I have found that all ugly things are made by those who strive to make something beautiful, and that all beautiful things are made by those who strive to make something useful. Oscar Wilde

# Supersymmetry: a missed opportunity to think twice

## Why naturalness is reserved to educated people…

June 18, 2021

Many of the world's best theoretical physicists have devoted their research to supersymmetry and some have devoted their entire careers to it. At the time of writing, none of the data collected and analyzed at the LHC show any sign of the existence of this long-awaited new symmetry. Even worse, the space of possibilities for building supersymmetric models is getting smaller and smaller. You never know what experiments can produce in physics, but wasn't this situation somewhat predictable? For me, it was. I will give some arguments that clearly show that supersymmetry physicists should have thought twice before embarking on this research program...

It is easy to criticize when those you attack are in a weak position. The series of remarks I am about to make here may therefore seem too convenient. However, many people in my circle of friends could testify that what I am going to talk about here is not new, and that I have already had the opportunity to make these points to them on several occasions, long before the supersymmetry physics community found itself in a weak position. Moreover, it is not my intention to rejoice in this situation. If I write today these arguments that I have already had the opportunity to mention (privately) before, it is, on the one hand, to share them with as many people as possible (although the audience of this site is rather small...) and, on the other hand, to look at the present situation with the history and philosophy of science in mind. It is a pity that the conclusion I reach with my arguments is that supersymmetry physicists are not educated enough in mathematics (that can be forgiven) but also in physics (that is more problematic).

Supersymmetry was conceived in the 1970s and has quickly attracted the best theoretical physicists. The basic idea, which will be essential for what follows, is to achieve a perfect symmetry between two types of elementary particles: the fermions and the bosons. A famous theorem in Quantum Field Theory, the Spin-Statistics Theorem, states that particles with integer spin are bosons and particles with half-integer spin are fermions.

Fermions obey the Fermi-Dirac statistics and obey the Pauli exclusion principle, for which it is forbidden for two instances of a (fermion) particle to occupy the same quantum state. All the quarks and leptons of the Standard Model of Particles Physics are fermions.

Bosons obey the Bose–Einstein statistics and a unique quantum state can be occupied by any number of instances of a (boson) particle. All the gauge particles that carry the interactions in the Standard Model of Particles Physics are bosons: the photon, the $W^+$, $W^-$ and $Z^0$, and the gluons. One has to add to this list the Higgs boson discovered in 2012.

Supersymmetry thus sets out to establish a new kind of symmetry that would associate any fermion with a boson and vice versa. The “superpartner” of any already known particle would thus be a new particle, which would still have to be discovered in a particle collider to confirm this new symmetry. This prediction doubles the number of known particles…

While expectations were high when the LHC, the world's largest operating particle collider, has started its first experiments in 2010 at CERN, none of these superparticles has yet been discovered. Worse still, the data accumulated over almost a decade is increasing the constraints on possible models of supersymmetry. Gradually, the supersymmetry physics community is losing hope.

This hope is however based on theoretical developments that are constantly presented to us as unstoppable: it would be the largest conceivable symmetry, it solves some “fine tuning” problems, in particular concerning the mass of the Higgs particle… The naturalness of the idea has been used as a strong argument in its favor. I refer the interested reader to the book "Lost in Math: How Beauty Leads Physics Astray", by Sabine Hossenfelder for more details regarding both the motivations for building supersymmetric models and to get an idea of the extent of the frustration with the non-discovery of superparticles… Another interesting, but quite pathetic, testimony can be found in the documentary film Particle Fever.

In what follows, I will develop three arguments, of a slightly different nature (I will shortly explain their technical and philosophical positioning) that show that the supersymmetry physics community did not really take the measure of the philosophical and technical scope of their approach: in other words, they should have thought twice before jumping headlong into this adventure which predicts twice the number of known particles… Defending supersymmetry with the arguments they have always displayed could not be enough to ensure the soundness of this research program, by far. So, for me, it is not a surprise that no superpartner of any known particle has been found yet.

### Do we need the Poincaré group anymore?

This argument relies on the role of the Poincaré group in supersymmetry. Any model in supersymmetry assumes the existence of a “super Lie algebra” in which there are commuting and anticommuting generators. These supersymmetry algebras are extensions of the Lie algebra of the Poincaré group: thus they contain at least the Poincaré symmetries.

In another text on this website, I raised the problem concerning the coexistence of the Poincaré group, as a fundamental group of Particle Physics, and the Higgs boson. Why? Simply because both give us a source of explanation for the masses of particles. I let the reader look at all the arguments concerning this problem in the mentioned text. Thus, since in science having two explanations for the same problem is one too many, one of the two must be set aside to make room for the other. This text was written before the discovery of the Higgs boson in 2012. Since then, the answer is clear: the Poincaré group should no longer be considered a fundamental symmetry of physics… The Higgs boson is sufficient to deal with the origin of masses.

But then, this has a direct impact on supersymmetry: since the Poincaré Lie algebra is at the heart of its construction, to do without the Poincaré group is to do without supersymmetry! This has already been stated in the mentioned text, so there is no need to repeat all the arguments here. Let me add that a physicist working on a supersymmetric extensions of the Standard Model of Particles Physics may be the kind of guy (sorry ladies, this argument doesn't apply to you...) who is wearing a belt and suspenders at the same time to support his pants. This is a delicate way of saying that such a physicist (this also applies to ladies) has not taken the physics she/he studies seriously. Does this make her/him a good physicist? I let you propose an answer…

### (Differential) Geometry draws a line between Fermions and Bosons…

The following argument mixes empirical data and mathematical constructions. So far, all fermions have been associated with so-called “matter fields” while all bosons have been associated with “interaction fields”.

Let me point out what looks like an exception to this empirical classification: the Higgs fields giving rise to the Symmetry Breaking Mechanism. To be precise, the Higgs fields are the fields introduced before the Symmetry Breaking Mechanism: they are not the remaining fields describing the Higgs boson after the application of the Symmetry Breaking Mechanism… I find the terminology on this point rather confusing in the literature, it deserves a clarification. From the strict point of view of the above classification, these Higgs bosonic fields should not be put in the “interaction fields” box, since, as they appear in the Standard Model of Particle Physics, they are rather “matter fields”-like. Remember that these fields are hand-crafted and many people have tried to understand what they are and where they come from… In fact, there is an elegant solution to this question, which is to consider these fields as the “purely algebraic part” of a “generalized gauge connection” in certain mathematical frameworks that extend the ordinary differential geometry of fibers bundles with connections. This can be seen for example in models constructed in Non Commutative Geometry or in models constructed using transitive Lie algebroids (take a look at this book or this paper for details). The idea is that one can define “generalized connections” that split into two parts: one contains the “ordinary gauge bosons” and the other contains the Higgs fields. Thus, I will consider Higgs fields, which behave like bosons, as part of the “interaction fields” box.

From a mathematical point of view, differential geometry, and its mentioned generalizations (Non Commutative Geometry, transitive Lie algebroids…), draws a strict line between fermions and bosons. The (classical) fermionic fields are smooth sections of vector bundles, which aggregate (by a tensor product) a gauge vector bundle, the one which implements the gauge symmetry, and a spinorial bundle, to take into account the spinor part (which is necessary due to their half-integer spin). The (classical) bosonic fields are then part of the connections, the mathematical structure that lifts the ordinary partial derivatives to vector bundles. In this respect, bosonic fields “act” on fermionic fields. In Non Commutative Geometry or on transitive Lie algebroids, these connections not only lift the ordinary (spatial) partial derivatives, but they also encode some “algebraic derivatives” which are added to space-time: this explains why we get new degrees of freedom which are naturally identified with the Higgs fields (just build a model and look at its content…): there is a potential for these fields that has the main characteristics of the Mexican hat in the Standard Model of Particles Physics, and these fields couple to the ordinary gauge bosons, through the curvature of the generalized connection, and to the fermionic fields, through the “minimal coupling”, i.e. the action of the generalized connection on these fields.

Now that the empirical boxes are matched with the mathematical boxes, imagine that the smartest idea of your life is to mix the contents of these boxes so patiently identified… This is the idea of supersymmetry! Implementing a symmetry between bosons and fermions implies adapting (or more simply “destroying”?) the mathematical structures that have been (empirically) associated with these two types of physical objects. How to manage this upheaval? This is the question I ask. Should we propose an entirely new mathematical framework? If the answer is yes, show me some compelling research papers in this direction…

### Bounded and unbounded: there are some hard borders to cross…

This is the most technical argument, since it is based on pure mathematics. It has been shown that the quantization of bosonic and fermionic fields does not follow exactly the same rules. Bosonic fields are quantized with commutators while fermionic fields are quantized with anticommutators. It is then easy to prove that bosonic fields become unbounded operators while fermionic fields become bounded operators. For more details, take a look at "Operator Algebras and Quantum Statistical Mechanics" by O. Bratteli and D. W. Robinson, where everything is clearly laid out. In fact, every physicist is more or less familiar with this property, since it is related to the fact that any number of bosons can occupy the same state, while at most one fermion can occupy a given state.

From this, we can clearly see that bosons and fermions do not share the same playground. Any educated researcher in physics knows that bounded and unbounded operators do not behave in the same way. Any mathematical book on operators in Hilbert spaces will convince the reader that the technical aspects are not the same. Unbounded operators require a great deal of care to treat, which is why they are surrounded by many difficult concepts and highly non-trivial results. Thus, from a mathematical point of view, trying to build a symmetry between bosons and fermions requires to take into account this highly non-trivial aspect of the associated quantum fields. As in physics our symmetries are built on representations, the challenge is great… Indeed, how to imagine “transmuting” by a linear transformation a bounded operator into an unbounded one? One solution would be to require that the coefficients of the linear transformation contain all the aspects of this metamorphosis. But then, the unbounded aspect would be encoded in the group (the Lie algebra if you prefer) and the fields would remain bounded by themselves. I would like to know the opinion of supersymmetric physicists on this delicate subject.

The previous argument about the differential geometry of bosons mentioned that bosons in Particle Physics (including Higgs fields), are degrees of freedom of (possibly generalized) connections. Since connections are some kind of derivations, and since derivations are unbounded operators, we see here the beginning of a convergence of structures: the fact that bosons are associated with unboundedness could be understood (and “justified”) from two independent mathematical origins (are they?). The first originates in the “physical behavior” of bosons (arguments from Quantum Statistical Mechanics), the second in the “minimal coupling” requirement in gauge field theories (arguments from Classical Fields Theory).

### Super! So what?

None of the arguments given above are strong enough to falsify supersymmetry. On the contrary, the last two are challenges for physicists since they open opportunities for very interesting research programs.

To be honest, as a mathematical-physicist, I really doubt that these research programs can succeed. This would require a large number of mathematical and conceptual disruptions and no current sign (mainly mathematical) allows us to hope to solve them. This is why I personally think that supersymmetry is in a bad way. Worse, it also means that the physicists who sold us supersymmetry for its elegance in solving (difficult) problems are wrong, almost from the beginning: supersymmetry opens up very difficult (mathematical) problems that current mathematics is not able to handle. There is no trace of a simple procedure to linearly change a bounded operator into an unbounded one (there are non-linear ways to do so, see for example the Cayley transform)… There is nothing that prevents a boson from being the section of an associated vector bundle. But since fermions are associated to spinors, which are themselves part of a very rigid and rich structure, it is difficult to think of a way to convert them into a “connection-like” structure, even in a hypothetical “generalized” way. Of course, one could say that we could abandon all this formalism. But then, it is up to physicists to find a new framework in which everything makes sense, at least as good, as elegant, as rich and as suggestive (yes, mathematics can be suggestive…) as that of differential geometry (or Non Commutative Geometry, or transitive Lie algebroids…). Have supersymmetry physicists launched a research program in this direction?

To make a comparison that I hope will illustrate the point, chemists have constructed the periodic table of elements to bring some order to the many chemical elements. Imagine declaring all the columns in this table to be meaningless! This may look very similar to a situation where physicists abandon the (mathematical) boxes mentioned above…

The first argument could be circumvented by a lazy non-technical approach: Why bother with the dual explanation? Why bother with redundancy? If the role of the Poincaré group overlaps with that of the Higgs fields, who cares? I agree. But if you look at the history of science, this kind of lazy proposal has never lived long… Yes, some of them have lived for a long time: Let's take the example of the two masses introduced by Newton, the inertial mass and the gravitational mass. Why two masses with the same values? The solution was proposed by Einstein: There is only one type of mass, et voilà! By et voilà, I mean there is a new theory to replace the old one, which is more precise… The principle of unification, on which supersymmetry relies quite heavily, is also a principle of “economy of means.” Thus, accepting two explanations for the same thing does not fit the way the supersymmetry program is envisioned, and certainly not past historical movements.

Let us comment on the epistemic values of these arguments.

• First argument: redundancy with other elements of the same research program… This is the famous Occam's razor argument, which at most can be used as a heuristic in the development of theories and models. But, since we are dealing here with the most fundamental laws of Nature, I strongly believe that this economy of means is a requirement for a good candidate for a “final theory”.
• Second argument: the coherent theoretical structure that combines empirical evidence, gauge principle, and the geometrical mathematical framework. The argument asks how to challenge these relationships, that are direct consequence of the application of the scientific approach that has been established long ago: relate empirical evidence with successful mathematical modeling and conceptualization. This is obviously the heart of the scientific process. Thus, if you want to shake this structure, you have to rebuild the entire edifice patiently put in place for decades. This requires the establishment of a new coherent theory (as coherent as the current one) that challenges some conceptions embedded in our minds as physicists. I think then that we are venturing more into a paradigm shift à la Kuhn than into a simple construction of a model within a well-established framework…
• Third argument: a mathematical argument, that is inescapable, based on the very principles of Quantum Field Theory. Playing with this mathematics is not easy and requires compatible implementation of ideas and concepts. Here we have to face the innocence of mathematical structures: We have no choice but to solve this problem using only pure mathematics…

I doubt that supersymmetry physicists have assessed the price to be paid, in terms of historical developments (examining the fundamental consequences of the discovery of the Higgs boson…), the paradigm shift (questionning the well-established geometrical framework that classifies all current empirical data…), and mathematical challenges (mixing apple and oranges…), that their research program induces. They focused on “naturalness”, forgetting that this rather ill-defined notion depends on culture, especially in mathematics and physics… So, maybe it is time to ask to them to wake up, learn more (good) mathematics, and either face these problems or change their research interests.