*Also in Spanish...*

Yesterday, I was trying to explain to Fabrizio Nesti at Tommaso Dorigo's blog why there can't be any "GraviGUT" symmetry, a hypothetical tool to unify gravity and other forces by extending the Yang-Mills group to a larger group that also contains the Lorentz group as a subgroup.

This "unification" violates the Coleman-Mandula theorem, of course, but there are easier ways to see why this particular concept cannot be right.

I will be talking about the SO(3,11) example although some other attempts may exist in the literature. People such as Fabrizio Nesti (a recent paper), Roberto Percacci (see his recent review), and also Garrett Lisi (explicitly in his newest surfer handbook), among others, have been trying to attack the problem of unification in similar ways. The discussion seemed hopeless but it's conceivable that at the end of the day, Fabrizio understood the simple points I was making.

**Examples of fake bigger symmetries**

The main problem is that if one is not strict enough about the rules of the allowed "symmetry breaking" mechanisms, one can always claim that a theory whose symmetry group is G actually has a larger symmetry, H, so that G is a subgroup of H, and that H is broken down to G.

Let me mention a trivial example that will be referred to as the key toy model in my discussion.

The Quantum Chromodynamics allows the quarks to have three colors; its gauge symmetry is SU(3). Imagine that we want to claim that the real symmetry is SU(390) but it is broken to SU(3). This is an obviously arbitrary, unphysical idea, isn't it? But let me show you how you could defend such an assertion.

All the colored fields that carry color indices, i = 1,2,3, could carry extended color indices, c,d = 1,2,...,389,390. Of course, you would predict too many colors, and so on. How can you get rid of the unwanted components of all the colored fields? Well, it's easy. You postulate that there is a "decoloring Higgs field", h_{cd}.

In the "correct" Lagrangian, all objects such as q_{c}, that came from the generalization of q_{i}, should be replaced by h_{cd}.q_{d}. And the decoloring Higgs field just happens to have the value

h_{cd} = diag (1,1,1,0,0,0,0,....,0).I can even write nice, SU(390)-invariant equations that have the solution above. For example, this is it:

h_{cc} = 3; h_{cd} h_{de} = h_{ce}Also, I require that h_{dc} is the Hermitean conjugate of h_{cd}.

Note that the two indices of "h" are conjugate to each other (much like in the Kronecker delta symbol), a detail I suppress to simplify the notation. The second equation above says that the "h" matrix must square to itself, so its eigenvalues must be 0 or +1. Assuming it is the case, the first equation says that the number of +1 eigenvalues must be three. Up to an appropriate local SU(390) transformation that I can perform at every point, the most general solution is therefore equal to the diagonal matrix above.

Physics of this SU(390)-invariant theory is indistinguishable from the ordinary SU(3) QCD; the additional components of all the colored fields are completely decoupled - they have no terms in the Lagrangian - and you can't operationally find out whether they exist or not. Fine, have I proven that QCD has a hidden SU(390) symmetry?

The answer is, of course, no. I needed the "h" field to have the vacuum expectation value it was given in order to obtain the right physics. And the vacuum expectation value breaks the SU(390) symmetry. Is there a point on the configuration space that unbreaks the SU(390) symmetry?

Yes, you can set h_{cd} to zero and the symmetry is restored. However, this choice doesn't solve our equations. In fact, you could rewrite the first equation in such as way that either 3 or 0 eigenvalues equal to +1 would be allowed:

(h_{cc}-1.5)^2 = 2.25The new laws for "h" would have two solutions: one that reduces the symmetry to SU(3) and one that keeps SU(390) unbroken. In this sense, you may say that the good old SU(3) QCD is a broken phase of the SU(390) theory which also has an unbroken phase. Note that there's nothing special about the SU(390) symmetry. I can do analogous things with any bigger hypothetical symmetry in any theory you can think of.

So what is really wrong about this proof of an underlying SU(390) or any other obviously fake symmetry?

The problem is that the symmetry breaking by "h" shouldn't be called a "spontaneous symmetry breaking". In fact, it is the best example of an "explicit symmetry breaking", something that guarantees that the symmetry is not there from the very beginning (and it's generically useless to talk about such a non-symmetry). It is about adjusting the coefficients of the different color fields in a color-asymmetric way - a clear example of an explicit symmetry breaking.

Why is this breaking explicit while the symmetry breaking in the electroweak, employing the conventional Higgs boson which also treats different SU(2) components differently, is spontaneous?

**The main difference is that the electroweak Higgs is dynamical while the decoloring h_{cd} Higgs field is not dynamical.**

What does it mean "dynamical"? It means that its derivatives appear in the action. Consequently, the equations of motion also contain terms with the derivatives of the Higgs and they imply some evolution for this Higgs field. Other parts of the dynamical system are able to influence the evolution of the Higgs field, at least in principle. This is true for the electroweak Higgs but not for our Higgs that broke the SU(390) symmetry down to SU(3).

If there is no dynamics for the symmetry-breaking fields, the bigger symmetry is unphysical. As a physicist, you shouldn't talk about it because by the physical criteria, it's not a symmetry of the physical system.

You may try to modify the theory by adding derivative terms for the h_{cd} field to our equations of motion - and by assuming that they will remain negligible. Indeed, if you did so, you could construct an SU(390) theory that is spontaneously broken down to the SU(3) QCD. However, the theory would also predict lots of new particles that are completely arbitrary - and haven't been observed. To agree with the experiments, you would have to postulate that these new dynamical components of the h_{cd} field are very heavy: they haven't been observed yet.

Such a picture would only influence your opinions about the very high energy limit of your physics and it would have no relevance for the observations of patterns at low energies. If I say the same thing inversely, observations of physics at low energies couldn't offer you any evidence for the particular structure of the new heavy "Higgses".

I could enumerate many more specific facts that make the SU(390) QCD contrived; for example, it's way too simple to gauge-fix the excessive symmetry and you don't gain anything by not gauge-fixing it which already suggests that the symmetry was fabricated. Moreover, spontaneously broken symmetries should produce massless (or light) Goldstone bosons for each broken generator of SU(390) which don't seem to appear in our world, further suggesting that the required new "Higgs fields" were unphysical. More generally, broken symmetries that are worth talking about always have some physical consequences - in some sense, they're as predictive as the unbroken ones.

Note that the original theory where the h_{cd} field had no kinetic terms at all is equivalent to the "infinite mass limit" of the massive h_{cd} theory - because that's the limit in which the coefficient of the kinetic terms is negligible relatively to the potential/mass terms. That's why it's "infinitely difficult" to go from one phase to another and unbreak the symmetry; if something requires an infinite amount of work, it's unphysical. Physicists should leave such things to theologians and other babblers.

**GraviGUT**

Now, the GraviGUT example is completely analogous. It is just another attempt that increases the symmetry that is unphysical, and if this symmetry is made physical, the new physics is neither relevant for the observations, nor interesting, nor justified.

What is the GraviGUT picture?

It starts with the observation that in SO(10) grand unified theories or GUTs, the fermions (leptons and quarks) are unified into the 16-dimensional chiral spinor representations of SO(10). These fields also transform as spinors under the Lorentz group, SO(3,1), and a tensor product of two such spinors may be understood as a spinor of a larger pseudo-orthogonal group. They like to choose SO(3,11) - something that you would get in a spacetime with 3 spatial and 11 temporal (!!!) coordinates.

This reinterpretation of a "tensor product of two spinor representations as a single spinor of a larger group" is actually the only semi-rational justification for the whole GraviGUT picture. Is that correct?

Of course, it is not correct. The fact that the term "spinor" appears at two places - SO(10) representations relevant for families; and the SO(3,1) representations for fermions - is a complete coincidence. In particular, you may also consider equally consistent SU(5) GUTs, where the fermions transform as 5+10bar (or its conjugate) - the fundamental representation and/or antisymmetric tensor - or, if you prefer irreducible representations, you may consider E6 GUTs where the fermions transform as the 27-dimensional fundamental representation.

There are not too many "kinds" of representations of Lie groups, so it's not shocking that the concept of a spinor appears at several places. You shouldn't view this fact as a sign from the Heaven.

Nesti and others of course try to sling mud at SU(5) and E6 GUTs because the quarks and leptons are not spinors in those theories so it's less natural (SU(5)) or impossible (E6) to extend the GUT group to a big pseudo-orthogonal one. But it's just a fact that the only thing that is "worse" about SU(5) or E6 GUTs in comparison with SO(10) - note that one gauge group is smaller than SO(10) while the other is larger - is that they're inconvenient for the GraviGUT program.

The statement that the SO(10) GUT is better because it produces spinorial matter, and it's good to expect fermions in the GUT spinor representations because that's what you see in SO(10) GUTs, is a classic example of circular reasoning. There is actually no independent argument supporting either of these two assertions. When all non-circular arguments are taken into account, all known representations in viable GUT theories are equally good, including 5+10bar of SU(5) or 27 of E6.

Moreover, the "consolidation" into an SO(3,11) multiplet only "works" for the fermions; the gauge bosons, the graviton, and other fields aren't completed to the full SO(3,11) multiplets; the breaking of this greater "symmetry" is much more obvious for the bosons.

Fine. This argument looking at the leptons and quarks didn't work. But is it possible to extend SO(3,1) and SO(10) to SO(3,11) that would be spontaneously broken?

The answer is No. The situation is analogous to the SU(390) colored case but the required physics would be much worse in this case. Why? Because SO(3,11) is a symmetry that mixes 14 directions of a space. 3+1 of them are equipped with actual directions in spacetime while 10 of them are not.

Having a direction in spacetime and not having it makes a lot of difference. In fact, that's about the most striking difference between two "directions" that you can encounter in physics. Consequently, there can't be any natural symmetry that rotates these two types of fields into each other: SO(3,1) and SO(10) can't be enhanced to SO(3,11). The kinetic terms in the Lagrangian only see the derivatives with respect to the dimensions that exist so they can't be rotated to derivatives that don't exist.

Of course, such an enhancement of a group would also violate the Coleman-Mandula theorem. This rigorous result, designed more than 40 years ago exactly to address attempts similar to Percacci's and Nesti's, says that interacting semi-realistic theories can only have symmetries that are the direct product of the spacetime symmetries such as the Lorentz group and internal symmetries such as SO(10): there can't be any half-internal half-external symmetry groups.

The Haag–Lopuszanski–Sohnius theorem extends the Coleman-Mandula theorem to the case of symmetries with Grassmannian generators - superalgebras - and it implies that supersymmetry is the only possible new "counterexample" to the naive generalization of the Coleman-Mandula theorem. That's also why Lisi, whose theory doesn't respect supersymmetry, can't possibly unify the laws for bosons and fermions. Only SUSY can link detailed properties of bosons to detailed properties of fermions.

Percacci et al. invent all kinds of propaganda why it's not a problem that their proposal contradicts the Coleman-Mandula theorem but all this propaganda is simply incorrect. They may invent the "SO(3,11)-symmetric phrase" and write various laws analogous to our h_{cd} fields that broke the SU(390) symmetry. But all of them will be equally fake, too. The symmetry breaking will be explicit in all cases, assuming a proper physical definition of "explicit" and "spontaneous".

It is not really possible to extend the SO(3,1) group that acts on the vielbeins in the first-order Palatini formalism of general relativity to a larger group simply because the very purpose of the vielbein is to provide us with an isomorphisms between the tangent bundle and the new abstract four-dimensional bundle.

The vielbeins have to be invertible because we need both upper and lower versions of the vielbein to formulate realistic Lagrangians. The vielbeins simply have to respect the dimensionality of the spacetime; they can't pretend that the dimensionality is something else than what it is. You should also appreciate that the SO(3,1) local group is an additional redundancy that you have to add if you use the vielbeins instead of the metric as the basic geometric degree of freedom; but you still have to guarantee that your theory is invariant under the diffeomorphisms which is the local symmetry that actually defines general relativity; the SO(3,1)-like local symmetries can never replace diffeomorphisms and they're not helpful to illuminate the origin of the diffeomorphisms (and gravity).

All attempts to mask the difference between the dimensions that exist and those that don't exist will be at least as artificial as the h_{cd} decoloring Higgs field - and the corresponding symmetry-breaking and symmetry-restoring phases will also be "infinitely far from each other" on the configuration space which is why the symmetry breaking should be called "explicit".

The actual "Higgs fields" that you will need to relate these two phases will be even more artificial than in the SU(390) case because they will need to mask the different spins of the relevant gauge particles - gravitons vs gauge bosons - among many other things.

To summarize, the SO(3,11) symmetry is complete bogus from any physical perspective. There are many other papers that try to extend symmetries and break them in related, comparably unusual ways. But the people who write such papers apparently can't sort the ideas in their heads - otherwise they would have known for years that the new hypothesized symmetries are not "real" and the fact that they can write a theory so that it pretends that its symmetry is larger is not surprising; it's always the case, whatever fake symmetry you invent.

**Bonus: why the tensor products of spinors can't be "consolidated" in the heterotic string**

If you know some basic string theory, there are many ways to see how different the "spinorial" character in spacetime and the internal SO(10) space is.

For example, take the E8 x E8 heterotic strings. The dimension of the gauge group is 496, a perfect number that coincides with the number of holy deniers on the recent black list by Schneider et al. :-) A single E8 has the 248-dimensional fundamental representation, transforming as 120+128 under its SO(16) subgroup - which also contains the SO(10) GUT group. The 120 is an adjoint while the 128 is the spinor of SO(16).

The latter, 128, is produced e.g. by quantizing the 16 left-moving fermionic degrees of freedom on the world sheet in the periodic-antiperiodic mixed sector, if you adopt the fermionic description of the extra heterotic degrees of freedom (32 extra real left-moving fermions). And indeed, you may also get spacetime spinors (which appear in the heterotic string, e.g. for the E8 x E8 gauginos) by quantizing the psi^mu spacetime RNS fermions on the world sheet in the periodic (Ramond) sector.

However, there is a qualitative difference between the 32 heterotic fermions lambda^a and the 10 RNS fermions psi^mu: the latter appear in the world sheet supercurrent - they have world sheet superpartners X^mu (and, correspondingly, they're also equipped with actual spacetime dimensions) - while the former don't.

Even more importantly, and this second point is related to the first one, the 32 heterotic fermions producing the E8 x E8 group are left-movers while the psi^mu fermions are right-movers (and only right-movers may enter the world sheet supercurrent, which is why it's related to the first difference). There can't be any symmetry that transforms purely left-moving fields into purely right-moving fields. There's no "phase" in which something like that would make sense. The only thing one can try is to "hide this qualitative difference from his own eyesight". But the difference is physical, crucial, and makes any interpretation of the heterotic string in terms of SO(3,11) or SO(25,1) invalid.

Note that the heterotic strings on tori allow you to continuously change the E8 x E8 gauge group to another 496-dimensional enhanced gauge group, SO(32): at generic intermediate points, the symmetry is U(1)^{16} only. Both of the 496-dimensional groups may be understood as subgroups of a larger gauge symmetry that becomes explicit in string field theory. But note that in all cases, the unification or transformation of the gauge groups implies the existence of new physical states at the Kaluza-Klein or string scale (which are reasonably light); if they only predicted new physics at "infinite energy scales", such changes of the symmetry and "breaking mechanisms" would be unphysical. In string theory, all symmetries we talk about are physical - they're linked to physical phenomena that are in principle observable.

The discussion of the heterotic string was just an example. In the heterotic case, everything is explicit and one may say exactly what is the big difference between the SO(10) directions - extended to SO(16) directions - and the SO(3,1) spacetime directions - extended to SO(9,1). They're left-movers and right-movers, respectively. But they're qualitatively different in any description of physics with the same low-energy limit.

It makes no sense to talk about new symmetry-restoring phases of physics if you actually need an infinite amount of energy to get from one phase to the other. Such attempts to "unify things" are just lame bookkeeping devices attempting to mask the actual, physically relevant difference between different kinds of indices, among other things. That's the main reason why the papers by Percacci, Nesti, Lisi, and others are meaningless sleights-of-hand.

And that's the memo.

**Bonus: Unification in string theory**

In heterotic string theory or any other background of string theory, the Yang-Mills forces

*are*unified with gravity. Doesn't string theory have to respect something like the GraviGUT, anyway?

The answer is No. The forces and their local symmetries are unified but they're not unified in an ordinary Yang-Mills group (and force) in the bulk. In perturbative string theory, you actually need the whole string to see the common origin of the gauge bosons and gravitons (aside from all other particle species).

A single string is a physical object that can be found in many vibrational states. These states may have different values of the spin in spacetime. One may identify the generators of the "enhanced stringy gauge symmetry" with the BRST-exact states of the first-quantized stringy Hilbert space. This point is obvious in string field theory.

However, these unphysical states - much like the physical states - are obtained by exciting the string's ground state with different kinds of excitations, different types of oscillators. In heterotic string theory, one can get e.g. the SO(16) x SO(16) gauge bosons and gravitons as

lambdaBoth of them are acting on the ground state of the Neveu-Schwarz sector where the right-moving fermions are antiperiodic. Both of them have a right-moving excitation by the -0.5th mode of "psi", the vectorial fermion.^{i}_{-1/2 (left)}lambda^{j}_{-1/2 (left)}psi^{nu}_{-1/2 (right)}|0>_{NS}

alpha^{mu}_{-1 (left)}psi^{mu}_{-1/2 (right)}|0>_{NS}

However, they differ in the left excitations. The gauge bosons use two -0.5th modes of the "lambda" degrees of freedom inherited from the 26-dimensional bosonic string (in this formalism, the 16 extra bosons were fermionized); however, the graviton contains the -1st oscillation of "alpha", a Fourier mode of the left-moving spacetime boson "X".

Both "lambda" and "X" - as well as "psi" - may be shown to be a necessary part of the heterotic string world sheet but they're not linked by any "simple" symmetry of the SO(3,11) type. The actual symmetries that string theory hides are much more subtle and one actually needs to do some proper work to determine what they are.

In braneworlds, Yang-Mills gauge symmetries boil down to the open strings' end points which are attached to D-branes (and carry the color labels, the so-called Chan-Paton factors in the open stringy context) while gravitons are still given by closed strings that have no end points. There is no "simple-minded" symmetry between open and closed strings because they are demonstrable different physically.

Still, one can show that a consistent theory with open strings must contain closed strings as well. The whole structure is hugely constrained - essentially determined by a few steps - but the "master principle" is not a simple symmetry of the SO(3,11) type. Instead, the basic consistency criterion of perturbative string theory is the conformal invariance of the worldsheet CFT, mutual locality of the physical vertex operators, and modular invariance (to get the right orbifolds, projections, and unitarity at loops).

These things are not excessively mathematical sleights of hand. They - or something equally mathematically potent - is absolutely necessary for physics to achieve the unification of gauge bosons with gravitons (that have a different spin). Physical theories should be made as simple as possible but not simpler.

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