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## 29 March 2013

## 21 March 2013

### Sasha Polyakov joins the Milner Prize winners

Sasha Polyakov joins the Milner Prize winners: Nine physicists won the inaugural $3 million Fundamental Physics Prize last summer and several winners were added in December.

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 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.)

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.

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.)

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.

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 \[

Q\ket\psi=0

\] 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.

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.

I just discussed a 1981 paper: note that Polyakov was a string theorist

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 tot he general public, especially because Sheldon Cooper thought that he 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. 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 through 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. That poses a challenge because the Dirac string effectively behaves as a thin solenoid and in quantum mechanics, such solenoids are visible by interference experiments. If an electron beam goes partly on the left side from the solenoid and partly on the right side, it interferes 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\[

\pi_2(G/H)=\ZZ.

\] 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, 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 still have observed a magnetic monopole yet. Cosmic inflation is generally believed to have diluted their density to an unmeasurably low number.

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 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 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.

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 string theory – 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, physics 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 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, too.

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

*When he was a bit younger. See his picture with a supplement to the prize. 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 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 MÃ¶bius 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.

**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 \[

Q\ket\psi=0

\] 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.

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. \(\exp(-C/g^2)\).

**Monopoles**But magnetic monopoles are more well-known tot he general public, especially because Sheldon Cooper thought that he 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. 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 through 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. That poses a challenge because the Dirac string effectively behaves as a thin solenoid and in quantum mechanics, such solenoids are visible by interference experiments. If an electron beam goes partly on the left side from the solenoid and partly on the right side, it interferes 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\[

\pi_2(G/H)=\ZZ.

\] 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, 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 still 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 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 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.

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 string theory – 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, physics 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 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, too.

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

## 20 March 2013

### "Signals from Before the Big Bang" --Were Telltale Patterns Glimpsed in the Afterglow?

"Signals from Before the Big Bang" --Were Telltale Patterns Glimpsed in the Afterglow?:

In 2010, University of Oxford theoretical physicist Roger Penrose made the sensational claim that he had glimpsed a signal originating from before the Big Bang working with Vahe Gurzadyn of the Yerevan Physics Institute in Armenia. Penrose came to this conclusion after analyzing maps from the Wilkinson Anisotropy Probe. The NASA WMAP satellite data reveals the cosmic microwave background, believed to have been created just 300,000 years after the Big Bang when the Universe cooled enough to allow electrons to link up with protons and form hydrogen atoms. The short gap between the two events means that the CMB captures details of the birth of our Universe, like its composition and the processes leading to its current state. So far, Nobel Prizes have been awarded both to its discovery and to the first detailed characterization of its properties.

However, critics such as Sean Carroll, a theorectical physicist and Senior Research Associate at the California Institute of Technology, counter that the Penrose circles are simply what you would expect from random alignments, not a new signal over and above the usual Standard Model predictions. The most obvious empirical fact about our observable universe, Carroll says, "is its temporal asymmetry — the early phase is very different from the late phase, even though no such difference is to be found in the fundamental laws of physics."

Regardless, Carroll points out, Penrose has been correct in insisting that the low entropy of our early universe is a crucial problem that is not well-addressed in modern cosmology.

Penrose's finding runs directly counter to the widely accepted inflationary model of cosmology which states that the universe started from a point of infinite density known as the Big Bang about 13.7 billion years ago, expanded extremely rapidly for a fraction of a second and has continued to expand much more slowly ever since, during which time stars, planets and ultimately humans have emerged. That expansion is now believed to be accelerating due to a scientific "X" factor called dark energy and is expected to result in a cold, uniform, featureless universe.

In 2010, University of Oxford theoretical physicist Roger Penrose made the sensational claim that he had glimpsed a signal originating from before the Big Bang working with Vahe Gurzadyn of the Yerevan Physics Institute in Armenia. Penrose came to this conclusion after analyzing maps from the Wilkinson Anisotropy Probe. The NASA WMAP satellite data reveals the cosmic microwave background, believed to have been created just 300,000 years after the Big Bang when the Universe cooled enough to allow electrons to link up with protons and form hydrogen atoms. The short gap between the two events means that the CMB captures details of the birth of our Universe, like its composition and the processes leading to its current state. So far, Nobel Prizes have been awarded both to its discovery and to the first detailed characterization of its properties.

However, critics such as Sean Carroll, a theorectical physicist and Senior Research Associate at the California Institute of Technology, counter that the Penrose circles are simply what you would expect from random alignments, not a new signal over and above the usual Standard Model predictions. The most obvious empirical fact about our observable universe, Carroll says, "is its temporal asymmetry — the early phase is very different from the late phase, even though no such difference is to be found in the fundamental laws of physics."

Regardless, Carroll points out, Penrose has been correct in insisting that the low entropy of our early universe is a crucial problem that is not well-addressed in modern cosmology.

Penrose's finding runs directly counter to the widely accepted inflationary model of cosmology which states that the universe started from a point of infinite density known as the Big Bang about 13.7 billion years ago, expanded extremely rapidly for a fraction of a second and has continued to expand much more slowly ever since, during which time stars, planets and ultimately humans have emerged. That expansion is now believed to be accelerating due to a scientific "X" factor called dark energy and is expected to result in a cold, uniform, featureless universe.

Penrose, however, as he reported in

The core concept in Penrose's theory is the idea that in the very distant future the universe will in one sense become very similar to how it was at the Big Bang. Penrose says that "at these points the shape, or geometry, of the universe was and will be very smooth, in contrast to its current very jagged form. This continuity of shape, he maintains, will allow a transition from the end of the current aeon, when the universe will have expanded to become infinitely large, to the start of the next, when it once again becomes infinitesimally small and explodes outwards from the next big bang. Crucially, he says, the entropy at this transition stage will be extremely low, because black holes, which destroy all information that they suck in, evaporate as the universe expands and in so doing remove entropy from the universe."

The foundation for Penrose's theory is found in the cosmic microwave background, the all-pervasive microwave radiation that was believed to have been created when the universe was just 300,000 years old and which tells us what conditions were like at that time.

The evidence was obtained by Vahe Gurzadyan of the Yerevan Physics Institute in Armenia, who analysed seven years' worth of microwave data from WMAP, as well as data from the BOOMERanG balloon experiment in Antarctica. Penrose and Gurzadyan say they have clearly identified concentric circles within the data – regions in the microwave sky in which the range of the radiation's temperature is markedly smaller than elsewhere.

The Cosmic Microwave Background (CMB) radiation is the remnant heat from the Big Bang. This radiation pervades the universe and, if we could see in microwaves, it would appear as a nearly uniform glow across the entire sky. However, when we measure this radiation very carefully we can discern extremely faint variations in the brightness from point to point across the sky, called "anisotropy". These variations encode a great deal of information about the properties of our universe, such as its age and content.

The "Wilkinson Microwave Anisotropy Probe" (WMAP) mission has measured these variations and found that the universe is 13.7 billion years old, and it consists of 4.6% atoms, 23% dark matter, and 72% dark energy.

According to Penrose and Gurzadyan, as described in arXiv: 1011.3706, these circles allow us to "see through" the Big Bang into the aeon that would have existed beforehand. They are the visible signature left in our aeon by the spherical ripples of gravitational waves that were generated when black holes collided in the previous aeon.

The "Penrose circles" pose a huge challenge to inflationary theory because this theory says that the distribution of temperature variations across the sky should be Gaussian, or random, rather than having discernable structures within it.

Julian Barbour, a visiting professor of physics at the University of Oxford in an interview with Physics World, says that these circles would be "remarkable if real and sensational if they confirm Penrose's theory". They would "overthrow the standard inflationary picture", which, he adds, has become widely accepted as scientific fact by many cosmologists.

*Physics World*, takes issue with the inflationary picture "and in particular believes it cannot account for the very low entropy state in which the universe was believed to have been born – an extremely high degree of order that made complex matter possible. He does not believe that space and time came into existence at the moment of the Big Bang but that the Big Bang was in fact just one in a series of many, with each big bang marking the start of a new "aeon" in the history of the universe."The core concept in Penrose's theory is the idea that in the very distant future the universe will in one sense become very similar to how it was at the Big Bang. Penrose says that "at these points the shape, or geometry, of the universe was and will be very smooth, in contrast to its current very jagged form. This continuity of shape, he maintains, will allow a transition from the end of the current aeon, when the universe will have expanded to become infinitely large, to the start of the next, when it once again becomes infinitesimally small and explodes outwards from the next big bang. Crucially, he says, the entropy at this transition stage will be extremely low, because black holes, which destroy all information that they suck in, evaporate as the universe expands and in so doing remove entropy from the universe."

The foundation for Penrose's theory is found in the cosmic microwave background, the all-pervasive microwave radiation that was believed to have been created when the universe was just 300,000 years old and which tells us what conditions were like at that time.

The evidence was obtained by Vahe Gurzadyan of the Yerevan Physics Institute in Armenia, who analysed seven years' worth of microwave data from WMAP, as well as data from the BOOMERanG balloon experiment in Antarctica. Penrose and Gurzadyan say they have clearly identified concentric circles within the data – regions in the microwave sky in which the range of the radiation's temperature is markedly smaller than elsewhere.

The Cosmic Microwave Background (CMB) radiation is the remnant heat from the Big Bang. This radiation pervades the universe and, if we could see in microwaves, it would appear as a nearly uniform glow across the entire sky. However, when we measure this radiation very carefully we can discern extremely faint variations in the brightness from point to point across the sky, called "anisotropy". These variations encode a great deal of information about the properties of our universe, such as its age and content.

The "Wilkinson Microwave Anisotropy Probe" (WMAP) mission has measured these variations and found that the universe is 13.7 billion years old, and it consists of 4.6% atoms, 23% dark matter, and 72% dark energy.

According to Penrose and Gurzadyan, as described in arXiv: 1011.3706, these circles allow us to "see through" the Big Bang into the aeon that would have existed beforehand. They are the visible signature left in our aeon by the spherical ripples of gravitational waves that were generated when black holes collided in the previous aeon.

The "Penrose circles" pose a huge challenge to inflationary theory because this theory says that the distribution of temperature variations across the sky should be Gaussian, or random, rather than having discernable structures within it.

Julian Barbour, a visiting professor of physics at the University of Oxford in an interview with Physics World, says that these circles would be "remarkable if real and sensational if they confirm Penrose's theory". They would "overthrow the standard inflationary picture", which, he adds, has become widely accepted as scientific fact by many cosmologists.

But Barbour believes that the result will be "very controversial" and that other researchers will look at the data very critically. He says there are many disputable aspects to the theory, including the abrupt shift of scale between aeons and the assumption, central to the theory, that all particles will become massless in the very distant future. He points out, for example, that there is no evidence that electrons decay.

Penrose and colleague Gurzadyn have answered the numerous critics who say that the circles do not contradict the standard model of cosmology in follow up paper, published on arXiv. In the short article, they agree that the presence of circles in the CMB does not contradict the standard model of cosmology. However, the existence of “concentric families” of circles, they argue, cannot be explained as a purely random effect given the pure Gaussian nature of their original analysis. “It is, however a clear prediction of conformal cyclic cosmology,” reports Physics World.

Related articlesPenrose and colleague Gurzadyn have answered the numerous critics who say that the circles do not contradict the standard model of cosmology in follow up paper, published on arXiv. In the short article, they agree that the presence of circles in the CMB does not contradict the standard model of cosmology. However, the existence of “concentric families” of circles, they argue, cannot be explained as a purely random effect given the pure Gaussian nature of their original analysis. “It is, however a clear prediction of conformal cyclic cosmology,” reports Physics World.

## 13 March 2013

## 03 March 2013

## 01 March 2013

### Surprising Third Radiation Belt Found Around Earth

Surprising Third Radiation Belt Found Around Earth:

Two giant swaths of radiation, known as the Van Allen Belts, surrounding Earth were discovered in 1958. In 2012, observations from the Van Allen Probes showed that a third belt can sometimes appear. The radiation is shown here in yellow, with green representing the spaces between the belts. Credit: NASA/Van Allen Probes/Goddard Space Flight CenterIn September of 2012, scientists with the newly launched Van Allen Probes got permission to turn on one of their instruments after only three days in space instead of waiting for weeks, as planned. They wanted to turn on the Relativistic Electron Proton Telescope (REPT) so that its observations would overlap with another mission called SAMPEX (Solar, Anomalous, and Magnetospheric Particle Explorer), that was soon going to de-orbit and re-enter Earth’s atmosphere.

Now, they are very glad they did, as something happened that no one had ever seen before. A previously unknown third radiation belt formed in the Van Allen Radiation Belts that encircle Earth. The scientist watched – in disbelief – while their data showed the extra belt forming, then suddenly disappear, like it had been cut away with a knife. They have not yet seen a recurrence of a third belt.

(...)

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Two giant swaths of radiation, known as the Van Allen Belts, surrounding Earth were discovered in 1958. In 2012, observations from the Van Allen Probes showed that a third belt can sometimes appear. The radiation is shown here in yellow, with green representing the spaces between the belts. Credit: NASA/Van Allen Probes/Goddard Space Flight Center

Now, they are very glad they did, as something happened that no one had ever seen before. A previously unknown third radiation belt formed in the Van Allen Radiation Belts that encircle Earth. The scientist watched – in disbelief – while their data showed the extra belt forming, then suddenly disappear, like it had been cut away with a knife. They have not yet seen a recurrence of a third belt.

(...)

Read the rest of Surprising Third Radiation Belt Found Around Earth (796 words)

© nancy for Universe Today, 2013. |

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