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Alternate memes drawn by hand. http://canv.as/p/vy9mo.

What a beauty. 

...aside from supersymmetry...

Apologies if the title was too dramatic. Supersymmetry and its possible looming discovery at the LHC has been discussed many times. But let's look at a different portion of modern model building in particle physics, Grand Unification. Grand Unified Theories (GUTs) are those that embed the Standard Model group\[

SU(3)_c\times SU(2)_W\times U(1)_Y

\] into a larger group, typically "simpler" (having fewer factors), ideally "simple" group in the technical sense (one factor). If it is possible, and it possible, the advantage is that the quarks and leptons may arise in a smaller number of multiplets (representations of the gauge group) and the lower number of factors in the gauge group implies a smaller number of adjustable coupling constants. So the GUTs are more constrained.

If they're verifiably right, they're more beautiful. But because they're more predictive, one may also be worried that they're less flexible and therefore less resilient towards falsification – a characteristic you may consider good or bad. However, the flexibility may be restored by adding various stuff and potentials at the GUT scale so the "qualitative difference" is somewhat debatable. At any rate, from a purely theoretical or aesthetic top-down perspective, it's rather natural to expect that Nature may want to unify the forces near the fundamental scale if She can.

And yes, She can. Many grand unified theories are naturally compatible with everything we may observe at low energies of doable experiments.




Grand unification has been discussed many times on this blog. In this text, I want to be somewhat more specific about the possible choices of the gauge groups and representations and possible particles that could show up at the accelerators (which is generally unlikely as the GUTs are associated with an insanely high energy scale, the GUT scale, which is not far from the Planck scale linked to quantum gravity).

\(SU(5)\): Georgi-Glashow model

The simplest gauge group we may pick is \(SU(5)\). Note that the rank – the maximum number of mutually commuting independent \(U(1)\) subgroups – is equal to four, much like for the Standard Model group. The Georgi-Glashow gauge group has dimension equal to \(d=5^2-1=24\). The adjoint representation, which is therefore 24-dimensional, may be decomposed into representations of the Standard Model subgroup:\[

24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}

\] In the parentheses, the first number represents the representation in which the piece transforms under the \(SU(3)_c\) colorful group of QCD. It may be a singlet (doesn't transform at all), a triplet (like quarks), antitriplet (like antiquarks), or octet (like gluons, the adjoint representation). More complicated representations may appear, too.

The second number in the parentheses is the representation under the \(SU(2)_W\) electroweak group. Note that the doublet \({\bf 2}\) is equivalent to its complex conjugate representation in this case – because the representation is pseudoreal (there's only one inequivalent spinor representation in 3 dimensions). The adjoint representation is 3-dimensional. The subscripts indicate the hypercharge i.e. charge under the \(U(1)_Y\) factor of the Standard Model gauge group.

You see that on the right hand side of the equation above, you find the adjoint representations of \(SU(3)_c\), \(SU(2)_W\), and \(U(1)_Y\). There are two new pieces that are complex conjugate to each other – they come with new gauge bosons that should however be insanely heavy (not accessible at the LHC) if the proton is supposed to preserve its longevity we know and love. The new bosons transform as color triplets as well as electroweak doublets. The latter fact implies that they have charges like an exotic quark doublet. We sometimes talk about new X-bosons and Y-bosons.

It's equally interesting to find out whether the leptons and quarks may be rearranged into representations of \(SU(5)\). The answer is YES and this fact is nontrivial. Let's look at all left-handed two-component complex spinor fields that are associated with a single generation of quarks and leptons (the whole field content or particle content is tripled at the end because there are three generations; and the Hermitian conjugate fields are added as well).

We find 15 such spinor fields in total: 2 (up, down: organized as an electroweak doublet) left-handed quarks, each in three colors (6 in total), 2 left-handed antiquarks (separate electroweak singlets), each in 3 colors (their Hermitian conjugates are right-handed quarks; again 6), left-handed electron and left-handed neutrino (they form a doublet), and a left-handed positron (an electroweak singlet). In total, \(6+6+2+1=15\). We may also add the left-handed antineutrino (so far unobserved and insanely feebly interacting if it exists) to raise fifteen to sixteen.

These 15 spinor fields are reorganized as the following representation of \(SU(5)\):\[

\mathbf{\bar{5}}\oplus\mathbf{10}\oplus\mathbf{1}

\] The decomposition of these pieces under the Standard Model group is\[

\eq{
\bar{5}&\rightarrow (\bar{3},1)_{\frac{1}{3}}\oplus (1,2)_{-\frac{1}{2}}\\
10&\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{-\frac{2}{3}}\oplus (1,1)_1\\
1&\rightarrow (1,1)_0
}

\] The last line, the left-handed antineutrino (whose Hermitian conjugate is the right-handed neutrino), is optional or uncertain, as we have mentioned. The first line contains the lepton doublet and the \(d\)-antiquark singlet (\(d^c\) and \(\ell\), to use some symbols) on the right hand side. The second line has \(q\), \(u^c\), and \(e^c\). The \(c\) superscript refers to the antiparticles (it stands for "charge conjugation"). Let me emphasize in advance that the clumping of the particles into these larger representations is inequivalent in flipped \(SU(5)\) models.

The \(SU(5)\) grand unified group may be broken down to the Standard Model if you realize that the Standard Model group is exactly the subgroup that commutes with the generator \(U(1)_Y\), the hypercharge. Up to an overall normalization (that sometimes omits the factor of \(1/2\)), it is given by the following traceless \(5\times 5\) matrix:\[

\frac{Y}{2} = \left(\begin{array}{rrr}
-\frac 13&0&0&0&0\\
0&-\frac 13&0&0&0\\
0&0&-\frac 13&0&0\\
0&0&0&+\frac 12&0\\
0&0&0&0&+\frac 12
\end{array}\right)

\] Equivalently, it is the subgroup that keeps a \(Y\)-like vev of a new GUT Higgs field transforming in \({\bf 24}\), the adjoint of \(SU(5)\), invariant. One may also achieve the breaking by Wilson lines (monodromies around non-contractible loops, to be more accurate) in stringy models with extra dimensions and by other means.

The minimum grand unified model predicted a rather slow but not too slow proton decay. Within a year or so, the prediction was ruled out. The simplest grand unified theory has been falsified. The broader concept hasn't been falsified and it has way too many positive features so that we shouldn't think we will kill it too quickly.

Flipped \(SU(5)\) i.e. \(SU(5)\times U(1)\)

The flipped \(SU(5)\) models are the "most comparable ones" to the Georgi-Glashow model. They're associated with the names Dimitri Nanopoulos, Stephen Barr, Ignatios Antoniadis, John Ellis, and John Hagelin and their research in the early 1980s. The realization of this previously overlooked possibility was linked to some string theory research in the 1980s and the 1990s: string theory arguably makes the flipped \(SU(5)\) models more natural than the Georgi-Glashow model.

Note that Sheldon Glashow had already omitted a \(U(1)\) factor once when he tried to construct a purely \(SU(2)\) theory of the electroweak force so it wouldn't be shocking if he has displayed the same excessive fanaticism for minimalism together with Howard Georgi again. ;-)

You could think that that \(SU(5)\times U(1)\) must be just a complicated extension of the original \(SU(5)\) in which we add possibilities and allow things to be deformed. You could think that the Georgi-Glashow model is a special case of the flipped \(SU(5)\) models. But you would be wrong. The flipped \(SU(5)\) models are qualitatively different from their unflipped cousins – and they're "equally robust", in some sense. Their arrangement of quarks and leptons into the \(SU(5)\) multiplets is permuted – therefore "flipped" – relatively to the Georgi-Glashow model.

To mention a visible example of the differences, note that the right-handed neutrino \(SU(5)\) singlet was optional in the Georgi-Glashow model. In the flipped \(SU(5)\) models you also have to use \(\bar{5}\oplus 10 \oplus 1\) but the last, singlet piece isn't optional at all. For a simple reason: it is not a neutrino. It is actually the left-handed positron! We surely need it. So the actual fermionic content we have to add in the flipped \(SU(5)\) models comes in the representation\[

\bar{5}_{-3}+10_{1}+1_{5}

\] where the subscripts indicate the charges under the \(U(1)\) factor of the flipped \(SU(5)\) gauge group. These subscripts are extremely natural from the viewpoint of an \(SO(10)\) or \(Spin(10)\) group in which \(SU(5)\times U(1)\) may be embedded. In that group, the three pieces combine to a 16-dimensional complex spinor representation; the \(U(1)\) charge is proportional to the sum of all the five weights. It is fair to say that the flipped \(SU(5)\) models become more natural than the ordinary Georgi-Glashow \(SU(5)\) if you look "beyond \(SU(5)\)" e.g. at the \(SO(10)\) models.

The decomposition of the relevant flipped \(SU(5)\) representations under the Standard Model group is\[

\eq{
\bar{5}_{-3}&\rightarrow (\bar{3},1)_{-\frac{2}{3}}\oplus (1,2)_{-\frac{1}{2}}\\
10_{1}&\rightarrow (3,2)_{\frac{1}{6}}\oplus (\bar{3},1)_{\frac{1}{3}}\oplus (1,1)_0\\
1_{5}&\rightarrow (1,1)_1\\
24_0&\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{\frac{1}{6}}\oplus (\bar{3},2)_{-\frac{1}{6}}
}

\] The subscripts on the right hand side are the values of the hypercharge. The last line is the adjoint representation again; there is an extra generator of the \(U(1)\) aside from this line, too. The first line describes the up-antiquark left-handed singlet and the lepton left-handed doublet. The second line describes the quark left-handed doublet, the down-antiquark left-handed singlet; and the left-handed antineutrino (unobserved). The third line is the left-handed positron singlet and can't be omitted.

Not too grand unified groups

Let me mention that there are other GUT-like gauge groups that are far from simple and that have been proposed with various justifications. The electroweak gauge group only treats the left-handed particles as doublets. One may add another \(SU(2)\) under which the right-handed particles are doublets. These left-right models have the gauge group \(SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}\).

The Pati-Salam model has the gauge group \(SU(4)\times SU(2)\times SU(2)\). The two \(SU(2)\) factors are analogous to those in the previous paragraph. However, \(SU(3)_c\) is embedded into an \(SU(4)\) where the "quark of the fourth color" is identified with a lepton.

Also, \(SO(10)\) models may be flipped to \(SO(10)\times U(1)\). The electroweak \(SU(2)_W\) may be extended to another \(SU(3)\) in the 331 models (Paul Frampton was a pioneer). \(SU(6)\) models have been constructed, too.

Trinification has a gauge group composed of three equal pieces, \(SU(3)\times SU(3)\times SU(3)\).

Truly large and simple GUT groups

However, the truly motivated "master gauge groups" are \(SO(10)\) and \(E_6\), an exceptional group. Note that the \(E_6\) group is the only exceptional group that has complex representations at all – something we need to treat left-handed and right-handed fermions separately.

This \(E_6\) is naturally embedded into the \(E_8\) that arises in the heterotic string and heterotic M-theory and it contains pretty much all the GUT groups above as a subgroup, including the group of trinification. The decompositions of all the representations under various subgroups are interesting and you should try to find all of them if you haven't done so. Also, \(SO(10)\times U(1)\) – where the \(U(1)\) may participate or not, depending on whether we have flipped models – may be embedded into \(E_6\).

The fermionic content of \(SO(10)\) models combines into the 16-dimensional spinor representation of \(Spin(10)\) I have already mentioned. But it's even more interesting to see what happens for \(E_6\) because the minimum nontrivial representation is the 27-dimensional fundamental representation, \({\bf 27}\). This inevitably contains some new spinor fields aside from the quark and leptons we know (and different from the right-handed neutrino, too)!

The decomposition of this representation under \(SO(10)\times U(1)\) is\[

27\rightarrow 16_{1}+10_{-2}+1_4.

\] Note that the trace of this \(U(1)\) in this 27-dimensional representation is \(16\times 1-10\times 2+ 1\times 4 = 0\). It has to vanish because it's the trace of a generic generator of a non-Abelian group, \(E_6\).

Now, the \(U(1)\) labels are funny. You may notice that only the 16-dimensional spinor of quarks and leptons has an odd label. If you correlated this \(U(1)\) charge with the R-parity in supersymmetric models (instead of using \(B-L\), and this is my idea), you could say that we will easily observe fermions from the 16-dimensional representation (we do) but bosons from the remaining, 10- and 1-dimensional representations!

So it's funny to ask what are the Standard Model charges of the additional representations. If you use the flipped \(SU(5)\) models, you will find out that \({\bf 10}\) decomposes into \({\bf 5}\oplus \bar{\bf 5}\) and these pieces contain both Higgs doublets of the MSSM (but this field content is tripled because there's a pair of Higgs doublets in each of the three generations!) plus a new electroweak singlet squark with the charges of the down-squark.

Symmetry has been a powerful tool in our successful efforts to understand Nature at least for a century. Supersymmetry tells us that we may expect superpartners of all known particles – although some of them may be much less accessible than others. But we could actually find traces of another, ordinary "bosonic" symmetry, by finding the (broken) symmetry partners of the known quarks and leptons.

For example, if the LHC ultimately started to find the new Higgs doublets predicted above as well as an exotic quark or three, it would be a huge hint of an underlying greater symmetry operating in Nature. Many of these models are heavily constrained and virtually all models with new light charged particles destroy the simplest "intriguing accident" of gauge coupling unification. But I view the simplest match for the MSSM gauge coupling unification to be just a 2-sigma intriguing bump. It may still be a misleading guide and there may be many more fascinating things we may discover if we sacrifice this one. Of course, it's up to us what we believe but on a sunny day in the future, Nature may very well force us to sacrifice the belief in the simplest scenario of gauge coupling unification.

So far, no official discovery of physics beyond the Standard Model has been announced by the LHC; just to be sure, the Higgs boson is almost as old physics as Peter Higgs himself. But this situation isn't guaranteed to last forever. It isn't even guaranteed to survive the Moriond 2013 conference in early March.

Stay tuned.

This is an extremely, extremely light topic. Popsci.com wrote a new article

5 Of Physics's Greatest Sex Scandals

A TRF guest blogger finds himself in a pretty good company.




Their list is the following:

• Paul Frampton and his sweetheart in Argentina, a cocaine-equipped fake Czech-born model Denise Milani (a recent news video at YouTube)
• Albert Einstein had a relationship with his cousin Elsa already when he was married to his hard-working first wife Mileva Marić
• Marie Curie who fell in love with his freshly dead husband's ex-student, Pierre Langevin. The French media called her a homewrecker and a Jew although she was neither and although the latter couldn't have possibly been insulting, anyway
• Erwin Schrödinger lived both with his wife and mistress, and he's had several examples of the latter over the years
• Stephen Hawking frequents sex clubs; any problem with that? And I think that Hawking must also be considered an extraordinary experimenter because he's been capable of having children despite his slightly constrained physical powers

Congratulations to the winners.

A press release from FSF Europe (issued November 20) welcomes a white paper from the German federal government on trusted computing and secure boot. "Another demand by the FSFE is addressed by the government's white paper. That before purchasing a device, buyers must be informed concisely about the technical measures implemented in this device, as well as the specific usage restrictions and their consequences for the owner: 'Trusted computing security systems must be deactivated (opt-in principle)' when devices are delivered… And 'Deactivation must also be possible later (opt- out function) and must not have any negative impact on the functioning of hard- and software that does not use trusted computing functions.'" The white paper is in essence a non-binding call to manufacturers, but is significant as a statement from a major national government against restrictions imposed via secure boot that may foreshadow more significant government action. The white paper is available in both English and German.

an image from fukung.net

Decades before de facto relaxation of cross border genre regulations, Serge strolled with protean disregard and somewhat arch insouciance through chanson, lounge jazz, world music (sic), ye-ye, pop psych, rock, dub, disco, funk et caetera.... Wanton profligacy did him no favours outside France: artistic fearlessness and a shrug of Gallic shoulders to misconceived Anglo-US notions of authenticity interpreted abroad as dilettantism and promiscuous cheese.


Actor, writer, director, producer, pop svengali, performer, musician, smoker, drinker and, at the end, dilapidated national treasure. Above all else, 20th century composer and lyricist nonpareil who considered writing songs a trifling art form compared to figurative art. Ever the accomplished painter and draftsman who gave up painting because he knew he would never equal Picasso or scatological comrade in arms, Dali. 


Franchement, Gainsbourg / Gainsbarre was a genius; not to be taken lightly.



High watermark must-have is Histoire de Melody Nelson. Go buy it. Cannabis soundtrack - also with Jean-Claude Vannier  - is something of a dry run for Melody Nelson (grooves, break beats, larger guitar riffs, orchestral drama, funked out bass...) and an essential gem in its own right.



Oddi wrth y brawd


[Bag Birkin Bonus in Comments]

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________________________________________

Just offices and cemeteries

It's refreshing that I may sometimes fully agree with a text by Matt Strassler:

Why Theories Don’t Go Into Hospitals

BBC's Pallab Ghosh has quoted Christopher Parkes of LHCb who has said "Supersymmetry may not be dead but these latest results have certainly put it into hospital."



Even if one (or two) gets into a hospital, it doesn't mean he's not a supersymmetric hero (a superhero for short). A shoulder surgery isn't the end of the world.

But nothing like that is possible in science.




Supersymmetry is a symmetry, a principle added to the list of conditions we expect from models. But it is not a particular model. There are many supersymmetric models or supermodels for short.



Some of them, like Petra Němcová, were named envoys for Haiti and they're much more than richly decorated hollow skulls. ;-)

But let's not get distracted too much. What I want to say is that one must carefully distinguish the confirmation or falsification of a particular model; and the confirmation or falsification of the whole principle or framework. These are totally different things. Moreover, in contrast with the opinion of Pallab Ghosh and his sources, the framework doesn't get falsified or "nearly falsified" if you falsify 80% or 90% of the particular models.

Of course, I have discussed the same issue many times, e.g. in Bayes and SUSY (May 2012).

If you falsify 90% of the models (SUSY models) in a framework (SUSY), and it's very hard to quantify the fraction because we deal with continuous, noncompact parameter spaces equipped with an ill-defined measure, the belief that the framework is still right requires the believer to think that he had a bad luck that only occurs in 10% of cases.

But it's not such a big deal to believe this modest "bad luck". It's equivalent to less than 2-sigma "deficit" arguing against your general point – in this case, I mean supersymmetry. But less than 2-sigma excesses and deficits are almost everywhere. They're just not terribly strong arguments for assertions either way. The other, partly theoretical arguments for and against SUSY are arguably much stronger than that.

The idea that theories – in this case SUSY – may be sent to hospitals is based on an application of "collective guilt" to models. One assumes that supersymmetric models represent a nation or a family and they share the pain of each other. So if some of the relatives – models – are killed, the others suffer.

But this ain't the case. Two models may share some properties – for example, they may respect the principles of broken supersymmetry – but their truth values are completely independent. In fact, they are negatively correlated because if one model is right, we know that the other, inequivalent model must be wrong! This negative correlation in the truth values is there regardless of any similarities in the assumptions or technical properties of the models.

Despite all the similarities between the models, the death of another model is always a good news for a model that hasn't been killed simply because the competition gets less severe. Moreover, the "clustering" of the models into the "families" that someone may prefer is artificial and there can be many other ways how to organize physical models into "families". The experimental signatures that models predict (e.g. lots of events with many leptons) may often be more important than their deepest assumptions (such as SUSY). No reliable conclusion may depend on an arbitrary way how one arranged the models into "families".



Car accident

Let me give you two similar examples showing why the "hospital" idea is logically flawed. You take a bus to go to a nice trip. In a car accident, 90% of the passengers in your bus get killed. You were a bit lucky and avoided all injuries. Now the question is:

Should you be taken to the hospital?

The answer is obviously No. You shouldn't be treated as an ill person. The past proximity to several people who were killed by a truck going in the opposite direction isn't a disease – let's ignore the psychological shocks you may have experienced (but frankly speaking, I don't really believe that the treatment of people with such shocks is too sensible or useful, either).

Your life goes on even though you have belonged to a group of people whose majority is gone. After all, all of us belong to a group – all people who have ever been born – whose majority is already dead. While the current world population is 7 billion people, the total number of people who have ever walked on the globe is significantly higher, over 100 billion. So over 90% of the people who have ever been born is dead by now. Does it mean that your existence and health is a contradiction? I don't think so. The whole life is about the circulation of the material between dead and alive organisms, it's about the selection.

Science is totally analogous to life. Evidence falsifies some theories – counterparts of life forms, species, and individual organisms – that were not sufficiently viable and it focuses the confidence and probability – a counterpart of the resources on the Earth – to those that have survived. In this way, the theories are getting more accurate, more sophisticated, more viable – much like the species and organisms.

It's completely incorrect to say that the people who live today are not viable just because they belong to the group of 100 billion people most of which are already dead.

Higgs search and elimination of possibilities

My second example is the search for the Higgs boson. Let's look at the situation we were experiencing months before December 2011 when the confidence level for the 126 GeV Higgs boson surpassed 4 sigma and sensible people became pretty much sure it was there.

Before December 2011, experimenters were only able to eliminate intervals of masses that the Standard Model Higgs boson couldn't have (let's assume the Standard Model is right – or at least a relevant approximate step in our improving knowledge).

A priori, the Higgs boson mass could have been anything between 0 GeV and 1,000 GeV. The prior probability that the mass would be above 600 GeV was already small, for various reasons. So let's shrink the window to 0-600 GeV. By 2011, a vast majority of this interval was eliminated. Around Summer 2011, only the interval 115-130 GeV remained viable. But no Higgs was discovered by a stronger-than-3-sigma signal yet.

Now you may think about it and say that it was strange. The Higgs had not been discovered yet and only an interval of width 15 GeV – 1/40 of the overall interval 0-600 GeV – remained as possible. Pallab Ghosh could have said that 39/40=97.5% of the Higgs boson idea had been falsified. The Higgs boson as an idea should have been taken to an intensive-care unit, the BBC should have written.

(If you decide that it's natural for the Higgs mass scale to be any number between 0 and the GUT scale, 97.5% could even be replaced by 99.99999999999999999%. Well, it should be over thirty digits "9" because the squared mass is what could be uniformly distributed)

But we know it would be a completely wrong conclusion. The Higgs boson was there, somewhere in the remaining interval. There has never been any good reason to doubt that some Higgs boson had to exist. The gradual shrinking of the "habitat" wasn't a sign of the Higgs boson's deteriorating health. Instead, it was a gradual improvement of our knowledge of Higgs' properties.

Elimination is easier than discovery

If you think about the numbers, you will easily understand why it's pretty ordinary that new particles are usually discovered after a big majority of the parameter space has been eliminated. The reason is simple: it's easier to eliminate a point in the parameter space (assuming that the point is really wrong) than to discover something in it (assuming that it's there). Why?

Well, the reason is simple. Physicists are usually satisfied with the 95% confidence (2-sigma) level exclusion but they demand a 99.9999% (5-sigma) level discovery. Now, 5 sigma is 2.5 times greater than 2 sigma but the number of collisions (or something else) scales as the second power so you need 6.25 times more "data" for a discovery than you need for the exclusion at the same point.

So if you assume that Nature sits at a generic point, you may make the following estimate. Find the moment at which about 50% of the parameter space is excluded. Multiply the amount of data collected by that moment by the factor of 6. And you will get an estimate of the amount of data that's needed for the discovery.

Now, this is just an estimate, not a strict rule, of course. The actual amount of data you may need may be 10 times smaller or 10 times greater than data and it's still not shocking. But if you apply those numbers to the Higgs boson or supersymmetry, you will realize that there was no reason to be "worried" about the general Higgs boson idea in the Fall 2011; and there's no reason to be "worried" about the general idea of supersymmetry today.

People should try to think a bit rationally and realize that the "collective guilt" principle can't be applied to physical models because the "clustering of theories into collectives" is completely artificial, man-made, and inconsequential for the validity of individual models. The fates of individual models are independent. And if you need some strong enough negative evidence against a whole framework, you need to eliminate 99.7% (3-sigma equivalent) or 99.9999% (5-sigma equivalent) of the parameter spaces. The elimination of 90% of a parameter space doesn't give us much useful information. It is only as powerful an argument as any other 1.5-sigma bump seen anywhere.

And that's the memo.


Brahe's health

Exactly two years ago, I described a Danish research focusing on Tycho Brahe's remains in Prague. He could have been murdered by mercury etc., perhaps even by Johannes Kepler himself. Today, BBC tells us that the Danish+Czech research is over. There was mercury in the beard but it was normal, not deadly. Moreover, Kepler's description of Brahe's declining health "matched a severe bladder infection".

Kepler has been great but I, for one, wouldn't consider the stories written by a prime murder suspect as uncritically as they did.

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Judy Funnie is a hipster…