis François Cauchy, i.e. Cauchy's father, had really nothing to do with this mess. He was a high official in the Parisian police before the revolution took over (during the "New Regime"). During the Reign of Communist Terror in 1794, when Cauchy was five, the family had to move to Arcueil. Things got safer when Robespierre – a guy who worked to transform bourgeoisie to a gang of leftwingers – was finally executed in 1794 and the family could return. When Napoleon took over 5 years later, Cauchy's father returned to police.

He was working directly under another high-tier policeman called Pierre-Simon Laplace who is today known, ehm, as a top mathematician. Joseph Louis Lagrange was well-known to the Cauchy family, too.

In fact, Lagrange advised Cauchy's father to enroll his son into the Central School of Pantheon. And no, Lagrange didn't want Cauchy to learn some proper mathematics. Due to uncle Lagrange's advices, Cauchy was supposed to learn classical languages and humanities. And he has won many prices in Latin and humanities, indeed. But despite the attempts by Lagrange to direct young Cauchy to this sissies' stuff, he chose an engineering career. In 1802, he scored among top 1% of the applicants to École Polytechnique and was accepted. He had some problems with the military-style rules in the school but he finished the school when he was 18, with the highest honors, and continued with civil engineering at the School for Bridges and Roads.

When he was 21, he already started to work as an engineer as well as a manager at something that Napoleon intended to become a naval base, Cherbourg. He had enough time to work on mathematical papers, anyway. His first two manuscripts on polyhedra were accepted; the third one on conic sections was rejected.

When he was 23, he realized he was overworked and the engineering job sucked, so he returned to Paris. He formally remained an engineer but was working for the ministry of interior and was on an unpaid sick leave etc. More importantly, he would work on higher-order algebraic equations, symmetric groups, symmetric functions, and the other Galois-like stuff (Évariste Galois himself was just an infant at that moment!).

In 1815, Napoleon was defeated in Waterloo – just to be sure, the place is very far from the Perimeter Institute – and Bourbon king Louis XVIII led the restoration attempts. So at the Academy of Sciences, mathematician Gaspard Monge and thermodynamics pioneer Lazare Carnot had to be fired for political reasons while Cauchy could have been hired for the same reasons. ;-)

Cauchy accepted but because a big part of the academic establishment was already composed of politically correct, left-wing activists, the reaction of his peers to his acceptance was harsh. He earned many enemies. In 1815, Cauchy could finally quit the engineering job and take the professor chair at École Polytechnique after Louis Poinsot who left for health reasons. Before that, Cauchy had already proven Fermat's polygonal number theorems. Many liberal activists could have been fired from the Bonapartist school while conservative Cauchy – whom Wikipedia calls "reactionary" – could be promoted.

He was living with his parents when he was 28 but his dad found a wife for him – a babe from a family that published most of Cauchy's writings. They had two daughters. Incidentally, Cauchy had brothers who became lawyers and one of them partly a mathematician, too.

As a mathematical factory, Cauchy flourished in the mid-to-late 1820s because he was lucky to live in a conservative political atmosphere. In 1824, Louis XVIII died and was superseded by an even more right-wing king, Charles X, which made Cauchy even more happy and more productive. He was also teaching at several schools simultaneously.

Things changed discontinuously in 1830. Charles X had to flee and the leadership was hijacked by a non-Bourbon king Louis-Philippe. Riots involving ignorant students took place near Cauchy's home. It had to be really annoying. Cauchy's lust to publish papers went nearly to zero – it's similar to my year around 2005 except that I had to be satisfied with a "relative conservative" in the form of Larry Summers who was finally removed in 2006. ;-)

Cauchy went to exile. In Switzerland, they still wanted him to endorse the new regime. He refused so he lost all positions he had in France. He went to Turin, Italy in 1831 and became a foreign member of the Royal Swedish Academy of Sciences.

Prague in 1830

In 1833, Cauchy moved to Prague in my homeland, the Austrian Empire ;-), to become a tutor of Henri of Artois, a spoiled brat from an aristocratic family. Back in Paris, Cauchy was already a bad lecturer and his teaching style resembled that of Sheldon Cooper. These difficulties escalated with young Henri who had no respect for Cauchy, mathematics, or anything of the sort, so even though Cauchy took the job very seriously, it was a disaster. The two main results of this tutoring was that Henri became a life-long math hater; and Cauchy hadn't done any research for 5 years. In 1838, he returned to Paris with his family that had been accompanying him in Prague since 1834.

He couldn't return to teaching – formally because he refused to endorse the new regime – but he badly wanted to be formally recognized by the science establishment in Paris again. He decided to get "there" though the Bureau to Determine the Longitude. He didn't need the oath so he was elected but the king was refusing to approve him for 4 years in which Cauchy was getting no money and had no academic rights (e.g. submitting papers). After that, Cauchy was eliminated altogether and replaced by Poinsot. Note that the opposite replacement was discussed above.

Throughout the 19th century, France was converging towards the separation of state and church, entities that Cauchy always wanted to unify. He became an enthusiastic Jesuit and officer in various Catholic and Jesus Society institutes. His colleagues would have full mouths of non-discrimination and so on but when this top mathematician of the French history applied for an ordinary chair in mathematics, he got just 3 out of 45 votes. Leftwingers have been a biased scum for at least 150 years.

The pan-European revolution of 1848 was mostly good news for Cauchy. The oath was removed from the law and he could regain the professorship again. He died on this day in 1857, sort of respected again. His name is one of the 72 names inscribed into the Eiffel Tower.

Research

In his early career, he made lots of advances about the Appolonius problem of circles touching three other circles; Euler's formulae for polyhedra; he introduced the notion of convergence, and other things. He was quickly becoming a pioneer of mathematical analysis – that's the part of maths dealing with functions, sums, integrals, derivatives, and... (this is why it's more advanced than just calculus – which is otherwise almost the same thing) with all kinds of limits.

He also made contributions to physics – wave mechanics (Fresnel's wave theory) and elasticity (Cauchy stress tensor). Add various things about membranes, vibrations, and so on. I have already mentioned the Fermat polygonal number theorem.

When it comes to mathematical analysis, he really laid the foundations of the holomorphic functions of complex variables, e.g.\[

\oint_C f(z)\dd z = 0

\] if there are no singularities inside the closed contour $C$. Seeds of this theorem already existed in 1814; the complete form was given in 1825. He figured out how to compute residues either from limits determining the Taylor expansion; or from the contour integrals. Pierre-Alphonse Laurent was the first man after Cauchy who contributed to this essential mathematical knowledge (Laurent series in 1843).

Cauchy tended to praise the mathematical rigor. Already in 1821, he was working with the infinitesimals – formally infinitely small numbers – but he was the first visible guy to have introduced the $\varepsilon$-$\delta$ gymnastics (games with limits) as the rigorous incarnation of the infinitesimal numbers.

It's an excessive task to enumerate everything that Cauchy has done in mathematics. I find it sort of funny to copy-and-paste a list of insights named after Cauchy:

Let me remind you: this guy was voted as unworthy the chair of an ordinary scholar in mathematics by 42 out of 45 self-described "pro-Enlightment" researchers in mathematics. Certain left-wing portions of the Academia have been rotten for centuries.

And that's the memo.

Intriguing spectra of finite unified theories (FUT)

2013-05-23T07:25:00.000+02:00

In November, I discussed FUTs (finite unified theories) which are $\NNN=1$ supersymmetric grand-unification-inspired versions of MSSM with the additional constraint that the divergences already cancel at the level of the effective field theory. This finiteness boils down to the vanishing of the beta-functions, some anomalous dimensions, and some relationships between the gauge and Yukawa couplings.

This condition doesn't seem to be a "must" – the divergences may very well be taken care of by the high-energy phenomena (string theory ultimately takes care of all divergences so its approximations don't have to be finite by themselves) – but it is an aesthetically intriguing condition, anyway. Now, the same authors released a new paper

Finite Theories Before and After the Discovery of a Higgs Boson at the LHC (S. Heinemeyer, M. Mondragon, G. Zoupanos)

where they calculate some new predictions and intriguing details.

They focus on the third-generation fermions and their superpartners, the Higgs sector, and the gauginos. The nicest FUTs they consider boast names such as FUTA and FUTB – the latter seem particularly attractive. They also take some LHCb results into account. In these models, $\tan\beta$ is typically rather large, $\mu$ is almost necessarily negative.

The spectra seem very intriguing and consistent with everything we know. Unfortunately, they're inaccessible to the LHC – or marginally accessible – and perhaps even inaccessible to ILC/CLIC. I like the representative table of a FUTB model here:\[

\begin{array}{|l|l||l|l|}
\hline
m_b(M_Z) & 2.74 &&
m_t & 174.1 \\ \hline
m_h & 125.0 &&
m_A & 1517 \\ \hline
m_H & 1515&&
m_{H^\pm} & 1518 \\ \hline
m_{\tilde t_1} & 2483 &&
m_{\tilde t_2} & 2808 \\ \hline
m_{\tilde b_1} & 2403 &&
m_{\tilde b_2} & 2786 \\ \hline
m_{\tilde \tau_1} & 892 &&
m_{\tilde \tau_2} & 1089 \\ \hline
m_{\tilde\chi_1^\pm} & 1453 &&
m_{\tilde\chi_2^\pm} & 2127 \\ \hline
m_{\tilde\chi_1^0} & 790 &&
m_{\tilde\chi_2^0} & 1453 \\ \hline
m_{\tilde\chi_3^0} & 2123 &&
m_{\tilde\chi_4^0} & 2127 \\ \hline
m_{\tilde g} & 3632 && {\rm masses}& {\rm in}\,\GeV
\\ \hline
\end{array}

\] You see that the LSP is the lightest neutralino below $800\GeV$. Staus are just somewhat heavier, $900\GeV$ and $1100\GeV$. Both sbottoms and stops fit the pattern that the lightest and heaviest one is at $2500\GeV$ and $2800\GeV$, respectively. The second lightest neutralino and the lightest chargino sit at $1450\GeV$, the remaining four faces of the God particle find themselves above $1500\GeV$ while the heavier chargino and the heaviest two neutralinos are above $2100\GeV$. Finally, the gluino is above $3600\GeV$.

Particularly the last figure is rather high (in a broader ensemble of models they analyze, the masses may go up to $10\TeV$ or so). We would have trouble to see such a gluino for years. But this model or at least similar models may be right. From a theoretical viewpoint, I see absolutely no preference when I compare models with gluinos at $1200\GeV$ and $3600\GeV$. Some people become very emotional and start to say that one of them has to be right or wrong or its rightness or wrongness means something a priori. Well, it just doesn't. Nature doesn't give a damn whether it's easy or hard for us to observe the superpartners. Once we observe them, many new things start to be clear. If we don't observe them, we are still extremely far from ruling out supersymmetry – and nice special supersymmetric models such as FUTB in this paper.

Its not my – or other humans' – job to rate the beauty of the values of particle physics parameters that emerge from Nature's decisions. It's Her job. Nevertheless, I must say that I would find a spectrum like the table above – or many other tables – elegant. It would probably mean that all these obnoxious idiots who like to say bad things about SUSY could remain loud for many more years. That's an annoying vision from a personal viewpoint but it can't change anything about the reality and it is less important than the actual beauty and physical near-inevitability that is carried by supersymmetry at some scale. If the known – mostly theoretical – evidence makes two models equally plausible and elegant, then one is obliged to love both of them equally, regardless of the fact that one of them may be much more accessible to the experiments. I view this commandment as a part of the scientific integrity.

A proof of the Riemann Hypothesis using the convergence of an integral

2013-05-22T15:30:00.001+02:00

Thursday morning update: After many hours, I decided that there is a critical error in the otherwise cleverly constructed proof. On page 138 (discussing Lemma 3), second part, he says "whence the function converges absolutely" essentially for any $z$ with a real positive part. But it seems he hasn't really established that (except for circular reasoning) because if RH is false, and it may be false, the numerator $|\psi(e^t)-e^t|$ goes like $e^{at}$ for some positive $a$ and the region of convergence is shifted by $a$. So the "absolute" part of the convergence isn't correctly proven, it seems to me. Maybe it's enough to prove the "ordinary" convergence but I suspect that there could be a similar error in the $g_1$ part of Lemma 3, too. Apologies if I am making a mistake.

Some people talk about the proof of "almost twin" prime integers separated by at most 70 million or something like that. I am not terribly excited by this result even if it is true. It's always more interesting to talk about somewhat promising proofs to the Riemann Hypothesis, not only because of the $1 million that will be given to the first person who solves the old puzzle.

Many people have thought that they had a proof but the candidate proofs have always failed so far. So you must understand it is extremely likely that we have another example of a failure here. But I am going to tell you, anyway. It would be great if some readers spend a sufficient time and energy by reading the paper. Please don't be repelled by the idiosyncratic Chinese English. Even I can recognize that it's not how a native speaker would formulate the ideas. ;-)

吴豪聪

That's his real name. Today, Hao-cong [first name] Wu [surname] of China sent me his new paper with a somewhat strange title (linguistically)

Showing How to Imply Proving The Riemann Hypothesis (PDF full)

published in the European Journal of Mathematical Sciences. How does the proof work?

It's likely that I won't quite reproduce everything that is needed for the proof in this blog entry even though I may try. Teaching things is the best way to learn them. ;-)

Wu elaborates upon some ideas initiated by Serge Lang, a famous mathematician. But that's the last comment about the sociological context. Now, let us look at the ideas which don't seem to require any esoteric new branches of mathematics.

The proof reduces the Riemann Hypothesis to a claim about the absolute convergence of an integral that is related to the Riemann $\zeta$-function in a simple way. Let's roll.

The function that Wu finds more convenient is called $\psi(x)$, pronounce "psi of ex". It is related to the Riemann $\zeta$-function by the following identities\[

\eq{
\phi(s) &= -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} =\sum_p \frac{\log p}{p^s-1}=\\
&= s \int_1^\infty \frac{\psi(x)}{x^{s+1}}\dd x = \frac{s}{s-1}+s\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x
}

\] where the sum over $p$ goes over the primes $2,3,5,\dots$. The first step you should be able to verify if you want to validate Wu's proof is that the identities above are satisfied if $\psi(x)$ is defined as the manifestly convergent sum\[

\psi(x) = \sum_{p^m\leq x} \log p = \sum_{n\leq x} \Lambda(n)

\] where $\Lambda(n)=\log p$ if $\exists m\geq 1: \,n=p^m$ for a prime $p$ and otherwise it is set to zero. Note that this $\psi(x)$ is defined in such a way that for a large $x$, it's expected to be very close to $x$ because the "probability to be prime" $1/\log x$ is cancelled by the factor $\log p$ from the definition of $\psi(x)$ – it's close enough already when we allow $m=1$ only.

The second step is to realize that the presence of a zero or zeroes of $\zeta(s)$ also implies (or would imply) a pole of $\phi(s)$, the [minus] "logarithmic derivative of the $\zeta$-function", at the same location of the complex plane. To prove the Riemann hypothesis, it is sufficient to prove that $\phi(s)$ has no poles for \[

\frac 12 \lt {\rm Re}(s) \lt 1

\] (in the "right half-strip", as I will call it) because the hypothetical "RH-violating" zeroes (and singularities) come in pairs symmetrically distributed relatively to the critical axis $s=1/2+it$ for $t\in\RR$. Note that $\phi(s)$ has a pole (or would have a pole) even for a higher-order zero of $\zeta(s)$.

The third step, and it's the only hard one, is to actually prove that one of the integrals involving $\psi(x)$ used to calculate $\psi(s)$ above\[

\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x

\] is analytic in the right half-strip so it has no poles over there. Consequently, the $\zeta$-function has no zeroes in the right half-strip and, by the left-right symmetry, no zeroes in the left half-strip, either.

Wu reduces the claimed analyticity of the integral above to the absolute convergence (convergence even if the integrand is replaced by its absolute value) and uniform convergence (the speed of convergence may be taken to be $\varepsilon$-independent), $\forall\varepsilon\gt 0$, of the integral\[

\int_1^\infty \frac{\psi(x)-x}{x^{3/2+\varepsilon}}\dd x.

\] It shouldn't be hard to see that the absolute and uniform convergence of the integral above (here) is enough for the analyticity of the previous integral, and therefore for the absence of the non-trivial zeroes. Note that the exponents $s+1$ for $s$ in the right half-strip and $3/2+\varepsilon$ for a positive $\varepsilon$ are the same objects.

So aside from the claims that should be straightforward, the beef of the proof should be the demonstration of the absolute and uniform convergence of the integral in the last displayed equation.

Note that Wu's approach is linked both to the "complex analytic" interpretation of the Riemann Hypothesis as well as the prime-integer-counting, "number-theoretical" interpretation. It's because sufficient experts know that the Riemann Hypothesis is equivalent to the statement\[

\forall \varepsilon\gt 0: \, \psi(x) = x+ O(x^{1/2+\varepsilon})

\] which says that if we accept that the probability for a "rough number $x$" to be a prime is $1/\log(x)$, then the estimated number of primes up to $n$ deviates from the actual one at most by a power law (that is producing the $O(\dots)$ term above.

Proving the convergence

OK, so how does Wu want to prove the uniform and absolute convergence? He offers some introduction to the theory of functions of real and complex variables together with some lemmas that are not quite well-known and that may even be new. Finally, the proof boils down to the existence (for any $s$ with a real positive part) of the Laplace transforms $g_{1,2}(s)$ of a function called $f_{1,2}(t)$ related to $\psi(e^t)-e^t$ for the subscript $1$ or its absolute value for the subscript $2$.

If you quickly want to focus on claims related to the $\zeta$-function and ignore various theorems and lemmas about completely general functions and their convergence etc. (assuming that these things are harmless and perhaps known to you, explicitly or intuitively), you may find it helpful for me to say that only Theorem 5 (among 7 theorems) and Lemma 3 (among 3 lemmas) is what you want to read. If there is some circularity in Wu's argument (secretly assuming RH), it's probably somewhere in Theorem 5 or Lemma 3.

In particular, I believe that Theorem 5 contains the main trick that allows us to show the convergence in the right half-strip. This theorem claims the absence of poles (except for the $s=1$ pole) of the function\[

\eq{
\Phi(s) &= \sum_p \frac{\log p}{p^s} = \phi(s)-\sum_p h_p(s),\\
|h_p(s)|&\leq B\frac{\log p}{|p^{2s}|}
}

\] On one hand, this capital $\Phi(s)$ is shown to be rather close to the lowercase $\phi(s)$, using an argument based on geometric series. On the other hand, the $2s$-th power of something appears in the difference between $\Phi$ and $\phi$ which makes $\sum\log n/n^{2s}$ converge for ${\rm Re}(s)\geq 1/2+\delta$. So the coefficient $2$ in $2s$ here is the ultimate reason why the meromorphic character of $\Phi(s)$ starts at ${\rm Re}(s)\gt 1/2$, how we get the one-half somewhere, and why the critical axis becomes a decisive boundary for the well-definedness of $\phi(s)$, too.

I don't see any mistake so far but I haven't really devoured all the beef of the proof yet, either, so no complete confirmation from your humble correspondent yet. But it is apparently making more sense every minute!

See the previous TRF blog entries mentioning the Riemann Hypothesis.

Ask questions to James Hansen

2013-05-22T13:18:00.000+02:00

Today, at 5 p.m. Boston Daylight Savings Time (11 p.m. Central European Time), James Hansen will give a talk over here.

Live Video streaming by Ustream

It's being claimed that you will be allowed to ask a question when he's finished.

First, people like James Hansen would make sure that the debate is over. And then they start the debate – so that no one inconvenient may really participate it.

I am not promising you anything, however. It's plausible that only convenient questions will be allowed. So Barbara Boxer will ask whether the Oklahoma tornadoes were caused by an SUV or by some beef steaks in McDonald's. Or some other sins against the glorious left-wing delusions that women and men of her caliber believe.

If you want to waste 73 minutes, feel more than free to watch a talk that Hansen gave yesterday in front of some people who think he is a "hero". Among other things, Hansen explains that he was skipping classes in the college because he didn't want to show how ignorant he was – which made him even more ignorant. But then he found the environmental movement and ignorance was transformed to a virtue.

Anthony Zee: Einstein Gravity in a Nutshell

2013-05-22T08:06:00.001+02:00

Škoda is not just a carmaker; it is producing happy drivers. And you may see that even the engines in the factory are having a great time.

In the same way, Anthony Zee – as Zvi Bern noticed – decided to make many readers fall in love with the physics of general relativity by having written this wonderful tome, Einstein Gravity in a Nutshell. Bern said that the goal wasn't to create new experts but Zee corrected him that he wanted to make the readers fall in love so deeply that they may dream about becoming experts, too. And the clearly enthusiastic Anthony had to enjoy the writing of the book, too.

I received this large, almost 900-page scripture on Einstein's theory yesterday. Obviously, I haven't read the whole book yet but I may have spent more time with it than most readers (more than zero) so that I can tell you why you should buy it and what philosophy, style, and content you may expect.

It's a book addressed to a wide variety of readers, including very young ones (perhaps college freshmen and bright high school students) and amateur physicists. Experienced physicists and professionals may find some gems or at least entertainment in the book, too. Because of this goal, the book starts with elementary things such as the units including $G,c,\hbar$ and Planck units, relativity even in classical physics, as well as basics of curved spaces, differential geometry, and so on.

The style is witty and somewhat dominated by words – and amusing titles. You may find lots of philosophical and historical remarks and stories from Anthony's professional life but the physics is always primary. And I mean physics, not rigorous mathematics. Tony is focusing on objects, phenomena, and their measurable and calculable quantities and the purpose of physics is to understand them and calculate them. So he spends almost no time with various picky issues – whether a function has to be smooth; whether one should use one fancy word from abstract mathematics or another. In fact, he considers the suppressed role of rigorous maths to be a part of the "shut up and calculate" paradigm that he subscribes to.

In some sense, you could say that the approach resembles the Feynman Lectures on Physics. It is very playful and the author is always careful to tell you think that are still fun and stop elaborating on details when he could start to bore you. So the book (probably) keeps its fun status at every place (it's true for the portions I have read). But Anthony Zee manages to penetrate much more deeply into general relativity with this strategy.

Once he goes through all the basics – which allow a beginner to start with the subject almost from scratch but which seem very entertaining for a reader who doesn't really need such introductions anymore – and he answers all the FAQs on tensors and lots of other things, he offers some of the simplest derivations of Einstein's equations and is ready to apply them.

It's useful to know what concepts are considered primary starting points by the author. I would say that Zee is elevating the concept of symmetries and the action – the latter allows us to formulate most dynamical laws in classical and quantum physics really concisely (although we know perfectly consistent quantum systems that don't seem to have any nice action; and the action always assumes that we prefer a particular classical limit of a quantum theory – and the classical limit isn't necessarily unique).

Concerning the applications, some of the historically important applications that were designed to verify the theory are suppressed. But you get very close to the cutting edge, including the general-relativistic aspects of topics that are hot in the contemporary high-energy theoretical physics and the cosmological/particle-physics interface. So you may actually learn advanced topics about black holes including some Hawking radiation (including the numerical prefactors of the temperature; but the author doesn't go extremely far here; note that amusingly enough, the Hawking radiation is even discussed in an introductory chapter); large and warped extra dimensions; de Sitter and anti de Sitter space including a discussion of conformal transformations (although it doesn't seem like a full-fledged textbook on AdS/CFT); topological field theories; Kaluza-Klein theory (with extra spatial dimensions) and braneworlds; Yang-Mills theory (there's lots of electromagnetism in the earlier chapters); even twistor theory; discussions on the cosmic inflation and the cosmological constant problem; and heuristic thoughts on quantum gravity (some of them are more heuristic than the state-of-the-art allows us; but Zee's philosophy is that textbook shouldn't be composed exclusively of the totally established stuff ready to be carved in stone).

Using lots of witticisms and clever analogies, Zee also proves some things you wouldn't expect – e.g. that Hades isn't inside the Earth. The equivalence principle is compared to the decision of all airlines, regardless of the size (and the size of their aircraft), to fly between two distant cities along the same path on the map. Witty and apt.

Anthony is convinced that most authors are explaining things in unnecessarily complicated ways – in some cases, perhaps, they want to look smart by looking incomprehensible. That's not Zee's cup of tea. He enjoys to simplify things as much as possible (but not more than that). And he loves to formulate things so that the reader is led to the conclusion that things are simple and make sense, after all. For example, there is a fun introduction to the least action principle (light isn't stupid enough not to know the best path) and we learn that "after Lagrange invented the Lagrangian, Hamilton invented the Hamiltonian". It makes sense, doesn't it?

There's a lot to find in the book. Some readers say that the book is less elementary than Hartle's book but more elementary than Carroll's. Maybe. Anthony is more playful and less formal but there are aspects in which he gets further than any other introductory textbook of GR.

The book is full of notes, a long index, and simply clever exercises. The illustrations are pretty and professional. If you are buying books to see photographs of attractive blonde women with toys, you won't be disappointed, either.

Because the book is really extensive and even the impressions it has made on your humble correspondent in the single day are numerous, I have to resist the temptation to offer you examples, excerpts etc. because that could make this blog entry really long by itself. Instead, I recommend you once again to try the book.

Tommaso Dorigo impressed by a cold fusion paper

2013-05-21T09:08:00.000+02:00

...but the paper is 100% crackpottery...

In his text "Is Cold Fusion For Real?", Tommaso Dorigo seems highly impressed by the following new Italian-Swedish preprint about cold fusion:

Indication of anomalous heat energy production in a reactor device

They claim that an Andrea-Rossi-style tube with nickel and hydrogen produced 10+ times more energy per liter of fuel than any known chemical reaction, as measured by thermal imaging cameras during 96- and 116-hour experimental runs.

Image credit: Rossi, Kullander, Essén and the e-Cat, retrieved from energydigital.com.

Dorigo says that "the conclusions of the tests are at the very least startling". He "continue[s] to believe in the scam hypothesis, but [he] must admit that this study impressed [him] for its reported result." Also, he must say that "[he] will from now on follow more closely the developing story of Rossi's E-CAT...".

Congratulations, Mr Rossi. You have clearly earned a fan who thinks that cold fusion is more realistic than supersymmetry! If you really believe so, Mr Dorigo, then you are an unhinged lunatic.

First, let's discuss the sociology. What I find remarkable is how easy it is for the authors to make folks like Dorigo repeat some self-congratulatory words such as the "third-party [investigation]" that appears in the first sentence of Dorigo's article.

If you look for a rather detailed, enlightened, entertaining, and completely reasonable criticism of the new paper (with discussions of mistakes, possible ways to do such an experiment to be convincing, and differences between science and pseudoscience, not to mention some useful links), see The E-Cat is back, and people are still falling for it! by Ethan Siegel (Starts With a Bang).

If you look at the previous publication record of the seven authors via the arXiv (click at the author names here), you will see that only two of them have previously published something. Bo Höistad appeared among dozens of authors in an (unknown) technical design report on a tracker in the KLOE-2 experiment. Hanno Essén of KTH Mechanics, Stockholm has published a dozen or two dozens of preprints on magnetism and fluid dynamics, none of which is really well-known and all of which look like papers by an engineer attempting to "do" physics as his hobby – which is what they are, after all.

Given the large amount of money that is circulating in the cold fusion business today, one must say that all the data are consistent with the seven authors' being inserted just as puppets who don't have to be afraid of their scientific credibility because they don't have much to lose (relatively to what they can be promised from the Rossi industry). So I have a problem with the adjective "third-party" in Dorigo's article. There doesn't seem to be solid evidence that this is independent from Rossi's organizations; the presence of the Italian co-authors makes the independence even less likely.

Update: Hours after this blog entry was written, Steven Krivit explained that this paper doesn't describe any independent test; instead, the authors were just witnesses of a Rossi demonstration. Essén, the only co-author with at least 2 other papers, admitted he didn't replicate the experiment. On the picture at the top of this blog entry, he is shown together with Rossi, in a position clearly indicating that Rossi is a teacher and Essén is an obedient disciple.

Dorigo himself may be rather positive because all the Italians form a family of a sort. A trailer for Mafia II, an amazingly realistic but too linear PC game.

A technical or sociological detail that doesn't prove that the preprint is rubbish but it's always a brightly shining and blinking "red light" for me is that the physics.gen-ph preprint was delivered as PDF only and it wasn't written in $\rm \TeX$ but rather in Microsoft Word (on a Mac). That makes it very likely that the authors don't actually know $\rm \TeX$ and most of such authors don't really know physics well, either.

Tommaso Dorigo often likes to paint himself as a person who is willing to think independently except that the extent to which he simply copied the self-congratulatory decoration that was added to this paper suggests that he isn't really thinking critically.

Now, somewhat more technically.

The authors are clearly ignorant not just about $\rm \TeX$; they seem to be unfamiliar with the basics of the scientific notation and the rudimentary methodologies that are needed in experiments. For example, the units of dimensionful quantities are always written as $[\rm W]$ including the square brackets. No, if we perform dimensional analysis and want to say something about the units of a quantity, we may use the symbol $[l]$ or $[{\rm length}]$ to describe the units of length. But we never actually write meters – the particular units themselves – into the square brackets! This is a detail but a strong indication that the authors haven't really gone through some relevant undergraduate freshman courses of physics.

Perhaps more importantly, the paper never discusses the error margins properly. You may easily check that they never talk about anything such a "systematic error" and when they talk about errors at all, they're just "optional". Sometimes they add them to the results, sometimes they don't, and if they do add them, it seems that they just make the numbers up. For example, on pages 22-23, they say that the effective power consumption has error of 10% because "errors of this extent are commonly accepted in calorimetric measurements". Wow. There's surely no "one-size-fits-all" error for calorimetric measurements. The error depends on what you measure, how you measure it, and almost all the details in your experiment. The magnitude of a systematic error of your apparatus actually has to be measured and calculated for your case specifically; you can't copy a "general figure" from completely different papers describing different apparatuses. This sentence by itself shows that they don't know how to do experiments well.

But on page 22, you see an example of the "key calculation" meant to show that the "reactor" produces lots of energy. Over the 116 hours of the experiment discussed here, the gadget was consuming 283.5 watts from the grid and producing 810 watts (much more), we hear. An anomalous production of heat, they happily announce. Those 810 watts are claimed to be determined from the thermal radiation that the reactor was emitting; the 283.5 watts are calculated as 35% of 810 watts.

Where does the figure 35% come from? The resistors (electric heaters of a sort) were on for "about" 35% of the time and off for "about" 65% of the time. Couldn't one just measure the precise time during which something was turned on and off? Are we supposed to think that they used a gadget to measure time that uses "one percent" as the unit of time? Don't you agree that this claim about "65% off" is just a potential lie to mask that the energy was coming in all the time? See also this comment for a convincing indication that they measured the incoming energy completely incorrectly (even though this should be a very easy task!).

OK, let's not think about details. It's plausible that the average consumption in those 116 hours was close to 283.5 watts. The produced average power 816 watts is much higher and it would be enough for a proof that some very strong anomalous production of energy is taking place. Where does the figure 816 watts come from?

It comes from equation (24); the power was calculated from the emitted thermal radiation. The numerical value 816 watts whose error is estimated as just 2% is the sum 741.3+17+58 watts. Clearly, the only large contribution that affects the overall qualitative result is the term 741.3 watts. Where does this come from?

It comes from Table 8 on page 21 where it's calculated as the sum of radiated 459.8 watts and 281.5 watts participating in convection. The latter figure, 281.5 watts, contradicts some calculation resulting in equation (17) on page 12 where convection was claimed to give about 466 watts. I was trying to check where various numbers were coming from but almost nothing seems to be consistent with anything else. The paper looks like an incoherent pile of rubbish to me.

Moreover, the gadget seems to depend on the power outlet and has a nonzero consumption. So you could expect that its power production will also depend on whether or not the resistor coils are turned on or turned off. However, this dependence of the produced power on time isn't discussed anywhere in the paper.

Let's ignore the differences between 281.5 and 466 watts. They still see lots of radiating energy that is produced, don't they? Well, they surely claim so. In equation (8) on page 10, they boast to have produced 1568 watts (during the other, 96-hour experiment) which is 1609 watts minus the small contribution from the room. The numerical value 1609 watts is computed in equation (5) and nearby equations as the Stefan-Boltzmann power corresponding to the (fourth-power-based) mean temperature around 723 kelvins (450 Celsius degrees) multiplied by the area of the Rossi cylindrical "black box" (whose bases are overlooked).

The emissivity is set to one i.e. they assume the "reactor" to be a black body. This choice is labeled "conservative". Except that the truth seems to be going exactly in the opposite direction. The actual emissivity is lower than one and it's the coefficient multiplying the fourth power of the absolute temperature to get the power. Because they seem to calculate the power from the measured temperature (the infrared camera is claimed to give the right temperature and automatically adjust the observed radiation for emissivity etc.; see page 7 of the paper), the actual power is actually much lower than [the calculated figure] 1609 watts. The emissivity of metals at similar reasonable temperatures seems to be 0.2 or so – something of this order – which reduces 1609 watts to something like 300 watts, pretty much equal to the consumption.

Pretty much every hint that I have looked at in the paper suggesting that they produce some energy that exceeds the electricity consumption from the power outlet seems to be plagued by similarly basic errors.

Recall that I believe that the error in virtually all Rossi's presentations is that he assumes that the boiling point of water is always 100 °C, regardless of the pressure, and he "takes credit" for the evaporation of lots of water that actually stays mostly liquid because it's below the boiling point which is above 100 °C if the pressure is elevated (and it is elevated in his setup); even visually, it's clear that liters of water can't be getting vaporized because the steam would look much more intense than the feeble traces of vapor coming from his "miracle gadget". All these errors seem completely elementary to me. It's hard to say whether the authors see them or not. I guess that they're training themselves to overlook them because they are afraid that by admitting they have believed and promoted something that so silly that clearly doesn't work, they would look like complete idiots – and indeed, they look like complete idiots even if they try to obscure the evidence.

To summarize, the preprint is complete rubbish and the authors are probably linked to Andrea Rossi personally but that doesn't prevent the loudest blogger of the LHC's CMS Collaboration to partially endorse this preprint – without even attempting to read it because "this is not [his] field of research" – and suggest even though he hasn't looked at this paper at least to see that it's pure trash (and it's very easy to see), he will more closely follow cold fusion because of that. It's so easy to propagate lies and stupidity in this world especially because most people are even more stupid, mindless sheep than Tommaso Dorigo.

At any rate, I am amazed by Dorigo's claim "not to be an expert" itself. He is an experimental subnuclear physicist and this is a claimed groundbreaking paper in experimental nuclear physics. I have not been an experimenter at all but I see nothing in the paper that I could be misunderstanding because of an insufficient background. You see that even people claiming to be "scientists" often don't behave as scientists. Without even trying to study something, they just uncritically endorse some ambitious claims. And in many cases, their "being a scientist" is exploited in the promotion of a nonsense even though they clearly failed to evaluate the issue scientifically. Note that it's enough to find a few dozens of such "scientists" who haven't performed even the basic checks and the media often claim a "scientific consensus" even though the strength of the scientific evidence behind the claim is exactly zero.

With this extremely sloppy attitude, you can't be surprised that Mr Dorigo and others have no problem to deny string theory or other basic pillars of modern science. This particular chap denies completely basic insights into nuclear physics as well – and never hesitates to use a paper not knowing how numbers with units are written as evidence that nuclear physics fails.

See also a dozen of previous articles on this blog that mention Rossi and fusion.

Concerning somewhat more realistic sources of energy, see this new wind turbine with funnels (via Joseph S.). They claim to concentrate the wind so that 1 kWh only costs $0.20 or so – almost competitive with coal etc. at the Czech prices. The gadget looks silly but I do think that the concentration of wind energy (and similarly, possibly, concentration of solar energy with mirrors) is a largely unused way to make these "renewable" sources cheaper and more productive.

Light Dirac RH sneutrinos seen by CDMS and others?

2013-05-21T06:50:00.000+02:00

What is dark matter made of?

We almost know that its mass should be dominated by a new light particle species that is heavy enough so that it moves rather slowly relatively to the speed of light ("cold" dark matter). Because dark matter isn't gone yet, such a particle must be stable or almost exactly stable – lifetime in billions of years, to say the least.

The lightest particle carrying a "new type of charge" is the best explanation why it's stable. By far the most popular clarification what this new charge is is the R-parity, a new "sign" introduced by SUSY. All the known particles in the Standard Model have the R-parity equal to $+1$ which is why the Standard Model interactions never produce individual superpartners whose R-parity is $-1$. For the Standard Model particles, the R-parity may be written as $(-1)^{3B-3L+2J}$ where the odd coefficients may be replaced by any odd integers (although some values may be more correct for the exotic particles).

The stable particle is then the lightest particle with the R-parity equal to $-1$, the lightest superpartner, the so-called LSP: it's almost all the supersymmetric models' candidate for the WIMP, the new dark matter particle. If you look just superficially, it may be the superpartner of any elementary particle in the Standard Model. The R-parity has just two possible values (signs) so it forms a group isomorphic to $\ZZ_2$. The number of R-parity-odd particles in a physical object is therefore conserved modulo two. In particular, they may be pair-created at the colliders and they're expected to pair-annihilate in the outer space.

However, if the LSP were a superpartner of a charged particle, it would be charged, too. And charged particles interact with the electromagnetic field. They may be seen. They're not dark. I am sort of convinced that one may construct viable theories of the observed data with non-dark matter as well – with "superheavy hydrogen atoms" that have a different charged particle as the nucleus, for example, pretend to be hydrogen, but store much more mass – but let's assume that the dark matter is dark, indeed.

In the literature, most models assume or conclude that the LSP is either the gravitino – the spin-3/2 superpartner of the graviton (in that case, the R-parity may be broken because the gravitino is still nearly stable due to the gravitational i.e. very weak character of its interactions) – or, much more frequently, the lightest neutralino. The neutralino is the spin-1/2 superpartner of an electrically neutral Standard Model elementary particles. Because there are several such particles (components of the W-boson, photon, and the Higgs field), there are several neutralinos. The mass of such neutralinos is given by a matrix and the "sharply defined mass" eigenstates are general superpositions of the photino, zino, and higgsinos – four neutralinos.

Physical technicalities usually force us to adopt the mass of hundreds of gigaelectronvolts for such a neutralino if it is the LSP. Such a "relatively heavy" neutralino couldn't explain the positive hints in the dark-matter direct search experiments that indicate the existence of a sub-$10\GeV$ WIMP. The SUSY phenomenological literature is really obsessed by the idea that the LSP has to be a neutralino. It almost looks like a Christian dogma and different cliques seem to fight whether the neutralino LSP is mostly a wino (Catholic) or a bino (Protestant) or a higgsino (Orthodox). I am afraid that the certainty that the LSP has to be a neutralino is one of the not quite justified assumptions that are adopted because of group think, not because of solid evidence.

Now, building on various previous papers such as this 2006 paper and the authors' 2012 paper (PRD), Ki-Young Choi (Korea) and Osamu Seto (Japan) propose an attractive label for this new hypothetical $8.6\GeV$ or so dark matter particle: a light Dirac right-handed sneutrino.

Light Dirac right-handed sneutrino dark matter

What does it mean? The adjective "light" means that its mass is supposed to be much smaller than hundreds of gigaelectronvolts. The word "sneutrino" says that it's the superpartner of a neutrino; the prefix "s-" may be interpreted not just as "the superpartner of" but also as "scalar" because these superpartners of spin-1/2 fermions are scalar fields and particles, i.e. spin-0 particles. The adjective "right-handed" means that it's the superpartner of a right-handed neutrino – all the neutrinos we directly observe today are left-handed (while the antineutrinos are right-handed but we are not talking about any antineutrinos with $L=-1$ here at all). Finally, the adjective "Dirac" means that the neutrino is assumed to have mostly Dirac (and not Majorana) masses, coming from the bilinear product of the left-handed and right-handed neutrino components. Note that the adjective "Dirac" says nothing about the sneutrino per se; we first say that the neutrino is Dirac and then add the "s-" prefix. How the neutrinos get their mass is important for the sneutrinos, too.

The Asian authors claim that they can write down a model of this sort that is pretty much consistent with all the data – positive and perhaps even negative (XENON100) direct search experiments; experiments measuring the invisible width of the Z-boson and the Higgs boson (including the "effective number of neutrino species").

It would be sort of fair if the superpartner of the most invisible Standard Model particle – a neutrino – became the first visible superpartner, albeit still a "dark one". ;-)

Don Lincoln of the Fermilab attempted to explain supersymmetry in 6 minutes today. See the video above. By the way, 10,000 papers on SUSY is probably a huge underestimate. It depends what you count but over 100,000 papers may be mentioning supersymmetry.

Investigation of the largest Czech credit union: assaulting the victims

2013-05-20T12:04:00.001+02:00

...including your humble correspondent...

Another annoying event occurred to me on Friday – and it's still happening and will be happening for quite some time.

I learned that the Czech National Bank, the supervisor of our financial markets, began to audit MSD (imsd.cz) or Metropolitan Credit Union (the largest Czech credit union) where I sent a very large amount of money on Tuesday. The transaction wasn't completed (which is why I started to be interested in the situation on Friday) and the money should have been returned but they remained in the air, invisible at both places. Today, I learned from the chairman of MSD that a reverse transaction (sending the money back from MSD) was ordered by MSD last Wednesday but it wasn't allowed because the authorities began to block outgoing payments (approximately) on that very day (?) and they didn't do it right.

Between Friday and today (Monday), I was gradually learning what was happening. The audit by the Czech National Bank that found it suspicious that the percentage of "loans given by MSD that need monitoring" is extremely low, was apparently combined with another (totally independent?) intervention, an investigation (by the Supreme Office of Prosecutors in Prague, whatever is the right English translation) of two large loans worth $50 million in total which are apparently fraudulent (attempts to rob most of the members of MSD such as myself by some individuals).

The credit union has 15,000 members – only members may have deposits and/or loans – and the balance sheet is about $750 million. You may calculate the average member's savings (well, the right number is actually higher than that because you should only count the members who are savers and not those who are members to borrow the money). Most of the growth of the members occurred in the last 2 years (I was added a month ago or so: a very unlucky timing, indeed), after years in which the credit unions regained their good name (damaged by failures in the 1990s; no insurance existed at that time).

About a quarter of that balance sheet may be available in cash. The financial situation of MSD seems very healthy according to their documents, much like the business model. It's the ultimate super-simple bank with a single savings account offering a reasonable interest rate; and just several types of CDs (three or four different maturity dates). They lend their members money for 10% a year in average (a large percentage of them is returned OK); and they can therefore afford to pay 2+ percent on the savings account and 2.5-4.5% or so on the deposits (the savers must be members of the credit union, too). No large expenses for complicated software (the internet banking is only passive), employees giving plush bears to clients, and so on.

I have always thought that a 10% interest rate for loans is reasonable. Near-zero interest rates are a pathological component of the postmodern age; they always amount to some government institutions' campaign to cram the banknotes down the people's throats. Such government institutions may think that they're helping the economy but they only help to distort it and screw it. It is just not natural to lend the money for near-zero interest rates. Such a habit completely ignores the (often unpredictable) risks as well as the pain associated with the "postponing of the consumption". Lenders wouldn't naturally demand that low interest rates; they're only willing to do so because the banking market has been heavily distorted by activists trying to "revive the economy" who actually just create one bubble after another, one crisis after another.

Moreover, in conservative nations such as Czechia (which inherited the conservative approach to finances from the Austrian Empire), the lowering of the interest rates really reduces the consumption because the economically most relevant people are "more savers than borrowers" and they just feel poorer when the interest rates are low, so they reduce their consumption. No one cares about these rather self-evident facts and one-size-fits-all dogmas about "good low interest rates" are parroted by pretty much all central bankers etc. in our country, too.

So I learned today that my large amount of money is "hanging in the air" today, from the chairman of MSD. I sent it while the Czech National Bank was already blocking the incoming payments. But the payment was still accepted and added to a shared "collecting" account of MSD but no longer moved to my account in MSD internally (because the people executing the block didn't do it at the right place).

Aside from that very large amount of money, I also have about 2.5 times the very large amount of money in MSD which is still far from my only assets but it's very, very large. I haven't had any contacts with any credit unions throughout my life but a month ago, I found the near-zero-percent (and less-than-inflation) interest rates in the large banks really obscene and moved a significant portion of my savings, encouraged by the full insurance of savings up to €100,000 (I wouldn't be attracted to a credit union without this insurance).

You may imagine that I wrote some number of e-mails to the Czech National Bank, to the prosecutors, to various banks etc. today, in order to fix the situation – especially with the not-fully-completed transactions (a problem that MSD is totally aware of but the government institutions never seem to care if they screw something). It's annoying. Well, it's not an existential thing for me, either. I've seen postings by a 60-year-old man who had all his life savings in MSD and can't pay his rent and food now because he didn't diversify at all. A possible millionaire-turned-homeless.

I can't resist to say a broader point: it's just another episode that is teaching me that the regulation and the government is a cure that is (much) worse than the disease. It seems to me that the prosecutors in particular are fanatics who are obsessed by the hope of catching a criminal – or a likely criminal – but they don't hesitate to shoot 15,000 innocent people (actually victims) along the way. More generally, the government institutions are bodies where one isn't really responsible for his or (in this case, mostly) her acts, in which he (or she) isn't really forced to compare costs and benefits. The costs of some potential achievement may be many times higher than the benefits (in this case at least 15 times) but these folks may still decide it's the right thing to do. They're ready to block a whole credit union because of two suspicious/bad loans. They're ready to destroy your national economy because of one totally bogus problem (e.g. "global warming") or anything of the sort. They have no sense of proportion because the "omnipotent" character of the government allows them to remain ignorant about all proportions.

Needless to say, there are always some people who support these more-harmful-than-useful activities by the government: it's the jealous voters who are more hateful towards other, successful people than the degree to which they want to enjoy their own life. So a government that does more harm than good is what they actively prefer.

So the members of MSD who are really victims of the two large possibly fraudulent loans are forced to spend at least a week or weeks (and maybe months or perhaps years, who knows) enhanced by some extra anxiety. Some of them probably become homeless because of that – the status of the credit union is probably going to remain uncertain for quite some time which makes it impossible to pay 100% of the savings from the insurance fund. The people in the government and state-organized independent supervising bodies aren't able or willing to distinguish criminals from their victims. They're not even able to properly block the transfer of money between a credit union and the rest of the world so that some money remain floating in the air (on the shared "collecting accounts"). If I am dissatisfied with a bank, I may try a competitor; if I am dissatisfied with some government bureaucrats or agencies, I am just screwed. Of course that the effectively two-choice elections have no chance to find the right recipes to improve the situation in hundreds of government bodies.

Speculations exist that the interventions are controlled by large banks that find the more effective competition from the credit unions inconvenient. I have nothing to say about these speculations except that I see no way to prove them invalid.

The investigators and regulators just don't seem too competent or helpful in any sense but all of us are paying these people (and they have obscenely high salaries) to "improve" the free markets. The free markets aren't, can't be, and shouldn't be perfect – but the government ain't a way to improve them. In the present crisis, government is not the solution to our problem; government is the problem.

And that's the memo (thanks, Mr Reagan 1981).

An article claims that MSD has been giving loans – up to 4 billion crowns (1/4 of the balance sheet) – to foreign firms with no history, probably represented by puppet representatives. These firms were probably supposed to rob the money out of the savings bank and never return them. If true, it's good that they stopped it, of course, if they did. If the fraudulent loans represent 40% of the currently available loans out there, it seems very likely to me that MSD will converge towards failure (or violation of the financial-situation conditions that savings banks have to satisfy) once the fraudulent loans are closed because the good loans can't produce enough profit to cover the interest rates etc. On the other hand, MSD should have enough cash to cover all the savers when it's closed so maybe no money will have to be used from FPV.

Ways to discover matrix string theory

2013-05-18T13:57:00.000+02:00

...more precisely screwing string theory...

The 5,250+ TRF blog entries discuss various topics, mostly scientific ones, including minor advances. However, there isn't any text on this website that would talk about matrix string theory (inpendently found 2 months later by a herald who inaugurated the new Dutch king and an ex-co-author of mine along with two twins).

If you search for the closest topic, you will find one article about Matrix theory published a year ago and a supplement about membranes in Matrix theory that was added a week later.

But now we want to talk about matrix string theory. It's a version of Matrix theory. Much like Matrix theory – or M(atrix) Theory – describes M-theory in 11 dimensions (which has no strings), matrix string theory describes type IIA or heterotic $E_8\times E_8$ string theory in $d=10$. So it's a stringy version of Matrix theory; or string theory formulated in a matrix form.

The discovery of matrix string theory was important for several reasons. First, it was an important confirmation of the ability of the Matrix theory concept to define the dynamics of string/M-theory in many situations; and it was the first time when we had a complete, non-perturbative definition of a string theory.

What do I mean by this comment? Before Matrix theory, all calculations in string theory would be organized as Taylor expansions in $g_s$, the string coupling. All amplitudes would be written as $A_0 + A_1 g_s + A_2 g_s^2\dots$, and so on. However, not every function may be expanded in this way and the general amplitudes in quantum field theory or string theory can't. For example, $\exp(-C/g_s^2)$ has a Taylor expansion whose terms vanish (because all higher-order derivatives of this function at $g_s=0$ vanish) even though the function was non-vanishing.

In this sense, a complete definition was absent. One could have even believed that the existence or consistency of string theory was just a perturbative illusion. Matrix string theory was the first "constructive proof" that string theory is well-defined even non-perturbatively. In the type IIA case, one had a definition for any $g_s$. In the $g_s\to\infty$ limit, one could easily show that the theory reduces to Matrix theory, the matrix model for M-theory; in the $g_s\to 0$ limit, one could prove – and this is the main achievement of the matrix string theory founding papers – that the dynamics reproduces the states and interactions of type IIA string theory as we had known them from the perturbative approaches.

Formal and informal derivations of the matrix string Lagrangian

Matrix theory is formulated in terms of the following Hamiltonian\[

H = P^- = \frac{N}{2} {\rm Tr}\zav{ \Pi_i^2 - [X_i,X_j]^2 +{\rm fermionic} }

\] which is interpreted as a light-cone component $P^- = (P^0-P^{10})/\sqrt{2}$ of the spacetime energy-momentum vector. Well, the original Matrix theory paper by BFSS (Banks, Fischler, Shenker, Susskind) talked about the "infinite momentum frame" and various "highly boosted limits". But one could easily go to the limit and rewrite the quantities in the light-cone gauge. I was always baffled how a paper by Lenny could have become well-known just because it made this self-evident point. My papers (written before Susskind) always took the light-cone gauge as an obvious fact, for granted, and I am confident that everyone who followed the Green-Schwarz machinery from the early 1980s (these physicists preferred to calculate things in the light-cone gauge at that time) had to immediately see that the more natural and more right way to interpret the BFSS model was the light-cone gauge and not just some half-baked "infinite momentum frame".

But let me avoid these discussions. I will assume that the reader has no problem with null combinations of spacelike and timelike components of the energy-momentum vector and realizes that they are often natural combinations to consider.

The Hamiltonian above also contains fermionic, Yukawa-like terms of the form ${\rm Tr}(\theta\gamma_i [X_i,\theta])$ needed for supersymmetry (and various related crucial cancellations) and all the fields are $N\times N$ matrices chosen for the matrix model to respect the $U(N)$ gauge symmetry; yes, all physically allowed states must be invariant under the whole $U(N)$ group.

In the previous articles, I tried to explain why this quantum mechanical model whose fields are "large matrices", generalizations of the usual non-relativistic operators $X_i,P_i$, contains multi-graviton states, their superpartners, and large membranes: it has all the objects it needs to agree with the physical spectrum of M-theory in 11 dimensions.

Now, we want to compactify M-theory on a circle. M-theory on $S^1\times \RR^{10}$ has been known to be equivalent to type IIA string theory in 10 dimensions (from the very first paper by Witten that introduced M-theory: the equivalence of the low-energy limits had been known for 10 years before that Witten's paper). What do we have to do with the matrix model to see all the physics of type IIA string theory?

There was some confusion about this question in the original BFSS paper on Matrix theory. The authors tended to believe that their exact Hamiltonian contains "the whole Hilbert space" of string/M-theory in all of its backgrounds. However, it wasn't the case. The moduli are modes with $P^-=0$ and they correspond to excitations of the $U(0)$ matrix model. The BFSS matrix model has no degrees of freedom for $N=0$ so there are no ways to change the moduli. Consequently, the model may only describe one particular superselection sectors – the states of string/M-theory that respect the asymptotic form of the spacetime that looks like one in 11-dimensional M-theory (with one light-like direction compactified on a "long" circle).

To see type IIA string theory, i.e. the states in a different superselection sector of string/M-theory, we need to construct a different matrix model. What is it?

At the end of 1997, Ashoke Sen and especially Nathan Seiberg proposed a straightforward way to derive the BFSS matrix model and its compactifications from a limiting procedure combined with some widely believed dualities in string/M-theory. It's a clever (and superior) derivation that allows us to derive matrix models that are gauge theories; as well as matrix models that aren't just "ordinary" gauge theories but their novel UV completions such as the $(2,0)$ theory in $d=6$ and little string theory.

However, if we want to find a matrix model for a compactification of M-theory on $T^k$ and the dimension $k$ of the torus isn't greater than three, it's enough to use the formal "gauge theory assuming" derivation I used at the beginning of 1997. How does it work?

One develops (your humble correspondent developed) a more general procedure to "orbifold a matrix model". The compactification on a circle is an orbifold by the group isomorphic to $\ZZ$ composed of translations by $2\pi R n$ in the direction of the circular dimension. To find the matrix description of the orbifold, we need to enhance $N$ sufficiently and constrain the matrices of this "enhanced BFSS model" in a way that says that "the matrices transformed by elements of the orbifold group are gauge conjugations of the original ones".

This may sound complicated but the example of the compactification, an important one, makes it rather clear what I mean. The BFSS model has matrices with elements such as $X^i_{mn}$ where $m,n=1,2,\dots N$ are the gauge indices. We need the set of values of these indices to be infinitely greater. So we replace these matrix degrees of freedom by $X^i_{mn}(\sigma,\sigma')$ where $\sigma\in(0,2\pi)$ with periodic boundary conditions (a circular set of possible values of this "index") is a continuous counterpart of the index $m$ and similarly for $\sigma'$ and $n$.

Now the group $\ZZ$ of the translations in the direction $X^9$ has a generator, a translation by $2\pi R_{9}$, and we identify it with the conjugation by $\exp(i\sigma)$, a gauge transformation matrix that only acts on the continuous $\sigma$ indices. Because the translation doesn't physically act on the bosons $X^1\dots X^8$ and their momenta $\Pi^i$, the condition "physical transformation equals gauge transformation" says that these matrices are simply functions of one $\sigma$ because they impose $\sigma=\sigma'$, or demand $\delta(\sigma-\sigma')$ in the kernel, along the way. Similarly, $X^9$ has an extra $\delta'(\sigma-\sigma')$ term on the right hand side so this matrix gets promoted to the covariant derivative $D_\sigma$. Again, what used to be the degrees of freedom in $X^9(\sigma)$ get reinterpreted as the component $A_\sigma$ of a gauge field.

It may sound incomprehensible or difficult or abstract but I don't find it constructive to spend too much time with that. When you do these operations properly, you will find out that the matrix model for type IIA string theory is a 1+1-dimensional gauge theory with the same group $U(N)$ as the BFSS model compactified on $S^1\times\RR$ where the $S^1$ part of the infinite cylinder arises from the $\sigma$ "continuous index" we had to add. This 1+1-dimensional gauge theory has a dimensionful parameter $g_{YM}^2$. The formal procedure "physical transformation defining the orbifold equals gauge transformation of the matrices" even tells us how the coupling $g_{YM}^2$ depends on the length of the circle $2\pi R_9$ in the compactification of M-theory. Together with some analyses of the interactions in the resulting matrix model, we may derive that $R_9/l_{Pl,11}\sim g_s^{3/2}$.

But let's not be too acausal. So far, we have derived the matrix model for type IIA string theory. It looks like the integral of the BFSS Hamiltonian over the circle $\sigma$ except that the component $X^9$ of the bosonic fields is replaced by the covariant derivative $D_9$ involving the 1+1-dimensional gauge field. The original BFSS matrix model may be viewed as the compactification of the 10-dimensional (non-renormalizable) supersymmetric gauge theory to 0+1 dimensions. When we're compactifying the dimensions of the M-theory we want to describe by a matrix model, we must decompactify the spatial dimensions that were dimensionally reduced in the BFSS matrix model to start with. For type IIA string theory in ten dimensions, we must decompactify one (add the single "continuous index" $\sigma$). This operation is the opposite of dimensional reduction and because in chemistry, the opposite of reduction is oxidation, this procedure to construct higher-dimensional versions of the BFSS model to describe lower-dimensional vacua of M-theory is sometimes jokingly called the dimensional oxidation. ;-)

Minimizing the energy

Just to be sure: we have "derived" that type IIA string theory in ten dimensions at any coupling is completely equivalent to the maximally supersymmetric $U(N)$ gauge theory in 1+1 dimensions whose "world volume" has one infinite timelike dimension and one circular, compact spacelike dimension. To get rid of the effects of the compactification of the light-like dimension, we need to take the large $N$ limit.

In some sense, this is a very modest generalization or variation of the original BFSS claim. I became totally certain that this matrix model is the right one. This certainty is probably necessary for one to be sufficiently motivated to study its physics a bit more closely. So I started with that.

If the 1+1-dimensional gauge theory is the full type IIA string theory, including its D-branes, type IIA supergravity at low energies, black holes, and many other things, it should contain what type IIA string theory is known to contain. For example, it must contain the strings. They must also be able to split and join.

Diagonal in a basis that may change

A general Hamiltonian defines the energy in a quantum mechanical model. All states may be written as superpositions of energy eigenstates. However, some states are more interesting than others: the low-energy eigenstates of the Hamiltonian. Because energy tends to dissipates, physical systems generally like to "drop" to their low-lying states. That's why the low-lying states, starting from the ground state (lowest-eigenvalue eigenstate of the Hamiltonian), are the most important ones.

In other words, the first step in trying to understand the physics of a Hamiltonian in a quantum mechanical theory is to try to help Nature to minimize the energy. How do we do it with the matrix model for matrix string theory?

Let's consider the bosons only; the fermions add additional degrees of freedom, terms in the zero-point energy (that mostly cancel some bosonic terms that would destroy a consistent spacetime interpretation of the physics if they remained uncancelled), and other details. If you assume that fermions play this peaceful, calming, generalizing role, you may say that the important physics is already contained in the bosons.

How do we minimize the energy carried by the bosonic parts of the Hamiltonian? The matrix string Hamiltonian contains $\int \dd \sigma\,{\rm Tr}(\Pi_i^2)$ times a coefficient. Clearly, this is minimized if the momenta $\Pi_i(\sigma)$ are zero. More realistically, these matrices may be approximately diagonal and the diagonal entries $\Pi^i_{nn}(\sigma)$ will behave as the degrees of freedom $\pi_i(\sigma)$ defined on a Green-Schwarz string. Soon we will see what happens with the extra $n$ etc.

The off-diagonal entries of $\Pi^i$ as well as the same entries of $X^i$ behave like W-bosons of a sort, massive degrees of freedom, and at low energies, the wave function is almost required to be proportional to the ground states wave function as a function of these off-diagonal entries.

More interestingly, we want to minimize the term ${\rm Tr}\zav{-[X_i,X_j]^2}$ in the energy, too. The minus sign has to be there because for each $i,j$, the commutator is anti-Hermitian so its square is negatively definite, not positively definite. How do we minimize it? Clearly, it will be smaller if the eight matrices $X^i$ commute with each other. (Quantum mechanically, the wave function will be concentrated near the points on the configuration space where they commute with each other.)

If they commute with each other, it means that we can simultaneously diagonalize them. In other words, we can write\[

X^i(\sigma) = U(\sigma) X^i_{\rm diag}(\sigma) U^{-1}(\sigma).

\] The matrix $U$ may be assumed to be unitary because Hermitian matrices are diagonalized in an orthonormal basis. The matrix with the "diag" subscript on the right hand side is diagonal. But an important detail is that $U(\sigma)$ must be allowed to be arbitrary because the energy minimization tells us nothing about the basis in which all the $X^i$ matrices are diagonal.

And that makes a difference because $U(\sigma)$ doesn't have to be periodic with the period of $2\pi$. Only the total field $X^i(\sigma)$ of the gauge theory has to be periodic. However, the transformation $U(\sigma)$ to the basis in which $X^i(\sigma)$ is diagonal may undergo a nontrivial monodromy if we change $\sigma$ by $2\pi$. The matrix $X^i_{\rm diag}(0)$, for example, was constrained by our rules to be diagonal but the matrix $U(0)$ that (via conjugation) brings a given $X^i(\sigma)$ to the diagonal form is "almost unique" but not quite. First, one may add some $N$ phases on the diagonal of $U$.

Second, and this is more important here, the matrix $U$ may be multiplied by a permutation matrix! If a matrix is diagonal in a certain basis, it is diagonal in a permutation of this basis, too! So we must consider more general matrices $U(\sigma)$ that are continuous functions of $\sigma$ but that obey\[

U(\sigma+2\pi) = U(\sigma) P

\] where $P$ is a permutation matrix. In combination with some continuous but also aperiodic diagonal matrices $X^i_{\rm diag}$, such a unitary matrix may still produce an energy-minimizing, periodic field $X^{i}(\sigma)$. This is the key subtlety not to be overlooked if you want to understand physics of matrix string theory.

What is this fact good for?

It's easy to see how the $U(N)$ matrix model, the two-dimensional gauge theory, contains $N$ "short strings". The degrees of freedom of each such short string is carried by the diagonal entries of $X^i(\sigma)$. There are $N$ such entries along the diagonal. However, we also need "long strings"; the length of the $\sigma$ coordinate space has been known to be proportional to the light-cone momentum $P^+$ to everyone who was familiar with the light-cone gauge string theory.

This $P^+$ is quantized, equal to $N/R$, because the null coordinate $X^-$ is compactified on a circle of radius $R$ (we want to send $R\to\infty$ to get rid of this semi-unphysical compactification which also forces us to send $N\to\infty$ to keep $P^+$ fixed). And we know how to find strings with $P^+=1/R$ i.e. with the $N=1$ unit of the light-like longitudinal momentum.

However, the permutation business tells us how to find the "long strings" with $P^+=N/R$ for any positive integer $N$. You pick an eigenvalue of $X^i$ along the diagonal; trace it as you continuously change $\sigma$ from $0$ to $2\pi$; and when you reach $\sigma=2\pi$, this eigenvalue doesn't connect to the original one at $\sigma=0$. Instead, it will connect to a different one and only if you increase $\sigma$ by $2\pi N$, you may return to the original function because $N$ basis vectors participate in a cycle of the permutation (used in the boundary conditions for $U(\sigma)$.

(The "long strings" were also called "screwing strings" by your humble correspondent because the monodromy bringing the eigenvalue to a new level every time you get around the circle looks like a screw. I didn't know what the verb "screw" had meant informally. But this informal meaning of "screwing" is one of the reasons why the incorrect name "matrix string theory" became more frequently used than the technically correct name "screwing string theory". Incidentally, note that "matrices" and "nuts [waiting for screws]" are translated by the same Czech word, "matice".)

Because every permutation may be decomposed into a product of circular cycles, we see that every low-energy state in matrix string theory is composed of several strings with arbitrary values of $P^+=N/R$. The permutation defines a "sector" of matrix string theory. The decomposition into the sector is just an artifact of the low-energy approximation; there is no sharp "barrier" between the sectors as they're continuously connected on the configuration space of the 1+1-dimensional gauge theory.

One may also derive the origin of some other subtle conditions. For example, the bosonic/fermionic states of the long strings obey the right statistics because the permutations that interchange the whole long strings are elements of the $U(N)$ gauge group that must keep all physical states invariant. However, one may also derive the $L_0=\tilde L_0$ condition for each separate string as the gauge invariance under the generator of the $ZZ_k$ cyclic group that defines the cyclical permutations associated with a given string. Well, this is really equivalent to $L_0-\tilde L_0 \in k\ZZ$ but for large values $k$, all values except for $L_0-\tilde L_0=0$ will correspond to string states of a high energy and will not belong to the low-energy spectrum.

Merging and splitting strings: jumping in between the permutation sectors

I have already said that in the low-energy limit, it looks like the Hilbert space is composed of sectors labeled by permutations in $S_N\subset U(N)$. Each cycle that such a permutation is composed of corresponds to one "long string" – an ordinary type IIA string – present in the configuration.

At the same time, matrix string theory allows you to continuously switch between different "sectors". This corresponds to changing the permutation or, equivalently, the decomposition of the total longitudinal momentum $P^+$ to the individual strings.

The most elementary operation changing a permutation is the composition of this permutation with an extra transposition (of two pieces of the string; or two eigenvalues). The low-energy approximation of the gauge theory's (matrix model's) Hamiltonian will involve the list of the allowed sectors and the free Hamiltonian for the individual strings that match the free type IIA string theory. However, the gauge theory isn't quite free so there will also be corrections and those may change the sector (the permutation). Those that only add one transposition will be the leading ones and they will correspond to nothing else than the usual splitting or merging of strings, a three-closed-string vertex.

We know that the gauge theory is supersymmetric so the interactions will have to preserve the same supersymmetry. DVV showed that the form of the splitting/merging leading interaction is essentially unique. But even without knowing its form, I could have derived – using a trick using the assumption that the large $N$ limit is universal and independent of $R$, the light-like radius – how the coefficient of the three-string vertex depends on the radius $R_9$ of the coordinate we compactified to get the matrix model of type IIA string theory out of the BFSS model for M-theory. (There are two radii compactified here which are often labeled as $R_9$ and $R_{11}$. People who don't understand the logic of matrix string theory may confuse them. The exchange of these two radii that is effectively used in the construction was also called the 9/11 flip and be sure that it was before my PhD defense on 9/11/2001.)

The DVV description of the permutations

In March 1997, DVV who were much more familiar with the standard machinery of two-dimensional conformal field theories described the free-string limit of the gauge theory by a concise term: the symmetric orbifold CFT. It means a CFT – a linear (not non-linear, in this case) sigma model on $\RR^{8N}/S_N$ where $S_N$ is the permutation group exchanging the $N$ copies of the 8-dimensional transverse space.

They also wrote down the explicit form of the three-string interaction vertex (leading interaction) emerging in this limit in terms of spin fields and twist fields, fixed a mistake in my not quite correct derivation of the level-matching $L_0=\tilde L_0$ condition, and added some comments about the appearance of the D0-branes (short strings with the electric field etc.).

Higher-order terms in the Hamiltonian

The transposition of two eigenvalues is just the simplest among the extra permutations that may change the sector. In reality, the matrix model for string theory predicts all the complicated permutations (cycles with 3 elements or any number of elements), too. One may guess a natural Ansatz how these terms look like at any order in $g_s$. We wrote these formulae with Dijkgraaf – a paper showing that the matrix string Hamiltonian is corrected at every order and how (these extra high-order terms produce contact terms interactions that are needed for the consistency of the light-cone gauge string theory but they may be largely circumvented in the usual covariant calculations based on moduli spaces of Riemann surfaces). This particular paper remained almost unknown, one of the numerous testimonies of the fact that in the 21st century, the interest in technical things such as "filling the gaps in the only non-perturbative definition of type IIA string theory we have" was dropping to zero. In 2003, people were already much more excited with philosophical gibberish such as the anthropic lack of principle and fabricated "technical evidence" that it applies in string theory.

I won't proof-read this text because I am afraid that its technical character will shrink its readership close to an infinitesimal number that can't justify the extra work needed for proofreading.

President is right to veto Martin Putna's professorship

2013-05-18T12:43:00.001+02:00

What is the most intensely discussed event in the Czech news these days?

Czech president Miloš Zeman decided to reject the recommendation of an academic council at the Charles University and not to name Dr Martin C. Putna as a full professor. The title "professor" is supposed to be somewhat more special in Czechia because the people with this proper title are named by the president of the country personally. In some sense, they're more analogous to the holders of the National Medal of Science. Like the amnesty, pardons, and members of the constitutional courts, the ability to influence the composition of the full professors is one of the traces of the power of the Czech president – a role that has become largely ceremonial over the decades.

Judging by the screaming in the media and comments and votes in various discussions, about 95% if not 99% of the people in the political parties, schools, and various intellectuals and pseudointellectuals criticize president Zeman for the decision. I can't even imagine how isolated I would feel if I belonged to that environment. In certain cases, one simply has to remain a dissident. When one dares to agree with such a decision by the president of the country, it's clearly one of these heresies.

It must be politically incorrect to point out that Mr Putna is a decadent moron and bigot who shouldn't be considered a good scholar – and who would clearly devaluate and humiliate the ring of the word "professor" if he were elected one. President Zeman must see it in a similar way and he wrote the justification of the refusal to the ministry of education. Many people are screaming that he must publish the justification except that 1) it's not the president's duty, 2) it would only lead to an escalation of the problems. How would it help if President Zeman pointed out that from a scholarly perspective, Mr Putna is just a pile of politically correct decadent crap? (Update, Sunday: Zeman suggested that the problem with Putna was his presence at Prague Pride, a gay parade.)

Let me begin in the early 1990s. I started to study at the Charles University (Faculty of Mathematics and Physics) and because I was always kind of interested in politics, I repeatedly ran in the elections to the student senate and was elected as one of the 8 rank-and-file members.

We would be solving lots of silly things – parties, management of noticeboards, our opinions about the projects to expand the faculty etc., and so on, and so on. One of the things I was proud about was that as a secretary of sports of a sort ;-), I made it possible for students to register for a "workout in the student hostels' gym" as a sport course that saved lots of people's time.

But I want to talk about a different vote that is relevant here. Someone proposed to create a new, 18th faculty of the Charles University. The original faculties would include the Faculty of Maths and Physics; Natural Sciences; Law; some medical and theological faculties, and so on, you get the point. But the proposal was to add a "faculty of humanities".

We were even honestly explained what sort of "researchers" would be there. Feminist studies, professional blacks, postmodern philosophers, homosexual actitivists, and other perverse "fields" that were destined to flourish in the coming decades. I think that Alan Sokal just managed to publish his hoax in the Social Text around the same time. Of course that the student senate (of our faculty of maths and physics) rejected the proposal to create a new faculty of the ultimate pseudointellectual trash that would pretend to be on par with maths and physics but these folks had had already so many supporters that the faculty was created, anyway. For a while, it was just the Institute of Liberal Education (since 1994) but became a full-fledged Faculty of Humanities in 2000.

Sow the wind, reap the whirlwind.

Martin Cyril Putna is already a representative of this new "faculty" that shouldn't have existed in the first place and he symbolizes pretty much everything that is wrong with these "humanities". I view him as the Czech counterpart of the feminist studies and professional whining blacks in the U.S., among others.

He is officially a historian of literature or a "comparative literature" expert. If you search Google Scholar for his name, you will find one extensive (800-page) 1998 monography about the Czech Catholic literature between 1848 and 1918 in the European context.

Several articles have closely related topics. The number of citations is 31 for the book and 7;4;4;3... for the several articles you will find. With two or three Polish and Russian exceptions, none of the papers referring to Putna was written by non-Czech authors. His work is not internationally competitive.

I have almost nothing against the history of the Czech Catholic literature except that it looks like an immensely narrow field and one shouldn't become a full professor if he only knows (or only knows how to copy and rearrange texts about) this narrow field (also, I believe that many amateurs who love to read books have the same if not more extensive knowledge of these special subtopics of the history and they don't demand to be even postdocs, surely not full professors). But this is not what I consider essential in this controversy. The essential thing is that he was recommended to become a professor for completely different reasons than several inconsequential articles about a topic no one really cares about.

He represents the ultimate political correctness of the most degenerated type.

In this video, Mr Putna tells us that he's been voting for the Green Party for 10 years or so because he views the party as the most profoundly spiritual one. Imagine that: these lunatics who are eating roots and earthworms, sleeping in the tree tops or in front of the power plants are "spiritual" from the viewpoint of a man who is supposed to become a distinguished professor.

You could expect Mr Putna to sort of support the Catholic Church. But he only supports a Catholic Church that would be led by homosexuals and all these things (that's why Zeman's refusal to promote him was praised by a Catholic official – a rare exception). So in the video above, we learn that the Green Party is spiritual because it supports homosexuals and other things! Holy cow. What is so spiritual about it? On the other hand, the churches are in no way spiritual because they are only interested about the church's assets and homophobia, we learn. A truly deep analysis of the church from someone who is supposed to become a distinguished professor focusing on the Catholic Church!

But that's not the main thing here. Putna was among the most intense critics of Mr Zeman during the presidential campaign. Zeman claims that he only learned about the fact yesterday, a long time after he wrote the refusal letter to the ministry of education, and if he had known about the criticism earlier, it could have increased (and not decreased) Putna's chances to be appointed. The most characteristic anti-Zeman tirade by Mr Putna is this would-be entertaining January 2013 video:

In a language that is supposed to resemble Russian but it's really the Russian language spoken by Czechs who don't really know Russian and who want to make fun of Russian, he pretends to be Mr Putin (their names are very similar, after all) who loved Mr Klaus and who will love Zeman, partly because Zeman also likes to drink alcohol. No other "ideas" may be found in the video.

(Well, I am not sure whether the hysterical support for Putna is mostly because he's anti-Zeman and anti-Klaus or because he is a gay activist but I am pretty sure that one of these two reasons is the key one. Let me clearly point out that one can't become a full professor just because he or she is gay: there are 250,000 gays in the Czech Republic which is still a rather high number.)

I just can't help myself. People who are satisfied with these extraordinarily shallow, oversimplified caricatures – Klaus is pro-Russian; Zeman is pro-Russian; Russia is always bad; someone's drinking habits are what politics is all about – are just indisputable imbeciles. I see no way how I could overlook this self-evident fact. It seems crazy to consider this moron a scholar, especially if he's supposed to be a scholar in the humanities, and the existence of other "scholars" who are ready to recommend this idiot for the highest academic titles must be viewed as a serious problem, not an example of the "academic freedom" which is how the would-be right of similar idiots to call themselves professors has been framed by many people in the media.

One could say that the video above is just a fun video, like my interpretation of the Tom and Jerry theme song. However, there's a difference: Tom and Jerry has manifestly nothing to do with theoretical physics, the reason why I had worked in the scholarly environment. On the other hand, Putna is a social scientist and videos such as the video above do reflect his opinions about his own field or closely related fields. This is really what his professorship would symbolize. (Also, the Russian in my YouTube videos e.g. in Jožin z Bažin in Russian is almost completely correct and the Russian viewers appreciate it for that reason, too. Putna is satisfied with the cheapest conceivable anti-Russian hatred and stereotypes.)

Let me avoid repetitiveness in the expression of my feelings and summarize. Most of the Czech nation – and not only Czech nation – is composed of gullible manipulated stupid sheep who have no idea about this would-be professor, his work, the functioning of the university system, and other things but they don't hesitate for a second if they can support a pseudointellectual wee-wee in his disagreement with the president of the Czech Republic who is actually a much more genuine scholar and intellectual than Mr Putna or anyone at the Faculty of Humanities, for that matter.

Ex-president Klaus hasn't used the right to veto the professors (he just said that in Putna's case, he would sigh but he has always approved the proposals) but I think that the president has always had this right. For Zeman, this part of the presidential rights seems more important than e.g. the amnesties and pardons. This is what the law says. So if people disagree about something, there is still something we should respect: the law. The situation seems completely clear to me. The president's right to veto such proposals isn't just legally sound: It's very healthy that an external authority that stands "above all the fields" is allowed to intervene into the decisions that would otherwise be made purely by cliques of friends who are often as intellectually inferior as the people at the Faculty of Humanities of the Charles University. I think that Larry Summers had played an analogous role at Harvard and everyone should have been grateful to him for that work.

A Prague Pride scene that, according to Zeman's weekend hints, may have decided about the veto of the professorship. The banner held by Mr Putna says "Catholic fagg*ts [uncensored on the picture] are greeting Mr Bátora [a conservative]". Putna's participation at this event wouldn't be my primary reason to reject his professorship but it's a negative issue, anyway, especially because these opinions about homosexuality aren't just his work-unrelated hobby. The promotion of the Catholic Church is a gay-promoting organization is what much of his work – or what they consider his present work – is all about. And sorry, this activism just isn't scholarly work compatible with the title of a distinguished professor.

William Happer on CNBC

2013-05-17T18:53:00.001+02:00

Things have improved a little bit in the attitude of the media to the climate debate.

Click here if you don't see a proper video above.

This is what Princeton physicist Prof Will Happer was allowed to point out on TV – and it wasn't even Fox News! ;-)

400 ppm of CO2 is nothing special, 1,000 ppm of CO2 would be beneficial for the productivity of agriculture – which has already gone up a little bit, thanks to the slightly increased CO2 levels. The Earth has seen concentrations as high as 4,000 ppm or higher. It wasn't during the era of humans – but the era of humans is a tiny fraction of the Earth's history and when CO2 was around 4,000 ppm, our primate ancestors – whose physiology and climatic preferences don't really differ from ours – were already alive.

Moreover, he was allowed to clarify that if the questions were neutrally enough asked, about a half of the scientists would be on his side.

Burn, baby, burn.

Via Bishop Hill.

Less optimistically, the media are full of the claims by "citizen scientists" around John Cook who claim that 97% of the relevant papers support the "global warming consensus". It's bizarre. I participated in the survey and in my random sample, there were 50% of papers that supported the consensus only.

Moreover, Cook promised me to send me the results but I was never sent them. For too many reasons, I just don't believe that Cook et al. possess elementary human honesty.

String theory = Bayesian inference?

2013-05-17T08:20:00.000+02:00

The following paper by Jonathan Heckman of Harvard is either wrong, or trivial, or revolutionary:

Statistical Inference and String Theory

I don't understand it so far but Jonathan claims that one may derive the equations of general relativity – and, in fact, the equations of string theory – from something as general as Bayesian inference by a collective of agents.

It sounds really bizarre because the Bayesian inference seems to be a totally generic framework that may be applied anywhere and that says nothing else about "what the theories should look like" while general relativity and string theory are completely rigid, specific, well-defined theories. How could they be equivalent?

Jonathan considers a collective of agents who are ordered along a $d$-dimensional grid. Each of them tries to reconstruct the probabilistic distribution for events that they observe experimentally. Collectively, these distributions define an embedding of a manifold in another manifold and Jonathan rather quickly states that various conditional probabilities we know from the Bayesian inference may be written as the Feynman path integrals with the actions that include $\sqrt{\det G}$, $\sqrt{\det h}$, and similar things!

Again, I don't understand it so far but needless to say, a proof that string theory is the same thing as rational thinking – and not just a subset of rational thinking – would be extraordinarily important. ;-) I will keep on reading it.

Valtr Komárek: 1930-2013

2013-05-16T16:01:00.000+02:00

U.S.: As predicted and discussed on TRF exactly 3 months ago, Ernest Moniz became the new U.S. secretary of energy.

Valtr Komárek died today (Fox News). He was one of the key minds behind the Velvet Revolution, in some sense a senior collaborator of the current Czech president and the previous one, and a left-wing politician whom I respected – and be sure they make up a very exclusive set.

His unusual biography reflects the dramatic history of Czechoslovakia and the whole world in the 20th century.

He was born out of wedlock to a Jewish family in Hodonín, Moravia, in 1930. Both parents died in a concentration camp. Valtr himself was saved from the Holocaust thanks to his foster parents, the Komáreks. As a young kid, he was educated as a "hip" anti-Semite, much like most kids in the region, before he learned he was a Jew and the parents were not biological ones when he was nine. After the war, when he was still a teenager, he became a member of the Communist Party of Czechoslovakia, partly because he was grateful to his poor adoptive parents who brought him up and who were communists, too.

He completed a degree in economics in Moscow, USSR, and in the 1960s, he worked in the State Planning Commission. What's even more remarkable for foreign readers is that he worked as an adviser to Che Guevara, an educated Argentine left-wing mass killer, between 1964 and 1967. So you would think that Komárek symbolized the ultimate extreme left wing in economics.

However, in the following year, in 1968, he returned to Czechoslovakia and became a key economist behind the Prague Spring, trying to construct a more viable i.e. more capitalist form of socialism. He remained in the top economics after the Soviet occupation but in 1971, he was moved to the Federal Bureau for Prices which was a somewhat "lower" institute that didn't affect the national economy this much.

In 1978, he returned to the Economics Institute of the Academy of Sciences and since 1984, he was the director of the Forecasting Institute. In his institute, the bulk of the Czechoslovak (in some cases) Chicago-school-like reformers including Klaus, Dyba, Ježek, Dlouhý, as well as a top communist Ransdorf and (after the fall of communism) Zeman have worked at various moments.

He was a highly visible character during the Velvet Revolution; see his name in 5 previous TRF blog entries. A well-known slogan "During [the reign of] PM Komárek, a crown will be like a mark" shows both enthusiasm as well as a slight shortage of realism (the slogan rhymes in Czech; the actual exchange rate was always above CZK 10 per DM). While he was pointing out his differences from Václav Klaus, he would do so in a friendly way. In this sense, he was unusual among the members of the Social Democratic Party that kind of attracted him in the following years. He left politics in 1993 when Czechoslovakia disintegrated. Three years ago or so, Komárek received the highest awards from ex-president Klaus.

He died after complications from heart surgeries. His son Martin Komárek is a highly productive and well-known journalist.

Incidentally, the Czech Republic became the 26th DAC country, switching from a recipient of development aid within OECD to donors as the first post-socialist country. Less optimistically, the Czech economy saw a 1.9% year-on-year decrease of the GDP. We lost to Switzerland (surprisingly undefeated so far) in the ice-hockey championship quarter finals.

The world media are most interested in the video of president Zeman shot during the traditional (every 5 years) ceremony opening the exhibition of the Czech royal crown jewels (7 top people of the state and the church have to lend their keys to open the treasure). Zeman has explained his strange motion by a virus infection ("virosis"). Most Czechs and the world media believe that he had gone through several glasses or bottles of virosis before the ceremony. ;-) I am not sure who is right.

Novim Group: "Just Science" AGW app

2013-05-15T09:58:00.000+02:00

Paul O. helped me to possess an iPod Touch, because of my modest contributions to his Our Climate app. I have downloaded about 500 applications on the device and the new addition today is called Just Science. This free app occupies about 50 megabytes on your iDevice.

It was created by the Novim Group led by Michael Ditmore at UC Santa Barbara; the Berkeley Earth Surface Temperature team led by Richard Muller belongs to the group.

The application does one thing only – it shows you a map of the globe with animated colorful maps showing how the temperature was changing between 1800 or so and today in various regions around stations that reported and on a monthly basis.

So it's really a mundane visualization of a single well-known public dataset. If you listen to this interview with Michael Ditmore, you will learn about a $40,000 award for the "best science app" from an unknown foundation.

You will probably agree that $40,000 is a somewhat excessive amount of money for having written a code that would take one line in Mathematica – an animated colorful map applied on a public dataset. You can zoom in or zoom out the globe, too. But that's probably not enough to justify the large amount, either.

However, the rest of the interview will inform you that the people who collaborated on this trivial, effectively one-line code have divided about one million of dollars among themselves. This is really painful. I can't understand how you could call these people "not corrupt" if they get paid $1 million for $100 of work. It doesn't matter whether they pretend to defend one side or the other side or the middle side; they're still corrupt. To say the least, you effective promise to scream that this work of yours is important (worth $1 million) even though it obviously isn't ($100), something that Mr Ditmore is clearly doing, and this confusion already contaminates the science and the scientific debate by lies.

The Novim Group seems to be linked to groups that promote geoengineering.

Needless to say, the app doesn't change anything whatsoever about the climate debate. You may see the graphs you have seen many times in a somewhat more localized, colorful form. I guess that you have seen similar visualizations, anyway. None of these animations answers anything about the cause of the temperature changes – which were mostly going in the positive direction but the imbalance is really compatible with (approximately pink) noise.

Moreover, these graphs not only fail to show the cause. They also fail to show the significance or, more appropriately, insignificance of the temperature changes since 1800. Whatever the temperature difference between two moments or two places is, you may visualize it in such a way that one temperature will be crimson and the other one will be light green or dark blue or anything else – two or three completely different colors. But this obscures the overall scale and the fact that we are talking about tenths of a degree of temperature change in two centuries – a temperature change most of us can't even detect when it takes place instantly or within a second.

Now divide, dilute, and slow down this temperature change from 1 second to 200 years, add it superimpose it with the constantly changing weather (where the outside temperature routinely jumps by ten degrees within a day) and try to answer the question whether this temperature change and its rate – whatever their cause is – is something that a rational person should become scared of or obsessed about or something that a rational society should spend billions or trillions for.

Richard Dawid: String Theory and the Scientific Method

2013-05-15T06:58:00.001+02:00

Richard Dawid is a philosopher of science who was trained as a high-energy theoretical physicist and his new book that you may pre-order – it will be released at the end of June – isn't another addition to the rants by endless rows of populist crackpots, jerks, and imbeciles who try to criticize string theory without a glimpse of a rational justification (those extraordinarily stupid and dishonest books peaked about 7 years ago).

Instead, it is a philosopher's attempt to identify and localize, name, summarize, articulate, and present the reasons why string theory could have become the definition of status quo in the state-of-the-art theoretical physics despite the fact that the most natural conditions that string theory has something "new and direct" to say about seem to be inaccessible far from the currently doable experiments.

For this reason and others, the book was endorsed by big shots such as John Schwarz and David Gross.

The expensive yet short, 210-page-long book is divided to 7 chapters. The first one is an extended introduction to string theory (technical; sociology of non-experts talking about string theory; three contextual arguments in favor of ST); the second one is on the general conceptual framework of physical theories; next one on underdetermination applied to string theory (including some Bayesian reasoning); dynamics in high-energy physics; underdetermination in physics and beyond; whether or not one may claim that ST is a final theory; changes proposed for "scientific realism".

In the segments about sociology, the author describes both the near-certainty of the practitioners about string theory's validity; as well as the cynicism by many of the non-experts. The more stupid and ignorant you are, the more cynical about string theory – the unifying pillar of the 21st century physics – you may become. The cynics appear in various adjacent, next-to-adjacent, and unrelated disciplines – despite the fact that, as Dawid points out, string theory has helped to transform the way how people think and talk about pretty much all of theoretical physics and all of high-energy physics.

The three reasons behind the near-certainty about the theory's validity are:

the non-existence of alternatives
the surprising emergence of coherent explanations within string theory
extrapolation of the previous successes in high-energy physics: the Standard Model was also conceived because of largely theoretical reasons, had no alternatives, led to a nontrivial, surprisingly consistent unification of our descriptions of many things, and therefore had to be right

Concerning the first argument, it is the actual explanation why the top bright theoretical physicists focus this high percentage of their intellectual skills on string theory. They simply divide their mental powers to all promising ideas, with the weight given by the degree to which they are promising. Because one may approximately say that there aren't any other promising "big ideas" outside string theory, people can't work on them.

It is easy to misunderstand – and deliberately obfuscate – these facts. There exist "trademarks" that are marketed as competitors of string theory. But nothing really works over there. There exist no signs that these theories are on the right track. The people associated with these directions know that but some of them try to mislead the laymen about these facts. Some of them simply want to help themselves personally; others may be less egotist but they want a "greater diversity of ideas" than what the available evidence suggests as the right degree of diversity. And yet another group is just incompetent.

Concerning the second argument, it is a theoretical argument but a very powerful one. If string theory were a wrong theory of Nature, one would have no explanation why it has taught us about so many mechanisms that unify previously different concepts in physics and that retain their complete consistency, despite all kinds of a diseases that would have surely killed a generic wrong theory many times. The deep association between string theory and the laws that everything in the Universe obeys seems to be the only explanation of this coherence and unifying power, the ability to produce unexpected links, relationships, and transitions while avoiding any inconsistency.

Of course, one could argue that string theory is this coherent, powerful, and "willing to teach us" because of a different reason: it could be just a coherent mathematical structure that doesn't form the skeleton of the foundations of physics. If you wish, it could be the Devil who is constantly tempting us rather than God. But such an alternative theory would apparently predict that there will already be a demonstrable incompatibility between the highly constraining principles of string theory and some of the numerous (understatement!) insights we have already learned about the physical Universe. There aren't any inconsistencies of this kind, either: at least as the first sketch, string theory agrees with all the general features (types of fields and interactions etc.) we know from particle physics and cosmology. There's a lot of evidence that string theory is both very deep and very physical.

The last argument is probably a good way to describe the actual reason why I disagree with the suggestions elsewhere in the book that one needs to redefine the scientific method or do similar things. It seems obvious to me that the reasons that make string theorists near-certain that string theory is the right description of Nature have been used by physicists at least for 50 years and, in some respects, much longer than that.

Around 1974, string theory was identified as a candidate theory of quantum gravity – the only consistent one in $d=4$ or higher so far. This already implies that its characteristic effects in which it shows its muscles in their full glory can't be directly measured in the experiments (already Max Planck was able to calculate that the Planck length was $10^{-35}$ meters or so). I knew this was almost certainly the case when I was 10 years old or so. This inaccessibility by direct experiments is a defining feature of any theory of quantum gravity. Despite this knowledge, I wasn't repelled by string theory. If we can't "touch" something, it doesn't mean that we can't scientifically study it. Atoms became a part of science well before people "saw" them (because of the mixing ratios in chemistry and many other reasons). Physics of the 20th century brought us many more examples like that. Physics is really working like that most of the time today! When I was 10, I didn't know that almost 30 years later, a new kind of Inquisition would hysterically try to prevent people from applying the scientific method to energy scales that can't be directly tested.

String theory is really using the same kind of thinking about the possible deeper levels of explanation that were employed – and turned out to be successful – in the advances associated with quantum field theory. Any criticism of these argumentative patterns seems totally unjustifiable to me: it's really the only way how to think about these matters scientifically. The only plausible alternative is not to think about the unification in physics and the fundamental scale at all. I just think that the mankind would become a horde of uncultural barbarian apes if it decided it doesn't want to think about these issues – if it wanted to prevent a fraction of its intellectual resources from thinking about these fundamental issues.

Some people love revolutions and permanent revolutions. Am I among them? It depends on what you mean a revolution. I surely oppose any attempt to replace rational arguments in science by irrational ones (e.g. ad hominem ones or slogans that have nothing to do with the actual technical research); or to "ban" any kind of an argument that is obviously rational. Every solid enough argument and line of reasoning or inference, however indirect, should be used when we are forming our opinions about scientific questions. When this is done correctly in the case of fundamental physics, we reach a near-certainty that string theory is a valid (and probably the final) theory of Nature. It's possible despite the experimental inaccessibility of the Planck scale because direct experiments are very far from being the only tool how we are learning the truth about physics in the 20th and 21st century.

If you prefer the cheaper books by crackpots, you may buy books by Faggott (early August) and Unzicker (late July) instead.

IRS was used to intimidate political opposition in the U.S.

2013-05-12T08:18:00.002+02:00

During my decade in the U.S., my tax returns got audited at least twice – both of them had to be fixed when I was already back in Europe and Obama was in charge (2009, for 2007); one was federal and the other one was a Massachusetts tax audit under Deval Patrick (related to 2006, done in 2010). The number seems high to everyone and I view it as rather strong evidence that it's no coincidence.

A scandal in the U.S. strengthens the case:

IRS official knew in 2011 of 'Tea Party' targeting: watchdog report

In 2011, the tax-collecting organization was specifically harassing tax-exempt social welfare charities with keywords indicating that they were Tea Party-affiliated or conservative in general. Their applications were selectively delayed, they were ordered to publish the names of all the sponsors, and so on, and so on.

I am amazed that this is not a much bigger scandal in the U.S. because it puts America almost on par with various semi-totalitarian regimes such as Belarus etc. But maybe it has become a mainstream belief in the U.S. that conservatives may be harassed in this way? Note that 80 years ago, the German Nazis began to burn "un-German" books and most of the population in that cultural nation supported those policies, too.

Everyone who values democracy in the U.S. should make his or her contribution to the hunting of the left-wing criminals behind the monstrous scheme so that they may spend many, many years in the prison. I have no idea how high in the hierarchy one has to go to see the main people who were making something like that possible but these highest-positioned folks should be punished to increase the chances that similar things won't be repeated.

Before 1989, communists were doing similar things in our country and while I was often proud about the Velvet that associated our anticommunist revolution, I was also often sorry that we couldn't punish the people responsible for the wrongdoings somewhat more visibly than we did. A few thousand lives in prison could have been more appropriate.

When your humble correspondent endorses every word (not only by Glenn Beck but also) by Dennis Kucinich who talks about the IRS, something special is going on.

Some revelations look like something that would be called conspiracy theories a few years ago. For example, it turned out that Noam Chomsky himself and a few comrades of him were behind Stephen Hawking's boycott of of the Israeli Presidential Conference. Wow.

Feynman, Schwarzschild: anniversaries

2013-05-11T07:57:00.002+02:00

Richard Feynman would celebrate his 95th birthday today.

BBC2 on Sunday: The Fantastic Mr Feynman to be aired; Telegraph review. Those who pay TV fees in the UK are probably allowed to download the video via this torrent. Well, they can watch it via iPlayer, too.

One of the most colorful and ingenious physicists of the 20th century would deserve much more than a blog entry – so just like in the cases of other giants, I will abandon all attempts to write a would-be comprehensive biography.

Instead, you may watch this 37-minute NOVA interview (above) filmed in 1973. You're also invited to remind yourself about the story of Feynman and feminists (the latter were clearly immensely obnoxious already decades ago). Interestingly, you may look what I wrote exactly five years ago.

Five years is a rather widespread unit of time, sociologically, so all the anniversaries are more or less equally important and round as they were five years ago. In that article, you may also learn that Eugene Dynkin is 89 years old today. Congratulations! May 11th is clearly the birthday of the folks who give us diagrams. ;-)

Karl Schwarzschild died on this day in 1916. He decided to serve in the German army even though he was over 40; I guess that you wouldn't find too many Jews who would be this enthusiastic German patriots after 1945...

The picture below, pemphigus (the rare disease that killed Schwarzschild), may be frustrating so please don't look at it if you're too sensitive.

Schwarzschild is most famously associated with the first nontrivial exact solution to Einstein's equations of general relativity. But he would also study optics, photographic materials, celestial mechanics, quantum theory, stellar structure and statistics, Halley's comet, and spectroscopy. According to Wolfgang Pauli, Schwarzschild was the first man who wrote the correct form of the action for the electromagnetic field coupled to charges and currents.

Amazon.com Widgets

Alexandre-Edmond Becquerel died on May 11th, 1891.

Off-topic: Czechs are the #1 beer consumers in the world. Our superiority has many aspects; for example, women know how to drink beer through their ears.

Why we should work hard to raise the CO2 concentration

2013-05-10T19:27:00.000+02:00

Many texts about the climate and related issues are highly, boringly repetitive. I believe that a typical person who regularly follows the research and debate about similar issues has heard 99% of the things that are written about the climate change or carbon dioxide etc. Even the research that claims to be new is often just rehashing some memes that have been around – and we usually have very good reasons to suspect that the results of the research were decided before the research was performed.

But there are some good exceptions. Two days ago, ex-moonwalker Harrison Schmitt and physics professor Will Happer of Princeton wrote an opinion article for the Wall Street Journal from which I could have learned some new things:

Harrison H. Schmitt and William Happer: In Defense of Carbon Dioxide

The demonized chemical compound is a boon to plant life and has little correlation with global temperature.

The basic theme of the article is simple and most of us learned it as fifth-graders: CO₂ is primarily the plant food while its other implications for Nature are negligible in comparison. Humanitarian organizations should work hard to help the mankind to increase the CO₂ concentration and it's surprising that virtually all of them are failing to do so.

Needless to say, the article was greeted with a highly nervous reaction from the anti-scientific left-wing extremist sources. They must think it's a blasphemy to remind anyone that CO₂ is the key compound that plants need to grow – and, indirectly, that every organism needs to get the food at the end. See, for example, the hysterical reactions by Climate Science Watch, the lousy and badly biased astronomer Phil Plait, anti-Fox-News attack dog organization Media Matters for America, and many others. Many of them seem to literally say that it's been disproven that plants need CO₂ to grow; I kid you not. The insanity of certain people who put ideology (and not just any ideology: I mean a highly pathological ideology) on the first place has no limits.

Mr Plait and others, please, try to gradually get used to the fact that we are past the peak global warming hysteria. The elementary parts of common sense, e.g. the realization that carbon dioxide's impact on Nature as a plant food is many orders of magnitude more important than the role of carbon dioxide as a greenhouse gas, will be returning to all segments of the society and you will be increasingly recognized as kooks if you deny such facts about 101 botany, 101 economy, and 101 ecology, even among the people whom you expect to be "ideologically obedient". Your position is totally unsustainable in the long run and even the medium run.

The (for me and others) new insights that the authors presented were differences between C₃ plants and C₄ plants. The latter evolved to cope with lower CO₂ concentrations but they still have to pay some price.

In both cases, plants absorb CO₂ through stomata in their leaves and they need a very large amount of water to grow. A higher CO₂ concentration allows them to reduce the number of stomata and save water, if I simplify things a bit. So one may say that a higher CO₂ helps the plants to deal with the shortage of water.

Today, the average CO₂ concentration reached 400 ppm or 0.04% of the volume (or, equivalently, of the number of the molecules in the air because the air is a nearly ideal gas) "today". The mankind will reach a maximum that is much higher, perhaps 600-1,500 ppm between 2050 and 2300. I don't know any details. No one knows them.

But if those future generations stop pumping CO₂ into the air, its concentration will drop dramatically. These days, Nature absorbs about 2 ppm worth of CO₂ every year from the atmosphere; it's because the "excess CO₂" above the equilibrium value which is about 280 ppm for our temperature is about 120 ppm. If the excess is gonna be 600 ppm, like in 880 ppm, it's very plausible that Nature will be eager to absorb five times more, i.e. 10 ppm from the atmosphere every year. That could be described as a nearly 1% drop of CO₂ in the atmosphere per year which could reduce the efficiency of agriculture by 0.5% or so per year.

It's not too much but it's not negligible, either. If their technological tricks are already maximized, they could easily find out that the dropping CO₂ is an order from Mother Nature that the population should drop by 0.5% a year. The Earth's ability to feed the mankind may start to drop at that point. 0.5% isn't a cataclysmic population decrease and it may be respected without mass starvation, by a lower birth rate. But it would still be annoying.

Nowadays, we enjoy CO₂ concentrations growing by 2 ppm i.e. 0.5% a year and this increase contributes a non-negligible part to the increasing efficiency of the agriculture. I hope that when people are forced to get used to the dropping CO₂, they will either find a way to mitigate this unwelcome evolution – e.g. by burning lots of biomass or something else – or they will have some other tricks.

For example, in 100 years, most of the agriculture may take place in some huge "greenhouses" (which are more useful if the CO₂ concentration in them is kept at elevated levels). What some nations are doing about beating Nature's limitations on agriculture is impressive. Open the new Google Earth Engine with the Landsat Annual Timelapse 1984-2012. You may see the satellite pictures of any region on the globe to check how it was changing during the last three decades. You are offered some real local catastrophes, like drying of the Aral Sea (or a lake in Iran), some changes that are overinterpreted as tragedies although they're really not, like Amazon deforestation and the retreat of a glacier in Alaska, but also examples of some impressive human activities.

They include expanding coal mining in Wyoming – I don't like it too much; and Dubai coastal expansion which is impressive at any rate. But my winner is the Saudi Arabia irrigation. What these Saudis managed to do with the desert is amazing. Look at the nicely ordered crop circles! The diameter of these circles is almost 1 kilometer. Cool. With enough energy, similar things can be built in Sahara and elsewhere, too. I see no reason why the average population density sometime in the future shouldn't exceed the current population density in the Netherlands – 500 people per square kilometer – which would mean that the world population may rather easily surpass 50 billion at some point (many centuries in the future).

I don't claim that I like the idea – large fields and forests where no one annoys me for hours in the afternoon usually seem more pleasing to your humble correspondent than overcrowded, loud areas ;-) – but liking something is a totally different issue than expecting something to take place!

In the honor of the heterotic string

2013-05-10T09:42:00.000+02:00

Heterosis or the hybrid vigor or outbreeding enhancement is the lucky event (and an important component of Darwin's evolution) in which the offspring has qualitites that surpass both parents, usually because it inherits the good characteristics from both.

The parents are on both sides.

If you search for "heterosis" or "hybrid vigor" via Google Images, you get lots of pictures of corn, puppies, cows, fictitious animal species, and Barack Obama, among other things.

In 1985, four Princeton physicists ignited the second part of the first superstring revolution (that began in 1984) when they discovered the cleverly named heterotic string in their two papers. These men, Gross+Harvey+Martinec+Rohm, are sometimes referred to as the Princeton String Quartet. You won't find any concert of theirs on YouTube but there are lots of pieces by the Brentano String Quartet playing at Princeton.

The heterotic strings represent two maximally decompactified (trying to have as few compactified dimensions as possible, in some sense zero) limits of superstring/M-theory among the six. The six limits are:

M-theory in 11 dimensions (added as a full member in 1995); all the vacua below are string-theoretical vacua in 10 dimensions (and were added in the 1980s)
type IIA string
type IIB string
type I string
heterotic $E_8\times E_8$ string
heterotic $SO(32)$ string

I wrote the type I string below the type II string theories because it may be viewed as a more derived, somewhat more contrived, example of the type IIB string with a consistent extra collection of objects added (an orientifold O9-plane and D9-branes on top of it).

So the heterotic strings are important, perhaps covering 1/3 of the approaches to the conventional configuration space of string/M-theory. Moreover, some people including myself believe that the $E_8\times E_8$ heterotic strings remain the most convincing and well-motivated incarnation of the real world and all the qualitative features we know about it within string theory.

What does the heterotic string have to do with heterosis?

What they have in common is that they are hybrids of two very different parents. Their father is the bosonic string theory that requires $D=26$ spacetime dimensions; their mother is the $D=10$ superstring. I am not sure whether I attributed the sex to the parents correctly. On one hand, SUSY is a female name and the supersymmetric side is prettier while the bosonic side is more unconstrained, a little bit like males; on the other hand, it's the superstring that contains both bosons and fermions and if you interpret bosons and fermions as the X and Y chromosomes, the side that has both of them (XY) should be male! ;-)

More seriously, how can you hybridize these two very different theories that don't even agree about the spacetime dimension? Very well, thank you for asking.

The fields on the heterotic string

Calculations in perturbative string theory are naturally not performed in the spacetime. They may be directly performed on the world sheet – the 2-dimensional surface or history that the 1-dimensional strings paint in the spacetime as they evolve in the 1-dimensional time. All the calculable quantities may be expressed from correlation functions in the world sheet theory – which is naturally a two-dimensional conformal field theory (conformal means that by rescaling all distances by a factor, even a factor that may depend on the location on the world sheet, doesn't have any physical impact; only the angles matter).

What does the world sheet theory look like? It remembers how the world sheet is embedded in the spacetime. Start with the bosonic string theory which is easier.

The coordinates of the world sheet may be denoted $(\sigma,\tau)$; the metric on this locally Minkowski space has signature $({+}{-})$. We may also Wick rotate $\tau\to i\tilde \tau$ which makes the world sheet Euclidean. Even more than in the spacetime, this trick makes many calculations much more well-defined. Such a Euclidean world sheet is very naturally described in terms of a complex coordinate $z$ and its complex conjugate $\overline z$. In fact, very many things are either holomorphic (or antiholomorphic) or they almost hermetically (not heterotically) segregate the dependence on $z$ and $\overline z$.

Bosonic theory has fields $X^\mu(z,\overline z)$ where the index $\mu=0,1,\dots ,25$ labels the directions in the 26-dimensional spacetime. At the beginning, the world sheet has a 2-dimensional coordinate reparameterization symmetry so two spacetime coordinates may be pretty much set to some standardized functions of $z$ and $\overline z$; something like that is done in the light-cone gauge, for example.

Alternatively, we may keep all the $26$ fields $X^\mu$ but we must also add Faddeev-Popov $bc$ ghosts to deal with the diffeomorphism and Weyl symmetry on the world sheet – much like we deal with the analogous gauge symmetry in Yang-Mills theories. These $bc$ ghosts add $c=-26$ to the "central charge", a quantum one-loop violation of the scaling symmetry of the theory we demand, and that's why we need to add $26$ bosons with $c=+1$ each to cancel the central charge and keep the theory scale-invariant and conformal at the quantum level. (There are many other, seemingly very inequivalent ways to derive the critical dimension. One of them, based on the light-cone gauge, produces $D=2-2/(1+2+3+\dots)$) which also gives the right value if you recall that the only meaningful finite number that the sum of integers may be equal to is $-1/12$.

So the bosonic string theory observables are calculated from some theory in 2 dimensions which contains $26$ Klein-Gordon fields $X^\mu$ and some extra fermionic fields $b,c$ (which are Dirac-like but we assign them a different spin than $1/2$; the spin in two dimensions is a bit more flexible and convention-dependent than in higher dimensions because the minimal multiplets are one-dimensional for any spin). Pretty simple. However, to calculate the scattering amplitudes of string states, you have to learn about all (even "composite") local operators in this 2-dimensional quantum field theory and be able to make the shape of the world sheet arbitrary and integrate over all shapes.

The case of the $D=10$ superstring – which gives us type I, type IIA, as well as type IIB string theory (those only differ by allowed boundary conditions and relative chiralities etc.) – is analogous. However, the ghosts are not just the fermions $b,c$ for the diffeomorphism symmetry but also $\beta,\gamma$ for the local world sheet supersymmetry; and, which is related, there are not just bosonic fields $X^\mu$ but also their fermionic superpartners $\psi^\mu$. The central charge from the $bc$-system is still $c=-26$. However, the bosonic $\beta\gamma$-system adds $c=+11$; note that all these central charges have the form $\mp(1-3k^2)$ where $\mp$ is the upper or lower sign for the bosonic or fermionic ghosts, respectively, and $k=1-2J$ where $J$ is the weight (dimension; the generalized number of lower indices) of a ghost (or the antighost). In total, the ghosts have $c=-15$ which may be cancelled by $c=+10$ from the fields $X^\mu$ and $c=10/2$ from their fermionic partners $\psi^\mu$.

Segregating left-movers and right-movers

We could study the bosonic string theory as a single theory and we could also study the superstring. In the latter case, we would be allowed to make several choices for the signs of the needed GSO projections; and allow or forbid unorientable and open strings (closed strings must always be included in a string theory and by default, they're orientable). In this way, we would get type I, type IIA, and type IIB string theories which could be compactified to get realistic theories in less than $D=10$ – this collection of steps and choices already exhausts all the basic possibilities in string theory.

However, we may also – perhaps shockingly – do something seemingly perverse but ultimately equally consistent. To construct the heterotic string.

To fully appreciate that this is actually an extremely natural and allowed procedure on the two-dimensional world sheet, you have to see how separated the left-moving and right-moving excitations on the world sheet are. For example, the mass Klein-Gordon fields in two dimensions obey the massless Klein-Gordon equation,\[

0 = \square X^\mu = (\partial_\tau^2 - \partial_\sigma^2) X^\mu = (\partial_\tau+\partial_\sigma)(\partial_\tau-\partial_\sigma)X^\mu

\] The box operator may be factorized to a product of two operators $\partial_\pm$! Consequently, the solutions to this equation are the configurations annihilated either by $\partial_+$ or $\partial_-$. In other words, they're functions of $\tau-\sigma$ or $\tau+\sigma$, respectively (or some linear superpositions of both). These two terms contributing to the general solution are called the right-moving and left-moving modes, respectively. When we switch to the Euclidean world sheet, "right-moving" and "left-moving" are translated to "holomorphic" functions of $z$ and the "antiholomorphic" functions of $\overline z$.

The general solution for $X^\mu(z,\overline z)$ may be written as a sum of terms that only depend on the former; and terms that only depend on the latter. No mixed dependence. Similarly, fermions may be completely separated so that one component may be fully required to be holomorphic or right-moving; the other component of the 2-component spinor may have a Dirac equation that says that it is a left-moving mode i.e. one that only depends on $\tau+\sigma$ or $\overline z$. (Sorry if my convention differs from someone else's; you have to be careful when you fully verify or study someone's papers and books.)

You may find it hard to write the world sheet action $S$ for the left-movers only (or the right movers only) and it may be hard, indeed. But the action isn't the final product we're after. We need the correlation functions of the operators and they can be calculated in the segregated way.

The hybrid, heterotic theory effectively uses $26$ bosonic fields $X^\mu(z)$, along with $b,c(z)$, and the $10+10$ bosonic and fermionic fields $X^\mu(\overline z),\psi^\mu(\overline z)$, along with $b,c,\beta,\gamma(\overline z)$. These fields with pretty much the same dynamics that can be determined from the parent theories control all the calculable quantities of the heterotic string.

What about the mismatch?

The first observation is that there's really no problem whatsoever with the separation of fields $\psi^\mu,b,c,\beta,\gamma$ into the left-movers and right-movers. They came as loose packages of the left-moving and right-moving part even in the non-hybrid superstring (or the bosonic string theory, in the case of $b,c$) simply because their field equations are first-order equations. Such field equations effectively say that a component of the field is holomorphic; or another component is antiholomorphic. And we may separate the components.

As you may see, the only potential subtleties of the segregation arise in the case of $X^\mu(z,\overline z)$ whose field equations are second-order equations (Klein-Gordon). How does the separation work here? The hybrid, heterotic string seems to think that it's embedded in the $10$-dimensional spacetime according to the (superstring-like) right-moving excitations propagating on the string; but in the $26$-dimensional spacetime according to the (bosonic-string-theory-like) left-moving excitations. (The question which is which depends on a convention, one independent from most of the similar binary conventions. Flipping the conventions leads us to equivalent constructions.)

And yes, there are subtle constraints. There seem to be $16$ bosonic fields $X^\mu$ on the bosonic, left-moving side that are completely erased on the superstring, right-moving side. How do such $16$ spacetime coordinates that half-exist, half-not-exist behave?

An interesting feature of these coordinates is that you may still compute the total momentum \[

P^\mu = \int_0^\pi\dd\sigma\,\partial_\tau X^\mu

\] (note that the $\tau$-derivative is the velocity which is proportional to the momentum density with a fixed coefficient) and the total winding, \[

W^\mu = \Delta X^\mu = \int_0^\pi\dd\sigma\,\partial_\sigma X^\mu= X^\mu|^\pi_0.

\] Well, that's true even in theories with both-sided $X^\mu$ – or in the heterotic string theory for those shared $10$ coordinates $X^\mu$ that exist on both sides. However, a special feature of the heterotic string is that\[

(\partial_\tau+\partial_\sigma)X^\mu = 0

\] which is the condition that only the left-moving parts of the sixteen coordinates are allowed. When this equation is integrated from $0$ to $\pi$ over $\sigma$, i.e. over the closed string, we realize that – with some normalization factors you must be careful about but I will simplify them a bit\[

P^\mu = W^\mu.

\] The momentum of the heterotic string in the direction of each of the sixteen "asymmetric" coordinates must be equal to the winding number – how many times the string winds around the given direction. That may seem bizarre but it makes a perfect sense.

You can't really prevent the string from having at least some nonzero values of $P^\mu$; but in combination with $P^\mu=W^\mu$, that implies that the windings must be allowed to be nonzero, too. So in some sense, these sixteen "lopsided" coordinates parameterize a $16$-dimensional torus.

Even self-dual lattices

Are all tori allowed? The answer is a resounding No. In fact, out of the infinitely many choices, only two completely rigid solutions solve the constraints and produce a consistent string theory (a string theory vacuum, to use the modern terminology in which string theory is already recognized as a unified theory with many solutions).

The torus may be represented as $\RR^{16}/\Gamma^{(16)}$ where $\Gamma$ is a symbol for lattices which are something like discrete groups $\ZZ^{16}$ in this case. However, the sixteen independent generators of $\ZZ^{16}$ don't have to shift the sixteen "lopsided" directions by the same distance; and as 16-dimensional vectors defining the translations, these generators don't have to be orthogonal each other. A sixteen-dimensional lattice is defined as this kind of $\ZZ^{16}$ group that may be tilted or stretched or shrunk in various ways.

The quotient means that the coordinates in $\RR^{16}$ become effectively periodic in some sense – but it's still some general sixteen linear combination of these coordinates that are periodic. The division by the lattice means that we're only interested in the coordinates "modulo integers" so we're interested in their fractional parts, kind of.

Fine, what are the allowed shapes of the lattice $\Gamma^{(16)}$? The lucky guys may be derived from the modular invariance of the one-loop toroidal stringy diagrams but this is too technical. We have this cool $W^\mu=P^\mu$ condition which may do pretty much the same job.

If you study quantum mechanics of particles propagating on a circle of radius $R$ and circumference $2\pi R$, you will be able to derive that the momentum $P$ has to be quantized – a number of the form $N\hbar/R$. Let's set $\hbar=1$; I just wanted to remind everyone that all those things may be written in everyday life units. This quantization emerges because the wave function $\psi(x)$ has to be single-valued on the circle. For example, when you study the orbital angular momentum, it's effectively a particle on a circle of circumference $2\pi$ (the coordinate $\phi$) and the dual "momentum" $L_z$ has to be an integer because of the single-valuedness of the wave function.

Now, if the particle were replaced by a closed (circular) string, it could also wind around the circle. The total winding would be a multiple of the circumference $2\pi R$ i.e. $2\pi R w$. Note that the momentum has units of $1/R$ which is inverse to the unit of the winding, $2\pi R$. If you make the circle shorter, the spacing of the winding will shrink but the spacing of the momentum will increase by the same factor.

How is this rule generalized for a general lattice? It's generalized by the statement that the lattice in which the momentum lives is dual to the lattice in which the winding lives. For example, the dual lattice to $k\ZZ$ is $(1/k)\ZZ$: both are effectively additive groups of integers but the physical sizes of the generators of these groups are inverse to one another. What does it mean to have a dual lattice in the general case?

It's not hard. If you have a lattice $\Gamma$, the dual lattice $\Gamma^*$ is composed of all the vectors $W\in\RR^{16}$ in a "dual vector space" (the space of linear forms) that obey $W\cdot V\in \ZZ$ for each $V\in \Gamma$; I wrote $W\cdot V$ as an inner product, assuming the usual ${\rm diag}({+}{+}\cdots {+})$ signature but I could have been more abstract and write it as $W(V)$, the action of a linear form on a vector. The factors of $2\pi$ must be dealt with somewhere but they don't change the qualitative message of all these constructions.

Because the momentum and the winding must belong to lattices that are dual to each other but, at the same moment, the momentum must be equal to the winding, the lattice of the allowed momenta must be equal to the lattice of the allowed windings i.e. to the dual lattice to the lattice of the allowed momenta. If a lattice is equal to its dual, $\Gamma=\Gamma^*$, we say it is self-dual. And that's a hugely constraining condition, it turns out.

For example, the simple $\ZZ^{16}$ lattice with the unit and orthogonal generators is self-dual because $W\cdot V\in\ZZ$ for every two vectors $V,W$ with sixteen integer-valued coordinates. The self-duality would surely disappear if you tried to deform and stretch the lattice in a generic way.

However, we may actually derive one more (significantly weaker) "even" condition from string theory: $V^2$ must be not just integer but it must be even: $V^2\in 2\ZZ$. This condition, arising from the need for $L_0-\tilde L_0=\dots + V^2/2$ to remain integer-valued for the whole spectrum or, equivalently, from the $\tau\to\tau+1$ part of the modular invariance, bans the simple $\ZZ^{16}$ lattice. Are there any even self-dual lattices?

Yes, there are.

Use the symbols $e_i$ where $i=1,2,\dots,16$ for the usual orthonormal basis of $\RR^{16}$. And take the lattice to be composed of all the linear combinations of $e_i+e_j$ for $i\neq j$, all the $e_i-e_j$ for $i\neq j$, and of \[

W_{\rm halfy} = (\frac 12, \frac 12, \frac 12, \dots , \frac 12)

\] where the same coordinate is repeated sixteen times. It's not hard to see that the inner product of any pair of the basis vectors (over integers) is integer. And because the inner product of every vector in the set above with itself is even – most nontrivially, the squared length of the last vector is $16/2^2=4$ – the lattice of all the integer combinations of the vectors I just described will be even.

It is also self-dual. Try to find the most general vector $W$ whose inner product with all the vectors in the "integer-based basis" above is integer-valued. Because it must hold for every $e_i-e_j$, you may see that $W_i$ and $W_j$ must differ by an integer. Because it must hold for $e_i+e_j$ as well, $W_i$ and $-W_j$ must also differ by an integer. It follows that the coordinates $W_j$ and $-W_j$ differ by an integer i.e. $W_j$ itself is an integer multiple of $1/2$ – it is either integer or integer plus $1/2$. And I have already justified that if one coordinate is an integer, all the others have to be integers; if one of them differs from an integer by $1/2$, all of them have to.

In the first (integer) case, you may show that the sum of the sixteen coordinates $W_j$ is even because the inner product with the vector with sixteen $1/2$ coordinates still has to be integer. But if the sum of the coordinates is even, it follows that you may express the vector as a combination of the $e_i\pm e_j$ vectors. Similarly, this holds for $W-W_{\rm halfy}$ if all the coordinates of $W$ are half-integral because the inner product of $W_{\rm halfy}$ with itself is an even integer.

The $SO(32)$ heterotic string

You have to go through this proof yourself to really understand it – you have to rediscover it – but the lattice I defined as the set of integer combinations of all the vectors is even self-dual. It's what we need for the heterotic string. If you "half-compactify" the sixteen purely left-moving excessive bosonic coordinates on this lattice, you will get a string theory producing an exact $SO(32)$ gauge symmetry in the spacetime.

The isometry of the torus is just $U(1)^{16}$ which is the "Cartan subgroup" of $SO(32)$ and it will be marketed as a part of the gauge symmetry because of the standard Kaluza-Klein mechanism. But there will be new gluon-like massless gauge bosons with a nonzero winding – corresponding to $4\times 16\times 15/2 = 480$ points in the lattice that obey $V^2=2$. And they will extend the symmetry group to a nice $SO(32)$.

More precisely, the symmetry group is $Spin(32)/\ZZ_2$ because the presence of the vectors such as $W_{\rm halfy}$ in the lattice means that some states transforming as a spinor under $Spin(32)$ will appear in the heterotic spectrum, too. Because the signs in $W_{\rm halfy}$ are sort of prescribed (and an even number of them will flip if we add some $-e_i-e_j$), we will only obtain one Weyl (chiral) spinor, not the other. That's the origin of the $\ZZ_2$ in the quotient. All the spinor states will give you massive states because $W_{\rm halfy}^2=4$ which is greater than $2$, the level where new massless states may still occur. The "spintensor" states with more general half-integral coordinates will be even heavier.

The spinorial states of the $SO(32)$ heterotic string have a nice interpretation in terms of D-branes in the dual type I string theory. The duality will be mentioned later.

The $E_8\times E_8$ heterotic string

Are there some other even self-dual sixteen-dimensional lattices? There is exactly one. In our construction of the lattice above – which is the "weight lattice of $Spin(32)/\ZZ_2$" – we used the vector $W_{\rm halfy}$ with the coordinates $1/2$ whose squared length was equal to $4$. This is not a minimum squared length for an even lattice; the squared length equal to $2$ would be just fine, too.

So you may also construct a totally analogous lattice $E_8$ – the "root lattice of $E_8$" – to the sixteen-dimensional lattice but in eight, not sixteen dimensions. For the heterotic string, you need to do something with sixteen coordinates. But it's easy to divide them to two groups of eight coordinates and compactify each group on an $E_8$ lattice.

The proofs that the $E_8$ lattice is even and self-dual are completely analogous to the proof for the $Spin(32)/\ZZ_2$ root lattice. But there's one really cool surprise. Because $W_{\rm halfy}^2=2$ for the $E_8$ lattice, it's the minimum allowed positive result, we may actually get new massless states (e.g. vector bosons and gauginos) from strings whose momentum and winding have half-integer coordinates, i.e. from the spinorial weights.

So some of the gauge bosons may transform as spinors of $Spin(16)$. The gauge group in the "smaller" construction could be expected to be $Spin(16)$, just like it was something like $Spin(32)$ in the first heterotic string. However, the spinors are actually massless so they must correspond to generators of the gauge group, too. And if you combine the $120$ generators of $Spin(16)$ with the $2^{8}/2=128$ generators that transform as a Weyl (chiral) spinor under $Spin(16)$, you obtain the $248$ generators of the group $E_8$. We don't get just $Spin(16)\times Spin(16)$ here; the gauge group is larger, $E_8\times E_8$.

A cute detail – which is seen to be no coincidence if you study anomalies in the heterotic string's spacetime – is that both groups have the same dimension\[

\frac{32\times 31}{2} = 248+248 = 496.

\] In fact, some gravitational anomaly in $D=10$ which must cancel is proportional to $(n-496)$. It's the $E_8\times E_8$ string that is much more promising as a starting point to realistic phenomenology. One of the groups $E_8$ may be broken to a subgroup such as $E_6,SO(16),SU(5)$ which are viable grand unified groups and, when some extra six dimensions are compactified on a Calabi-Yau-like manifold, you get theories with realistic spectra and interactions (grand unificiation and SUSY is automatically built upon the Standard Model).

I should mention that within the purely Euclidean signature spaces, even self-dual lattices only exist in $8k$ dimensions. I have discussed the unique eight-dimensional even self-dual lattice, $E_8$, the lattice producing the equally named Lie group, and the two possible sixteen-dimensional even self-dual lattices. The next dimension where even self-dual lattices exist is $D=24$. Aside from $E_8\oplus E_8\oplus E_8$ and $E_8\oplus \Gamma(Spin(32)/\ZZ_2)$ and an analogous $Spin(48)/\ZZ_2$, one finds new examples, in particular the cool and less trivial Leech lattice which is crucial for the string-theoretical explanation of the monstrous moonshine (more).

Fermionization

A cool feature of the two-dimensional conformal field theories is that the same theory (physically) may often be expressed in many different ways. Instead of using sixteen left-moving bosons (on the bosonic side), we may "fermionize them". A free real boson in 2D CFTs is equivalent to two free real fermions whose operators may be expressed \[

\psi = \exp(+i\phi/2),\quad \bar\psi = \exp(-i\phi/2)

\] in terms of the boson $\phi$ or, equivalently, the boson may be written as $\partial_+\phi = \bar\psi\psi$. It may sound crazy: How could a tensor product of two fermionic Fock spaces look like a single bosonic Fock space? For the states of a single point-like particle species occupying one one-particle state, they're different. But if you include all the factors of the Fock spaces corresponding to all the harmonics along the string and you pick the right boundary conditions, it just works.

So a funny thing is that the same conclusion – there are exactly two possible heterotic string theories in ten dimensions – may be derived from the $32$ real fermions that may be used instead of the $16$ bosons above. You must only be careful about their allowed boundary/periodicity conditions around the closed heterotic string; and, which is related, about the GSO-like projections that tame the spectrum a little bit.

I won't discuss the details but the $Spin(32)/\ZZ_2$ heterotic string may be constructed out of $32$ real left-moving fermions $\lambda^a$ that are either simultaneously periodic; or simultaneously antiperiodic (there are two sectors). Because all the fermions are treated in the same way (their friendship and common fate isn't perturbed, not even by the boundary conditions), you get the $SO(32)$ symmetry: the symmetry currents are simply $\lambda^a \lambda^b$ which is $ab$-antisymmetric. The spinorial states arise from "spin fields" or from the "highly degenerate" sector (because it has zero modes) with periodic, and not the simpler antiperiodic, boundary conditions for $\lambda^a$. There's one GSO-like projection you have to impose.

Similarly, the $E_8\times E_8$ heterotic string may be obtained if you split the set of $32$ fermions $\lambda^a$ to two groups of sixteen fermions and allow the periodic or antiperiodic boundary conditions for each group separately (there are four sectors, AA, AP, PA, PP). The multiplicity of the sectors is inseparable from two independent GSO-like conditions. Again, you could think that this breaks the group to $Spin(16)\times Spin(16)$ but you will find the extra spinor states and the gauge group gets enhanced to $E_8\times E_8$. The resulting theory may be shown to be equivalent – even at the level of string interactions, not just degeneracies in the free spectrum – to the heterotic string theories we obtained via the bosonic construction.

T-duality as a unification of both heterotic string theories

If you pick a direction in the "large" ten-dimensional spacetime and compactify it on a circle as well, the two heterotic string theories become smoothly connected into "one heterotic theory". Why?

The bosonic construction makes it a bit easier to explain. In the construction of the heterotic string theories above, we discussed lattices in a $16+0$-dimensional Euclidean space. I added the zero to emphasize that the signature was purely positive, Euclidean, and there were no time-like dimensions.

If you compactify the "non-chiral" boson $X^{9}$ on a circle, it will have both left-moving and right-moving parts. In the physically most natural inner product, these two parts will behave as coordinates with the opposite signature. In effect, we added $1+1$ dimensions to the $16+0$-dimensional lattice. The result is $17+1$-dimensional.

A funny fact is that a $17+1$-dimensional even self-dual lattice that we may still require for a consistent compactified heterotic string theory of this sort exists and it is... unique. In fact, the only Minkowskian even self-dual lattices exist in $p+q$ dimensions where $p-q$ is a multiple of eight and they're unique whenever $pq\neq 0$. How can it be unique if we had two solutions to start with? Well, it's unique because\[

\Gamma(Spin(32)/\ZZ_2)\oplus \Gamma^{1,1} = \Gamma(E_8)\oplus \Gamma(E_8)\oplus \Gamma^{1,1}.

\] The lattices that you obtain from the two ten-dimensional heterotic string theories' lattices by adding a simple $\Gamma^{1,1}$ from the compactified $X^9$ are the same lattices, just rotated by a "Lorentz transformation" in $17+1$ dimensions.

All these things may be fully proven but the implications are sort of remarkable. You may start with the $Spin(32)/\ZZ_2$ heterotic string in ten dimensions. You compactify one more dimension so that only $D=9$ coordinates remain noncompact. You break the group to some $U(1)^{18}$ by changing the Wilson lines and other moduli and when you adjust these scalar fields to some right values, the gauge symmetry will suddenly start to get enhanced again. But you will get $E_8\times E_8$ instead of $Spin(32)/\ZZ_2$. An unexpected feature of the construction is that you can't view either of the two groups as a "broken phase" of the other. In fact, they're two equally large groups (when it comes to their dimension, $496$). They should be treated democratically; string theory allows you to break groups to smaller ones but also enhance groups to larger ones (at special points of the moduli space where some stringy states happen to go massless) and these two processes seem to be equally fundamental.

If string theorists weren't forced to see that such transitions exist, much like limited experimenters are hit by Mother Nature to their faces when She forces them to see something they should have seen for quite some time, they (or philosophers) would probably never "invent them" themselves. Nature and mathematics are smarter than us, even the smartest among us.

Heterotic-K3 duality

If you compactify the heterotic strings at least on a two-torus, you will get the vacua that, at strong coupling (but with the volume of the tori kept at a certain value in certain units), may be equivalently described by a totally non-heterotic string theory compactified on a seemingly highly nontrivial manifold, the four-dimensional K3 surface.

For example, a heterotic string on $T^3$ deals with $19+3$-dimensional lattices, and exactly this $22$-dimensional lattice, with the right signature (if extracted from the intersection numbers of pairs of two-cycles), may be identified in the cohomology of the K3 surface. The heterotic string arose from a particular "freedom to hybridize", a loophole in the regulations who can have offspring with whom. But this "freedom to hybridize" is actually the same loophole as the possibility to find one more hyper-Kähler, real-four-dimensional manifold aside from the torus, the seemingly nontrivial and curved K3 surface. They're really "the same thing" visualized with the help of different geometric pictures and different degrees of freedom. But in the heterotic and K3 case, we're just thinking about the physics in two different ways – it's a difference in our impressions or visualization or conventions for symbols, sort of – but the underlying mathematics and physics is completely isomorphic.

String theory is full of such unifications of things that naively look completely different.

Different fates of heterotic string theories at strong coupling

String theory and its vacua are living organisms that always prepare lots of surprises for us. Already in $D=10$, we may ask what the physical phenomena look like if we send the string coupling constant (or the string dilaton) to infinity?

Before the second superstring revolution in the mid 1990s, people would guess it's some uninteresting mess they don't want to talk about. For both heterotic theories, we get the same kind of mess.

However, it was shown that string theory – perhaps because it's such a perfectionist consistent theory – never leads you to a mess. Any simply describable legitimate limit of the moduli space must be a theory with so many special properties that it looks as natural as the starting point.

Even if I told you before the mid 1990s that the strong coupling limit of a string theory must be equivalent to some other string theory, you would probably make many wrong guesses about the theories you actually get from the two $D=10$ heterotic string theories above. The $Spin(32)/\ZZ_2$ and $E_8\times E_8$ heterotic string theories are qualitatively the same structures, up to a difference in "technical details", so they should give you similar strong coupling limits, up to a difference in some other "technical details".

But this argument or expectation would be wrong, too. The strong coupling limits of both theories look very different from one another.

If you try to turn the string coupling constant $g_s$ in the $SO(32)$ heterotic string theory to a value much larger than one, you must still get a ten-dimensional supersymmetric theory whose spacetime gauge group includes $SO(32)$. The ten large dimensions couldn't disappear. The supersymmetry couldn't disappear. The gauge group couldn't disappear. What the limit could be?

Well, there's one more supersymmetric string theory with the $SO(32)$ gauge group, namely type I superstring theory; the $SO(32)$ group arises from 32 possible half-colors of "quarks" at the end points of open strings or, equivalently, from the 16 spacetime-filling D9-branes and their mirror images (behind the orientifold mirror). This theory is not heterotic. It's purely "fermionic"; no hybrids. But its strings are unorientable and may be open or closed. (Heterotic strings must be orientable because the left-moving and right-moving parts of it have "inequivalent guts" so they can't be confused. For a similar reason, heterotic strings can't be open because an end point would have to "reflect" left-moving waves to some right-moving waves but the allowed waves in the two directions are inequivalent and can't be mapped to one another.)

Still, the theories are completely equivalent. Type I theory with the coupling $g_s$ is equivalent to the heterotic $SO(32)$ theory with the coupling $1/g_s$. Incidentally, the spinorial states of the $SO(32)$ heterotic strings appear as states of a particular non-supersymmetric (non-BPS) D0-brane in type I theory. The number of such D0-branes is conserved just modulo two, i.e. as an element of $\ZZ_2$. A single D0-brane of this kind is stable because it's the lightest object/state that gets mapped to minus itself under the 360° rotation in $SO(32)$ gauge group (the lightest spinor-like object in the theory).

The fate of the $E_8\times E_8$ heterotic string is completely different and it was only understood by Petr Hořava and Edward Witten at the end of 1995, months after lots of similar discoveries. Again, the supersymmetry can't disappear. The ten large dimensions can't disappear. The $E_8\times E_8$ gauge bosons can't disappear. So what the other description with these properties may be if I can assure you it's not the same description (the theory can't be S-self-dual)?

The answer is that while the ten dimensions can't disappear, a new dimension can appear. The strong coupling limit of the $E_8\times E_8$ heterotic string is an 11-dimensional theory, M-theory, whose new dimension has the shape of the line interval of length $L$. The value of $L$ is an increasing function of the string coupling $g_s$. But M-theory seems to have no non-Abelian gauge bosons.

The heterotic string itself becomes a cylindrical membrane, M2-brane of M-theory, stretched between the two ends of the world.

Well, it has gauge bosons if the 11-dimensional spacetime has boundaries. In fact, at the boundaries of such a spacetime, the gravitinos must be constrained to chiral fields. That creates 10D anomalies near the boundaries and they have to be cancelled. The gravitational anomalies may be cancelled by some other chiral fermions you may add, the gauginos, but that may create new gauge anomalies and mixed anomalies etc. At the end, it turns out that you may cancel all of these anomalies but only if $E_8$ gauge bosons (and their superpartners, gauginos) live on each boundary of the 11-dimensional spacetime! The fact that it works at all boils down to some nontrivial properties of $E_8$, some difficult and seemingly "very lucky" identities relating traces of products of generators of the group in the adjoint representation.

Because a line interval has two endpoints – a thick "desk" has two surfaces on both sides – you will get two $E_8$ factors of the gauge group for each point in the remaining 10-dimensional "large" spacetime (yes, the gauge bosons and gauginos are confined to the end-of-the-world boundaries, much like some gauge fields are confined to D-branes or singularities): the gauge group will be $E_8\times E_8$. The 11-dimensional M-theory with two boundaries – the heterotic M-theory – is also a good starting point to build realistic compactifications of string theory.

(Physicists describe the line interval as $S^1/\ZZ_2$, a quotient of circle by the group mirroring it from the left to the right. This quotient is a line interval because e.g. the left half of the circle is a "fundamental domain" while the right part of the circle is just a $\ZZ_2$ copy of it. And yes, one-half of a circle is the same thing topologically as a line interval. In my conventions, the end points of the line interval are the fixed point under $\ZZ_2$, i.e. the uppermost and lowermost points of the circle.)

You may see that the heterotic string shows the remarkable uniqueness and interconnectedness of string/M-theory. There are just two possible heterotic string theories in $D=10$ linked to two fully non-Abelian groups that may satisfy difficult anomaly cancellation conditions in the $D=10$ spacetime. These two precious solutions may be derived in several languages that use different mathematical toolkits – bosons and lattices; grouping of fermions etc. – and they're connected after a compactification of another dimension. Further compactifications may be equivalently described in terms of M-theory or type II strings on K3 surfaces and the strong coupling limits are also equivalent to type I theory or M-theory with boundaries.

String/M-theory always knows what it's doing and it won't leave you in trouble. The maximally supersymmetric vacua of string/M-theory are arguably "prettier" but they're too symmetric and too sterile. The heterotic strings have 1/2 of the maximum supersymmetry (inherited from the mother, the right-moving supersymmetric side of the hybrid) but that's more than enough for SUSY to play its maternal protective role (and to add fermions into the spectrum and eliminate tachyons from the spectrum, the crippling vices that the offspring didn't want to inherit from the bosonic dad). The realistic vacua require even less supersymmetry.

As the amount of supersymmetry decreases, we are losing the ability to compute everything interesting easily but we're gaining lots of structures and new twists. The heterotic string vacua have so much supersymmetry that they still allow us to compute the answers to almost all simple (BPS) questions essentially by classical calculations but the freedom is already high enough to allow us to connect both versions of the heterotic string theory with one another and with type I, type IIA/IIB and M-theory on K3, with M-theory spacetimes that possess boundaries, and with other vacua I haven't discussed: already at this level, we may see the (more than minimal) interconnectedness of the diverse web of solutions to string theory.

Nassim Haramein: science as religion

2013-05-09T07:10:00.001+02:00

It is the second time when I was contacted by someone who seems to be a fan of Nassim Haramein. Who is that? Another surfer dude in Hawaii, a self-taught supergenius, we are told, who will give us unlimited free energy according to the green optimists (no, there has never been anything remotely rational about the environmentalists), who has an impressive website called The Resonance Project, who will unify the mankind, and do tons of other wonderful things.

In fact, when you search for YouTube videos with him, you seem to get over 75,000 hits, videos that cover not only his unified theory, physics and spirituality, the pyramids and orion belt, but also everything else that some folks could find deep and important.

The people who believe that there's something – anything – in this stuff (it's a relatively small group, because of the small number of viewers per video, but they're real cultists, because of the number of videos) must feel happy all the time, perhaps probably because they're permanently high.

I just can't possibly get it. I can't understand how someone may overlook that this is a continuous stream of complete nonsense occasionally interrupted with isolated words taken from the physics jargon. Some aspects of it are unoriginal. This Gentleman will give us a perpetual-motion machine of the first kind, we're promised, and he's not the first one.

But for example, the trailer above – which is a trailer for a documentary about a paper called Quantum Gravity and the Holographic Mass that he managed to publish somewhere (in a junk journal whose name tries to sound like Physical Review; imagine: a documentary about a single paper) – combines mundane things about the spin (the spin is everywhere in Nature, indeed: but what exactly does he claim to have learned about it? Do his fans ever ask simple questions like that?) with some remarks about quantum mechanics, holography, anything.

And sometimes you're pushed towards a punch line after which you want to explode in laughter and some of you even may explode. For example, he says that all the miracles may be done with the help of the most fundamental particle in the world. There is a pause for you to think what is the most fundamental particle according to this chap: the Higgs boson, the graviton, the photon, the neutralino, or something completely new? After the pause, you're told the answer:

It's the proton!

Holy cow. The proton is the messiest and non-fundamental particle that has ever been called "elementary" by people in the 20th century. It's a bound state of 3 valence quarks glued together with so much glue and excess kinetic and potential energy that the 3 quarks only add about two percent to the rest mass of the proton. The rest is mess, gluons, quark-antiquark pairs, and so on.

Moreover, and this fact is of course related to the compositeness, the proton isn't unique at all. It has all the siblings in the multiplets – the neutron is the closest relative which belongs to the same isospin doublet (=couple) with the proton. One may extend the doublet to an $uds$ baryon octet (=eight elements). More generally, all strongly interacting particles – hadrons – are proton's relatives and there are hundreds of them.

The proton is the most stable hadron but this stability is just a result of many unimportant coincidences and accidental inequalities satisfied by the masses of the objects etc. The neutron decays to the proton and other things (electron plus antineutrino) simply because it just happens to be a bit heavier, enough to decay. If the neutron were a bit lighter, it could be the other way around: the proton could decay to a neutron, a positron, and a neutrino.

Haramein's idea that the proton is fundamental is reflected in various papers he wrote, e.g. one about the Schwarzschild proton that offers some preposterous claims about the proton's being a black hole or something like that. (Just to be sure, a proton has an extremely low density relatively to what is required for small black holes. The black hole of the proton mass would have to be more than 20 orders of magnitude smaller than the proton.)

People believing these things – perhaps including Haramein himself – must have no clue whatsoever about the difference between things and concepts that are fundamental according to the modern scientific arguments and those that are not (or those that are extremely far from it). I am inclined to think that this inability to distinguish is the root of all the spiritual misinterpretations of physics and any religious cult that claims to be compatible with modern physics.

For millennia, people would talk about anthropomorphic gods. They were fundamental in the scheme of things. Well, science was going in a very different direction. Fundamental things in the Universe can't resemble humans because humans are extremely far from being fundamental themselves. They're demonstrably composed of smaller co-operating parts – at many hierarchically arranged levels – that just teamed up to create many composite objects. Organisms are among them, all organisms are relatives of each other, and humans are probably more skillful than others due to many other accidents and random mutations and happy coincidences in their history. The difference isn't truly "fundamental" in the sense of "totally qualitative". A human may be skillful or smart but it's just nothing else than a slightly improved monkey.

The incorrect notion that humans are "fundamental" was the basis of religions for quite some time and people are already familiar with the fact that science surely claims otherwise. But the claim that the proton, for example, is fundamental is "original". People should have known that the proton isn't fundamental for almost 40 years as well. Similar findings are clearly not widespread yet – even (and perhaps especially?) among the people who claim that "physics" (they mean Haramein's physics in this case) has changed their lives. What these people see behind the word "physics" has simply nothing to do with the actual physics and its results obtained by the scientific method. They believe that the key content of physics is a secret occult art they don't have to fully understand – it's being discovered by shamans with special, almost supernatural skills such as Mr Haramein.

Amazon.com Widgets

You might say that this gap is just a property of a crazy cult. But it's not. To a certain extent, perhaps a smaller extent, all the people reading and believing the mass media belong to a similar cult. When you uncritically read the articles about physics in the media, especially the truly theoretical or fundamental physics or some socially sensitive fields such as the atmospheric physics, you will be filled with an amazing amount of totally nonsensical gibberish while the legitimate, interesting, and sometimes groundbreaking science will remain almost invisible.

The problem isn't just in the frequency. The nonsensical and downright wrong and childishly wrong statements are generally presented with a much higher level of enthusiasm which is why they probably have the capacity to send a higher number of new people on the wrong track. The ultimate underlying reason is that the average people – and journalists are average people – represent just a subset of the relatively stupid monkeys. There's no easy fix.

Bonus, via Preposterous Universe: Neil deGrasse "Mike" Tyson is defending manned spaceflights by yelling at Lawrence Krauss and attempting to break Brian Greene's mouth into thousands of tiny pieces. Rest of debate: this piece is at 32:50 here. (Warning: lots of cheap tendentious crap about climate change and women's being 50% of the science community is voiced there.)

One more comment. Stephen Hawking decided to boycott an Israeli presidential conference although his spokesman tried to claim that health concerns were the reason why he won't come. It's very unfortunate for Hawking to fight against this country which is one of the world's main science powerhouses – if not the strongest one (per capita, among comparably large areas or segments of the population).

Short questions often require long answers and proofs

2013-05-07T09:25:00.000+02:00

Several debaters as well as complexity theorist Boaz Barak religiously worship their belief that it must be that $P\neq NP$ and that the question whether the proposition holds is extremely deep because $P=NP$ would revolutionize the whole world.

Most of their would-be arguments are examples irrational hype, fabricated justifications of the limited progress in a field, and group think. I will primarily focus on a single major wrong thesis they promote, namely the idea that a mechanical or polynomially fast or efficient algorithm to solve a problem specified by a short prescription must be short, too.

So let me begin.

$P$ and $NP$

$NP$ is the class of problems whose solution, if you're told one, may be verified to be correct by $O(C\cdot N^\ell)$ operations or fewer (the number of operations is effectively the time!) for some $C,\ell$ if $N$, an integer specifying the number of objects in the problem or its size, is asymptotically large.

$P$ is the class of problems whose solution may be found, and not just verified, after $O(C\cdot N^\ell)$ operations. Verifying a solution can't ever be harder than finding one so we see that $P\subseteq NP$ but we don't know whether $P=NP$ or $P$ is a proper subset.

There are lots of examples of $NP$ problems that don't seem to be in $P$ – but, with some surprise, they could be. The provably "most difficult problems in $NP$", the so-called $NP$-complete problems, are those whose fast solution could be translated to a fast solution to any $NP$ problem. So $NP$-complete problems are on the opposite (hard) side of the $NP$ set than the $P$ problems if this "polarization" can be done at all.

Examples of $NP$-complete problems include the the clique problem i.e. the decision whether a combinatorial graph has a complete graph with one-half of the nodes that are completely connected with each other; the travelling salesman problem (find the shortest closed route that visits all nodes/cities in a graph), and so on.

I think that a physicist would find the computation of the permanent (the determinant-like sum of products over permutations in a matrix but without the minus sign) to be a more natural representative of the not-in-$P$, difficult, class because it's more "explicit" – it's given by a compact formula. However, it's not in $NP$ – there's no way to quickly verify the resulting value of the permanent, either: an example of the fact that $NP$ is a constraining, narrow-minded class. On the other hand, it's known how a fast calculation of the permanent may be used to solve $NP$-problems; it's known that the permanent is $NP$-hard (which is a larger class than $NP$).

Computer scientists' most favorite difficult $NP$-complete problem is probably 3SAT or SAT – to decide whether a proposition constructed from logical operations AND, OR, and binary variables $x_i$ (obeying a certain constrained format, if you add the digit "3" or another one at the beginning) is a tautology (a proposition equal to TRUE regardless of the values of variables $x_i$) but faster than going through all the $2^N$ possible choices of the truth values of the variables. This very preference favoring 3SAT already shows some "unnatural bias" – they're focusing on things that are easier to write on the existing computers (including the preference for binary computers instead of e.g. ternary ones that we could be using equally well) which is not really a good, invariant criterion for the mathematical depth.

A reason why all these – effectively equivalent (equally hard, convertible to each other with minimal losses) – problems probably don't have an exponentially fast resolution (why they're not in $P$) is that at least for some of them, you would expect that if a solution exists, it would already have been found. So you use the same "mathematical induction" – if New York skyscrapers haven't been demolished by an airplane in 100,000 previous days, chances are that it won't happen tomorrow, either; instead, there may be a law that this will never happen and this conjecture has passed some somewhat nontrivial test. Such an argument is risky but it has at least some logic. Or at least, people would say it had before 9/11.

However, many complexity theorists prefer completely different arguments – or modifications of the valid yet inconclusive argument above – which aren't rational.

The key problem why $P=NP$ wouldn't be far-reaching even if it were true is that the algorithm to quickly solve 3SAT could be "fast enough" so that it obeys the polynomial criterion but it could still be "slow enough" to make it completely impractical. The number of operations could go like\[

10^{100}\times N\quad {\rm or}\quad N^{100}

\] or some other function. The first Ansatz above is linear – very slowly growing with $N$ – but it has a large coefficient. The second Ansatz has a modest coefficient, namely one, but the exponent is large. Already for $N=2$ or $N=10$, respectively, the two expressions above give you at least a googol and no computer will ever do this many operations.

One could conjecture that mathematics isn't this malicious and both the exponent and the prefactor are likely to be of order one for all problems; a physicist would call this argument "an argument from naturalness". However, the experience with many highly natural problems in mathematics suggests that this application of "naturalness" fails much more often than many people might expect.

Let me emphasize that even if $P=NP$ and a fast algorithm to solve the 3SAT problem or even to compute the permanent only takes $N^4$ operations, a natural polynomial scaling, there is no contradiction. The discovery of this fast algorithm would make a significant number of calculations of the "clique" or "traveling salesman" type vastly more efficient but the rest of the world would stay the same. You couldn't automatize the search for proofs or deep discoveries in maths or physics, music, and so on because these activities haven't been reduced to 3SAT and not even to the harder permanent without "big losses".

But in the rest of the text, I want to provide you with some evidence and context showing that $P=NP$ is still very likely to produce impractically slow algorithms in practice.

Forums: short questions, long answers

If you were ever answering questions on forums such as Physics Stack Exchange, you must know that a very short question often requires an extremely long answer. Well, questions such as "explain string theory to me" may require hundreds or thousands of pages and people usually don't try. But even if the question is realistic, a long answer is simply needed.

But we may talk about a more rigorous incarnation of the same question: the length of proofs needed to prove a theorem. Half a year ago, John Baez wrote about insanely long proofs and about the way how some of them may dramatically shrink if we're allowed to assume the consistency of an axiomatic system etc.

However, I don't want to play these Gödelian games because in most cases, you end up talking about propositions you wouldn't be interested in to start with – propositions that are not always human-readable and that are only interesting because of the length of the required theorems. On the other hand, even if we talk about very practical, important math problems, we often end up with the need to write down very long proofs and very long books on the classification of things.

Before I will mention a few examples, it is important to mention that they're examples of proofs and classifications that we have already found. There may exist many interesting theorems with even longer proofs, perhaps much longer proofs, and we haven't found them yet – partly because it becomes harder and harder to search for longer and longer proofs. So it's totally plausible that extremely long proofs of interesting theorems are almost omnipresent in maths. In fact, it seems sensible to assume that the average length of the proof that the mathematicians and others are finding or mastering is an increasing function of time.

I am talking about proofs but the same comment may apply to speedy algorithms to solve certain problems. These algorithms may still be extremely complicated and the fact that we haven't found them yet doesn't really prove that they don't exist. This caution is promoted in the Wikipedia article on $P$ vs $NP$. While it offers us the would-be impressive slogan by Scott Aaronson,

If $P = NP$, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss...

(I've already explained that this fairy-tale suffers from many serious illnesses: one of them is that it completely overlooks the fact that the main ingenious thing about these famous men is that they can find – and not just follow – some de facto step-by-step procedures to achieve something) it also quotes two researchers as examples of those who believe that it's wrong to be biased and the experts in that field should comparably eagerly study the possibility that $P=NP$ and try to search for proofs of that:

The main argument in favor of $P \neq NP$ is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. [...] The resolution of Fermat's Last Theorem also shows that very simple questions may be settled only by very deep theories.

—Moshe Y. Vardi, Rice University
Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required.

—Anil Nerode, Cornell University

Very wise. Aaronson really represents the unlimited prejudices, retroactive rationalization and hyping of these prejudices, and even bullying those who are showing that the opposite possibility is also compatible with everything we know and who are working to add some progress in that direction. Aaronson's approach to $P=NP$ is a similar skewed approach to science as the climate alarmism or Lysenkoism – after all, Aaronson has openly come out of the closet as a climate alarmist himself, too.

I need to emphasize that in physics, we also claim that various marvelous hypothetical engines – like the perpetual-motion machines – don't exist. But in physics, we actually have a much stronger case for such no-go theorems. We can really prove the energy conservation from the time-translational symmetry of Nature, via Noether's theorem. We may prove the second law of thermodynamics in the form of the H-theorem and its variations. These results only depend on assumptions that have been heavily tested in millions of experiments, assumptions that we really consider the most rock-solid foundations of all the natural science. Aaronson's (and not only his, of course) prejudices are rooted in no comparably solid foundations.

Fine, now the examples of the long proofs.

Fermat's Last Theorem

I have discussed some history of the proof last month. The statement of the theorem is extremely easy to formulate. Almost all the well-known mathematicians of the last centuries spent hundreds of hours per capita in attempts to prove the conjecture. At most, they succeeded for some individual exponents $n$.

Suddenly, in the 1990s, Andrew Wiles succeeded. Many people had already switched to the lazy philosophy saying that "if it hasn't been found for several centuries, it can't be found". They were proved wrong. Wiles' proof was rather long and, even more importantly, it depended on highly advanced, hierarchically abstract concepts that "seemingly" have nothing to do with a simple statement about integers, their powers, and their sums.

In this random episode, Pat and Mat try to bring a painting through the door and hang it in the living room. They spent quite some time with it – more than 8 minutes on the video – and tried many methods which still doesn't prove that they have proven that no better solution exists.

In some formal language, Wiles' proof would still require at least thousands of characters. A brute force search for proofs would require at least something like $O(256^{1,000})$ operations, if you understand me, but even if $P=NP$ were true and allowed us to search for proofs polynomially quickly, the power could still be what it is in $1,000^8$ which would make it impossible to find the proof in practice. The hypothetical proof of $P=NP$ would still be very far from providing us with proofs of every theorem or a fast solution to any problem.

And of course, all the comments about Wolfgang Amadeus Mozart, Bobby Fischer, or Carl Gauss are completely silly and unrelated to the technical question above, except by verbal tricks. After all, these and other ingenious men had superior skills due to some kind of a brute force that others didn't receive from God. It may be great to believe that something divine, qualitatively different, intrinsically creative is behind these men's superiority but it ain't so. Almost everyone has gotten some memory, CPU, and creativity when he or she was born but some people just get much more of it and it – along with some training, hard work, and lucky circumstances – allows them to do things that others apparently can't do or at least don't do.

Four Color Theorem

The four-color theorem says a very simple thing: areas on a map may always be painted by four colors so that no two adjacent areas share the same color.

This is how you may paint the U.S. map with four colors.

Sometimes three colors are simply not enough. Divide a disk to three 120° wedges. They need to have 3 distinct colors. An extra area surrounding most of the disk touches all the three wedges so you need a fourth color.

And five colors is safely enough; it was proved in the 19th century and the proof is elementary.

However, four colors are enough, too. But there is almost no "wiggle room" left. Consequently, the existence of a four-colored painting of the map often looks like a good luck. Equivalently, the known proof is very complicated.

The first proof was constructed with the help of computers. They needed to verify thousands of possible "submaps". In 1996, the number of submaps that require a computer-enhanced verification was reduced to 633 but it's still large. It's been argued that no human can really check the proof by himself.

Boaz Barak tries to argue that 633 may perhaps be a bit larger than one but it will decrease to a number of order one. Well, it doesn't have to. It may be that 633 or 450 will be proved to be the minimum number of submaps that need to be verified in a similar proof. It's still an extremely large number.

Moreover, we're talking just about the four-color theorem, a kindergarten kid's playing with pastels. It seems pretty much obvious to me that more structured problems will require a much larger number of individual cases that have to be checked by the brute force. For example, there may exist an algorithm that searches for length-$N$ proofs of a theorem in a polynomial time, $C\times N^\ell$. Such an algorithm may still require an astronomical large memory for the algorithm itself and/or an astronomically large (either smaller or larger or the same) memory for the intermediate data. The fact that we don't know of such an algorithm today doesn't mean that it doesn't exist.

On the other hand, it's still plausible that the four-color theorem also has a different, elementary proof. Two years ago, I was sent something that looked rather promising by a chap and spent many days with that. At the end, there was nothing that looked like a proof. Today, I would probably be able to see that "it can't possibly be a seed of the proof" much more quickly – simply by seeing that too elementary a proof can't possibly know about the "wisdom" or "good luck" that are needed to four-color some difficult enough maps.

Classification of finite groups, largest sporadic groups

A group is a set with the associative multiplication\[

(fg)h = f(gh), \quad f,g,h\in G

\] and with a unit element and inverse element for everyone. A group may be a finite set. How many finite groups are there? Of course, infinitely many, e.g. the cyclic groups $\ZZ_n$ already form an infinite subset. Moreover, you may always construct Cartesian product groups and other constructions. Can you list all the mathematically possible groups in terms of some well-understood building blocks such as $\ZZ_n$, $S_n$, $D_n$, $A_n$, and their direct or semidirect products and/or a few extra operations?

Just for two decades, we have known the answer to be Yes.

The achievement – the classification – is written as the union of hundreds of mathematical papers by a hundred of authors. Thousands and thousands of pages are needed to solve the simply formulated problem above. Aside from the easy-to-understand groups above – some of which are coming in understandable infinite families – the classification forces you to discover 26 truly unusual examples of groups, the sporadic simple groups. (Sometimes the Tits group, albeit a bit simpler, is counted as the 27th sporadic group.)

The largest sporadic group is the monster group. Its number of elements is about $8\times 10^{53}$, almost a million trillion trillion trillion trillions. It's a very particular, fundamental, and important integer in maths that arises "out of nothing". The dimension 248 of the largest exceptional Lie group, $E_8$, is analogous except that the number of elements of the monster group is vastly larger.

The laymen could also intuitively guess that objects described by very large integers such as this one are contrived, unimportant in practice, and so on. But if you read some articles about the monstrous moonshine, you will see that just the opposite is true. The largest sporadic group is, in some sense, the most elementary sporadic group. An algebra of vertex operators that respect this huge finite symmetry carries an explanation for the coefficients in the expansion of the $j$-function, a very natural function (basically unique if holomorphic) mapping the fundamental domain to a sphere. The AdS/CFT dual of the conformal field theory with the monster group symmetry is the "simplest" gravitational theory you may think of, the pure gravity in the 3-dimensional AdS space, at least at the minimal curvature radius.

The latter relationship suggests some kind of a duality. If the CFT has some group with many elements, its dual is simple and has a few fields. There is some complementarity between these two things and the product could be something like $10^{54}$, if I describe the relationship in a "moral" way that isn't quite accurate.

When we approach some really natural structures in maths – especially the deep maths that has links to the foundations of quantum gravity etc. – we easily and naturally get prefactors that may be as large as $8\times 10^{54}$ or other large numbers. Of course that if an algorithm gets slowed down by this factor, it doesn't matter that it's still "polynomial": it will be impractically slow and useless in practice.

And those things surely do happen in maths often. Even though our current knowledge is almost certainly biased towards knowing the shorter proofs and methods and be ignorant about the longer ones (because the latter are harder to be found), we already know that there are many important proofs and algorithms and classifications that are extremely long and hierarchically complex.

The string-theoretical landscape

String theory is the richest theory that the mankind knows and one may enter the realm through many entrances. The evidence strongly suggests that it's a completely unique structure with no deformations or siblings. However, it may have many solutions.

In the class of semi-realistic stabilized flux vacua, people have tried to estimate the number of solutions to string theory's equations. The "stabilized" adjective implies that there are no moduli left; no continuous parameters can be freely adjusted. With these extra conditions, the number of vacuum-like solutions has to be finite or countable and if one imposes some additional semi-realistic criteria, he ends up with a finite number. But the number is often quoted as $10^{500}$ although this estimate is far from a rigorous enough calculation.

At any rate, it seems very likely that powers of a googol are a good estimate of the number of string-theoretical vacua obeying certain very natural (and sort of realistic) constraints that make them very promising as descriptions of the real Universe around us. Whether there exists a principle that picks one of them or some of them according to some non-anthropic criteria remains unknown.

But even if we ignore these vacuum-selection and anthropic questions, it seems true that string theory produces a high number of solutions. If you accept that string theory describes the laws of physics in the most general sense, you may ask whether the laws of physics allow some particular phenomena or creatures. Describe the creature in some way. To find out whether they appear anywhere, you may be ultimately forced to search through those $10^{500}$ different universes.

The $P\neq NP$ complexity theorists usually talk about a few dozens of their pet $NP$-complete algorithms, not about the problems for "mature scientists" such as the searches for some stringy vacua with some properties. So they think that the traveling salesman and equivalent problems are "almost everything" that is worth thinking about. Well, it surely ain't so. But in the "traveling salesman" type of problem, they know what $N$ is.

What is $N$ in the case of the problem "Find a stringy vacuum with some properties"? Needless to say, really fundamental physics problems usually don't come with instructions such as "use the label $N$ for this and that". In some sense, we are dealing with a single string theory and the finite number of solutions is just an order-of-one extra complication. In some counting, we have $N=1$. Nevertheless, it makes the search $10^{500}$ or so times more time-consuming, at least if you have to rely on the brute force method (which isn't clear). It's an example of the huge prefactor that Boaz Barak believes not to occur in practice. It surely does occur – it occurs in the case of many fundamental math problems such as those above as well as in the case of the most natural and fundamental problems in physics, those asking about the basic properties of string theory.

More generally, I want to say that there are lots of "subfields in maths" that may be completely mastered – in the sense that you may turn a previously seemingly creative activity to a mechanical process – but the number of concepts and algorithms that you sometimes need to learn to make these things mechanical is often very large. Think about all the tricks and rules you need to know to integrate large classes of functions or anything else. Think about the huge amount of code hiding in Mathematica which allows you to solve so many things. Mathematica and Wolfram Alpha are examples of projects in which people actually made a significant progress in their efforts to "automatize" many things. While I find it unlikely that Wolfram or others will ever find a gadget that "automatizes everything" – e.g. the tasks "find the shortest proof of a valid proposition" or "find the fastest algorithm that solves something" etc., I don't really have a proof (and not even legitimate, strong enough evidence) and it's just wrong to promote big ideologies that try to "frame" everything so that it's compatible with the predetermined conclusions even though the truth may be different and the alternatives might be "framed" as well, if you tried a little bit.

I feel that people such as Boaz Barak are extremely narrow-minded in their focus on 3SAT and several other, comparably down-to-Earth, problems; extremely biased against the existence of more sophisticated algorithms and proofs than they know now (partly because they don't really want to learn something difficult and new and they don't want to be shown to have been unable to find out something remarkable that they should have found if they were ingenious enough); and they spend way too much time by rationalizations of the limited progress in the search for much better algorithms to solve various things.

Moreover, their experience with the really nice and important problems that have some "beef" is limited, otherwise they couldn't possibly claim that large prefactors and very long minimal proofs or algorithms can't appear in maths. They surely can and they often do. At least in this sense, "naturalness" (in the sense of the preference for order-of-one coefficients in answers to questions) fails in maths: the number of concepts and the amount of time you need to solve a short problem formulated as a length-$N$ string is often much larger than $N$ or a small multiple of its modest power. As Barbie correctly said, math class is tough. But it should still be taught at schools.

Mathematics is a complicated network of things that are easy, things that are hard, things that look hard but they're easy, things that look easy but they are hard, and the hardness has very many aspects. The idea that all problems deserving to be a part of maths may be divided to easy and hard ones in a binary way is a childishly naive concept. Very simple questions often require one to discover and (in the case of others) master very complicated bodies of knowledge to understand the answers. And very rigid and in this sense simple laws – like, in the most extreme case, the laws of string theory – are often able to produce an amazingly rich set of phenomena, implications, and relationships, i.e. pulsating and intelligent worlds with lots of fun. Complexity theorists who effectively assume that solutions and straightforward algorithms have to be as short and easy-to-learn as the questions themselves are effectively assuming that "what comes out had to come in". But this just isn't true in maths and science. It's the whole point of physics, for example, that we can explain diverse things with a limited canon of laws – but these in principle simple laws have many consequences and create many patterns that have to be understood if you want to master the implications of the laws.

The idea that "proofs and solving algorithms are about as short and as deep as the formulation of the problem" may be described as the "garbage in, garbage out (GIGO)" paradigm and this is what the research may easily look like if you insist on similar prejudices.

Comparing the depth of the millennium problems

2013-05-06T10:29:00.001+02:00

The Riemann Hypothesis is probably the deepest one

In 2000, the Clay Mathematics Institute offered 7 times $1,000,000 for the proofs of the Millennium Prize Problems. Is it possible to compare which of them are deeper than others?

Needless to say, such a comparison depends on personal preferences, emotions, and there is probably no rigorous way to "prove" that one problem is deeper than others. However, that doesn't mean that one can't have an opinion; and it doesn't prevent some opinions from being more well-informed than others.

So let me offer a sketch of the seven problems and some evaluation of their depth.

The Poincaré Conjecture

I began with this one because it has been proven by Grigori Perelman. The statement of the theorem is simple:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

For three-dimensional manifolds, Perelman had to prove that everything that smells like a sphere is a sphere. The "smelling" means that the manifold is "closed" i.e. free of boundaries etc.; and it is "simply connected" which means that you can't "laso it". Every topological circle within the manifold may be gradually and continuously shrunk to a point.

You may ask why the problem talks about three-dimensional spheres only. Roughly speaking, it's because less-than-three-dimensional manifolds are too simple to be classified (and to prove the claims about the classification). Among the two-dimensinoal surfaces, you have Riemann surfaces with $h$ handles – and perhaps $b$ circular boundaries and $c$ crosscaps, too.

And higher-dimensional manifolds become sort of simpler, too. At least those that "smell like a sphere" are easy to be put under control. Needless to say, the diversity of complicated manifolds such as the Calabi-Yau manifolds becomes more extreme for higher dimensions. But when you talk about "truly special, highly constrained" manifolds, there is a sense in which $d=3$ maximizes the "depth of the mathematical insights" needed to understand the set of manifolds.

Perelman solved it and in some sense, the technology behind the proof is sort of "easy" and the proof allows us to say that there was "nothing totally new" hiding behind the conjecture. The proof is easy to formulate in terms of concepts that are natural in quantum field theory and especially string theory, namely the so-called Ricci flows (this idea was first proposed as a path to the proof by Richard Hamilton).

One may study strings propagating on a 3-dimensional manifold and use this "non-linear sigma-model" as an effective theory. One may try to change the characteristic world sheet length and look what it does with the whole theory. This is about the flows of the renormalization group. Effectively, such a transition to lower world sheet energies or longer distances has the effect of "rounding the target three-manifold". So if it is a potato close enough to the sphere, it will increasingly resemble the sphere with the standard metric.

Pretty much constructively, you may actually end up with the sphere. The flow of the renormalization group, the Ricci flow, may betray you because in some cases it may produce a singularity instead of making your theory ever smoother. But these possible evolutions may be controlled by some extra "surgeries" and a careful supervision of the "fundamental group" of the manifold at each stage. At the end, Perelman showed that a rather natural strategy – a method to make the spherical character of the original manifold manifest – really works and may be made totally rigorous.

There are of course many other special questions about the geometry in $d=3$ – in some sense, many problems of knot theory could be included here. The Poincaré conjecture is rooted in the "totally continuous" questions of geometry and it should therefore not be surprising that the proof sounds like a proof in mathematical physics. Such "purely continuous proofs" with very limited links to any auxiliary discrete structures etc. are very important – especially in physics – but one could argue that at the end, they're sort of straightforward once the right "physical intuition" is used.

It was great when Perelman completed the proof. But if we ignore the historical coincidence that the proof is a relatively recent event, I think that many "comparably difficult" problems and proofs in the purely continuous geometry are much more important and deeper – classification of Lie groups, classification of possible holonomies of manifolds, and so on.

Birch and Swinnerton-Dyer conjecture

This conjecture proposes a way to count the diversity of the arithmetic data defining an elliptic curve (a two-torus written as a quadratic-cubic equation in complex variables; named in this way because of links to the "elliptic integrals") by the Hasse-Weil $L$-function.

I don't want to go into details here but it's a counting problem that is very similar to Fermat's Last Theorem ultimately proved by Andrew Wiles. That's also why Wiles was the natural person to ask for a description of the problem. Recall that Wiles proved Fermat's Last Theorem by trying to count certain things.

It could be difficult to compare FLT and the Birch and Swinnerton-Dyer conjecture when it comes to their depth – I am no real number theorist, after all. However, let me mention that most of us loved FLT more than this particular Millennium Prize Problem. But we should acknowledge that much of this preference boils down to a relatively shallow criterion – Fermat's Last Theorem may be explained to a schoolkid. The elliptic curves are about a bit more advanced maths.

I am not sure whether this difference should be counted as evidence that FLT is "more deep". It's surely more popular because it's more accessible. But when one starts to master the actual maths that is needed for the full proof of FLT, it turns out that it's the same kind of maths that is (probably) needed to decide about the fate of this Birch-and-someone conjecture. The problems may be comparably deep and there could exist similar "truly comparable" other problems in number theory, too. FLT only differed by its being extremely accessible. I mean the proposition, not its proof which is very advanced and complicated.

FLT and this Millennium Prize Problem are examples of number theory and its connections with "geometry over discontinuous fields". One generalizes some natural concepts we know from the continuous geometry and they turn out to be relevant for problems that sound like problems about integers, especially primes and factorizations etc. (number theory). This is always a deep link. However, I will argue that the Riemann Hypothesis is probably deeper than these FLT-like problems because in the RH case, the flow of wisdom goes in both directions, more symmetrically. When we solve FLT or this Birch-and-someone problem, we learn something about integers and solutions of equations with integers etc. using methods that we also routinely use in the continuous context but we don't seem to learn much about the continuous maths.

The Hodge conjecture

The Hodge conjecture talks about the de Rham cohomology – cohomology is about the independent antisymmetric tensor fields (forms) that are closed by not exact (and, nearly equivalently, homology is the space of topologically inequivalent and non-trivial, non-contractible submanifolds of a manifold) – and it says that if you talk about manifolds that may be described via complex algebraic equations, there's also a similar way to describe all the cohomology classes as combinations of the classes of submanifolds that may also be specified by complex algebraic equations.

In some sense, this problem says that if a manifold seems to be equivalent to complex algebraic equations, then complex algebraic equations are also enough to describe all the cohomology classes. There's nothing "new" that would force you to go outside the complex algebraic "box" if you started inside the box in the first place.

It's interesting that this problem hasn't been proven but it's hard to predict the apparent depth that this problem will have after a hypothetical proof is found. Such a proof may turn out to be constructive and somewhat straightforward – in some sense, analogous to Perelman's proof of the Poincaré Conjecture. It may differ e.g. from Wiles' proof of Fermat's Last Theorem by remaining "simple" for all dimensions. In order to prove FLT, Wiles had to use qualitatively more advanced mathematical techniques than the techniques you need for the proof of FLT specialized to a particular exponent such as $n=4$. We don't know but I find it somewhat plausible that the future proof of the Hodge conjecture will be more uniform – a more straightforward generalization of a natural proof you could construct for a particular dimension $d$.

But of course, the proof may be much deeper than that. One must always remember that we should better talk about "a proof" because there can exist many proofs, even many qualitatively different proofs. Some of them may be much shorter than others; some of them may be much deeper than others. Only the first proof of the Millennium Prize Problems earns a million of dollars but it's always possible that another, inequivalent proof that is still in the pipeline is actually deeper and more important (but unrewarded).

The Yang–Mills existence and mass gap problem

This problem is most closely linked to quantum field theory, the main "technology" in which I was sort of professionally trained, but that doesn't mean that I consider this problem to be too deep. In fact, I don't.

The real reason is that using physical arguments, sometimes very deep ones (including wisdom hiding in the AdS/CFT), we have acquired answers to much deeper and more detailed questions than just the existence of the mass gap. To get the prize, you need to present an argument that is pretty much rigorous – and define a QFT sort of rigorously along the way. And once you complete this "paperwork", you also have to use your framework to prove a rather basic claim that pure QCD doesn't contain any "arbitrarily light massive particles"; the lightest massive particle has a strictly positive mass (comparable to the QCD scale) and nothing (i.e. gap) in between this mass and zero.

I don't think that this is a natural way towards profound insights. Quantum field theories and string theory are sort of pulsating, living animals that may be studied with the help of many techniques and perspectives that admit very precise rules and that work but that don't fit the mathematicians' fully elaborated "straightjackets" that have already been formulated in the very rigorous mathematical way. In some sense, this problem wants you to kill the pulsating animal and decompose it into some boring components that admit a totally rigorous mathematical definition.

When you do it, you will only recover some physical insights that are known to be true to the physicists – in fact, physicists may answer much more detailed and quantitative questions of this kind. I would say that the importance of the problem is in rigorous mathematicians' desire to declare that quantum field theory is "fully incorporated" to the rigorous maths, within a basic mathematical framework that has been completely found and should no longer evolve in the future. I am not sure whether I would subscribe to this thesis. QFT and string theory are already "the cutting-edge physics" tools and as physicists are discovering their new properties, they're also learning about new ways to mathematically formalize them. To write down "a definition" of a quantum field theory and to prove some basic properties of the corresponding mathematical structure is a settled subject but "all interesting knowledge about quantum field theories" is surely not a completed subject.

Navier–Stokes existence and smoothness

This problem is close to physics, in fact it is the only obvious "classical physics" problem in the list. The Navier-Stokes equations describe fluid mechanics which displays complicated phenomena, especially turbulence that looks "cool" on the pictures. That's a reason why this topic is popular. To win the prize, you must:

Prove or give a counter-example of the following statement:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

You potentially don't have to learn the answer to any interesting "physical" question. The reason why the smooth, globally defined solution to the Navier-Stokes problem exists – or doesn't exist – may be highly technical details you have to be picky about. It is not guaranteed at all that a winner of this prize must achieve some genuine progress in the "puzzling physics properties" of turbulent fluids etc.

Turbulence leads to interesting conceptual phenomena such as a self-similarity of the relevant pictures. It's been believed for quite some time that all the relevant statistical distributions are pretty much self-similar. There is now clear evidence that this ain't the case. The self-similarity is broken at some point.

There is another issue – the real-world fluids with a cutoff at the atomic length scale may differ, when it comes to some short-distance behavior, from the fluids idealized by the Navier-Stokes equations. The problem is a mathematical one about the equations so the winner of the prize may also miss many actual real-world subtleties associated with the existence of atoms, and so on.

There are numerous interesting problems and patterns related to fluid mechanics in general and turbulence in particular that may be discussed or proved. I am just not sure whether the rigorous mathematical formulation above is a natural way to direct the researchers towards the most profound parts of these problems of mathematics of classical physics. The problem was probably incorporated in order to state that "there are still interesting math problems related to classical physics" – a thesis I would almost certainly subscribe to – but I am not sure whether the representative problem was formulated usefully.

P versus NP problem

This computer-science problem was the original reason that made me write this blog entry. Scott Aaronson didn't like that I interpreted the "P = NP problem" as a comparably messy problem to the question whether chess is a draw (whether two ideally clever chess players attempting to reach the best results inevitably end up with a draw).

Just to be sure, I do think – like most experts – that chess is a draw and that P isn't the same thing as NP.

In this computational complexity business, you have an integer $N$ that measures the "size" of the objects in the problem you want to solve and the main question is whether the number of computer operations scaling like a power of $N$ for large $N$ is enough to solve such a problem in a class. The polynomially fast solutions are considered "fast"; the "slow" ones need exponentially or factorially (etc.) long timescales to be solved.

As the diagram above shows, there is a large set of NP problems. They're "quickly checkable". NP stands for "non-deterministically polynomial". They're problems such that if you're told a solution, you may verify it is a solution in a polynomial time. Verification is generally believed to be easier than the search for solutions, so that's why people generally believe that "P is not NP".

In the NP class, you have the P subclass – problems that can be (polynomially) quickly solved, not just verified. And then there is the NP-complete subset. Well, the NP-complete subset is the intersection of NP with the set of "NP hard problems" which are those that are as difficult as the hardest problems in NP. In practice, NP-complete is a limited class of problems that have been proved equally complicated as other NP-complete problems by various ways to convert one problem in the class to another (plus all the other problems that may be shown equivalent in the future). So I can just give you a representative: quickly compute the permanent of a matrix (the determinant-like sum over permutations but with no minus signs).

If P were the same as NP, which is believed to be false, then NP-complete problems, because they're still elements of the NP problems, would be solvable in a polynomial time as well, so NP-complete would be the same set as P and NP, too. In the more likely case that P isn't NP, NP-complete and P problems are two completely disjoint subsets of NP ("opposite" in their complexity).

P vs NP problem is sort of interesting because it's rather easy to be formulated but if you look closely, it's really a messy problem and there isn't a good reason why there should exist an accessible enough resolution. We're asking whether problems are polynomially quickly solvable. This very adjective is sort of contrived. From a practical viewpoint, there is usually a big difference between polynomial and longer-than-polynomial time requirements but it doesn't seem to me that this distinction is too fundamental from a deeper theoretical viewpoint. In other words, I believe that this "polynomial technicality" is already enough to classify the P vs NP problem as a "mostly practical messy problem", not a mathematically natural fundamental problem.

At the same moment, I think that some hype promoting the P vs NP problem resembles TV commercials way too much. In Wikipedia, Scott Aaronson is quoted as saying:

"If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."

That sounds great. We surely want to know whether everyone may be a Mozart; whether Mozart's work may be automatized. Well, it probably can't. But even if P were equal to NP, it could still be difficult to construct the "straightforward algorithm" that solves an arbitrary problem in NP. P = NP would say that a fast solving algorithm exists for each problem in NP; it doesn't say that we actually know a method how to find such an algorithm for any NP problem. For the known mutually equivalent NP-problems, you could perhaps do it but it could still be hard to find the equivalence of other NP-complete problems with the known ones (recall again that if N equals NP, then all problems in NP are NP-complete, too). The translations to the "already solved" NP problems could become "increasingly difficult" for problems that deviate from the initially solved subclass. Things could be very hard in practice even if P were equal to NP.

I may describe the same problem in the following, nearly equivalent way: the P vs NP problem is asking about the existence of an algorithm for each problem in NP and whether such an algorithm is polynomially fast. So in some sense, it's not a question about some important mathematical structures themselves but about higher-order "texts" (algorithms) that discuss the mathematical structures and their not-too-fundamental properties such as the polynomial speed.

The brute-force way to decide would be to look at all candidate algorithms and check whether they work and whether they are polynomially fast. Clearly, this can't be done in a finite time because the potential algorithms, although polynomially fast, may still be given by an arbitrarily long program.

"Finding all nicest and fastest algorithms that may solve a difficult problem" looks like an extremely creative, Mozart-like problem and it's plausible that this creative achievement is needed to decide about the P vs NP problem. In this sense, I think that not only the proof of P = NP but also the disproof probably require something extraordinarily difficult and Mozart-like. If P isn't NP, as most experts expect, it still seems natural to assume that to find the proof of "P isn't NP" is an NP-complete problem of a sort. Such a proof may be extremely long or unnatural or whatever. To decide about P = NP, it seems like you need to place yourself about "all the Mozarts" whose superioty you want to prove if you want to prove that P isn't NP. And that's a contradiction, at least a morally or practically. You shouldn't prove that all of us are inferior insects relatively to Mozart by becoming the superior supervisor and guru of Mozart and every other genius. ;-)

So I don't expect any proof of P = NP and I don't expect any proof that P isn't NP, either. One of the proofs, probably the latter, probably exists but it's probably extremely complicated and doesn't illuminate much.

The Riemann hypothesis

The Riemann Hypothesis seems to be the deepest problem to me although it may ultimately turn out to be just about one physics-related problem/insight among many.

It may be phrased as a problem on analytic functions of a complex variable: the Riemann $\zeta$-function has no roots away from the real axis and the $1/2+is$ vertical line. But it may also be phrased as an equivalent problem about number theory, namely the distribution of primes. For example, an equivalent way to phrase the Riemann Hypothesis is to say\[

|\pi(x) - \operatorname{Li}(x)| \lt \frac{1}{8\pi} \sqrt{x} \, \log(x), \qquad \text{for all } x \ge 2657.

\] The number of primes smaller than $x$ may be approximated by the integral of $1/\ln(t)$ – this inverse logarithm function quantifies the probability that a number comparable to $t$ is a prime – integrated up to $x$ and the error is very small, given by the right hand side of the inequality above. (Similar but weaker statements with a larger allowed error on the right hand side may be proved, e.g. by the prime number theorem; they're equivalent to the proved non-existence of the zeroes of the $\zeta$-function outside the real axis and outside the width-one critical strip symmetrically located around the critical axis.)

Because of its relationships to integers and primes, the Riemann zeta function is an extremely natural function of a complex variable. In some sense, it's the most natural "non-elementary" function of a complex variable that has something to do with integers in general and primes in particular. The function may be defined in many ways, it clearly has infinitely many non-trivial zeros on the $1/2+is$ vertical axis, and it sounds crazy we're not able to prove that it has no extra zeroes (which would be paired symmetrically with respect to this vertical axis) in the bulk of the complex plane (well, they would be in the width-one strip around the vertical axis).

The values of $s$ for which $\zeta(1/2+is)=0$ are very interesting, non-trivial real numbers. The Hilbert–Pólya conjecture is a general approach to the proofs of the Riemann Hypothesis that many, including myself, find extremely promising and natural. It says that the set of such real values $s$ is the spectrum of a Hermitian operator. This operator may be defined in a way that makes the Hermiticity manifest; and that makes it manifest that the eigenvalues are linked to these zeroes i.e. special values of $s$; that would prove that $s$ is real and there are no other non-trivial roots (away from the axis).

In some sense, the non-integer values of $s$, the zeroes of the Riemann $\zeta$-function, are "inverse" to the set of primes on the real axis. I won't tell you the exact sense but if the allowed momenta were primes, the allowed windings would be the roots $s$ of the $\zeta$-function. This vague statement also implies that the zeroes of the $\zeta$-function become more dense, as $2\pi\ln s$, if you go further away from the real axis; note that the probability that an integer is a prime was decreasing as $1/ \ln n$.

I have tried many times to go one step further (or several steps). In one approach of mine, I decided that for each such zero $s$, there is actually a whole unitary representation of $SL(2,\RR)$. There should exist a natural physical system with the $SL(2,\RR)$ symmetry whose states include unitary representations of this group. One has to find the physical system, prove that the second Casimirs are given by $x(x-1)$ where $x=1/2+is$, and the Riemann Hypothesis follows from some known results about five classes of the unitary representations of $SL(2,\RR)$. I find it utterly natural to use the whole $SL(2,\RR)$ and not just a generic single Hermitian operator, especially because of the role that $x(x-1)$ plays in the $\zeta$-function and the role that the $1/2+is$ vertical line plays in the theory of representations of that group.

Also, for some time, I believed that the non-existence of the physical string states above Martin Schnabl's tachyon-condensation solution to the open string field theory proved the non-existence of other non-trivial zeroes – because the form of Schnabl's solution may be written in terms of the $\zeta$-function, too. I have never been able to complete the proof, not even at the physicist's level of rigor, and I am not sure that a proof of this sort exists today.

At any rate, I tend to think that because the $\zeta$-function is so unique in the complex calculus, its very visible and manifestly correct property – the location of zeroes on two straight lines only – is something important that we should understand. The Riemann Hypothesis seems much more unique, special, and fundamental to me than any other problem discussed above.

And that's the memo.

Aaronson's anthropic dilemmas

2013-05-04T17:59:00.000+02:00

This text has been expanded and covers the rest of the book... Originally posted on May 1st

If you read my previous observations on Scott Aaronson's book including all the comments, you will see my remarks about all the chapters up to Chapter 15 about the quantum computation skeptics – where I agree with almost everything Aaronson writes although he seems to focus on the dumb criticisms and writes too little about the more intelligent ones (and e.g. about the error-correcting codes).

Chapter 16 is about learning; perhaps too much formalism if we compare it with the relatively modest implications for our understanding of the process of learning.

Chapter 17 is the most hardcore "computational complexity" part of the book and hopefully the last one that is intensely focusing on the complexity classes. It's about interactive proof systems. Aaronson often wants to present all of computer science as a "fundamental scientific discipline" so he tries to apply these superlatives to aforementioned "interactive issues", too.

I have a lot of trouble to get excited about these problems.

In the interactive proof system, two beings – a verifier and a prover – are exchanging messages whose goal is to ascertain whether a given string belongs to a language or not. The prover cannot be trusted while the verifier only has finite resources. It looks like an immensely contrived game – from game theory – to me. Detailed questions about such a game seem about as non-fundamental to me as the question whether chess is a draw.

The only true reason why I would want to prove $P=NP$ or its negation (or even the numerous less important results of this sort) would be to get a million of dollars.

Needless to say, I think that Scott Aaronson is a world's top professional in the computational complexity theory – and I think that the quantum aspect is an optional cherry on a pie for him, an extra X-factor that he adopted to feel rather special about the computational complexity theorists themselves.

But for me, this is a portion of mathematics that is completely disconnected from fundamental problems of natural sciences. I like to think about important scientific problems. But the complexity papers aren't really about the beef, about particular problems. They are thinking about thinking about problems – and they don't really care what are the "ultimate" problems and whether they're true (e.g. in Nature). In this sense, suggesting that this is a fundamental layer of knowledge about the world or the existence is as silly as the proclamations of anthropologists who study dances of wild tribes in the Pacific but who also try to study the interactions among scientists. These anthropologists are trying to put themselves "above" the physicists, for example, even though in reality, they are inferior stupid animals in comparison – people who completely miss the beef of physics and who may only focus on the irrelevant, superficial, sociological makeup on the surface. In some sense, Scott as a computational complexity theorist is doing the same thing as the anthropologists but with more mathematical rigor. ;-)

Moreover, the computational complexity theory seems to be all about a particular "practical" quantity I don't really care about much – namely computational complexity. I am probably too simple a guy but I primarily care about the truth, especially the truth about essential things, and I don't really care how hard it is to find or establish the truth. So the whole categorization of problems to polynomially or otherwise easy ones – and Aaronson defines dozens of complexity classes and discusses their relationships – is just something orthogonal to the things I find most important.

But let me stop with these negatively sounding remarks about the discipline. Computer science is surely a legitimate portion of maths and Aaronson is talking about it nicely.

Chapter 18 is about "fun with the anthropic principle".

This part of the book doesn't need any physics background – because this principle used by some physicists isn't about any scientific results, either. It's about their emotional prejudices and unsubstantiated beliefs in proportionality laws between probabilities and souls (which boils down to the fanatical egalitarianism of many of these folks).

The chapter is at least as wittily written as the rest of the book. The end of the chapter talks too much about complexity again but let's focus on the defining dilemmas in the early parts of the chapter. After a sensible introduction to Bayes' formula and its simple proof, Aaronson talks about some characteristic problems in which people's attitudes to the anthropic reasoning dramatically differ.

Hair colors in the Universe

At the beginning, God flips a fair coin. If the coin lands heads, He creates two rooms – one with a red-haired person and one with a green-haired person. If it lands tails, He creates just one room with a red-haired person.

You find yourself in a room with mirrors and your task is to find the probability that the coin landed heads. Well, you look into the mirror that's a part of each such room. If you see you are green-haired, the probability is 100% that the coin landed heads because the other result is incompatible with the existence of a green-haired person.

What about if you see you are a redhead?

A natural (and right!) solution, one mentioned at the beginning, is that the probability is 50% that the coin landed heads. The existence of a redhead is compatible with both theories (heads/tails) so you are learning nothing if you see a redhead in the mirror. You should therefore return to the prior probabilities and both theories, heads and tails, have 50% odds by assumption.

In my opinion (LM), this is really the most correct calculation and justification one may get. I tried to "improve" Aaronson's justification a bit.

Now, one may also (incorrectly!) argue that the probability of heads is just 1/3 instead of 1/2 if we see a redhead. The tails hypothesis is twice as likely, 2/3, than the heads hypothesis because – and again, this is an explanation using my language – it makes a more nontrivial, yet correct, prediction of the observed hair color. The heads hypothesis allows both colors so the probability that "you" will be the person with the red hair color is just 1/2.

But I believe this argument is just wrong. It doesn't matter how predictive the hypotheses are! By assumption, the prior probability of heads and tails were 50% vs 50%. The tails hypothesis is more predictive because it allows you to unambiguously predict your hair color – it has to be red because you're the only human in that Universe. But we know that this doesn't increase the probability of heads above 50%.

For that reason, we also don't need an additional "adjustment" of the argument – and this adjustment is wrong by itself as well – that returns the value 1/3 back to 1/2. We may return from 1/3 to 1/2 if we give the Universes with larger numbers of people – in this case, the heads Universe – a higher "weight". There is no reason to adjust these weights. The point is that the prior probabilities of heads and tails are completely determined here by an assumption so any inequivalent "calculation" of these prior probabilities based on the number of people in the Universe is wrong. We just know it to be wrong. We're told it is wrong!

Aaronson "calculates" the value 1/3 of the probability by Bayes' formula. But the calculation is just conceptually wrong because the prior probabilities of heads/tails are given as 1/2 vs 1/2 at the very beginning and the observation of a redhead provides us with no new data and no room to update the probabilities of hypotheses. The observation of a greenhead does represent new data. The arguably invalid update in the case of the observation of a redhead plays one role: to counteract the update from the green observation so that the probability of heads weighted-averaged over the people in the Universe will remain equal to the probability of tails. But it's not the redhead's "duty" to balance things in this way. By seeing his red color, he just learns much less information about the Universe than the greenhead (namely nothing) so he has no reasons to update.

Using slightly different words, I may point to a very specific error in the Bayesian calculation leading to the result 1/3, too. Aaronson says that the probability $P({\rm redhead}|{\rm heads})$ is equal to 1/2 – probably because in the two-colored heads Universe, there are two folks and they have "the same probability". But that's a completely wrong interpretation of the quantity that should enter this place of Bayes' formula. The factor $P(E|H)$ that appears in the formula should represent the probability with which the hypothesis $H$ predicts some property of the Universe we have actually observed, $E$ i.e. the evidence. And what we have observed isn't that a random person in the Universe is a redhead. Instead, we have observed that our Universe contains at least one redhead; in particular, the predicted probabilities $P({\rm redhead}|{\rm heads})+P({\rm greenhead}|{\rm heads})$ don't have to add to one because both "redhead" and "greenhead" refer to the observation of at least one human of the given hair color so these two colorful observations are not mutually exclusive. (You should better avoid propositions with the word "I" because this word is clearly ill-defined across the Universes; there's no accurate "you" or "I" in a completely different Universe than ours because the identification of the right Universe around you is a part of the precise specification of what "I" or "you" means; you should treat yourself as just another object in the Universe that may be observed, otherwise you may be driven to spiritually motivated logical traps.) The probability of this actual observation – evidence – is predicted by the heads hypothesis to be 1, not 1/2. With the correct value 1, we get the correct final value 1/2 that the heads scenario is right!

I must mention the joke about the engineer, physicist, and mathematician who see a brown cow outside the train. The first two guys say some sloppy things – cows are brown here (engineer); at least one cow is brown here (physicist) – but the mathematician says that there's at least one cow that's brown at least from one side in Switzerland. This is the correct interpretation of the evidence! The situation in the previous paragraph is completely analogous. (There's a difference: people are less afraid to say unjustifiable and/or wrong propositions that are probabilistic in character, e.g. "I am generic", than Yes/No statements about facts that are "sharply wrong" if they're wrong. But probabilistic arguments and conclusions are often wrong, too!) I am surprised that even Scott Aaronson either fails to distinguish the different statements; or deliberately picks one of those that actually don't follow from the observations! This is the kind of the elementary schoolkid's mathematical sloppiness that powers most of the anthropic reasoning.

At the end, the error of the Bayesian calculation may also be rephrased as its acausality. It effectively assumes that the probabilities of different initial states are completely adjustable by some backward-in-time notions of randomness even though they may be determined by the laws of physics – and by the very formulation of this problem, they are indeed determined by the laws of physics in this scenario!

Madman

A madman kidnaps 10 people, puts them in a room, throws 2 dice, and if he gets 1-1 (snake-eyes), he kills everyone. If he gets something else, he releases everyone, kidnaps 100 other people, confines them, and throws again. Again, 1-1 means death for everyone, another result means that 100 people are released and 1,000 new people are kidnapped. And so on, and so on.

You know the rough situation and you know that you're kidnapped and confined in the potentially lethal room (but you don't know whether some people have already been released). What's the probability that you will die now?

Obviously, you know the whole mechanism of what will happen. He will throw dice. The probability to get 1-1 is obviously 1/36. That's the chance you will die.

Aaronson presents a different, "anthropic" calculation telling you that the chances to die are vastly higher, essentially 8/9. Why? Well, the madman almost certainly releases the first 10 people and then probably the 100 people as well etc. but at some moment, he sees snake-eyes so, for example, he kills 100,000 people and releases the previous 10,000+1,000+100+10 = 11,110 people. Among the folks who have ever been confined to the scary room, about 100,000/111,110 = 8/9 of them die. So this could be your chance to die; the ratio doesn't seriously depend on the number of people who die as long as it is high enough.

Which result is correct? Aaronson remains ambiguous, with some mild support for 8/9. I think that the only acceptable answer is 1/36. The argument behind 8/9 is completely flawed. It effectively assumes that you're a "generic" person among those who are kidnapped on that day – there's a uniform distribution over those people. But that's not only wrong; it's mathematically impossible.

The average number of people who will die is\[

\sum_{n=1}^\infty 10^n \zav{\frac{35}{36}}^n \frac{1}{36}

\] but this is divergent because $q=350/36\geq 1$. Chances are nonzero that the madman will run out of people on Earth and won't be able to follow the recipe. At any rate, the reasoning behind $p=8/9$ strongly assumes that the geometric character of the sequence remains undisturbed even when the number of the hostages is arbitrarily large. It effectively forces us to deal with an infinite average number of people and there's no uniform measure on infinite sets because there exists no $P$ such that $\infty\times P = 1$.

I think that this is not just some aesthetic counter-argument. It's an indisputable flaw in the calculation behind $p=8/9$ and the latter result must simply be abandoned. In this case, we know very well it's wrong. If the madman tries to causally stick to his recipe as long as it's possible, the probability for each kidnapped person to die is manifestly $p=1/36$.

The wrong, anthropic results often make unjustified calculations based on the "genericity" of the people – assumptions that some probability measures are uniform even though there is absolutely no basis for such an assumption and in our scenario, this uniformity assumption explicitly contradicted some assumptions that were actually given to us! And the anthropic arguments also tend to make acausal considerations.

Doom Soon and Doom Late

This is also the case of the "doomsday is probably coming" argument. Imagine that there are two possible worlds. In one of them, the doom arrives when the human population is just somewhat higher than 7 billion (Doom Soon). In the other one (Doom Late), the population reaches many quintillions (billions of times larger than the current population).

Again, just like in the hair color case, if we have reasons to expect that the prior probability of both worlds are equally or comparably large, then we have no justification to "correct" or "update" these probabilities. The existence of 7 billion people is compatible both with Doom Soon and with Doom Late. So both possible scenarios remain equally or comparably likely!

The totally irrational anthropic argument says that Doom Soon is 1 billion times more likely because it would be very unlikely for us to be among the first 7 billion – one billionth of the overall human population throughout the history. This totally wrong argument says that we're observing something that is unlikely according to the Doom Late scenario – only 1/1,000,000,000 of the overall history's people have already lived – and our belief that we live in the Doom Late world must be reduced by the factor of one billion, too.

That's wrong and based on all the mistakes we have mentioned above and more. The main mistake is the acausality of this would-be argument. The argument says that we are "observing" quintillions of people. But we are not observing quintillions of people. We are observing just 7 billion people. If the Doom Late hypothesis is true, one may derive that the mankind will grow by another factor of one billion. But if we can derive it, then it's not unlikely at all that the current population is just 1/1,000,000,000 of the overall history's population. Instead, it is inevitable: $p=1$. So the suppression by the factor of 1 billion is completely irrational, wrong, idiotic, and stupid.

The only theory in which it makes sense to talk about quintillions of people – the Doom Late theory – makes it inevitable that the people aren't distributed uniformly over time. Instead, they live in an exponentially growing tree. So there's manifestly no "intertemporal democracy" between them that could imply that we're equally likely to be one of the early humans or one of the later ones. We're clearly not. It is inevitable that in most moments of such Universe's history, the number of people who have already lived is a tiny fraction of the cumulative number of the people in the history (including the future).

Aaronson offers another idiotic argument that may sometimes be heard. A valid objection to the Doom Soon conclusion is that it could have been done by the people in the world when the population was just 1 million or another small number – e.g. by the ancient Greek philosophers. And they would have been wrong: the doom wasn't imminent. Aaronson says that it doesn't matter because "most" of the people who make the argument are right.

But again, this is completely irrelevant. Whether most people say something is an entirely different question from the question whether it's right. And indeed, in this particular case, we may show that the probability is very high that the "majority" that uses the anthropic arguments is wrong! What's important is that the methodology or logic leading to the "doomsday is coming" conclusion is invalid as a matter of principle. It doesn't matter how many people use it! One can't or shouldn't invent excuses why these arguments are flawed by saying that some quintillions of completely different (and much less historically important, per capita) people at a different time would reach a valid conclusion. I don't care. I want to reach a correct conclusion myself and I don't give a damn whether some totally different people are right. Of course that they're mostly wrong.

Anthropic principle and a loss of predictivity: what is the real problem?

At the end, it's mentioned that the anthropic principle is often criticized for its inability to predict things. It's indeed unfortunate if a theory makes no prediction. But it's not a valid logical argument against a theory. The correct theory may make much fewer or much less accurate or unambiguous predictions than some people might hope!

The actual problem – one that may be used as an argument against the anthropic principle – is sort of the opposite one. A valid argument is that the alternative explanations that are more accurate, tangible, and predictive have not been excluded. There may be an old-fashioned calculation of the value of the cosmological constant, $\Lambda\sim 10^{-123}$. And science proceeds by falsification of the wrong theories, not by "proofs" of correct theories.

We know that the anthropic explanation would have been wrong as an explanation of – now "materialistically" understood – features of Nature simply because we have better explanations that we know to be much more likely to be true than the anthropic one. And the same thing may happen – and, I think, it is likely to happen – in the future, too. If you can't really show that this expectation is wrong, you shouldn't pretend that you have proved it!

Perhaps, science will be forced to switch to anthropic arguments because beyond a certain point, there just won't be any old-fashioned explanations. Maybe quintillions of people will live in that future world and the claim that the "open problems are explained anthropically" will therefore be true for a "majority" of the mankind that will have lived throughout the history. But that won't change the much more important fact that the anthropic principle will have been wrong throughout the whole previous history of physics.

Aaronson is clearly close to all the anthropic misconceptions discussed above – which may be correlated with herd instincts, mass hysterias, "consensus science", and other pathologies. This is also manifest in his humiliating comments about the role of Adam and Eve. Well, I don't want to discuss the literal interpretation of the Bible which I don't believe, of course. But he wants to suggest that there is some uniform measure that makes it less likely to "feel that I am an early human".

But this is just totally wrong. There is absolutely no justification for such a uniform measure and because the population was growing pretty much exponentially (demonstrably so), this fact indeed pretty much allows us to prove that each early human was exponentially more important than the current ones and we're more important than the future ones.

In recent years, I got sort of interested in the history, e.g. the local history, and I studied the villages etc. that existed on the territory of Pilsen and in its vicinity. There were just hundreds of people and a few lousy houses and the folks didn't have almost anything but they were clearly very important because the hundreds of thousands of people who live here today have arisen from the small number of ancestors. So each of the ancestors is just much more important than an average contemporary human in the overall historical scheme of things. "Adam and Eve" were clearly even more important, if I express it in this way.

If we divide some consciousness or soul or something based on the spiritual importance, it's totally plausible to say that Adam and Eve (plus Jesus and His close relatives, or whoever counts) have 50% of it and the rest is divided among later humans, if you allow me to express the point more concretely than what is really possible. The argument "I can't be special or early because it is unlikely due to some uniform measure on the history's humans" is completely wrong. It is acausal, it uses mathematically non-existent measures, and it uses uniform measures that have no justification and that sometimes contradict legitimately calculable measures.

So I agree with Scott Aaronson that the anthropic reasoning may be defined as some part of probability theory that is more about feelings and opinions than about solid results. Well, most of the people – including Aaronson himself – clearly end up with completely wrong arguments and results which is just another way to disprove the anthropic principle. ;-) I can't be generic because almost all people seem to be morons. In fact, even these would-be generic people may use the same arguments because almost all of these stupid folks are still much smarter than generic insects and bacteria that are far more numerous. The whole idea of "considering oneself generic in a set" is just a way to contaminate a correct or rational argument or result by an incorrect or irrational one that is believed by the inferior life forms.

Chapter 19 is about the free will.

I found it amusing and agreed with what he had to say. He starts by pointing out some errors of free will supporters as well as foes – the "absence of free will means that all criminals have to be liberated" (silly: we sometimes punish toxic machines even though they don't have a free will!) and "undetermined implies random" (not the same thing).

Then he discusses childish examples with a Predictor who knows how you will act (that shouldn't be possible if the free will really exists) – Robert Nozick has played with such things. And finally, he gets to the Conway-Kochen free-will theorem which I was waiting for. There's an elementary, useful explanation what it says and how it guarantees quantum-certified random numbers.

Chapter 20 is on time travel.

Time machines are primarily wanted because they could speed up computation, something that isn't too important for your humble correspondent, especially because time machines and (macroscopic) closed time-like curves are impossible. The complexity chatter is legitimate maths but it doesn't turn me on. I still see that discipline as a conglomerate of many largely disconnected results with connections that are at most ad hoc. It must be easy to get lost in that jungle.

I must say: it's crazy that people like Peter Shor listen to these talks, how you compute something quickly by constructing a time machine and allowing your grandfather to have sex with your grandmother. Then he leaves the MIT seminar room and criticizes string theorists for not being sufficiently down-to-Earth and connected with the observations. Holy cow! Who is disconnected here?

The closed time-like curves are impossible for various deep reasons and one could discuss it. But Scott Aaronson chooses a different attitude, one of a spoiled frat who screams "I want them I want them I want them because I want fast computers!" The links to physics – which are an important motivating theme of the book – seem mostly bogus to me because he's ready to ignore the physics insights whenever it's appropriate to study abstract problems about "computation in the worlds with totally different laws of physics than ours".

Chapter 21: cosmology

This chapter shows that Aaronson has a pretty good background in cosmology. In most cases, it's pretty manifest that he got this knowledge from conversations with physicists and cosmologists (in many cases folks I know rather well) but that doesn't change my feeling that his presentation of the energy density in the universe, expansion, entropy of the Universe, and the holographic principle including things like Bousso's light sheet is more accurate, complete, and meaningful than what most actual experts would be able to write down.

In this chapter I noticed, perhaps even more strongly than in the previous chapters, that the switching between these physics topics and the omnipresent topic of complexity classes is somewhat unnatural – that Aaronson must also realize that even this chapter is constructed out of two largely disconnected topics. Physics and cosmology impose certain laws and limitations on all the objects inside – humans, computers, cucumbers, and everything else. Everyone must respect them and someone who proposes new computers or cucumbers should better learn about the laws. But one can't learn a sufficient amount about the laws just by discussing what kind of a computer we would like. The limitations imposed by the mathematical insights summarized in computer science belong among the limitations but they don't exhaust the full list because there's also physics that imposes constraints on what the mathematicians and computer scientists label as indisputable (and pretty much arbitrary) axioms.

As long as we talk about computers in the real world, the laws of physics/Nature are always primary and fundamental. Aaronson seems to implicitly ignore this fact at many places.

Chapter 22: answering all students' questions

The last lecture in 2006 – when he began to write the book – had the same format as the last lectures in courses by Feynman: the instructor could have been asked any question by students and was turned into an oracle. There are fun questions in the list. In some of them, Aaronson just reiterates his opinions about unsettled conjectures on the complexity classes (will they be proved or disproved in the future?). But there are also speculations on laws transcending quantum computers and their limits and so on.

In his answer to the last question, Aaronson suggests that computer scientists could be working in physics departments. It's just a historical accident, we hear, that they're not there. Well, I don't think so. It's applied maths. They're not really learning mechanics, field theory, and so on, because it's not needed. And they don't really care much whether some axioms they build upon may be realized in the Universe around us. So it's not physics. QCD or string theory are very far from mechanics but people doing it still start by learning and then are building on foundations that do include the characteristic subdisciplines in physics. They have to because the subdisciplines of physics are tightly connected to a compact whole. Computer scientists are doing something else – not trying to find out the ultimate underlying laws ("axioms" in the language of mathematicians) but choosing arbitrary axioms, regardless of their agreement with the empirical data, and getting interesting results and conclusions out of them. So it's maths, not physics.

Of course that with some very inclusive definition, all quantitative thinkers or all scholars or perhaps all employed people are doing some "generalized physics". But I don't think it's right to promote this inflation and degradation of the word "physics".

My (undergraduate) Alma Mater, the Faculty of Mathematics and Physics of the Charles University in Prague, is vaguely divided to the sections Physics – Mathematics – Computer Science – Teaching of M/Ph/CS. So computer science is de iure put on par both with mathematics and physics as an independent branch. Still, it shares the floors and buildings with some kind of mathematicians (and especially philosophers of mathematics and set theorists), not with physicists, and for good reasons.

Will you help John Cook "quantify the consensus"?

2013-05-03T06:05:00.001+02:00

An Australian man concerned with climate change, John Cook, has sent me, because of this "one of the more highly trafficked climate blogs on the web" you are just reading, the following link:

Survey of Peer-Reviewed Scientific Research (University of Queensland)

I suppose that the gibberish characters at the end of the URL will be used to identify the TRF readers.

If you have fifteen minutes or so, you may try to be asked about 10 randomly chosen (unverified) abstracts of 1991-2011 papers from "Web of Science" and whether or not (at the scale 1-7) they confirm the "consensus" on the hysterical man-made global warming.

John Cook boasts to be the leader of this "research" and to have passed some ethical guidelines at the aforementioned university (which are likely to be either ignored in spirit or unethical by themselves).

I suppose this is an important part of his campaign meant to distinguish himself from a similar "researcher" Stephan Lewandowsky who had to flee Australia after his fraudulent papers attempting to identify climate skeptics with conspiracy theorists and kooks.

You may also leave your e-mail at the bottom of the survey if you want to be sent results and you may send a message to the survey's overlords.

I have rated my abstracts. The grades were 3,3,4,3,4,5,3,6,3,5: the average is exactly 4 which means neutral. As far as my ensemble goes, even the highly biased and sometimes low-quality literature – with lots of social sciences and other pseudosciences just blindly looking for a problem, taking the natural scientific claims about AGW for granted – is undecided when it comes to the question whether CO2 is dominant.

Some papers in my list discussed a variety of external forcings. The last paper showed a significant contribution of cosmic rays on the climate. At any rate, the idea of a pro-AGW consensus in the literature seems preposterous to me.

By the way, two weeks ago, NASA showed that CO2 and NO are cooling the thermosphere (upper atmosphere), at least during solar storms, by reflecting some of the solar radiation back to space. I think that Sullivan et al. misinterpret the findings, however. It's about the thermosphere – 85 km above the surface – so it has no direct implications for the temperature near the surface (it has more impact on visual fireworks we can see). In fact, it's rather typical that even the stratosphere (more than 10+ km above the surface) has the opposite temperature trends than the troposphere (where the weather takes place).

A bonus off-topic picture, via Wolfram Blog. The four images show the imaginary part, real part, argument, and absolute value of the Roger-Ramanujan function (given by an infinite continued fraction expression) in the unit disk where it's naturally analytic. Mathematica may be used to explicitly calculate some values of this elegant function – both numerically (e.g. by testing 10,000 digits of the numerically calculated result with a conjectured compact form) as well as symbolically prove that it's the right result, thus going beyond what Ramanujan (and Hardy etc.) could accomplish.