"Magical" supersymmetry
To close this exploration of the Grand Unification epoch, we need to evoke supersymmetry. We’ve already introduced this important concept in our discussion of supergravity and superstrings. The reason there is such high conviction that our reality is somehow “super” in that way, despite the lack of convincing observations, is that supersymmetric theories find themselves at the nexus of our two main concerns right now. They could be the key to the biggest issues faced by “regular” GUTs. They also offer, as we’ve seen, a natural way to have gravity emerge in a quantum field or quantum string framework.
It is not pure chance that supersymmetry finds itself in this enviable position. Indeed it acts very much as a bridge between internal and spacetime symmetries. Let’s review briefly the different type of continuous transformations pertaining to these symmetries:
Spacetime transformations are part of the Poincaré group \(ISO(3,1)\) (the group of isometries in Minkovski space), which is an Abelian Lie group, and itself is assembled from two types of transformations:
Lorentz group transformations preserve the Minkowski metric \(\eta\), so they verify \(\Lambda^T \eta \Lambda = \eta\). They belong to the group \(SO(3,1)\), which is isomorphic to \(SL(2,\mathbb{C})\), which itself can be seen as \(SU(2) \otimes SU(2)\) (that is why the Dirac spinor is made of two Pauli spinors). Concretely speaking, the group has 6 generators: three of them generate the spatial rotations, while the other three are the “boosts”: the switch of Galilean referential along a specific axis (which can be seen as rotation in spacetime). The six generators are represented as the components of an anti-symmetric 4-tensor \(M_{\mu\nu}\).
4 spacetime translations \(P_\mu\).
We know special relativity to be valid in all circumstances, so all of our theories need to be Poincaré-invariant. Interestingly, all of these transformations were known and accepted as symmetries in the pre-Einstein era. But because the maximum speed was assumed to be infinite, their most general form was not found until the twentieth century.
Internal transformations do not change the spacetime properties of the objects they transform at all. The internal symmetry group \(G\) is the Lie group corresponding to the Standard Model or to any other GUT we are studying. We can call the generators of the corresponding Lie algebra \(g_l\).
The overall symmetry group is the direct product of the Poincaré group and the internal symmetry group. This means that two generators from different groups always commute: there can be no simple groups that would incorporate all these types of symmetry together. That has been shown to be rigorously true, under a number of desirable circumstances (like causality, locality, finiteness…) as the Coleman-Mandula theorem.
Therefore the group of continuous symmetries of the Lagrangian ought to look something like \[ISO(3,1) \otimes G\] And the following commutation relations must be true: \[[P_\mu,g_l] =0 \quad [M_{\mu\nu},g_l]=0\]
However, there is one of condition of Cole-Mandula that can be weakened without jeopardizing the underlying physics. It is the assumption that all symmetry groups must be Lie groups (and thus that all their generators define a Lie algebra). It is actually simpler to focus on the algebra picture. What defines a Lie algebra is the existence of a "bracket" operator \([\cdot,\cdot]\). When we worked with matrix representations we just accepted that \([A,B]=AB-BA\), for brevity sake, but that is only true (by definition) in those matrix representations. Rigorously speaking the Lie bracket of a Lie algebras \(L\) verifies the following properties:
\[\begin{aligned} &[l_1,l_2] \in L &\quad (\text{stability})\\ &[l_1, (l_2+l_3)] = [l_1,l_2] + [l_1,l_3] & \quad ( \text{linearity})\\ &[l_2,l_1] = -[l_1,l_2] &\quad (\text{anti-symmetry})\\ &[l_1,[l_2,l_3]]+[l_2,[l_3,l_1]]+[l_3,[l_1,l_2]]=0 & \quad (\text{Jacobi identity}) \end{aligned}\]
There is a way to extend any Lie algebra through grading. In particular, \(\mathbb{Z}_2\)-graded Lie algebras are called superalgebras. Let’s call \(S\) the superalgebra or \(L\). Then there is some vector space \(V\) (which is not a Lie algebra) such as: \[S=L \oplus V\]
What we will do is define a linear superbracket \([\cdot,\cdot\}\) over \(S\), with the following properties.
\[\begin{aligned} [l,l'\} &\in L & [l,v\} &\in V & [v,v'\} &\in L\\ [l,l'\} &= - [l',l\} & [l,v\} &= -[v,l\} & [v,v'\} &= [v',v\} \end{aligned}\]
In addition, it verifies some generalized Jacobi identities which we don’t reproduce here. The main point is that the restriction of the superbracket over \(L\times L\) is simply the Lie bracket. \(L\) is stable under the superbracket, however \(V\) is not, so \(V\) cannot be a Lie algebra.
Besides, the antisymmetry/symmetry laws show that over \(L\times L\) as well as \(L \times V\), the superbracket behaves like a commutator, but over \(V \times V\) it behaves like an anticommutator (and indeed in matrix representations, it can be realized as \(\{A,B\}=AB+BA\)). We will colloquially write:
\[\begin{aligned} [l,l'\} &=[l,l'] \\ [l,v\} &=[l,v] \\ [v,v'\} &={v,v'} \end{aligned}\]
We can apply that procedure to a specific Lie algebra, like the Poincaré algebra, and obtain its super algebra, often called the SuperPoincaré algebra.
As we’ve said before, Coleman and Mandula have shown that the group of continuous symmetries governing our universe must be of the form \(\text{Poincaré} \otimes G\). There is no larger Lie algebra than Poincaré that we could use instead (it needs to be larger so that it includes the Poincaré transformations, since we know special relativity to be true).
However Coleman-Mandula has nothing to say about spaces that are not Lie algebras. We could show that the SuperPoincaré algebra could a larger entity that we could use.
The Haag–Łopuszański–Sohnius theorem does just that, and goes even further: it shows that using the "super", graded version of the Poincaré algebra, is the one and only way to extend the symmetry group. That’s a beautiful mathematical result that could be promising from a physical perspective.
So what do the commutation relations that characterize this SuperPoincaré algebra look like? There are different options already: we can show that generators of the "\(V\)" space (the new ones) come in pairs. The simplest case: 2 generators, is called \(N=1\). Since we’re looking at the classic Poincaré algebra in (3,1) spacetime, the simplest SuperPoincaré is labelled \(D=4,N=1\), even though other versions are used in more complex theories, including superstrings of course.
We’ll call these new generators \(Q_\alpha\) and \(\overline{Q}_{\dot{\alpha}}\). The underlying mathematics leads to the following superbracket relations:
\[\begin{aligned} [M_{\mu\nu},M_{\rho\sigma}]&= i\eta_{[\mu[\sigma}M_{\nu]\rho]} \quad & (1)& \quad & [M_{\mu\nu},Q_\alpha] & = i(\sigma_{\mu\nu})_\alpha^\beta Q_\beta &\quad(7)\\ [P_\mu,P_\nu]&=0 \quad &(2) & \quad & [P_\mu, Q_\alpha]&=0 &\quad(8)\\ [M_{\mu\nu},P_\rho]&= i \eta_{\rho[\nu}P_{\mu]} \quad &(3) & \quad & [Q_\alpha,g_l]&=-Q_\alpha &\quad(9)\\ [P_\mu,g_l]&=0 \quad &(4) & \quad & \{ Q_\alpha , Q_\beta\} &=0 &\quad(10)\\ [M_{\mu\nu},g_l]&=0 \quad &(5) & \quad & \{ Q_\alpha , \overline{Q}_{\dot{\beta}} \} &= 2\sigma^\mu_{\alpha \dot{\beta}}P_\mu &\quad(11)\\ [g_l,g_m]&= f^n_{lm} g_n\quad &(6) & \\ \end{aligned}\]
(1) and (2) define respectively the Lorentz (rotations and boosts) and the Euclidean (translations) groups. (3) completes the overall Poincaré algebra.
(4) and (5) is a consequence of the Coleman-Mandula theorem. Poincaré generators and internal symmetry group generators commute.
(6) is the generic definition of \(G\)’s structure constants.
(7) shows that the \(Q_\alpha\) generators transform like spinors under the action of the Poincaré group. That is why they are called fermionic generators. It is not altogether surprising: these generators are characterized by anticommutation laws, just like the Dirac matrices, and elements from higher-dimensional spinor representations.
If we remember that \(M_{12}=J_3\), we also get the momentous consequence that \([J_3,Q_1]=+\tfrac{1}{2}Q_1\) and \([J_3,Q_2]=-\tfrac{1}{2}Q_2\): the supersymmetry generators can change a particle’s sping by a half integer, meaning it turns a boson into a fermion and vice versa!(8) implies that, like standard model generators, supersymmetry transformations preserves mass.
(9) shows that unlike the Poincaré generators, SUSY generators don’t commute with the internal symmetries of the theories. In practice this results in another \(U(N)\) symmetry called R-symmetry.
(10) and (11) are the anticommutation relations governing the fermionic sectors of the SuperPoincaré algebra. At \(N=1\) they imply that each particle has a single superpartner.
A more abstract, but perhaps simpler way to look at the SuperPoincaré algebra is to introduce the concept of superspace. It starts from the idea that since the generators \(P_\mu\) correspond to translation along spacetime coordinates, the generators \(Q_\alpha\) and \(\overline{Q}_{\dot{\alpha}}\) could similarly correspond to translations along some new coordinates \(\theta_\alpha\) and \(\overline{\theta}_{\dot{\alpha}}\). Of course, because of the anticommutation relations, these are not "normal" real coordinates; instead they are Grassmann numbers. Ultimately, it appears that the SuperPoincaré algebra is just the analog of the Poincaré algebra over the SuperMinkowski space, with coordinates \((x_\mu,\theta_\alpha,\overline{\theta}_{\dot{\alpha}})\).
Let’s go back to the most concrete feature of supsersymmetry so far: it introduces a group of transformations that can turn a fermion (a matter particle with half-integer spin) into a boson (a force-mediating particle with integer spin), and vice versa. It is definitely a spacetime symmetry, since changing the spin of a particle does change its spacetime properties. At the same time, it is closer to the way we think of internal symmetries: we consider those as able to “change” a particle into another, for instance a neutron into a proton, or a blue quark into a red quark. Supersymmetry seems to generalize this by allowing fermion-boson transformations, and thus to constitute one more step in the unification process. But it does that by elegantly integrating the symmetries of special relativity with those of quantum physics (pursuing what Dirac had started by formalizing spinors).
The Minimal Supersymmetric Standard Model (MSSM) is then a theory whose symmetry group is \(\text{SuperPoincaré} \otimes G_{SM}\) instead of \(\text{Poincaré} \otimes G_{SM}\). The theory predicts that each particle predicted by the Standard Model msut have a superpartner:
The superpartners of fermions (spin 1/2) are spin 0 bosons (scalar bosons) called sfermions (including selectrons, sneutrinos, squarks, etc.).
The superpartners of the gauge bosons (spin 1) are spin 1/2 fermions and are called gauginos (including photinos, winos, bino, gluinos).
The superpartner of the Higgs bosons (spin 0) are spin 1/2 fermions, the Higgsinos (unlike the SM which can make do with a single Higgs, the MSSM requires a Higgs doublet to generate fermion masses).
The theory predicts that both superpartners have the exact same mass. Obviously we know that is not the case since we’ve never observed any superpartner in the observable mass range. We also know that no observed fermion-boson couple can be a superpartner doublet, since they would need to share all quantum numbers (except spin), which is never the case. This means that supersymmetry is broken, so the MSSM must include a symmetry breaking process as well.
Since supersymmetric theories aim to improve upon the Standard Model it also makes sense to apply that concept to Grand Unified theories. We can work with \(\text{SuperPoincaré} \otimes G_{GUT}\) where the GUT group could be \(SU(5)\), \(SO(10)\), \(E_6\), or any of the other groups we’ve encountered. All the extra bosons or fermions predicted by those theories would then all have their own superpartners.
By the way, since the Poincaré algebra has been rendered "super", could \(G_{GUT}\) also become a Lie superalgebra? That is indeed allowed by the Haag–Łopuszański–Sohnius theorem. However, it seems to be decisively proved that the resulting theories are not renormalizable: none of our calculations could converge and yield actual physical quantities.
Now to conclude, the perceiving reader might wonder if we haven’t gone on a flight of fancy. Supersymmetry is mathematically pleasing, but it introduces numerous extra particles and phenomena. Can it give us anything in exchange? The answer is a resounding "yes!", and then some.
The hierarchy problem is neatly solved by supersymmetry. As we’ve seen, that problem relates to the un-naturalness of perturbative calculations relative to the Higgs boson. Pesky quadratic terms would force us into picking highliy un-natural combinations of free parameters for the quadratic terms to cancel and leave us only with finite results. However the quadratic terms pertaining to each fermion (resp. boson) would be exactly opposite for a fermion (resp. boson) with the exact same quantum numbers, save for spin. That is the exact definition of a superpartner! So, in a supersymmetric theory, the cancellation of the quadratic terms happens in the most natural way. This was actually the initial motivation to explore theories with a fermion/boson symmetry.
Gauge unification is also magically facilitated by supersymmetry. We’ve explained that in GUTs such as \(SU(5)\), the coupling constants of interactions would meet at a single point, however that was a shortcut: although the situation is much improved in GUTs versus the Standard Model, the couplings only approximately meet. Thus, large, quite unnatural corrections would be needed to enforce exact gauge unification. However, supersymmetry seems to always result in much more precise unification, requiring small corrections only even in the MSSM (that is, without Grand Unification). This is even truer for supersymmetric GUTs.
We have seen that the prediction of \(SU(5)\) theory for the proton decay were not compatible with experimental observations. One way to make that decay more unlikely, and therefore to push up the GUT scale, is to use a larger GUT group like \(SO(10)\). Another one is to use supersymmetry: the GUT breaking scale in supersymmetric \(SU(5)\) is already much higher.
Dark matter is more of a cosmological problem than one for quantum physics. Yet it is a fact that most of the universe’s matter (non-luminous, non-baryonic matter which we know is there from studying the large-scale properties of the observable universe) does not appear to be made of our commonplace particles. Light, semi-stable superpartners seem to be good candidates to play that role.
Supergravity is by far the most tantalizing prospect that is encouraging research into supersymmetry. Remember that: \(\{ Q_\alpha , \overline{Q}_{\dot{\beta}} \} = 2\sigma^\mu_{\alpha \dot{\beta}}P_\mu\). Everything we’ve done so far is talk about global supersymmetry. What happens if we try to develop a local gauge theory that incorporates supersymmetry? This means that we will make the theory invariant with regards to supersymmetry transformations that are functions of spacetime parameters \(\epsilon(x_\mu)\). Because of the anticommutator bracket above, this means that the theory is also invariant under translations that depend on some \(\epsilon'(x_\mu)\). That is the definition of a theory that invariant under general coordinate transformations: in other words: general relativity! We can show that conversely, if we start from a global supersymmetry and impose invariance under general coordinate transformation, we get local supersymmetry as a consequence! \[\text{Global SUSY} + \text{Gravity} \leftrightarrow \text{Local SUSY}\]
Now we can at last seize back the chronological thread that we had introduced. At the end of the Planck epoch, around \(10^{-43}\text{s}\), the Grand unification era starts, with a compactified spacetime over which the perturbative approach works: in other words, identifiable quantum fields are recognizable, and this brand new spacetime is rife with GUT denizens: leptoquark bosons, all the additional content that higher-ranked GUT groups may yield, various Higgs scalars, and all of the superpartners that accompany these exotic creatures.
The stable, (very) far future
From the post-Big Bang compactification, a meta-stable state has emerged: the expanding (3+1)D curved spacetime that we now live in.
Over a long period of time, all the matter content of the universe will have decayed, leaving only the ultimate stable particles (potentially only photons). Even with no matter, gravitational structures could still persist, chiefly black holes (spacetime singularities).
It is well accepted, though, that black holes evaporate, emitting Hawking radiation. Hawking radiation is mostly made of photons, but the smaller the black hole, the likelier it is to take the form of particle-antiparticle pairs. Over time, those will either meet or decay further, reducing the universe’s contents to photons, plus a tiny amount of stable fermionic matter, reflecting the matter-antimatter asymmetry.
Post-black hole evaporation, we then get a uniformly curved universe with uniform energy density, and no complexity whatsoever! There is no telling at which point exactly this would happen, but this will likely be north of \(10^{100}\) years (1 googol years): a small black hole with a mass like our Sun’s takes \(10^{64}\) years to evaporate, and it is believed that the future collapse of galaxy clusters will give rise to black holes \(10^{14}\) as heavy as the Sun.
At the end of a black hole’s lifecycle, the temperature is such that it would actually appear to be luminous. The Universe may then get a last luminous hurrah... But unlike the light that we see from the stars, this will not be the harbinger of glorious supernovae, creators of complexity, but a dying light.
Once the universe is entirely uniform, two futures are possible, depending on the dark energy content. In one scenario, the "Big Freeze", expansion continues forever but at an ever-slower pace. The temperature converges to zero, and the Universe ends up as a static, forever-frozen fabric. However an alternate scenario can be envisioned: the "Big Rip". If the expansion rate is instead increasing, the curve of expansion will one day become exponential. Every distance will thus tend towards infinity, and the observable universe from any spacetime point will have a radius equal to at most the Planck length.
Absent an accepted theory to unify quantum physics and general relativity, it is difficult to envision how these cosmological scenarios will impact the behavior of quantum fields. What we know however is that at extremely low energies, quantum effects can manifest at larger and larger scale. A Big Freeze universe might then witness macroscopic quantum fluctuations on larger and larger scales, until they become Universe-wide, coming full circle, since this was also the case right after the Big Bang. The familiar picture of fields and particles would then become again meaning less. It may turn out that the Big Freeze might look more like the paradoxical image of an infinite ocean of boiling water at the absolute zero...