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Sasha Polyakov joins the Milner Prize winners

  • Off-topic, cosmology: Planck, the WMAP's successor, has released the data: press release, BBC, published papers, real time, data, ESA live TV. The cold spot is much larger than previously thought and almost certainly not an irrelevant fluke. The composition of the cosmic energy (dark energy / dark matter / baryonic) was modified from 73-23-4.5 to 68-27-5 percent. Jesus' father was born 13.82 billion years before Christ, up from the previous age 13.73 billion years. ;-)

    The Universe when it was 380,000 years old (via Planck)

    High-\(\ell\) variations as well as polarization measurements support the standard ΛCDM theory impressively, too. The theory seems to overestimate \(20\leq\ell\leq 40\) modes; the total discrepancy may be just 2.5 sigma, however. Pure scalar invariance \(n=1\) is already excluded at 5.4 sigma: \(n\lt 1\). See an image summarizing the paper on cosmological parameters. Non-gaussianities, \(f^{NL}=2.7\pm 5.8\), and altered complex inflationary models are being strongly constrained. More conclusions.

  • AMS-02: If you're waiting for the dark matter results of AMS-02, it's plausible that we may have to wait until it's fully published in a journal – a significant delay.
Nine physicists won the inaugural $3 million Fundamental Physics Prize last summer and several winners were added in December.

When he was a bit younger. See his picture with a supplement to the prize, a trophy designed by world-renowned Danish-Icelandic artist Olafur Eliasson. The ceremony at the Geneva International Conference Center took place yesterday and resembled the Oscars, too.

Yuri Milner wants to make everyone who matters rich. But given the finiteness of his wealth, he may be forced to switch to a lower pace at some point and only add one $3 million winner a year. So far, it seems to be the case in 2013. The new winner is spectacular, too: it's Alexander Markovich Polyakov (*1945 Moscow).

I officially learned about the new winner from a press release a minute ago (see also Russia Today, Nude Socialist, CERN, and realtime) but during that minute, I was already thinking what I would write about Polyakov and here is the result.

Polyakov's INSPIRE publication list shows 197 papers with 31,000 citations in total. That's truly respectable but in some sense, it may still understate his contributions to the field.

Up to the mid 1970s, he would work at the Landau Institute in Chernogolovka (Blackhead) near Moscow. You know, Lev Landau was a great physicist and the training that his institute gave to younger Soviet physicists was intense.

At that time, he would already work on two-dimensional field theory although the physical motivation often had to something with the ferromagnets, IR catastrophe caused by 2D Goldstone bosons, and other things. However, around 1975, he would move to the West: he started with Nordita that was founded by Niels Bohr in Stockholm in 1957 and continued with Trieste, Italy. It's my understanding that he has kept peaceful and productive relationships with the Soviet scientific institutions.

For decades, we've known him as a top physicist at Princeton University in New Jersey, of course. Ironically enough, I haven't spent any substantial time talking to him although I was a grad student just 15 miles away, at Rutgers. But I have talked a lot to virtually all of his recent collaborators and I did spend some time in his office. In the late 1990s, they turned it into my waiting room before a seminar. So I looked at the science books, papers, and some materials about the candidates for another prize he was evaluating as a member of the Nobel committee.

(If this information was secret, it should have been hidden more carefully! Just to be sure, I haven't touched any of those confidential papers and I did promise myself to be silent for 10 years about everything I could have seen without touching, too. This commitment wasn't breached.)

Polyakov's physics: AdS/CFT

More seriously, Sasha Polyakov is a giant because he is a string theory pioneer and because he has cracked many phenomena in gauge theories – and be sure, gauge theory and string theory are the two key theoretical frameworks of contemporary high-energy physics – especially when it comes to the phenomena that can't be analyzed by perturbative techniques (by simple Feynman diagrams that work at the weak coupling).

In general, these discoveries allowed the physicists to attack questions that could previously look inaccessible and many of these insights helped to reveal the logic that implies that gauge theory and string theory, at least in some "superselection sectors", are equivalent. These days, we understand many of these insights as aspects of the AdS/CFT correspondence.

The most cited paper (one with over 6,000 citations: this shows how much the holographic correspondence has been studied) he co-authored along with Steve Gubser and Igor Klebanov (GKP) is one about the correlators in the AdS/CFT correspondence. It came a few months after Maldacena's paper and Edward Witten published similar results independently at roughly the same time.

Maldacena's claim that string theory in the AdS space was equivalent to the maximally supersymmetric gauge theory or another theory on the boundary was true but it was missing some "teeth" needed to calculate "anything" we're used to calculate on both sides of the equivalence. The GKP results created a dictionary that allowed to translate the most typical observables – correlation functions – to the other description. Note that the paper talks about the "non-critical string theory" etc. even though it immediately says that this "non-critical string theory" is linked to or derived from the "ordinary" 10-dimensional critical string theory. This is clearly an idiosyncrasy added to the paper by Polyakov himself; the co-authors would surely avoid references to "non-critical string theory" if everything in the physics was fundamentally "critical string theory".

This Polyakov's handwriting is manifest in most of his other papers on string theory in the last 20 years. He had spent quite some time with "general strings" – such as the "flux tubes" in QCD – 20 years earlier and these "formative years" have affected his thinking. But you should have no doubts that these days, he understands that the dualities relate gauge theories to the 10-dimensional string theory today. He just got to that conclusion and other conclusions differently than many other physicists.

For years, Polyakov has been a visionary and he has outlined many visions. They often turned out to be rather accurately reproduced by the actual technical revolutions in the field. But some details of Polyakov's visions were slightly incorrect or insufficiently ambitious, we could say. It's perhaps inevitable for a great visionary that even these aspects of his visions that weren't quite confirmed (the importance of "non-critical string theories" in the holographic dualities is a major example) sometimes leave traces in his papers and talks.

BPZ and two-dimensional CFTs

The second most cited paper he co-authored (with over 4,000 citations) was written in the Landau Institute in 1983 (and published in 1984) along with Belavin and Zamolodchikov; yes, all the three authors were Sashas (I know Zamolodchikov very well, from Rutgers).

When you're learning string theory these days, you must master various methods to deal with the two-dimensional world sheet of the propagating (and joining and splitting) strings. The world sheet theory may look like "just another" quantum field theory, with its Klein-Gordon and other fields. Its spacetime (I mean the world sheet) has a low dimension but you could think that this difference is just a detail. Many beginners who want to learn the world sheet methods expect that they will do exactly the same things as they did in the introductory QFT courses but \(d^4 x\) will be replaced by \(d^2 x\) everywhere. ;-)

However, it's not the case. The theory on the world sheet is conformal. Its physics is scale-invariant (no preferred length scales or mass scales) and it is, in fact, invariant under all angle-preserving transformations, the so-called conformal transformations. When you look at these symmetries in the context of a \(d\)-dimensional space or spacetime, you will find out that they form the group \(SO(d+1,1)\) or \(SO(d,2)\), respectively. (The increased argument in \(SO(x)\) is an early sign of holography.)

This cute music video shows the "ordinary" conformal transformations of the two-dimensional plane that form the group \(SO(3,1)\sim SL(2,\CC)\), the so-called Möbius transformations. They're the only one-to-one conformal transformations of the plane (including one point at infinity) onto itself.

In two dimensions and only in two dimensions (and this is the uniqueness that makes strings special among the would-be fundamental extended objects), the conformal group is actually greater and infinite-dimensional as long as you allow transformations that can't be extended to one-to-one transformations of the whole plane onto itself (if you allow the plane to be wrapped, the maps to be non-simple, and so on). You should remember from your exposure to the complex calculus that any holomorphic function \[

z\mapsto f(z)

\] preserves the angles. This fact may also be expressed by the Riemann-Cauchy equations and has been extremely helpful in solving various two-dimensional problems, e.g. differential equations with complicated boundary conditions in fluid dynamics in 2 dimensions. In string theory, the conformal symmetry allows us to show the equivalence of the plane and the sphere, the half-plane and the disk, the annulus and the cylinder, and the infinitely long tube added to a Riemann surface to a local insertion of the (vertex) operator, among other things. The symmetry is also the reason why the loop diagrams in string theory remain finite-dimensional despite the infinite number of shapes of Riemann surfaces.

When the two-dimensional quantum field theory is conformal, one discovers lots of new laws, identities, and methods to calculate that don't exist for generic quantum field theories and BPZ discovered many of them. Because the theory is invariant under an infinite-dimensional conformal group, or at least its Lie algebra (the Virasoro algebra), one may organize the states into representations of this algebra. Much like in the Wigner-Eckart theorem (where the rotational group plays the role we expect from the Virasoro algebra here), the correlation functions of the operators (which are mapped to states in a one-to-one way) may also be expressed using a smaller number of functions summarized by the so-called "conformal blocks". These conformal blocks are associated with individual representations of the Virasoro algebra and their products' sums yield the correlation functions.

BPZ also realized that the Virasoro algebra is very constraining which makes some 2D conformal field theories fully solvable. So they also pioneered the analysis of the minimal models, using the modern terminology, and other things. I shouldn't forget to make a trivial point. We also often talk about the BPZ conjugate which is a "dual state" that may be defined with the help of some conformal transformations and has a "different logic" but a similar value to some Hermitian conjugates of the corresponding operators. This concept was established by BPZ, too.

Another triplet of well-known authors were Knizhnik-Polyakov-Zamolodchikov. A profound paper on the renormalization of 2D CFTs as theories of gravity and their fractal structure. Vadim Knizhnik, a student of Polyakov, has contributed a lot to theoretical physics even though he died when he was just 25.

March of the high-energy physics students, by Vladimir Vysotsky

Modern covariant BRST treatment of string theory

You may be rightfully unimpressed by someone's being in teams with 3 co-authors. Fine. Already the third most cited paper by Polyakov (one with nearly 3,000 citations) was written purely by himself and it's a truly foundational one: Quantum geometry of bosonic strings (1981). I have already mentioned it in the blog entry 25 years of the Polyakov action in 2005. Yes, I wrote that blog entry 7.5 years ago; this blog has become a museum of a sort by itself. ;-)

Some string theorists (e.g. Michael Green and John Schwarz) loved to use the light-cone gauge to define the theory. (I used to count myself in that group, too. That was one reason why Matrix theory was so familiar to me from the beginning.) While it makes the absence of ghosts (negative-norm states) totally obvious as it has no unphysical degrees of freedom whatsoever, it obscures a part of the Lorentz invariance (although the symmetry is still there and it is still exact, of course). Most string theorists prefer to define (and calculate) string theory in a framework that makes the Lorentz symmetry (the postulates of special relativity in the spacetime) manifest, the so-called covariant approach.

However, covariant formulations of theories almost inevitably lead to various unphysical states (e.g. time-like and longitudinal polarizations of a photon, to mention the simplest QED example) that have to be removed from the physical spectrum in some way. An ad hoc method to do it (even in string theory) was generalizing the Gupta-Bleuler quantization in QED. In string theory, the Virasoro algebra has an extra "central charge", energies have to be shifted by properly adjusted constant additive terms, and unphysical polarizations have to be removed by two additional conditions.

These days, we do all these things in a "more systematic way", using the modern covariant quantization or "BRST quantization". One defines the BRST charge \(Q\) that is nilpotent i.e. obeys \(Q^2=0\) and physical states are cohomologies of \(Q\) i.e. classes of states obeying \[


\] identified by the equivalence\[

\ket\psi\approx \ket\psi + Q\ket\lambda

\] A particular representative of each physical state – each cohomology class – may be identified with a physical state in the "old covariant quantization" which proves the equivalence of the old and new covariant quantization. However, the new one is much more sensible, unified, and systematic. One has to add the \(b,c\) Faddeev-Popov ghosts but when it's done, the conformal algebra is free of anomalies and central charges once again.

(Did you notice how huge a percentage of the aforementioned discoveries was made by the Russians? They sometimes know how to answer questions that the top Western physicists have no clue about.)

Polyakov didn't invent the BRST methods in general but he did realize that they may be applied not only to Yang-Mills symmetries whose symmetry generators are spacetime scalars but also to the conformal symmetry whose generators are spacetime (well, world sheet) vectors (more precisely and locally, the stress-energy tensor on the world sheet). This gave him a new, streamlined derivation of \(d=26\) for the critical dimension of bosonic string theory and a convenient formalism that is preferred by most string theorists to do the perturbative calculations in string theory.

Needless to say, the formalism may be generalized to the \(d=10\) superstring, too. Polyakov himself derived the \(d=10\) critical dimension of the "fermionic string" (usually called "superstring") in another famous 1981 paper. This much for comments that he would dismiss the special status of 10-dimensional string theory.

Most people who learn how to perturbatively calculate things in string theory start with the Polyakov action. It's partly a matter of subjective choices and a popular formalism among several equivalent ones (much like in the case of the Feynman path integrals in general) but it's still very important for the real-world physics to proceed.

Topologically non-trivial solutions in gauge theories

I just discussed a 1981 paper: note that Polyakov was a string theorist before the first superstring revolution (1984), and be sure that the number of people writing important papers about string theory was extremely low at that time, so he should arguably be counted as an early pioneer almost in the same category as John Schwarz and others.

But we return by a few years, to the 1970s, to discuss some insights about gauge theories that Polyakov made at that time. I have already mentioned that Polyakov was very important in cracking some problems in non-perturbative – i.e. strongly coupled, inaccessible using the Feynman diagrams – physics of gauge theories. New solutions such as solitons and instantons are a very important part of this wisdom.

Instantons are named after an "instant of time" because they're objects that are localized not only in one place of the space but also in one moment of time. We mean the Euclidean time. We usually present them as solutions in the Euclidean spacetime whose signature is e.g. \(({+}{+}{+}{+})\). BPST instantons (see Wikipedia) co-found by Polyakov in 1975 are an example. These configurations of the fields are still important because they're still solutions – stationary points of the action – which is why they still importantly contribute to the Feynman path integral although the contribution is suppressed by the factor of \(\exp(-S)\) i.e. \(\exp(-C/g^2)\).


But magnetic monopoles are more well-known to the general public, especially because Sheldon Cooper thought that he had discovered them at the North Pole. I discussed these matters in the article about David Olive but let me explain some of these issues here, too.

Maxwell's equations say that the magnetic monopole sources can't exist because\[

\nabla\cdot \vec B = 0

\] in the vacuum. The divergence is what defines the charge density (recall that \(\nabla\cdot \vec D=\rho\) but it's simply zero in the magnetic case. So only dipoles may exist. However, one may imagine that the right equations are generalized and they have some nonzero right hand side in the equation above, too. In classical field theory, the impact of such a generalization is obvious.

Paul Dirac was the first man who was thinking about the magnetic monopoles in quantum mechanics. In quantum mechanics, we usually describe the magnetic fields in terms of the vector potential \(\vec A\) where \[

\vec B = \nabla\times \vec A.

\] However, if \(\nabla\cdot \vec B\neq 0\), we can't find a globally well-defined, single-valued field \(\vec A(x,y,z)\) that would produce the right magnetic field \(\vec B\). On the other hand, when the magnetic monopole is localized at the point \((0,0,0)\), we may define \(\vec A\) almost everywhere except for a semi-infinite string starting at the origin and going (along an arbitrary path) up to infinity. This semi-infinite string is known as the Dirac string.

Effectively, we are replacing the magnetic "pure North" monopole source by a dipole whose "pure South" side is sent to infinity. We want the "pure South" side to be invisible, along with the Dirac string itself. If you want to make something invisible, sending it to infinity may be a good way to start. However, this trick still poses an extra challenge because the Dirac string effectively behaves as a thin solenoid (the magnetic monopole may therefore be emulated by "normal physics" with the help of thin solenoids which may be produced by electric currents) and in quantum mechanics, such solenoids are visible by interference experiments. (These interference experiments may reveal some information about \(\vec A\) that goes beyond the values of \(\vec B\) in the regions that the interfering particles may access: in this sense, \(\vec A\) is more real in quantum mechanics than it is in classical physics.) If an electron beam goes partly on the left side from the solenoid and partly on the right side, it interferes with itself and the phase of the interference pattern depends on the magnetic flux through the solenoid.

That's the Aharonov-Bohm effect. Dirac basically discovered it long before (1931) Aharonov and Bohm (1961). If you want the Dirac string to be invisible to these interference experiments, the magnetic monopole charge \(q_M\) must obey\[

q_M\cdot q_E\in 2\pi \ZZ

\] for all electric charges \(q_E\) of particles that may be used as probes in the interference experiment (a special discussion would be needed to decide whether quarks may qualify or not). The condition above is the Dirac quantization rule. It says that if magnetic monopoles exist, their magnetic charges have to be multiples of an elementary magnetic charge that is pretty much inversely proportional to the elementary electric charge.

Now, the Dirac monopole has a singular field that goes like \(\vec B\sim 1/r^2\). In 1974, Gerard 't Hooft and independently Polyakov constructed their monopole as a non-singular solution generalizing the Dirac monopole that replaces the "core" by some smoothly varying fields. To do so, the electromagnetic \(H=U(1)\) gauge group has to be embedded to a larger group \(G\) such that the coset \(G/H\) contains topologically nontrivial spheres \(S^2\) i.e. such that the second homotopy group is nontrivial\[


\] This group \(G\), e.g. \(SU(2)\), is broken to the \(U(1)\) and the non-single-valuedness of \(\vec A\) in the \(U(1)\) gauge group along the Dirac string may be represented by a variable embedding of \(U(1)\) into \(SU(2)\). In particular, while the \(SU(2)\) group is spontaneously broken to a \(U(1)\) almost everywhere in space, the very core of the monopole solution at the origin is non-singular and the \(SU(2)\) symmetry is fully restored over there.

These solutions and their generalizations play an important role in grand unified theories – and cosmological models that depend on them, in string theory, and elsewhere. A special puzzle is posed by the fact that neither Sheldon Cooper nor we have observed a magnetic monopole yet. Cosmic inflation is generally believed to have diluted their density to an unmeasurably low number.

Polyakov loops

The Polyakov loop may be equivalently described as the \(SU(2)\) holonomy around the Euclidean time circle in the thermal calculations of the gauge theory. It's great to start with a definition if we can – that's what the previous sentence was all about – and I can even give you the formula for the loop:\[

U = {\mathcal P} \exp \zav{ i\oint_0^\beta A^{SU(N)}_0 d\tau }

\] Its generalization, the Polyakov-Maldacena loop, would have multiples of the scalar fields in the exponent, too.

The letter \({\mathcal P}\) refers to the path-ordering along the path (it's mathematically almost the same operation as the time-ordering \({\mathcal T}\) you know from the interaction picture in quantum field theory). You may see that the expression above is pretty much the same thing as the Wilson loop except that the path is (Euclideanized) time-like and closed; the delimiters were written as \(0\) and \(\beta\) only to remind you of the length of the closed path. These subtleties – closedness and the Euclidean time-likeness – equip the loop with some properties that Wilson wouldn't have thought of.

(Even though 't Hooft and Polyakov were on the same side of the barricade just a minute ago, when we talked about monopoles, you shouldn't confuse the Polyakov loop with the 't Hooft loop. The latter also modifies a general Wilson loop but it requires us to use the electromagnetically dual gauge field: it reverse the role of electric and magnetic fields.)

As every holonomy, the loop \(U\) is an element of the group \(SU(N)\). But what's interesting is what is the expectation value \(\langle U\rangle\) in a gauge theory calculated at some inverse temperature \(\beta=1/T\). It turns out that this expectation value decides about the "activity" of another characteristic phenomenon in strongly coupled gauge theories – about confinement. We say that the Polyakov loop \( U\) and its eigenvalues detect the deconfinement transition. When the temperature is low enough, the theory is confining and the eigenvalues of \(U\) are distributed in one way. At higher temperatures, the behavior is qualitatively different. There is a phase transition in between.

(See a famous paper on confinement by Polyakov.)

I can't resist to mention a 2003 paper by Aharony, Marsano, Minwalla, Papadodimas, and Van Raamsdonk that gave a new, dual, secretly totally equivalent interpretation of the temperature where this phase transition occurs. If you know some string theory, you must know that the number of states is exponentially growing with their mass and the partition function therefore diverges above a critical temperature, the Hagedorn temperature (comparable to the string scale). Above this temperature, the ever more excited strings would dominate. These authors argued that the deconfinement transition is actually equivalent to the Hagedorn transition in the corresponding string theory. And be sure that the Polyakov loop served as a key litmus test in their work.

(Some of these younger authors have interacted with Polyakov at Princeton for years and they consider him a guru.)

As you can see, much of Polyakov's spectacular work has something to do with the new qualitative phenomena that the flux tubes – seemingly composite objects in gauge theory (the "electric field lines" that tend to clump in QCD because the interactions in that theory are nonlinear: the gluons carry the charges, too) – exhibit at strong coupling. In fact, they behave as strings in string theory even though strings in string theory were thought to be very different: unlike the flux tubes, they are "fundamental" i.e. "elementary". However, while the elementary/composite status of an object seems to lead to two logically very different situations for physicists who construct the theories, physics itself doesn't really care about this difference between elementary and composite objects. The two descriptions – gauge theory and string theory – may be exactly equivalent and at least in some cases, we know that they are actually equivalent.

Quantum field theory has been our "theory of nearly everything" (Lisa Randall's term) for most of the 20th century. Some people who learn its techniques – especially the Feynman diagrams and everything you need to calculate them – end up feeling that at the end, quantum field theory is a somewhat shallow idea that isn't too rich when it comes to the ideas it includes and produces. I don't want to decide whether this sentiment is justified. But whether it's justified or not, a part of it is an artifact of the perturbative approximation.

New organizing ideas, qualitative phenomena, and mathematical methods become useful, relevant, paramount when the coupling constant is large and Alexander Polyakov has discovered an unusually high portion of these ideas, phenomena, and methods. By doing so, he has greatly helped to unify the network of ideas that underlie the contemporary theoretical physics, discover new phenomena, and link phenomena that used to be thought of as disconnected. The unification of quantum field theory and string theory is a key achievement in this direction and Polyakov was a major visionary who envisioned that breakthrough and who added quite a lot of technical beef to it, too.

Along with other reasons, this is why he surely deserves to join the club of the Milner Prize winners.

If you want to listen to some pleasing Russian accent, have a look at this Strings 2012 talk by Polyakov on the quantum instabilities of de Sitter space.

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snail feedback (8) :

reader Peter Fred said...

Lubos will you stop ranting and raving about the wonders and beauties of string theory. The party is over. Experiments have come to light--my experiments--that show that the very foundation of General Relativity and hence your beloved string theory is untenable in much the same way geocentric theory was proved untenable with the Galileo's observation of the phases of Venus. Of course, Galileo had the near impossible task to get his fellow high-profile scientist to look through a telescope to see the phases of Venus. You have been generous enough to allow a picture of my experimental results to be posted your blog. Now is the time to actually look at the picture that has been allow to be posted. In a simple, inexpensive table-top experiment such as the one previously posted on you blog, not one of the legions of your fellow believers in General Relativity would be able to would be able to get a 7 lb test mass to loose 99% of its normal terrestrial surface weight as has been observed in my posted unnoticed experiment. I have lots of other experiments which show the same general result so do go on and on about who I trained under or the low cost of my equipment.

reader anna v said...

Good choice.

I had heard him lecture on strings in QCD back in the late 1970s, very impressed I was.

His name has not been prominent in all the new stuff, and I thought he had disappeared below the horizon. Glad to see he has been well known and respected by the theoretical community.

reader Luboš Motl said...

Since 1995, he still has many papers with 100+ cits,

That's a lot. They include the O(N) models - the Vasiliev theory toy model for AdS/CFT - and many other things. Nevertheless, the feeling that the physicists focused on the present feel some 1970s nostalgia from Polyakov is probably substantiated. ;-)

reader Trimok said...

Off-Topic Cosmology : Planck 2013 results pre-print could be read here:

reader duley said...

``However, around 1975, he would move to the West:''

he did not move anywhere, he was probably just traveling

reader Dilaton said...

Thanks for this very nice, article Lumo, I guess Alexander Polyakov would like it too, if he could see it :-)

Why is he sometimes called Sasha ...? There is obviously something I am missing ...

This great text belongs to the TRF articles I always enjoy iteratively: the first read is to gain an overview, the second one to take my time to really get and understand out of it what I can, and again later after having learned some more stuff...

The funny thing is that TRF articles that somehow catch and excite me seem the stick in my head even when I was not able to understand every detail. When reading things that have any relationship to what I have seen here, it ofteh happens to me that I remember having read about it already here on TRF, and then I come back to reread the corresponding article :-)

A slightly other thing (hoping that I am not a pain in the neck ;-) ...):

Dear Lumo, could you help this seemingly seriously interested in string theory student once more?

He obviously really wants to know and he has set a bounty on the question, because it has gone unanswered and with low views etc ...


reader Luboš Motl said...

Thanks for your interest and smile, Dilaton!

Sasha is a normal colloquial form of "Alexander" just like "Joe" or "Pepa" are colloquial forms of "Joseph". Sasha and Alexander sound very different but so do Pepa and Joe. ;-) It's the same name.

reader antonio carlos pocobi said...


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