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.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.
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.
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.
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