From Polyakov to Perelman: History of the Ricci Flow (and a Millennium Prize)

March 19, 2010

It has just been announced by the Clay Mathematics Institute, that G. Perelman has won the Millennium prize for “resolution of the Poincare conjecture”.

Previously I’ve written about the Poincare conjecture and Perelman’s celebrated proof of it, which was based on the so-called Ricci flow equation. Now let me talk more about how this story is related to physics, and where this equation first appeared in history.

The Ricci flow is an analog of the heat equation for geometry – it is a diffusive process acting on the metric of a Riemannian manifold. It is a special case of so-called geometric evolution equations, that describe the deformation of the metric of Riemannian manifolds driven by their curvature.
Actually, what is usually called the Ricci flow equation, constitutes a very simple equation. Let M be a n-dimensional compact manifold with positive definite metric g_{ij}. The Ricci flow equation is

dg_{ij}/dt = -2 R_{ij}(g)

where t is an external parameter defined on some interval and i,j =1, 2, …, n; the Ricci tensor with components R_{ij} is calculated from the Riemann tensor as the trace R_{ij}=R^k_{ikj}.

In physics it was introduced by Friedan in the 1980’s in the renormalization-group (RG) flow of two-dimensional sigma models. Such sigma models describe the propagation of strings in curved backgrounds. A variant of the Ricci flow appeared already in the 1970’s in the renormalization group studies of non-linear sigma models; the original computation was performed by Polyakov who considered the class of O(n) sigma models with target space metric that of the round sphere on S^{n-1} for n>2. Polyakov (1975) originally considered the O(n) nonlinear sigma models. Here the radius of the sphere is the only parameter of the theory, and as such, it serves as the inverse of its coupling constant, i.e. R ~1/g. These two-dimensional models are not conformally invariant in the quantum regime, as it was found that the coupling constant runs with respect to the world-sheet renormalization scale parameter \Lambda to lowest order in perturbation theory.

The computation of the perturbative beta function was later extended by Friedan (1980) to encompass two-dimensional non-linear sigma models with generalized coupling given by the target space metric g_{ij} of arbitrary Riemannian manifolds. Here

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}-\frac{\alpha'^2}{2}R_i^{\, klm}R_{jklm}+...

where the dots stands for higher order curvature corrections, and the RHS is a perturbative expansion in \alpha'. 
The first-order term in \alpha' is

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}.

Now, to see that the Ricci flow is essentially the same as RG flow, we identify the “time” t with the energy scale \Lambda through t=\log(\Lambda^{-1}), then \Lambda \sim e^{-t} and so large positive t corresponds to small energy \Lambda: flow in t is therefore a flow from the ultra-violet (UV) regime to the infra-red (IR) regime, or from small to large distance scales. The calculation above of Friedan led to the development of the sigma model approach to string theory, since the requirement of conformal invariance for the world-sheet quantum field theory gives rise to the vacuum Einstein equations in target space, R_{ij}=0 plus higher order corrections. Further generalizations can be considered by including other massless modes of the string, such as the dilaton \phi and the anti-symmetric torsion field B_{ij} to the action of the world-sheet sigma model.

Perelman’s proof (2003) of the Poincare  conjecture is related in an interesting way to this physical model. Perelman’s proof uses an “entropy” functional {\cal F}. Let {\cal M} be the space of smooth Riemann metrics on M. Define {\cal F}:{\cal M}\times C^\infty (M)\rightarrow R by

{\cal F}(g,f) =\int_M\left(R+|\nabla f|^2\right)e^{-f}dV

this is basically the Einstein-Hilbert action in general relativity, where f is a “dilaton” field; roughly speaking we go from Perelman to Polyakov by setting f=\phi. To my knowledge, the full extent of the connection between string theory and Perelman’s proof has not really been understood yet.

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Unparticles, Unpolitics, and Their Possible Signatures

December 11, 2007

“Unparticles” and “unpolitics” are two seemingly unrelated concepts which you might never have heard about before, so let me start by explaining the first one.

So, what is an “unparticle”? In particle physics it has recently been suggested by Howard Georgi, that there exists “stuff” which cannot be thought of as particles:

Unparticle Physics 

but nevertheless could be observed at the LHC accelerator in CERN, due to start in 2008. He calls this stuff “unparticles”. This is an intriguing and controversial idea, since our world seems to be well-described in terms of particles.

The idea of unparticles comes from the principle of scale invariance, meaning that the physics of a system remains the same regardless of a change of length (or equivalently energy). Such a scale transformation looks like x -> x’ = (e^s) x. A theory of particles can only be scale invariant if the particles have zero mass and charge: A scale transformation multiplies the mass with a rescaling factor raised to the mass dimension. The standard model of particle physics is surely not scale invariant; the photon, for example, is massless, but its charge is non-zero. However, it is possible that there is another sector of the standard model, the “unparticles”, which interacts so weakly with the known particles of the standard model that they have not been observed; and which is exactly scale-invariant. It is difficult to describe the detailed physics of such a sector, but important characteristics at low energy can be derived from scale invariance. One important consequence is that unparticle stuff will look in the detector like a non-integer number of invisible particles. For example, it could happen that 3/7 particles were missing in the detector. Such an observation would be a very clear sign of something interesting going on!

While you might need a 2 billion EUR detector like LHC to discover unparticles, “unpolitics” is easy to recognize. But, what is “unpolitics”?

While following the general election in Denmark in Nov this year, I thought that a new term, unpolitics, should apply to one of the parties, called New Alliance (Ny Alliance). However, I later realized that such a term already existed, but used as meaning “apolitical”, or “not being concerned with politics”. This is not exactly how I am going to define it. 

 Ny Alliance (New Alliance) is a danish political party which was founded in May 2007 by Naser Khader and two others. Naser was a member of the Social Liberals Party, but wanted to counter the influence of the right-wing and xenophobic Danish People’s Party. At first, this project gave New Alliance a lot of momentum, and early opinion polls indicated that they could secure 12 out of 179 seats in the Parliament. In the November election they only managed to get 5 seats, and a times it was uncertain if they would be able to be represented at all. Why was this so? One of the main reason, I think, is that New Alliance is a typical representative of what I will call “unpolitics”.

Unpolitics is “stuff” in the world of politics, which is represented by political persons, but which not really can be thought of as politics. In unpolitics, the most important elements are often popular persons, but with no, or just very few, really new ideas. One idea of New Alliance was to reduce the income tax to 40%; a member even suggested that the 40% could be experimentally implemented on Denmark’s third-largest island. This proposal was quickly abandoned. Another idea is free food for school children. New Alliance has been notoriously slow in formulating a detailed party program. When asked about concrete political questions, the typical answer was that such an answer could not be given, since they represented a “new” approach towards danish politics. Their main reason of existence just being to counter the influence of another party. In reality this did not happen. 

Therefore, a possible signature of unpolitics is this. Unpolitics is scale invariant: at every scale – large and small – you don’t find any “stuff” of politics, just popular persons.

References: Howard Georgi’s two papers on unparticles, hep-ph/0703260, and 0704.2457 [hep-ph]


The Nobel Prize in Physics 2007

October 9, 2007

And the prize goes to…. 

Albert Fert (France), and Peter Grünberg (Germany), “for the discovery of Giant Magnetoresistance” (GMR). More details about the physics of GMR over at Clifford Johnson’s blog.

Via: nobelprize.org


String Theory: Crash Course

April 21, 2007

A Tool for Living in the 21st Century:

The excellent Seed magazine offers a cribsheet [PDF] which includes a basic introduction to string theory.

Prof. Clifford Johnson over at Asymptotia.com was an adviser.

Other interesting cribsheets are one on stem cells, and one on climate change.


Global Temperature, Global Warming?

March 17, 2007

What is global warming? Most people would answer this seemingly simple question with something like the following (see the article at wikipedia.org):

Global warming is the observed increase in the average temperature of the Earth’s near-surface air and oceans in recent decades and its projected continuation. […]

One would think that all scientist agree on this definition. However, actually they don’t.

Some scientist would say that it does not even make sense. The June 2007 issue of the Journal of Non-equilibrium Thermodynamics includes a paper of Christopher Essex (U. of Western Ontario), Ross McKitrick (U. of Guelph) and Bjarne Andresen (Niels Bohr Institute), with the interesting title:

Does a Global Temperature Exist? [PDF]

In this paper it is argued that the concept of a “global temperature” is thermodynamically as well as mathematically meaningless. First of all, you cannot just add local temperatures on the Earth and then take the average to define a single “global” temperature of the Earth. Secondly, the average is not canonically defined. For example, taking a box of air with temperature 0 degrees and an identical one with temperature 100 degrees would lead to an arithmetic average of 50 degrees (add the two numbers and divide by two). However, the geometric average in this case, obtained by multiplying the two numbers (in degree Kelvin) and taking the square root is 46 degrees. Thus claims of distaster – or not – maybe a consequence of the averaging method used.

So, what is Global Warming? Can it be defined in a sound way, both from a physics and mathematics viewpoint?

Eli Rabett over at Rabett Run thinks that this paper is “a bowl of steaming crap”; I guess Lubos Motl thinks otherwise.

Update: The climate-friends at RealClimate.org thinks that this paper is irrelevant.


Fermi’s Paradox and Galaxy Probes

January 21, 2007

The Fermi paradox (1950) can be formulated as follows

If there are extraterrestrial civilizations out there, then where are they?

In a recent paper, Rasmus Bjork of the Niels Bohr Institute argues that finding other life in the Galaxy by using space probes, and possibly solving the paradox, might take extremely long time. A time which is comparable to the age of the universe. More precisely, he estimates that with 8 probes each having 8 subprobes around 4% of the Galaxy can be explored in 9.6 billion years.

So, if we sometimes feel lonely in the Universe it is because Aliens still haven’t had the time to visit us…

See also the discussion at Mangan’s Miscellany.


Deconstructing Strings?

November 14, 2006

“Deconstruction” is a method of critical analysis of philosophical and literary language that emphasizes the internal workings of language and conceptual systems, the relational quality of meaning, and the assumptions implicit in forms of expression.

Todays most fascinating paper is without doubt the one by Bert Schroer entitled:

String theory deconstructed (a detailed critique of the content of ST from an advanced QFT viewpoint)

In this paper Bert (BS) supposedly gives “a detailed and comprehensive critique of claims and methods of string theory from an advanced quantum field theoretical viewpoint.” BS starts out by listing nine claims of string theory “which afterwards will be shown to be fundamentally flawed”. These nine claims are:

1 ) The Kaluza-Klein argument can be used in QFT (or ST) to encode compactified spatial coordinates into inner symmetries
2 ) In ST supersymmetry is spontaneously broken
3 ) Holography is a construct which needs quantum gravity as a prerequisite
4 ) The Maldacena conjecture is about a AdS—CFT holography
5 ) The counting zero mode degree of freedom estimate about the cosmological
constant is consistent with the principle of local covariance
6 ) String theory solves the “information paradox”
7 ) Strings are quantum objects with a localization in spacetime which is
string- instead of point-like
8 ) It has been shown that ST contains QFT in the limit of low energies.
9) The S-matrix of ST has the properties of a particle physics S-matrix

For one thing, BS does not like KK compactifications (claim #1), since “I recently red that already Pauli had shown that this is impossible, but there was no reference given.” And concerning claim 3), BS states that “I think that anybody who knows the framework of particle physics (say beyond the level of recent QFT texts which where written by string theorists) would agree that holography from d+1 to d dimension and its possible inversion cannot be anything else than a radical change of the spatial encoding of a specified algebraic substrate; using this word for anything else would be a misuse and lead to misunderstandings.” This seems to debunk the idea that holography should be related to quantum gravity.

But BS’s arguments against “the Maldacena conjecture” are even stronger. For example, he says that “I do not know any competent quantum field theorist who does not accept Rehren’s work as the correct formulation of AdS—CFT holography (Hollands, Wald, Brunetti, Fredenhagen, Verch, Buchholz, …)”. It is – at least to me – unclear what the …’s stand for here; but even more staggering are BS’s adventures into advanced psychoanalysis: “For psychologically understandable reasons it was this metaphoric QG connection which attracted the attention of string theorists (QG is the raison d’etre for string theory) and which led Maldacena to formulate a conjecture involving a vague idea of supersymmetric string under the KK curling (with its even more vague idea of its QG content) on the dual AdS side in case one starts from a (supersymmetric) conformal field theory”.

For some reason also, quantum mechanics seems to be enough to understand black hole physics: “Of course one can use Bekenstein’s classical formula and equate it with this microscopically computed entropy to determine epsilon (I have not done this, but there can be no doubt that at this point the Planck length enters and determines the size of the vacuum polarization cloud). The calculations are in two papers […]”.

With Maldacena (and … and ….) literally on his knees, an alternative resolution of the apparent clash between quantum mechanic and general relativity was put forward by Wald: “His proposed solution was the start of the modern theory of QFT in CST in which the Lagrangian formalism is abandoned in favor of the adoption of the dichotomy of AQFT between the algebraic structure of QFT and the admissible states on such algebras.”

Numerous other advanced arguments seem to kill the claims 5) and 6) above. And for 7) I learned, that: “The localized algebras are monades with very different properties from algebras one meets in QM. There can be no doubt that the understanding of their positioning in a common Hilbert space will be an important step on the long way towards QG.”

But of course the “monad” (or, in biological terms, flagella) point of view also call into question whether string theory contains quantum field theory in its low-energy limit (8): “The message from this last case is that metaphoric arguments (e.g. looking at functional representations without actually doing the functional integrals) may turn out to lead to wrong results. Take for example the case of 2+1 dimensional QFT which have braid-group statistics. If the spin is anyonic (i.e. not semi-integer) the statistics is plektonic and the upholding of the spin-statistics theorem in such a case prevents the nonrelativistic limit to be a (second quantized) QM; it remains a nonrelativistic QFT. Only if one relinquishes the plektonic commutation relations, but preserves the anyonic spin one finds Wilczek’s anyons in the form of quantum mechanical Aharonov-Bohm dyons […]”, and then “The message from this illustration is that a theory can only be asymptotically (e.g. for long distances) contained in a more fundamental one if their structures harmonize.”

But the flagella (monads) also kill the S-matrix arguments 9): “A much more detailed correspondence of Leibniz’s image of reality in terms of indivisible monades to the conceptual structure of particle physics is provided by the algebraic setting of QFT (AQFT). If one identifies Leibniz’s monades with copies of the unique hyperfinite type III_1 factor algebras then it can be shown that any QFT permits a faithful encoding into the relative positions of a finite number of monades”.

At this point I started thinking: is this all a joke? Was I being fooled? Staring a the screen I was wondering whether or not I had really been fooled. On the one hand, if I wasn’t fooled, then this paper was serious, hence I was fooled by my understanding of physics. But if I was fooled, then I did get what I expected from BS, so in what sense was I fooled?.

Then suddenly I realized, that I had seen this text before, but just in another (isomorphic) disguise. It was the famous “Sokal hoax”, a hoax paper published by physicist Alan D. Sokal in 1994, entitled “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity”. (Sokal’s hoax served a public purpose, that of attracting attention to what Sokal saw as a decline of standards of rigor in the academic community; for this reason, Sokal’s text was “liberally salted with nonsense”). For example, quoting from Sokal:

“In mathematical terms, Derrida’s observation relates to the invariance of the Einstein field equation […] under nonlinear space-time diffeomorphisms (self-mappings of the space-time manifold which are infinitely differentiable but not necessarily analytic). The key point is that this invariance group “acts transitively”: this means that any space-time point, if it exists at all, can be transformed into any other. In this way the infinite-dimensional invariance group erodes the distinction between observer and observed; the [pi] of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.

and

“It is still too soon to say whether string theory, the space-time weave or morphogenetic fields will be confirmed in the laboratory: the experiments are not easy to perform. But it is intriguing that all three theories have similar conceptual characteristics: strong nonlinearity, subjective space-time, inexorable flux, and a stress on the topology of interconnectedness.”

.

So, was I fooled or not?