Saturday, November 09, 2013

Kaluza and Weyl: adding \(U(1)\) to GR

Exactly 10,000,000 years ago, on November 9th, 1885, two mathematical physicists were born, Hermann Weyl and Theodor Kaluza. Click the hyperlink for some 2007 TRF biographies. Yes, "ten million" is the translation of 128 from decimal to binary. ;-)

It may seem remarkable for two mathematical physicists to be born on the same day and the same year. It may look even more remarkable if you realize that both of them were German-speaking although the surprise decreases once you acknowledge that in the late 19th and early 20th century, most of the mathematical elite was German-speaking.

But you should be stunned that these two men who were born on the same day did make two very similar contributions: adding the \(U(1)\) gauge symmetry to Einstein's general relativity with the goal of unifying gravity with electromagnetism. The two men differed in many ways – Hermann Weyl was a broad, prolific mathematician while Theodor Kaluza was a typical one-hit wonder (you might say that this is an excessively unflattering description of a man who spoke 17 languages and claimed to prefer Arabic) – but when it comes to the birthday and the unification of gravity and electromagnetism, you would have a hard time to look for two mutually non-interacting people who were closer to one another.

(The only man I know who was born exactly on the same day and year as myself, the Czech politician Vít Bárta, was arrested three days ago. It seems he was pushing a BIS agent to commit crime and leak some information. What about your birthdaysakes?)

Let me spend some time with the attempts to unify gravity and electromagnetism around 1920.

It was a nice time, some quantum-like observations were already known but quantum mechanics had not been born yet. Einstein had completed GR by 1916 and it looked sensible to claim that electromagnetism and gravity were the only two forces in Nature.

Well, today we know it wasn't quite right. We usually say that there are four basic forces in Nature: the strong nuclear and the weak nuclear forces have to be added to the list. The following two paragraphs are a bit technical so let's write them as a quote.
If we restrict our attention to down-to-earth phenomena that are perceived by an average Hollywood star every day, the strong force keeps the quarks together inside protons and neutrons and produces a residual force that keeps protons and neutrons together inside the nuclei, despite their electrostatic repulsion. The strong force is "caused" by the gauge bosons of the \(SU(3)_{\rm color}\) gauge symmetry, the so-called gluons of QCD.

The weak force is known to low-brow Hollywood stars as the interaction responsible for the beta-decay (and the decay of the neutron as the simplest example of it). Let us kindly remain silent about Hollywood stars who are even more low-brow than that. ;-) The messenger particle is a massive W-boson (its nonzero mass is why it's a short-range force) and these bosons guarantee the \(SU(2)_{\rm weak}\) symmetry and mediate the corresponding interaction. Well, they do so along with another gauge boson, the neutral Z-boson (coupling the neutral currents), and the electromagnetic \(U(1)_{\rm em}\) actually isn't just added as a factor on top of this \(SU(2)_{\rm weak}\). Instead, the extra decoupled factor is \(U(1)_Y\) whose generator is a linear combination of the hypercharge and the third (Z-boson-affiliated) component of the \(SU(2)_{\rm weak}\) group.
Fine. The life was simpler a few years before 1920. Electromagnetism and gravity seemed to be the only known forces in Nature. People could sort of reasonably hope that a good enough description of these forces would produce all the atoms and nuclei, too. This belief became silly in the 1930s if not 1920s while Einstein (because of his blindness to the quantum revolution) continued to believe the two-force myth for decades but that's another story.

Great. Let's believe that there are only the two long-range forces, electromagnetism and gravity. We want to unify them because the theory of everything should be unified. How should we do that?

Gravity is described by general relativity. At each point of the spacetime, there is a "gravitational gauge field", the metric tensor \(g_{\mu\nu}(x^\lambda)\). It locally defines the geometry i.e. the squared proper length of an infinitesimal line interval \(dx^\mu\):\[

ds^2 = g_{\mu\nu} dx^\mu dx^\nu

\] We adopted the Einstein summation rule. We demand that the theory is described by the same equations even if we transform the coordinates by a general coordinate redefinition (diffeomorphism)\[

dx^\mu \to dx^{\prime\mu} (x^\mu).

\] Great. This leads us to consider tensor fields and the simplest new tensor field constructed out of the metric tensor is the Riemann curvature tensor and its contractions, the Ricci tensor and the Ricci scalar. The action may be written as the integral of a multiple of the Ricci scalar (the action has to be a scalar) i.e.\[

S_{EH} = \frac{1}{16\pi G}\int d^4 x\,R\sqrt{-g}

\] which produces Einstein's equations as the equations of motion.\[

R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}

\] I avoided any censorship of the cosmological constant \(\Lambda\) term, something that Einstein has incorrectly believed to have been the greatest blunder of his life, because since the late 1990s, it has seemed clear that we need this term for observational reasons.

Physicists who endorse special relativity commonly work in the \(c=1\) units. Sometimes, general relativists also set \(G=1\) or, even more naturally, \(8\pi G=1\), while particle physicists usually avoid the latter because gravity is so weak in the world of their phenomena. (In perturbative string theory, we sometimes pick a convention inconsistent with \([8\pi]G=1\), namely \(\alpha'=2\) or \(\alpha'=1/2\) or \(\alpha'=1\).)

On the other hand, electromagnetism has its own gauge field, too. It is the vector field \(A_\mu(x^\lambda)\). In analogy with the diffeomorphism symmetry, we enforce the \(U(1)_{\rm em}\) gauge symmetry. This condition implies that the equations of motion have to be constructed from the gauge-covariant objects such as the field strength \[

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.

\] Well, for an Abelian group like \(U(1)\), the object above is gauge-invariant, not just covariant (the word "covariant" means "transforming nicely, like the simplest objects with the same indices", while "invariant" means "not changing at all when you make a gauge transformation").

The equations of motion are nothing else than Maxwell's equations. One-half of these equations is guaranteed automatically if we choose \(A_\mu\) as the basic set of elementary fields. The other one-half of Maxwell's equations says\[

\partial_\mu F^{\mu\nu} = j^\nu,

\] perhaps with some extra universal prefactors. The right-hand side contains the charge density and the corresponding current. It's straightforward to write this equation within GR as well, pretty much in the same form. (You have to replace the partial derivatives by the covariant ones. Note that this is not necessary in the Bianchi-identities part of Maxwell's equations because the connection terms cancel after the antisymmetrization.) Note that Maxwell's equations and Einstein's equations are second-order (second derivatives) in the "potentials" \(g_{\mu\nu}\) and \(A_\mu\).

All these things seemed clear by 1916 but physics was still (or already) separated to two different interactions that were based on different fields, different symmetries, and so on. Shouldn't they be unified much like electricity was previously unified with magnetism?

As a high school student, I read most of these papers by Einstein about "unified field theories". Well, I didn't really speak much German so the word "read" may be too strong a word. But I had a dictionary and the most important objects were the mathematical symbols, anyway. ;-)

A few years later, it was obvious to me that most of Einstein's attempts were naive and misguided. For example, he wanted to "efficiently combine" the gravitational and electromagnetic fields by defining an asymmetric metric tensor\[

G^T_{\mu\nu} = g_{\mu\nu} + F_{\mu\nu}.

\] So simple. You just add the symmetric metric tensor of gravity and the antisymmetric field strength tensor of electromagnetism. It may look natural but it's just a physically unnatural administrative trick, a bookkeeping device. The equations of motion are second-order in \(g_{\mu\nu}\) (they depend on the Riemann/Ricci curvature) but first-order in \(F_{\mu\nu}\). So neither first-order nor second-order equations for \(G^T_{\mu\nu}\) will be natural. We are really adding apples and oranges here.

(Perturbative string theory sort of allows you to add the symmetric metric tensor with the antisymmetric field \(B_{\mu\nu}\) but the latter is a potential, not a field strength. Well, on D-branes this \(B_{\mu\nu}\) gets mixed with an \(F_{\mu\nu}\) field strength but due to the new fields, it's still nonsensical to create a theory that only adds the metric tensor and the normal field strength. Quite generally, we may say that it's no progress to add symmetric and antisymmetric tensors – the right direction is exactly the opposite, namely to decompose all tensors and representations to the irreducible parts because those may enter the most general allowed equations differently.)

In other papers, Einstein wanted to switch from \(g_{\mu\nu}\) as the elementary field to the Christoffel symbol \(\Gamma^\alpha_{\beta\gamma}\) and add various antisymmetric parts to this non-tensor (add the torsion) and identify this part with the electromagnetic fields in various ways. All of these attempts were really misguided for various reasons.

Well, we might argue that the very program to unify electromagnetism and gravity was misguided because this unification only occurs at the Planck scale, at energies higher than the energies of other unifications (i.e. after other steps of the unification). So any unification of gravitation and electromagnetism could be thrown to the trash bin if you were strict.

But there were two classes of papers that were natural, at least at some moral level. The Weyl-style papers and the Kaluza-inspired papers (papers on the Kaluza-Klein theory, as we call this direction today). Weyl was trying to add the \(U(1)\) gauge symmetry among the local symmetry principles of GR (add it to the diffeomorphisms) just like in the "seggregated" treatment of gravity and electromagnetism. But he wanted to interpret this \(U(1)\) as the noncompact form of it, \(\RR^+\), which was also proposed to have a geometric visualization as the (spacetime-dependent) change of the scale for distances.

The need to use the noncompact version of \(U(1)\) is actually a problem but just a "global one". At the level of Lie algebras, Weyl's thoughts were OK. Well, the problem I mentioned two sentences ago was still preventing Weyl from constructing realistic models. Moreover, despite the geometric interpretation of the Weyl scaling, he didn't really naturally unify the local groups.

However, the most sensible way to incorporate electromagnetism into general relativity in a unified way was kickstarted by Theodor Kaluza in 1919. For two years, Einstein was the slow referee who was preventing Kaluza's paper from being published. But that changed in 1921 and Einstein became a fan of the idea of a sort, although a less enthusiastic fan than what would be appropriate for the only sensible "GR-EM unification idea" that Einstein has ever worked on.

The right way to combine the metric tensor and the electromagnetic field isn't to work with the torsion; it isn't to add symmetric and antisymmetric tensors to general tensors. The right way to combine them is to realize that the spacetime is a bit more complicated than we thought and it doesn't have just 4 spacetime dimensions. It has at least 5 spacetime dimensions. So the full metric tensor becomes \[

g_{MN}, \quad M,N=0,1,2,3,4

\] The values \(0,1,2,3\) correspond to the usual indices \(\mu,\nu=0,1,2,3\) which label the well-known time and three spatial coordinates. The coordinate \(M=4\) or \(N=4\) is new and it supplements the metric tensor with some new components \(G_{\mu 4}\) and \(G_{44}\). Under the reparameterizations of the four well-known coordinates, \(G_{\mu 4}\) is a vector just like \(A_\mu\) while \(G_{44}\) is a new scalar field (that I will call the dilaton: hi Dilaton).

The new scalar field may look unwanted but the vector field \(G_{\mu 4}\) is apparently exactly what we need to explain electromagnetism. It's not just "the right number of new fields". Kaluza showed that when we decompose the five-dimensional Einstein's equations to the equations with one index equal to \(4\), we do get Maxwell's equations, up to some harmless nonlinear and/or dilaton-dependent corrections. He reached this conclusion by doing several lines of the boring algebra – by deriving the curvature tensor from the Christoffel symbol and by decomposing the components according to the number of indices that are equal to \(4\).

It shouldn't be surprising that he had to derive the gauge-covariant Maxwell's equations out of the diffeomorphism-covariant Einstein's equations in five dimensions. Why? Because the electromagnetic \(U(1)\) gauge symmetry itself has been incorporated into the group of five-dimensional coordinate transformations (diffeomorphisms!). And because Maxwell's equations pretty much follow from the gauge symmetry and the right collection of fields and because we have the right fields and the right gauge symmetry, we must also derive Maxwell's equations from the good equations that obey the more general principles, the five-dimensional ones.

Klein's clarification

Kaluza didn't "actively know" that the fifth coordinate in his picture was compactified. It had to be clear to him that while he claimed to unify the known four dimensions with a new, fifth one, he was treating the fifth dimension a bit differently. I actually think that he was always assuming the "strict dimensional reduction". It means that none of the tensor fields are allowed to depend on the fifth coordinate at all:\[

T_{\mu\dots} (x^0,x^1,x^2,x^3,x^4) = T_{\mu\dots} (x^0,x^1,x^2,x^3).

\] There's no dependence on the new dimension of space. After Klein's contributions, we know that this condition ("dimensional reduction") was equivalent to banning all electrically charged fields and particles. So Kaluza could have derived the equations for the electromagnetic waves but he couldn't have incorporated any electromagnetic sources. Without sources, electromagnetism is a bit incomplete.

Imagine that you have a five-dimensional theory obeying the principles of general relativity as well as the independence of the tensor fields on \(x^4\). Now, you may ask: what are the coordinate redefinitions that preserve this special form of the tensor fields? Clearly, you're not allowed to mix \(x^4\) with \(x^\mu\) too much because the tensor fields would acquire an \(x^4\)-dependence from the mixing and their \(x^\mu\)-dependence.

For a while, you might think that the special form only allows the four-dimensional diffeomorphisms\[

dx^\mu \to dx^{\prime\mu} (x^\mu), \quad x^4 = x^{\prime 4}.

\] But this is actually not the most general five-dimensional coordinate redefinition that preserves the \(x^4\)-independence. We may also redefine \(x^4\) but only in a very limited way:\[

x^4 \to x^{\prime 4} = x^{4}+\lambda(x^\mu).

\] We may change \(x^4\) by an additive shift. However, this additive shift, \(\lambda\), may depend on the well-known 3+1 spacetime dimensions. Because we're only shifting in the fifth direction, the independence of the fields on this direction is untouched. The metric tensor components \(G_{\mu 4}\) remember the "stress" or tilt so they're sensitive on \(\partial_\mu \lambda\) but that's OK because \(\partial_\mu \lambda\) is \(x^4\)-independent, too.

Everything seems to be alright. In other words, the \(U(1)\) gauge symmetry at each point of the spacetime is "geometrized" as a simple additive shift of the new coordinate by an amount that depends on the point in the 4-dimensional spacetime. As Klein realized, we want the \(x^4\) dimension to be compact i.e. the coordinate to be periodic. That's why we automatically get the compact, one-dimensional \(U(1)\) group: this group is nothing else than the isometry of the circle (the isometry of a manifold is the group of diffeomorphisms that preserve the values of the metric tensor at each coordinate point) and the circle is the shape of the compact dimension.

A simple rescaling is enough to choose a particular convention for the period of \(x^4\). We may consider \(x^4\) and \(x^4+2\pi\) to describe "the same point" in the five-dimensional spacetime. Alternatively, at least if we assume \(G_{44}\) to be constant for a while, we may allow the periodicity of \(x^4\) to be arbitrary but it is fixed by the condition \(G_{44}=-1\) (in the mostly minus convention for the metric), just like we have \(G_{33}=-1\) etc.

This choice of coordinates is particularly intuitive. We simply have a 5-dimensional spacetime but the coordinate \(x^4\) is periodic. The periodicity may be called \(2\pi R\) and \(R\) is the "radius of the circular dimension".

Oskar Klein understood quantum mechanics very well so he of course also knew that the wave function ("de Broglie wave", to use a very old language) depended on the coordinates as\[

\exp(i p_4 x^4 / \hbar).

\] When we demand this wave function (we really want to talk about a field!) to be single-valued, it implies that \(x^4\to x^4+2\pi R\) isn't allowed to change the phase above. But the exponent changes by \(2\pi i p_4 R/\hbar\). The exponential doesn't change if this exponent is a multiple of \(2\pi i\) because \(\exp(2\pi i)=1\). It is a multiple of \(2\pi i\) if \(p_4\) is an integer multiple of \(\hbar / R\).

If the five-dimensional theory contains no tachyons (superluminal particles or particles with spacelike momenta), it guarantees that \(E\geq |\vec p|\). In particular, \[

E\geq |p_4|.

\] The energy of a particle can't be smaller than the momentum in the new direction. But this momentum \(p_4\) is either zero or at least equal to \(\hbar/R\). If \(R\to 0\), all the new particles with \(p_4\neq 0\) are automatically very high-energy states that may be overlooked in an approximation.

Well, we can be more specific. The five-dimensional dispersion relation relates the energy, momentum, and the rest mass as (\(c=1\))\[

E^2 - p_1^2-p_2^2-p_3^2-p_4^2 = m_0^2.

\] However, it's equivalent to an equation where \(p_4^4\) is moved to the right hand side:\[

E^2 - p_1^2-p_2^2-p_3^2 = m_0^2+p_4^2=\dots

\] This is a particularly natural step if we don't want to treat the fifth dimension on par with the normal four dimensions anymore. When we downgrade it in this way, the momentum component \(p_4\) becomes a contribution to the rest mass as understood by an effective four-dimensional theory:\[

\dots =m_0^2 + p_4^2 = m_{0,\text{4-dim eff.}}^2

\] Note that \(m_0\) without extra subscripts was the rest mass as naturally defined in the five-dimensional spacetime.

But now we may summarize all these things. We previously said that the single-valuedness implied that \(p_4=Q\hbar / R\) where \(Q\) is an integer. Because the momentum is nothing else than the change of the phase induced by a translation of the spatial coordinate, \(x^4\) in this case, and this translation was exactly what we identified with the \(U(1)\) gauge transformation, one that is generated by the charge, it's clear that the integer \(Q\) in \(p_4\) is nothing else than the electric charge!

Moreover, the formula for \(m_{0,\text{4-dim eff.}}\) above tells us that the squared electric charge contributes to the mass of the charged particle (as understood in four-dimensional terms: the five-dimensional field was able to produce particles with many different momenta \(p_4\) – many different electric charges in four dimensions – and all of them had the same "five-dimensional mass").

The only question was how large \(R\) was. Kaluza was wise enough to conclude that \(R\) should be comparable to the Planck scale. That implies that the lightest charged particle should have a mass of order \(\hbar/R\) in the \(c=1\) units, i.e. the Planck mass. That's overwhelmingly heavier than the electron, and that's why the theory was in trouble.

The conclusion that the radius of the new dimension is Planckian was mostly a matter of intuition – or, if we use more self-confident words, a result of a dimensional analysis argument. Klein could have also invented "old large dimensions" 70 years before Arkani-Hamed, Dvali, and Dimopoulos. It would be cool but not too shocking because he wrote down an "almost complete Standard Model" 30 years before Glashow and pals, too (he really did understand massive vector fields as messengers of short-range forces in the 1930s). Klein was well ahead of his time and the natural maths of the Kaluza-Klein theory was what was leading him in the right direction.

However, the radius of the new dimension comparable to the Compton radius of the electron was apparently leading to other problems – one of them was, of course, that Klein didn't know how to explain why the proper radius of the new dimension stabilizes at this unnatural value (the question is just a little bit less mysterious to us today: the only natural length scale seems to be the Planck scale as long as we avoid complicated enough mathematical expressions but at least everyone understands that the tiny electron/Planck mass ratio is an empirical fact so the theory has to include this small number somewhere or explain it in some way).

Stringy twists to the Kaluza-Klein theory

String theory resuscitated all these ideas sometime in the 1970s and especially in the mid 1980s when it began to build realistic models that are "completely correct as the first sketches", using Edward Witten's words. However, the incorporation of the Kaluza-Klein theory into string theory brought us some new improvements and generalizations. What are they?

First of all, we don't have just one extra dimension but 6 or 7. This allows the compactification manifold to have more complicated isometry groups than \(U(1)\). In principle, you could imagine that such an isometry group is enough to describe the whole Standard Model group. However, it may be shown that 6 or 7 dimensions isn't enough for that.

This is related to the second "modernization" of the old Kaluza-Klein theory. String theory generalizes many notions of geometry itself. For example, in perturbative string theory, it splits spacetime coordinates \(X^\mu\) to "almost independent" left-moving and right-moving coordinates on the world sheet. The separation is possible thanks to the conformal symmetry on the world sheet which makes many quantities holomorphic or antiholomorphic (or their simple sums and products as opposed to most general functions of the world sheet coordinate and its complex conjugate).

Because of this separation, string theory also allows you to compactify "purely left-moving coordinates". That's the miraculous virtue of the heterotic string. Moreover, one may get new currents – symmetry generators – not just from old-fashioned geometric isometries but also from their complicated CFT primary field counterparts. That's why sixteen left-moving periodic bosons don't allow the Kaluza-Klein gauge group to be just \(U(1)^{16}\) in the 10D spacetime, as we could naively expect from the old Kaluza-Klein theory. Instead, the whole gauge group gets enhanced to a rank-16, dimension-496 group, either \(SO(32)\) or \(E_8\times E_8\). These gauge groups may still be viewed as the "stringy generalized isometry" of the self-dual 16-torus that is used in the role of the compact dimensions according to the rules of the (stringy modernized) Kaluza-Klein theory.

Another important stringy modification of the Kaluza-Klein theory is that even in the maximally decompactified spacetime, which is 10-dimensional in perturbative superstring theory, \(m_0\) is a more complicated quantity than a simple numerical constant. Recall an equation written above\[

m_{0,10}^2 + p_{4-9}^2 = m_{0,\text{4-dim eff.}}^2

\] where I added a subscript "ten" (formerly "five") to identify the rest mass according to the maximally decompactified i.e. ten-dimensional theory; and where I replaced \(p_4\) by \(p_{4-9}\) to emphasize that the momentum has six compactified components instead of just one. You see the generalization?

So far, I have made the obvious generalizations. However, the extra new point of string theory is that even \(m_{0,10}^2\) is something else than a simple number. The ten-dimensional rest mass of a string is determined by the amount of its internal oscillations. The more the string oscillates, the more massive particle it resembles (even in ten dimensions).

I hope that you remember the problem experienced by Klein – namely that if the radius of the new dimension is Planckian, the electron (the lightest charged particle) should have a mass comparable to the (huge) Planck mass, too. Where does string theory find a loophole in this crippling conclusion?

The stringy loophole is that for the models with Planckian sizes of the extra dimensions etc., the electron effectively arises from a 10-dimensional field whose \(m_{0,10}^2\) is negative and almost precisely cancels the contribution from the compact momenta \(p_{4-9}^2\). That's why the electron's \(m_{0,\text{4-dim eff.}}^2\) observed in the four-dimensional spacetime is so much smaller than the Planck mass.

Such cancellations may look like a fudge but they're actually not fudges at all. Supersymmetry as well as the natural structures of perturbative string theory make such cancellations natural and sometimes more or less inevitable. To discuss which of these adjectives is right and why, we would have to focus on particular classes of realistic stringy compactifications. The detailed reasons why we naturally get light charged particles differ between the type IIA braneworlds, heterotic string theory models, M-theory compactifications, F-theory, and so on. But the broad conclusion holds: the extra stringy features – the properties of string theory that go beyond "ordinary field theory in 10 dimensions" – provide us with exactly the required loopholes that invalidate Klein's conclusions that made the Kaluza-Klein theory so unpopular sometime in the late 1920s or in the 1930s. There exist natural string theory models, both with the Planckian and larger extra dimensions, which generate realistic spectra of particles including light charged particles such as the electron.

Even though the program to unify "just the electromagnetism with gravity" has been known to be outdated for many decades, Theodor Kaluza has found and Oskar Klein has clarified an important conceptual trick that Nature is using to unify gravity with the non-gravitational forces described by the gauge fields. Because Kaluza and Klein didn't know string theory, some features of their models looked unsatisfying. But I am sure that if they were taught string theory today, they would agree that their visions suddenly make complete sense.

And that's the memo.


  1. Theodor was born in schliessen like my Grandfather Johann Kaluza, I think

  2. kashyap vasavadaNov 9, 2013, 3:45:00 PM

    Great summary Lubos. If I understand, cosmology may require replacement of Lambda by a time dependent function (some field) , which would destroy covariance. Have string theorists succeeded in replacing Lambda by an appropriate function? If ST unifies forces then this must happen.If you have talked about this before, please give a reference. Thanks.

  3. Thanks, Kashyap, and nope, the cosmological constant isn't time-dependent. That's why the word "constant" is in the name. Up to some highly questionable 2-sigma flukes claimed by someone, everything in cosmology is compatible with the claim that the dark energy causing accelerated expansion *is* the cosmological constant i.e. it is the Lambda term above where Lambda is spacetime-independent.

    In theory, the value of Lambda depends on which vacuum you are in. The vacuum may be visualized as a minimum in a complicated landscape. The elevation of each minimum gives you the cosmological constant (the vacuum energy density, in particle physicists' preferred interpretation).

  4. kashyap vasavadaNov 9, 2013, 5:53:00 PM

    OK. But at some point in the history of cosmos, perhaps at the time of inflation , there would have to be space time dependent field in place of Lambda. It is not clear to me how this field (say Inflaton) would become constant Lambda or Lambda is a different quantity which comes up just after the inflation is over. Don't they serve similar purpose, repulsive expansion?

  5. Strominger and others (arXiv:0907.2562, 0811.0181) consider left-moving versus right-moving coordinates. Vacuum mirror symmetry is broken toward matter to default compactify the added dimension. Absolutely discontinuous symmetry parity is outside Noether. Conserved currents have a footnote engaging parity violations, chiral anomalies, symmetry breakings; Chern-Simons parity repair of Einstein-Hilbert action in quantum gravitation .

    String theory assuming the Equivalence Principle (EP) offers 10^500 vacua. Given trace EP violation by opposite shoes, how many chiral vacua remain? Geometric parity is an emergent phenomenon. The smallest build scale is atoms - a periodic crystal's unit cell. Crystallography's 320 periodic space groups include 11 pairs of enantiomorphs. Anything so crystallizing has maximum geometric parity divergence, left versus right. Tonnages of alpha-quartz single crystals are annually grown for crystal oscillator chips, enantiomorphic space groups P3(1)21 and P3(2)21.

    String theory can be made empirical. Drop single crystal quartz enantiomorphs, and look.

  6. Haha Lumo, how did you know that I can not avoid reading such an immensely nice article ... :-D ?!

    This text about exactly at my level, there are only a few things left I dont unterstand (in the last paragraph). For example how does conformal symmetry allow to partition the X^{\mu} into right and left movers? I dont see this relationship ...

    Anyway, reading this article is really what I needed to distract me from certain things that did not go so well last week :-)


  7. The Burning Bush was picked as the Czech candidate for the foreign Oscar this year but the proposal was killed by some Americans who didn't find the format to be OK or whatever so a film by Menzel became the representative instead.

    Commies rule Hollywood. They didn't want their heroes trashed in public, much less winning awards.

  8. Thanks a lot for this excellent article!

    I especially like the section "Klein's clarification" (haven't read the "Stringy" section yet), since I didn't know that Kaluza just meant that the tensor fields don't depend on x^4 at all!

    I always used to wonder what the meant when hey says that the dimension is only a "mathematical"/"abstract" thinng, etc.

    Thanks for the clarification, and as to how this basically is what Klein said because the symmsetry along x^4 is the same as a symmetry that results from the relevant (circular) compactification.

  9. thejollygreenmanNov 10, 2013, 4:44:00 PM

    Recent event in the USA - during the government shutdown - showed that not country in the world has a copyright on cunts, they are everywhere.

    Judging by the way American civil servants behaved at Mount Rushmore - on private property - in Yellowstone park towards the elderly, and at the Washington, we can clearly see that they will cart their own off to concentration camps, just as quick, if not quicker, than the German, Austrian, Polish or Czech Civil Servants did, the minute they get word to do so by their glorious leader in the White House.

  10. I believe Scott's claim rests on a narrow definition of what constitutes a scientific forecast. Given a particular definition of the term, the claim can easily be true. I'm not sure I understand why a particular definition should be considered the "right" one, though. I am open to having it explained to me, though.

  11. You expressed much more eloquently how I feel about Scott's proposition.

  12. But when the WW II veterans on their walkers and in their weelchairs simply went past the baton-wielding uniformed thugs to visit their monument, it was an uplifting moment. Sure tugged on my heart strings. And while you may be right about how U.S. civil servants may not be any better, I believe that civil disobedience and plain physical courage are still more prevalent in the U.S. than elsewhere.

    Well, maybe not. The curious lack of outrage over NSA blanket spying on all citizens seems to disprove my point. Depressing thought.

  13. You use the word loophole for -m2 . That suggests something not quite legitimate or at least not the best answer. To the contrary it is a feature essential to allow us to conserve the charge and supercharge etc. of different particles at varying external momentum. One can externalize the momentum and angular momentum and by this conserve energy. But inside the particle we must have some way to identify its charge , etc. .. How else but additional dimensions could we conserve charge or for that matter have a zero mass photon if there were not degrees of freedom that are not externally seen as mass-energy?
    So to me it seems not a loophole but compelling evidence for string theory.

  14. Lubos:

    Some years ago my coworker Przemek Klosowski gave me this Polish tonguetwister:

    W Szczebrzeszynie chrzaszcz brzmi w trzcinie

    (In Szczebrzeszyn the beetle rustles in the reeds)

    For this poor American he provided the pronunciation guide

    V Shcheebjesheenee hjonshch bjmee v tjcheenee

    and promised that if I could repeat it 10 times fast down at the Polish embassy, they might grant me honorary citizenship.

    I'm still working on it.

  15. Dear Jeffrey, LOL, I granted you a white list status, perhaps because their medieval cz-rz-ie etc. spelling makes it look even more twisted. ;-)

  16. Brian G ValentineNov 12, 2013, 6:15:00 PM

    "you might say that this is an excessively unflattering description of a man who spoke 17 languages and claimed to prefer Arabic

    I don't know if I speak "17" of them but I think I prefer Arabic as well

    What about your birthdaysakes

    Constantin Caratheodory and I share the same birth day but not the same year, of course.

    Of the 10 nontrivial scalar differential equations comprised by the contracted Riemann-Christofel tensor of spacetime, two are redundant, four represent the gravitation potential, and the other four are left undefined unless additional conditions are imposed on them; Weyl and others identified these with the 4 components of the scalar electromagnetic field, which he could do, only if length was an inexact differential.

    Few subscribe to a theory of nonintegrability of length today but I like like the idea anyway.

    Dirac could not tolerate reading anything written by Weyl evidently.

  17. Wow, Brian, Arabic. Have you dated an Arab woman? ;-)

    Just the date but not the year is trivial, I can emumerate a dozen of such people in my case.

    It is simply *not* correct to call the component equations of a tensor equation "scalar equations". A component and a scalar may represent 1 number in both cases but they're different - mutually exclusive. A scalar is something whose value doesn't change under the coordinate transformation. A component of a tensor (except for a scalar) does change.

  18. Dirac was a physicist and Weyl was (in spite of Lubos calling him a "mathematical physicists") a mathematician.

    Once "proof" of that is his famous (but joking) remark " my work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful". I will not try to suggest that this shows that he as an early string theorist ;-)

    The second proof is Michael Atiyah's remark that whenever he thought of some fundamentally new idea he invariably discovered that Weyl had thought about it first. This is certainly excessively modest of Atiyah, but it also shows that the principal interests and achievements of Weyl were in mathematics.

    The third and to me the most convincing argument is Weyl's famous book on invariant theory "The Classical Groups", which was, by the way, the first book my supervisor told me to read when I started my graduate work at Oxford. It was very hard to read, thoroughly algebraic but in the end I learned more from it than any other book on this subject. I strongly suspect that Dirac would have hated it.

  19. Brian G ValentineNov 12, 2013, 7:22:00 PM

    I dunno, if y is a variable and satisfies some equation with values in R^1 I call it a "scalar equation," and if similarly y has components with values in R^2 I call it a "vector equation," even though it might come from the same original equation (as a 2nd order d.e. for example), so I guess I'm abusing language here like I abuse everything else.

    No I have never dated an "Arabic woman" or anybody except my wife for that matter

    I am still proud to share the same birthday with Caratheodory, so I advertise it. Sorry.

  20. Brian G ValentineNov 12, 2013, 9:29:00 PM

    The first book by Weyl I read was "Theory of groups and quantum mechanics," anyway I learned a lot from what both Michael Atiyah and Hermann Weyl contributed, and if Atiyah had that much regard for Weyl - then everybody ought to.

  21. For Dr. Armstrong and his colleagues, the definition of what constitutes a "scientific forecast" is implicit in a check list that is referenced by them in their paper. The model that is claimed by Armstrong et al to be "scientific" has features which, according to the authors of this check list, are characteristic of a model that makes "scientific forecasts." In this post, I address the claim by Armstrong et al that their model makes "scientific forecasts."

    As noted by Werdna and Motl, the term "scientific" is polysemic, that is, it has more than one meaning. That it is polysemic supports an argument in which a term changes
    meaning in the midst of this argument. In logical terms, an argument in which a term
    changes meaning is an example of an "equivocation."

    An equivocation appears to be an example of a syllogism, that is, an argument whose conclusion is true. This appearance is, however, false and misleading. Thus, by rule, to
    draw a conclusion from an equivocation is logically improper. To draw such a conclusion is an "equivocation fallacy."

    Commonly climatologists employ the equivocation fallacy in making arguments about global warming ( ). In doing so, they draw improper conclusions from equivocations.

    The possibility of drawing improper conclusions from equivocations may be avoided through disambiguation of the terminology in which arguments about global warming are made. Thus, through disambiguation, the stage can be set in which logically proper conclusions are drawn from arguments about global warming.

    In the description of a climate model, a logically and scientifically important distinction is between a model that makes a predictive inference and a model that does
    not. A "predictive inference" is an extrapolation from an unspecified state of a system to a specified state; conventionally, the former state is called the "condition" while the latter state is called the "outcome." A condition and outcome describe an event.

    Events are a property of the former type of model but not the latter. For the IPCC climate models of AR4, there are no events. Hence, the climate models of AR4 are of the
    latter type. They make no predictive inferences or predictions.

    In statistical jargon, the count of statistically independent events of a particular description is called the "frequency." The ratio of the frequency of events of a particular description to the frequency of events in general is called the "relative frequency." The relative frequency of an event of a particular description is the empirical counterpart of the probability of an event of this description. Given that it has no empirical counterpart, the probability of an event of a particular description does not
    exist under the methodology of science.

    In the classical logic, the permissible values of the probability of an event of a particular description are limited to 0 and 1. The former of the two values corresponds to "true" for the proposition that an event of this description occurs and the latter to
    "false" for the same proposition. The non-existence of the probability as a concept wipes out the notion that a proposition may be true or false thus wiping out logic.

    The model of Armstrong et al differs from the models of AR4 in possessing underlying
    events. However, these events lack the property of independence. The lack of independence wipes out the notion of a frequency hence wiping out logic. Also, claims made by this model regarding the outcomes of these (non-independent) events are falsified by the evidence. In these respects, the model of Armstrong et al is not what I would call a "scientific model."

  22. Lubos, if you have not seen it, please consider watching this life story by Jan Klein, famous Czech / US immunologist. Pay special attention to his experiences with Communism and the State Police. He was fortunate to be in Japan at the time of the Warsaw Pact invasion and decided to go the the US.

  23. If you sort through the topics he has some very critical comments re science publishing, fashionable science, mitochondrial Eve hypothesis etc.

  24. See 101. The importance of avoiding politics in science
    Duration: 5 minutes, 7 seconds.