# What is Supersymmetry?

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Just very recently I was looking into the subject of supersymmetry. Consider expressions of the form

$$\mathcal{L} = \frac{1}{4\pi} \int_{M^4} d^4x d^2\theta \tau_0(\Lambda_0) \mathrm{Tr} W_\alpha W^\alpha \ + \frac{1}{4\pi} \int_{M^4} d^4x d^2\theta \ \mathrm{Im} \ \tau_0(\Lambda_0)\mathrm{Tr} \Phi \bar{\Phi} + \ c.c.$$

$$\mathcal{L} = \frac{1}{2} \left(-t \int d^2 \widetilde{\theta} \Sigma \ + \ c.c. \right)$$

Despite the beauty of such formulas, it seems that we lack a fundamental understanding of the principle that underlies supersymmetry. For example, in general relativity, we have the equivalence principle. What is the principle behind supersymmetry, decades after its discovery? Some physicists say that it is just a book keeping device. Is this true?

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No. In its core, it's a symmetry between particles of different spin, i.e. spinorial Noether charges, such that the Coleman-Mandula theorem is circumvented.

Because {Q,Q}~P, the gauging of SUSY necessarily introduces gravity in the form of GR (with torsion), i.e. the equivalence principle.

ohwilleke, Demystifier, arivero and 1 other person
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No. In its core, it's a symmetry between particles of different spin, i.e. spinorial Noether charges, such that the Coleman-Mandula theorem is circumvented.

Because {Q,Q}~P, the gauging of SUSY necessarily introduces gravity in the form of GR (with torsion), i.e. the equivalence principle.

I agree with your thought that supersymmetric theories, like D=11 supergravity, have their own underlying principles. For example, string theory obviously has its own underlying principle, and therefore, type IIA supergravity which is the low energy limit of the type IIA string, probably has some underlying principle which is as far removed from ordinary physics principles as string theory is. To give another example, the decompactification limit1 of M-theory doesn't even contain strings, so the physical principles that underlie M-theory will be even more arcane.

But what I said about supersymmetry before, I was talking about the notation. It is possible that it is some kind of book-keeping device.

[1] Becker, Becker, Schwarz. String theory and M Theory. Chapter 8.

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I don't get that. Isn't every kind of notation a "book keeping device"?

What could be possible, is that susy will play a similar role in QFT's like complex numbers do in classical mechanics: a very usefull and insightfull mathematical tool without direct empirical confirmation. Is that what you mean?

judiefletcher
The history of advance in physics is one of unification and generalization.

Newton unified terrestrial and celestial mechanics.

Maxwell unified electricity, magnetism, and optics.

Special relativity unified Maxwell's electromagnetism and Galilean invariance.

Special relativity generalized Newtonian mechanics to include speeds close to the speed of light.

General relativity unified special relativity and Newtonian gravity.

General relativity generalized special relativity to include non-inertial frames in addition to inertial frames.

General relativity generalized special relativity to include curved space in addition to flat space.

Quantum field theory unified special relativity and quantum mechanics.

Electroweak theory unified electromagnetism and the weak force.

String theory generalized particle physics to include particles with more than 0 dimensions.

String theory unified quantum field theory and general relativity.

Supersymmetry is a generalization that allowed anti-commutators in addition to commutators.

Supersymmetry combined fermions and bosons.

Supersymmetry unites the two main categories of particles, bosons and fermions, solves the hierarchy problem, leads to the unification of the coupling constants at high energy, allows for fermions within string theory, automatically leads to supergravity, predicts the lightest supersymmetric particle as a candidate for dark matter, and through the Affleck-Dine mechanism, explains baryon asymmetry. In addition, some people think it is beautiful because of the underlying symmetry, which has always played a fundamental role in physics, such as in Noether's theorem. One of the reasons for postulating magnetic monopoles is that they would make Maxwell's equations for symmetric. For some people, that is a legitimate motivation.

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String theory generalized particle physics to include particles with more than 0 dimensions.

String theory unified quantum field theory and general relativity.

Supersymmetry is a generalization that allowed anti-commutators in addition to commutators.

Supersymmetry combined fermions and bosons.

Supersymmetry unites the two main categories of particles, bosons and fermions, solves the hierarchy problem, leads to the unification of the coupling constants at high energy, allows for fermions within string theory, automatically leads to supergravity, predicts the lightest supersymmetric particle as a candidate for dark matter, and through the Affleck-Dine mechanism, explains baryon asymmetry. In addition, some people think it is beautiful because of the underlying symmetry, which has always played a fundamental role in physics, such as in Noether's theorem. One of the reasons for postulating magnetic monopoles is that they would make Maxwell's equations for symmetric. For some people, that is a legitimate motivation.

To be clear, this is what is supposed to happen. None of the matter described in the quoted material above has actually happened.

String theory doesn't actually unify quantum field theory and general relativity, even though it is supposed to do so. Supergravity doesn't actually work. None of the particles or forces predicted by it have been observed, which they would have to in order for it to solve the hierarchy problem.

Supersymmetry does not in any form ever formulated to date, actually solve the hierarchy problem, doesn't actually lead to the unification of the coupling constants at high energy, doesn't actually produce a candidate for dark matter that isn't observationally excluded, and does not actually explain baryon asymmetry.

It is all a very beautiful toy model but it is not a good description the real world and strict experimental exclusions make it very challenging to devise any variation upon it that could describe the real world.

There is absolutely zero observational evidence for either String Theory or supersymmetry.

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I don't get that. Isn't every kind of notation a "book keeping device"?

What could be possible, is that susy will play a similar role in QFT's like complex numbers do in classical mechanics: a very usefull and insightfull mathematical tool without direct empirical confirmation. Is that what you mean?

Possibly. I have heard Witten say that M-theory is so difficult because it involves thinking on a plane of abstractness that we are not accustomed to. We can talk about the "configuration space" of a mechanical system for example, but we are not used to thinking about entire theories at a level where we can compare them, i.e. we are not used to thinking about the "space of theories." For example, consider something like the N = 2 superconformal algebra. N allows us to classify theories, and hence move in the "space of theories". So this classification is made possible by supersymmetry, and maybe in this sense also it is some kind of book keeping device. Without supersymmetry, we would have only the bosonic string, but with supersymmetry we have several theories. This was originally interpreted as a bad thing, but maybe it is just suggesting that we have to think at a more abstract level than we are used to.

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There is absolutely zero observational evidence for either String Theory or supersymmetry.

@love_42 There were hopes of observational evidence at the LHC and I am pretty sure @mfb has posted on this

Mentor
There were hopes, but so far nothing. The LHC experiments should increase their datasets by more than a factor 10 over the LHC lifetime, so we might find something in the future. Something that could end up with e.g. a solid 6 sigma evidence could look like a 1-2 sigma fluctuation today that no one cares about.

PhDeezNutz, ohwilleke and pinball1970
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There were hopes, but so far nothing. The LHC experiments should increase their datasets by more than a factor 10 over the LHC lifetime, so we might find something in the future. Something that could end up with e.g. a solid 6 sigma evidence could look like a 1-2 sigma fluctuation today that no one cares about.

Yes and no. I don't think that the LHC is capable of producing any 6 sigma signals that are thusfar not even a 1-2 sigma signal. You'd need to be talking about a generation or two of colliders post-LHC to get that.

If Supersymmetry were a valid theory with parameters that are experimentally accessible at some foreseeable future collider, the magnitude of the fluctuations might be modest if it was a high enough energy scale phenomena, but you'd expect to see lots and lots of anomalies in plausibly correlated phenomena.

You'd also expect signals in observables that have global sensitivity (like muon g-2, ultra high energy cosmic ray decay observations, https://arxiv.org/abs/2102.07797 and a few others such as Higgs boson decay channels https://arxiv.org/abs/2102.13429). Increasingly, those aren't panning out.

Some of the very generic signals that we aren't seeing, like diphoton decays not predicted by the Standard Model, that would be generated by BSM particles such as a new scalar or pseudoscalar Higgs, scalar supersymmetric particles that are unstable, Z' bosons, and massive spin-2 particles are also very clean and not very sensitive to particular SUSY or String Theory variants. See https://arxiv.org/abs/2102.13405

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Mentor
I don't think that the LHC is capable of producing any 6 sigma signals that are thusfar not even a 1-2 sigma signal.
You misread my post. I said what could end up being 6 sigma might be 1-2 sigma today - that's what you get from the simple sqrt(luminosity) scaling, ignoring future improvements that could make the difference even larger. There are tons of 1-2 sigma deviations. No one cares about them because they are everywhere.
This is true both for individual signals and global fits.

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What is the principle behind supersymmetry, decades after its discovery?

The principle is that the laws of nature should have symmetries that unify bosons and fermions. Since bosons describe forces - generally speaking - while fermions describe matter, this would mean a unification of forces with matter.

It's surprisingly tricky to come up with laws that unify bosons and fermions in this way, so if you write these laws in typical physics notation the formulas tend to look scary and complicated.

ohwilleke, Fra and Klystron
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The principle is that the laws of nature should have symmetries that unify bosons and fermions. Since bosons describe forces - generally speaking - while fermions describe matter, this would mean a unification of forces with matter.

It's surprisingly tricky to come up with laws that unify bosons and fermions in this way, so if you write these laws in typical physics notation the formulas tend to look scary and complicated.

Hi John,

Yes, I suspected that it is something like that. It could be some notational device to keep track of things. BTW, we have spoken before. I sent you an email when I was a math/physics undergraduate at Warwick. I said.

something like "how can a finite set of points have a 'topology'?" or something like that, wondering about the axioms for topological spaces and assigning things like discrete or indiscrete topologies to sets and how to think about such topologies and the axiomatic approach and you replied telling me that it takes some time to see why this kind of approach can be very powerful. Thanks for taking time to reply to an undergraduate, at a different university!

I was just beginning to study axiomatic/rigorous mathematics with axioms and proofs and theorems etc, and at the time it seemed like a pointless thing, being used to the more intuitive approach of physicists. I do now see how powerful and meaningful this kind of thing can be.

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