Friday, November 19, 2010

What grand unification can and cannot do

Among the physics fans, there's still a lot of confusion - sometimes deliberately propagated confusion - about the basic questions surrounding gauge unification.

So what is grand unification and what it can do for us?

Aside from gravity described by Einstein's general relativity, there are three non-gravitational forces: the electromagnetic interaction - unified in the 19th century; the weak nuclear force; and the strong nuclear force.

The last two interactions include the adjective "nuclear": that's why you need to look at the microscopic world and to use quantum mechanics to appreciate what they really mean. And that's the reason why Einstein never understood anything about them. He thought that the goal was to unify electromagnetism and gravity only; his attempts to construct a unified field theory had no chance to succeed because gravity is the last force that may join the other forces - but only a long time after one understands the unification of electromagnetism with the two nuclear forces.

The physicists who continued to care about the detailed facts revealed by the experiments have eventually discovered the underlying character of the two non-gravitational forces and realized what they share with electromagnetism. What they share is very powerful. The formalism is ultimately closely analogous for all the three forces. And grand unification that will be discussed below is the idea that one shouldn't describe the three forces just by "three analogous pieces" of formalism but by "one piece" only.

Already in the 19th century, people would be able to describe electromagnetism in terms of the 4-vector potential, "A_mu". It wasn't uniquely determined, however: there was a redundancy. "A_mu" may be changed by a gradient "partial_mu lambda" of a parameter field "lambda(x,y,z,t)". Such a change - called the gauge transformation - modifies neither the electric fields "E" nor the magnetic fields "B" which are the only things that can be measured (in classical physics).

When quantum mechanics was appreciated, it was clear that the charged fields have to be multiplied by "exp(i.Q.lambda)" for the symmetry to remain a symmetry. It was the first time when this "lambda" occurred without any derivatives. This absolute value of the exponential equals one. The multiplicative group of all such complex number is called "U(1)", the simplest unitary group. That's how electromagnetism was identified as a "U(1) gauge theory", using the modern language.

What about the other non-gravitational forces?

As recently as in the 1930s, there was a complete confusion about their number etc. As David Gross described in a highly entertaining talk, people were confused not only about the validity of quantum mechanics at short distance scales. Most of them didn't quite understand that quantum field theory applied both to bosons and fermions. And they also thought that there was only "one kind" of a nuclear force that had something to do with particles called "barytron" and "mesotrons" which were bizarre hypothetical hybrids of pions, muons, and perhaps many other very different actual particles. ;-)

Recall that there is a very sharp difference between the strong force and the weak force. For example, the strong force is very strong. It's stronger than electromagnetism. That's why it can beat the repulsive electric force between two protons and keep the protons together inside the nuclei. On the other hand, the weak force is very weak and only causes some very rare processes such as beta-decay of the nuclei - including the decay of a neutron that takes 15 minutes or so.

In the article by David Gross above, it's explained that Oskar Klein was perhaps closest to the "Standard Model" back in the late 1930s. He didn't quite understand the non-Abelian generators - but the structure of his "theory of nearly everything" was remarkably similar to the electroweak theory that was found 30 years later.

In 1953, Yang and Mills finally wrote down their equations for non-Abelian gauge theories. In the early 1960s, Glashow successfully proposed the SU(2) and, more importantly, SU(2) x U(1) as a description of the weak force - that was previously described by the effective four-fermion interaction initiated by Fermi in the 1930s and clarified (especially when it comes to the Lorentz indices) by Feynman and Gell-Mann in 1958.

Glashow's theory was completed by Salam and Weinberg in the late 1960s when they used the correct Higgs mechanism to spontaneously break the SU(2) x U(1) symmetry down to the electromagnetic U(1). The Higgs mechanism itself had been "ready" and, much like (previously) the idea of Yang-Mills theories themselves, waited in the general toolkit of neat mathematical physics ideas that wanted to be applied in practice.

On the other hand, the SU(3) gauge theory for the "color charge" was found to be asymptotically free and relevant for the attraction between quarks or partons. The quarks were seen to be confined in protons and neutrons and the remnants of this strong force are also able to keep the nuclei together. Unlike the SU(2) x U(1) electroweak gauge group, the SU(3) strong gauge group is not broken. It's just confined.

To summarize, all the non-gravitational forces were described by Yang-Mills theories based on certain gauge groups. It just happens that John Iliopoulos was the first man to write a paper that put all these terms together - in what we currently call the Standard Model (including QCD). The term "Standard Model" itself was previously coined by Steven Weinberg himself but he mostly meant the electroweak theory only. Of course I would find it insane if Iliopoulos were actually called the father of the Standard Model - as some writers have argued.

Spectrum of the Standard Model

So the Standard Model forces are derived from the group which is the direct product of three factors,
SU(3) x SU(2) x U(1).
Here, SU(3) is the symmetry rotating three possible colors of quarks - that may be playfully called "red, green, and blue" although there is of course no relationship with the actual colors (frequencies of photons) that our eyes can see. It's just helpful for the analogy that people can see three independent colors with their eyes and their "uniform mixture" is perceived as grey or color-neutral. The SU(3) group is confining but becomes weaker at shorter distances which makes the quarks and gluons "asymptotically free" at very short distances.

The SU(2) group is the main group of the weak force, linked to the so-called isospin. The U(1) - the generator of the hypercharge - has to be added to distinguish the properties of the partners of the isospin doublets (e.g. electrons and their neutrinos) after the electroweak symmetry breaking. The SU(2) x U(1) group is broken by the Higgs mechanism to a subgroup, another U(1) which generates the ordinary electromagnetic charge: it's the sum of the hypercharge and the third (z) component of the SU(2) triplet of generators, in the standard convention.

The breaking of the symmetry makes the W and Z bosons massive. The photon is the combination that remains massless because it's associated with the generator of U(1) - the electric charge - that is not contained in the Higgs field condensate that broke the symmetry in the first place (the SM Higgs bosons are neutral, too). That's why the photons remain massless, the electromagnetic force remains a long-range force, and the electric charge continues to be conserved.

All these three forces are mediated by spin-one gauge bosons - the photon, 8 gluons, Z boson, W+ boson, W- boson. There are 8 gluons (counting the color-related degeneracy) because the SU(3) has 3^2-1=8 generators and each generator produces its own gauge boson. And there are 4 electroweak gauge bosons because SU(2) x U(1) has (3+1=) four generators.

When it comes to the representations of the rotational or Lorentz symmetry (well, the little group), each gauge boson has 2 or 3 polarizations. The photons and gluons have 2 polarizations because their symmetry generators remain unbroken. For photons, the 2 polarizations may be visualized as the transverse "x,y" linearly polarized or as "clockwise,counter-clockwise" circularly polarized photons (which is just a different basis of the same 2D space). This limited number of polarizations (which is only possible for massless gauge bosons) is what follows from the conservation of the symmetries: they're unbroken. The original potentials "A_mu" would have 4 components but 2 of them are killed by the gauge symmetry.

(That's needed to eliminate the negative-normed time-like components but there are actually 2 killed polarizations and not just 1 - one of the four is "pure gauge" and zero-norm and decouples from everything; and the other is forbidden by the "Gauss's law, div D = rho, or essentially div E = 0" restricting the initial conditions.)

On the other hand, the W+, W-, and Z bosons correspond to the (4-1=) three broken generators of SU(2) x U(1). The bosons become massive and massive vector particles have to have 3 polarizations (x,y,z) instead of just two to form the representation of the SO(3) little group. The additional polarization that was absent for the photons and gluons - the longitudinal polarization - is a new degree of freedom that isn't composed out of the gauge field only (the gauge symmetry still kills two). Instead, the component only arises because there also exists the Higgs field whose "Goldstone boson" components are "eaten" to become the new, longitudinal polarizations of the W,Z bosons. This "eating" is a basic insight of the Higgs mechanism.

The simplest Higgs field is a 2-component complex doublet of scalar fields which is equivalent to 4 real polarizations. 3 of them are "eaten" by the W,Z bosons, giving them the extra polarizations (3 in total), so only 1 real Higgs scalar is left after the Higgs mechanism. Correspondingly, in minimal supersymmetric models, there have to be two Higgs doublets - twice the minimal amount. That is equivalent to 8 real polarizations. Again, 3 of them are eaten by the W,Z bosons which is why 5 Higgs bosons are left in the MSSM.

So the Standard Model presents all spin-one carriers - gauge bosons - as the messengers of the three forces. A Higgs sector (composed of scalars, in the most canonical case) has to exist to break the electroweak symmetry.

What about the fermions?

Aside from photons, gluons, W,Z bosons, and the Higgs sector, there are a few important elementary particles. One of them is the graviton but we only discuss non-gravitational forces at this point. The remaining particles we haven't yet mentioned are the elementary fermions - leptons and quarks. Fermions obey the Pauli exclusion principle which is why they're the main components of the "matter that can't penetrate other matter" that we know and love.

The leptons are the electron together with its two heavier cousins (with the same charge and spin), the muon and the tau. All particles also have their antiparticles I won't discuss separately; only for "totally neutral" particles such as the real Higgs boson, photons, gluons, and graviton, the antiparticle coincides with the original particle.

All three charged leptons have their electrically neutral siblings under the SU(2) isospin partnership, namely the neutrinos. The neutrinos would have the same properties as electrons and friends if the electroweak group remained unbroken. The breaking - the Higgs mechanism - is what makes the electrons so much more interacting, visible, and lively than their neutrinos. Note that there are "six flavors" of the leptons - three generations of charged leptons plus their neutrinos. The charged leptons have twice as many components than the neutrinos but we will discuss this point later.

There are six flavors of quarks, too: the upper-type quarks (up, charm, top), and their SU(2) lower-type partners (down, strange, bottom). The masses are increasing from a generation to the next generation. Don't forget that the actual number of "quark species" is 3 times higher than that because each of these 6 flavors of quarks can come in 3 different colors under the SU(3) group.

I have already enumerated all the elementary particles of the Standard Model. It's somewhat contrived but one can master this game of Mother Nature. If you become certain what are the charges (and/or the representations) of the leptons and quarks, you may write the most general renormalizable Lagrangian for these fields. It's called the Standard Model. Once you fine-tune its 19-30 parameters or so (depending on whether you include the small and usually undetectable but nonzero neutrino masses), you may describe all non-gravitational phenomena that have ever been observed, with the only uncertain exception of some newest observations of the dark matter (and other possibly imminent observations of new physics at the LHC).

Unifying the multiplets

The Standard Model is based on the group SU(3) x SU(2) x U(1). Each of the three factors has a separate coupling constant - there are two additional friends of the "fine-structure constant". There is no simple yet universal way how the couplings of three different factors in a group could be linked to each other. So the number of parameters is higher than it would be for a single factor.

Can't we have a simpler group - namely, a "simple group" in the technical sense (one factor) - that would also describe all the phenomena? There is no immediate obstacle. And indeed, things work very well even after you spend many hours with the checks. In January 1974 when your humble correspondent was 1 month old and Howard Georgi was a Harvard Junior Fellow, Howard Georgi and Sheldon Glashow wrote a neat and very important paper,
Unity of all elementary-particle forces.
You may actually memorize the abstract which simply says:
Strong, electromagnetic, and weak forces are conjectured to arise from a single fundamental interaction based on the gauge group SU(5).
Great, especially because the model in the paper could actually achieve what it claimed. The strict model in the paper - the simplest model of grand unification - also made some predictions that were falsified. But before you're obsessed with new predictions, you should check whether your model can agree with the existing data. That's what the two physicists did very well.

How does grand unification work?

Well, grand unification simply acknowledges that the SU(3) x SU(2) x U(1) group of the Standard Model is a subgroup of somewhat larger groups. SU(5) is enough. It has 5^2-1=24 generators which is more than 8+3+1=12 generators of the Standard Model. Of course, you can't just compare the dimensions when you check whether a group is a subgroup of another.

But the Standard Model group is a subgroup. It's not hard to see why. The SU(3) x SU(2) group elements may be written as block-diagonal 5x5 matrices with two blocks (3x3 and 2x2). And there is a room for an extra U(1), too. Its generator may be written as a multiple of diag(+2,+2,+2,-3,-3). Note that the trace (sum of diagonal elements) could be made to vanish, as needed for "S" (special) in the "SU(5)".

So the 24 gauge bosons of the SU(5) gauge theory should be split to the 12 gauge bosons that are known from the Standard Model - plus 12 extra gauge bosons that remain very heavy, because of some new Higgs mechanism that operates at a much higher energy scale than what is accessible to the colliders (not too far from the Planck scale, it actually seems).

You actually need some "Higgs sector" to break the grand unified SU(5) group. In field theory, the Higgs fields should transform as representations of SU(5). You need pretty big ones to be able to break the SU(5) symmetry in the right way. Modern developments, especially those in string theory, indicate that it is very unlikely that the grand unified symmetry is actually broken by a mechanism that may be described by field theories too well. It's more likely that the symmetry is broken by some characteristic stringy effects (e.g. Wilson lines around compact dimensions - or various fluxes in extra dimensions etc.). So you don't need to organize the Higgs sector into SU(5) multiplets.

However, the fermions have to be arranged into multiplets of the new gauge symmetry, SU(5), because they can actually be seen at low energies. They are so light that they cannot possibly depend on purely stringy effects. Do the leptons and quarks fit into SU(5) families? Happily, the answer is Yes. The fermionic content of the Standard Model is composed of three generations. Each of them transforms in the same way under the Standard Model group - and probably also the same way under the grand unified group. So it's enough to check the fields in 1 generation.

For fundamental theories, it's more natural to describe fermions as 2-component spinors. A Dirac spinors may be decomposed into two 2-component spinors. How many left-handed 2-component spinors one generation of fermions stores (besides their right-handed complex/Hermitian conjugates describing the antiparticles)? Well, there are 2 from the electron and 1 from its neutrino (only the left-handed part of the electron interacts via the weak force, and is linked to the neutrino; the right-handed electron is a weak force singlet and the right-handed neutrino has never been seen at all). There are also 2 from the up-quark and 2 from the down-quark - but all these numbers have to be multiplied by 3 because there are three colors for each quark. So we have
2 + 1 + 3 x (2 + 2) = 15
fifteen 2-component fermions per generation. Again, it's not just about the counting of the dimensions but even if you check all the other things, these fifteen 2-component spinors may be written as
5 + 10bar
which are two representations of SU(5). Here, 5 is the fundamental 5-dimensional representation - the normal one that transforms under the 5x5 matrices. And 10bar is the complex conjugate representation to the antisymmetric tensors with two indices which have (5x4)/(2x1) = 10 components.

If you break the SU(5) group into the Standard Model group and you watch what's happening with the formerly irreducible pieces, 5 and 10bar, under the smaller group, you will exactly reconstruct the representations of a single fermionic generation under the Standard Model - with each piece having the same possible colors, isospins, and the right value of the hypercharge.

If you are bothered by the two pieces, 5 and 10bar, and you would also like one piece for the fermions (e.g. because you only want "one kind of Yukawa [Higgs-fermion-fermion] couplings" etc. - even though there will inevitably be different Yukawa couplings for each generation, anyway), then you may prefer an even bigger unification group such as spin(10). SU(5) may be embedded into it by writing the complex number "a+bi" as a 2x2 matrix, "((a,b),(-b,a))".

This gauge group, usually written somewhat carelessly as SO(10) because the Lie algebras coincide, admits a 16-dimensional complex spinor representation that decomposes as 5 + 10bar + 1 under the SU(5) subgroup. The extra singlet "1" is inevitably neutral under SU(5) - and the Standard Model, because it is a singlet - and it may be interpreted as a new neutrino component. Nothing prevents it from interacting with the ordinary neutrino component (even via a simple bilinear interaction), so for general couplings, they form a massive 4-component Dirac neutrino together and the new singlet is referred to as a "right-handed neutrino".

Anomaly cancellation is simple in GUT

The "16" representation of SO(10) also makes it most straightforward to prove that all the "triangle anomalies" cancel among the quarks and leptons. The triangle anomalies for three SO(10) generators may be written as
Tr(Γab Γcd Γef + Γab Γef Γcd)
times factors unrelated to the Yang-Mills gauge group. The gamma matrices are matrices relatively to the 16-dimensional chiral spinor of SO(10). The expression is antisymmetric in the "ab", "cd", and "ef" pairs of indices: the pairs of indices, such as "ab", represent the generators of spin(10).

The symmetrization comes from Feynman diagrams. The trace is easily seen to vanish for any combinations of "a,b,c,d,e,f". The only way how the trace could be nonzero would be if the operator were proportional to the unit operator - other products of gamma matrices have vanishing traces. That means that each value of the index has to appear twice among "a,b,c,d,e,f".

But the only "not manifestly vanishing" choice of "a,b,c,d,e,f" is then equivalent to "1,2,2,3,3,1" but just somewhat less manifestly, the trace will vanish anyway simply because gamma_{23} and gamma_{31} anticommute. So the anticommutator included in the symmetrized expression is zero. ;-) QED, GUT, GOOD. :-)

The result we just obtained may be immediately used in the "less unified" theories, too.

The right-handed neutrino doesn't contribute anything to the traces that only involve SU(5) generators - because the right-handed neutrino is neutral under SU(5). So the trace over 16 states may be reduced to the trace over 15 states. That's why the cancellation also takes place for the 15 two-component spinors only, as long as we only check the SU(5) generators. That's enough to check all the cancellations in the Standard Model (which is inside SU(5)), too. Note that without grand unification, one needs to check the cancellation of about 6 different types of anomalies (electric charge cubed and many combinations of isospin, hypercharge, color etc.).

That's a general theme in unified theories - and in fact, in string theory, too: unified theories often have a bigger number of objects but they are more closely linked together by new symmetries or their other common physical ancestry (in string theory). This common ancestry or symmetries make many proofs using the unified theories much more straightforward even though we are actually proving a much stronger statement.

What the grand unified groups cannot unify

It's important to realize that the grand unification only unifies particles with the same spin. The generators of SU(5) are Lorentz scalars: they carry no indices that would know about directions in spacetime (or space). The SU(5) is totally independent of the Lorentz or Poincaré group. They commute with one another.

(The Coleman-Mandula theorem guarantees that the two commuting factors - spacetime symmetries times internal symmetries - is the only possible realistic bosonic symmetry group for an interacting theory; supersymmetry is a loophole - and the only loophole - allowing us to mix the geometric and internal symmetries a bit because it is not a bosonic symmetry. But all other "mixed/geometric" symmetries would be so constraining that they would force all the interactions to de facto vanish.)

The vanishing of the commutator is the reason why if you take a particle, measure its spin, and transform it under SU(5), you get the same result as if you take the same particle, transform it under SU(5), and measure its spin afterwards. In other words, the spins of the transformed and untransformed particles have to coincide.

In particular, SU(5) or another gauge symmetry cannot ever unify bosons and fermions in unified multiplets because bosons and fermions always have different spins: recall that it is integer-valued for bosons and half-integral for fermions. Everyone who claims to unify bosons and fermions by an ordinary bosonic group such as SU(5), E6, E7, E8, or anything like that is a hack without a basic understanding of groups in physics.

Yes, Garrett Lisi belongs to this set.

You need supersymmetry if you want to unify bosons and fermions into the same multiplet - i.e. if you want to relate their properties to each other. Supersymmetry and grand unification are independent concepts (based on independent, mutually commuting symmetries) - although their combination is even more powerful (and more consistent with the observed values of couplings) than each concept separately.

Also, it is impossible to unify gravity with the non-gravitational forces by ordinary groups. The graviton's spin is two so it cannot be transformed by SU(5) transformations to the spin-one gauge bosons, either. The gauge symmetries for gravity (diffeomorphisms) are vaguely analogous to the Yang-Mills symmetry but they're not the same thing and they cannot be unified (except for theories with extra dimensions known as Kaluza-Klein theory or its extensions in string theory which does present gravitons and gauge bosons as different modes of the same string).

But at the level of field theory in 3+1 dimensions, the gauge symmetry simply preserves the spin, so the fermions will always remain separated from the gauge bosons. And the graviton remains separated as well. There's a lot of confusion among the interested laymen - and bad physicists - about these basic points.

Some people - and it is not just Garrett Lisi - think that if there is a group such as SO(3,1) that occurs somewhere in a description of gravity, it is the same type of a group as Yang-Mills groups and the unification of gravity with other forces becomes possible. And indeed, general relativity may be written in terms of tetrads (or vierbeins) in which SO(3,1) is an additional local symmetry that rotates the local tetrads (without affecting the metric tensor).

However, the local SO(3,1) symmetry is never the whole local symmetry of general relativity. Aside from the local SO(3,1) group, even if it is included by choosing the tetrad formalism, there is also the diffemomorphism group and the diffeomorphism group (in the bulk) simply cannot be obtained as a subgroup of a Yang-Mills group (in the same bulk). That's related to the graviton's spin which is higher than the gauge bosons' spin.

The higher spin of the gravitons makes them "more fundamental", in a sense. So it is the Yang-Mills theories that may be deduced as their simplification in theories with extra dimensions (the Yang-Mills group is the isometry group of the additional Kaluza-Klein internal manifold), not the other way around. And in string theory, all particle species may be considered to be versions of the "stringy generalization of the graviton" - the meaning of this "generalization" is any string, after all. ;-)

What the grand unified groups may be

The group SU(5) was just the smallest "simple group" in the technical sense that could include the Standard Model. Other groups were immediately proposed - especially SO(10) - which has an SU(5) subgroup - and E6 - which has an SO(10) subgroup. There are also "flipped SU(5)" models which are really SU(5) x U(1) models with an extra U(1) - analogous to the extra hypercharge U(1) in the electroweak theory.

And then there are various groups that are not simple (so they're not really "grand unification") but have several factors. Among them, the Pati-Salam models based on SU(4) x SU(2) x SU(2) and trinification based on SU(3) x SU(3) x SU(3) are the most famous ones. While these groups are not simple (they have several factors), the couplings for the individual factors may still be partly or fully linked if you impose additional discrete symmetries (permutations of the factors).

However, there is a key condition here. The groups must admit complex representations - representations in which the generic elements of the group cannot be written as real matrices. Why? It's because the 2-component spinors of the Lorentz group are a complex representation, too. If we tensor-multiply it by a real representation of the Yang-Mills group, we would still obtain a complex representation but the number of its components would be doubled. Because of the real factor, such multiplets would always automatically include the left-handed and right-handed fermions with the same Yang-Mills charges!

That's unacceptable because the left-handed fermions' properties differ from the right-handed ones. That's necessary for the parity violation in the weak interactions, the odd number 15 of the 2-component fermions in each generation that we found, surely for CP-violation, and so forth. So the complex representations of the groups are totally necessary.

That has many consequences. For example, there are five exceptional Lie groups,
G2, F4, E6, E7, E8.
Only the last three are large enough to play the role of a grand unified group. But among these five groups, only E6 actually has any complex representations at all - starting from its 27-dimensional fundamental representation (and its unitarily inequivalent 27-dimensional complex conjugate "antifundamental" representation).

This simple fact automatically means that E6 is the only viable grand unified group among the exceptional groups. The other groups are inconsistent with the observed parity violation in Nature - e.g. with the fact that the neutrinos have to be left-handed.

There are many other limitations. SO(10) has a complex 16-dimensional spinor. But many other orthogonal groups, e.g. SO(11) or anything that fails to be of the form SO(2+8k), fails to have complex spinors. The spinors are either real or pseudoreal (quaternionic) in the 7/8 of the cases which would actually lead to the same problems as the real representations.

So anyone who claims that he has a grand unified theory based e.g. on E8 is a hack who misunderstands exceptional Lie groups in physics, too.

The 1985 heterotic string theory admits the gauge group E8 x E8 - or SO(32) - in ten dimensions. For a decade, the E8 x E8 version was the only known realistic incarnation of the Standard Model in string theory. One of the E8 is a "hidden sector" while the other one is nicely broken to an E6-like subgroup.

(At least four additional large classes of viable vacua in string theory, different from the E8 x E8 heterotic string, were identified in the recent 2 decades - namely Hořava-Witten's heterotic M-theory which is the strong coupling limit of the E8 x E8 strings that produces a new, 11th dimension whose shape is a line interval; intersecting type IIA braneworlds; M-theory on G2-manifolds; F-theory local models together with various type IIB orbifold/orientifolds and perhaps the KKLT vacua.)

How is it possible that E8 works as the "original grand unified group" in string theory even though it has no complex representations? Well, it's because the E8 group only applies to the ten-dimensional physics. The compactification on the 6-dimensional Calabi-Yau manifold is - and has to be - such that the original E8 group(s) is (are) broken by the "geometric effects", namely by a gauge field configuration on the Calabi-Yau manifold. The E8 symmetry is not broken by 4-dimensional Higgs fields but by more general, 10- or 11-dimensional stringy/M objects.

That's why the light fermions only form representations of the unbroken 4-dimensional group, e.g. an E6. So the larger group, E8 x E8, is only visible if you use a complete description with extra dimensions - and perhaps other objects in string theory. However, if you want to remain confined in the realm of four-dimensional effective field theories, E8 can never appear as a grand unified group.

Most of these things were understood immediately as soon as Georgi and Glashow published their paper in 1974 (if not earlier). Of course, the things that depend on heterotic strings could only be understood after the heterotic string theory was constructed in 1985. ;-) But none of these things is new and physics graduate students learn it in standard advanced courses. Let me repeat three obvious facts that are often violated by physicists who are not serious or who simply don't understand the basics of the unification game:
  1. only gauge groups with complex representations, and not e.g. E8, may be the grand unified groups
  2. grand unified groups cannot include fermions and bosons in the same multiplets, or link their detailed properties and interactions in any way
  3. grand unified groups cannot include gravitons and gauge bosons in the same multiplet.
Georgi's and Glashow's overreaction

In 1974, Georgi and Glashow wrote their paper - which, despite lots of amazing things that these two men have contributed, remains the most important paper by Georgi, to say the least. However, their simplest model predicted a relatively speedy proton decay (lifetime around 10^{30} years or something like that) that wasn't observed within a year as expected.

So the two physicists overreacted and decided to abandon top-down physics altogether, focusing on rather mundane bottom-up ideas instead. Nature has slapped them in their face so they told themselves, "Screw you Nature, I will never try to see you topless or from any top-down perspective again."

I think it is a manifest overreaction, especially if you realize that in the last 35 years, the top-down physics has remained the main source of genuinely conceptually new and valuable ideas in bottom-up physics, anyway.

The speedy proton decay was obviously a wrong prediction but it remains uncertain whether the proton is exactly stable or not. Because there seem to be no theoretical reasons why the baryon conservation law should be exact - and not just the accidental approximate symmetry it seems to be in the Standard Model - it's legitimate to appreciate that the proton is relatively likely to be unstable. Just the lifetime has to be longer and of course, the suppression of the "excessively speedy proton decay" became a key engineering challenge for grand unified model builders.

Obviously, it also means that the minimal model by Georgi and Glashow couldn't be right. But if one doesn't get the perfectly final and right answer immediately, it doesn't mean he's not on the right track - as Glashow (with his SU(2) models without U(1)'s etc.) and others should know very well.

So other physicists who don't feel the urge to "overcompensate" their previous would-be sins can still freely realize that the grand unification is still a damn excellent idea that is justified by many "coincidences" about the spectrum and the values of the coupling constants of the Standard Model. The grand unification is likely to be right at some level even though we surely don't know the exact content and details of the right grand unified theory. However, we also know many detailed incarnations that can't be used in the world around us.

And that's the memo.

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