# Learn About Supersymmetry and Deligne’s Theorem

Common Topics: group, spacetime, symmetry, groups, particle

In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here I give an informal explanation what this theorem says and why it has profound relevance for theoretical particle physics:

Deligne’s theorem on tensor categories combined with Wigner’s classification of fundamental particles implies a strong motivation for expecting that fundamental high energy physics exhibits supersymmetry.

I explain this in a moment. But that said, before continuing I should make the following

Side remark.

Recall that what these days are being constrained more and more by experiment are models of “low energy supersymmetry”: scenarios where a fundamental high energy supergravity theory sits in a vacuum with the exceptional property that a global supersymmetry transformation survives. Results such as Deligne’s theorem have nothing to say about the complicated process of stagewise spontaneous symmetry breaking of a high energy theory down to the low energy effective theory of its vacua. Instead, they say something (via the reasoning explained in a moment) about the mathematical principles which underlie fundamental physics fundamentally, i.e. at high energy. Present experiments, for better or worse, say nothing about high energy supersymmetry.

Incidentally, it is also high energy supersymmetry, namely supergravity, which is actually predicted by string theory (this is a theorem: the spectrum of the fermionic “spinning string” miraculously exhibits local spacetime supersymmetry), while low energy supersymmetry needs to be imposed by hand in string theory (namely by assuming Calabi-Yau compactifications, there is no mechanism in the theory that would single them out).

End of side remark.

Now first recall the idea of Wigner’s classification of fundamental particles. In order to bring out the fundamental force of Wigner classification, I begin by recalling some basics of the fundamental relevance of local spacetime symmetry groups:

Given a symmetry group ##G## and a subgroup ##H \hookrightarrow G##, we may regard this as implicitly defining a local model of spacetime: We think of ##G## as the group of symmetries of the would-be spacetime and of ##H## as the subgroup of symmetries that fix a given point. Assuming that ##G## acts transitively, this means that space itself is the coset

$$X = G/H$$

For instance if ##X = \mathbb{R}^{d-1,1}## is Minkowski spacetime, then its isometry group ##G = \mathrm{Iso}(\mathbb{R}^{d-1,1})## is the Poincaré group and ##H = O(d-1,1)## is the Lorentz group. But it also makes sense to consider alternative local spacetime symmetry groups, such as ##G = O(d-1,2)## the anti-de Sitter group. etc. The idea of characterizing local spacetimes as the coset of its local symmetry group by the stabilizer of any one of its points is called Klein geometry.

To globalize this, consider a manifold ##X## whose tangent spaces look like ##G/H##, and such that the structure group of its tangent bundle is reduced to the action of ##H##. This is called Cartan geometry.

For the previous example where ##G/H## is Poincaré/Lorentz, then Cartan geometry is equivalently pseudo-Riemannian geometry: the reduction of the structure group to the Lorentz group is equivalently a “vielbein field” that defines a metric, hence a field configuration of gravity. For other choices of ##G## and ##H## the same construction unifies essentially all concepts of geometry ever considered, see the table of examples here.

This is a powerful formulation of a spacetime geometry that regards spacetime symmetry groups as more fundamental than spacetime itself. In physics literature, it is essentially known as the first-order formulation of gravity.

Incidentally, this is also the way to obtain super-spacetimes: simply replace the Poincaré group with its super-group extension: the super-Poincaré group (super-Cartan geometry) But why would one consider that? We get to this in a moment.

Now as we consider quantum fields covariantly on such a spacetime, then locally all fields transform linearly under the symmetry group ##G##, hence they form linear representations of the group ##G##.

Given two ##G##-representations, we may form their tensor product to obtain a new representation. Physically this corresponds to combining two fields to the joint field of the composite system.

Based on this, Wigner suggested that the elementary particle species are to be identified with the irreducible representations of ##G##, those which are not the direct sum of two non-zero representations.

Indeed, if one computes, in the above example, the irreducible unitary representations of the Poincaré group, then one finds that these are labeled by the quantum numbers of elementary particles seen in the experiment, mass and spin, and helicity for massless particles.

One may do the same for other model spacetimes, such as (anti-)de Sitter spacetimes. Then the particle content is given by the irreducible representations of the corresponding symmetry groups, the (anti-)de Sitter groups, etc.

The point of this digression via Klein geometry and Cartan geometry is to make the following important point: the spacetime symmetry group is more fundamental than the spacetime itself.

Therefore we should not be asking:

What are all possible types of spacetimes over which we could consider Wigner classification of particles?

What are all possible symmetry groups such that their irreducible representations behave like elementary particle species?

This is the question that Deligne’s theorem on tensor categories is the answer to.

To give a precise answer, one first needs to make the question precise. But it is well known how to do this:

1. A collection of things (for us: particle species)
2. which may be tensored together (for us: compound systems may be formed)
3. where two things may be exchanged in the tensor product (two particles may be exchanged)
4. such that exchanging twice is the identity operation,
5. and such that everything has a dual under tensoring (for every particle there is an anti-particle);
6. such that the homomorphisms between things (for us: the possible interaction vertices between particle species) form vector spaces;

is said to be a linear tensor category.

We also add the following condition, which physically is completely obvious, but necessary to make explicit to prove the theorem below:

• Everything consists of a finite number of particle species, and the compound of ##n## copies of  ##N## particle species contains at most ##N^n## copies of fundamental particle species. Mathematically this is the condition of “subexponential growth”, see here for the mathematical detail.

A key example of tensor categories are categories of finite-dimensional representations of groups, but not all tensor categories are necessary of this form. The question for those which are is called Tannaka duality: the problem of starting with a given tensor category and reconstructing the group that it is the category of representations of.

The case of interest to us here is that of tensor categories which are ##\mathbb{C}##-linear, hence where the spaces of particle interaction vertices are complex vector spaces. More generally we could consider ##k##-linear tensor categories, for ##k## any field of characteristic 0.

Deligne studied the question:

Under which conditions is such a tensor category the representation category of some group, and if so, of which kind of group?

Phrased in terms of our setup this question is:

Given any collection of things that behave like particle species and spaces of interaction vertices between these, under which condition is there a local spacetime symmetry group such that these are the particles in the corresponding Wigner classification of quanta on that spacetime, and what kinds of spacetime symmetry groups arise this way?

Now the answer to Deligne’s theorem on tensor categories is this:

1. Every  ##k##-linear tensor category is of this form;
2. the class of groups arising this way is precisely the (algebraic) super-groups.

This is due to

Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

based on

Pierre Deligne,  Catégories Tannakiennes , Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.

reviewed in

Victor Ostrik, Tensor categories (after P. Deligne) (arXiv:math/0401347)

and in

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 9.11 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)

Phrased in terms of our setup this means

1. Every sensible collection of particle species and spaces of interaction vertices between them is the collection of elementary particles in the Wigner classification for some local spacetime symmetry group;
2. the local spacetime symmetry groups appearing this way are precisely super-symmetry groups.

Notice here that a super-group is understood to be a group that may contain odd-graded components. So also an ordinary group is a super-group in this sense. The statement does not say that spacetime symmetry groups need to have odd super graded components (that would evidently be false). But it says that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of possible supergroups. Not more. Not less.

Hence Deligne’s theorem — when regarded as a statement about local spacetime symmetry via Wigner classification as above — is a much stronger statement than for instance the Coleman-Mandula theorem + Haag-Lopuszanski-Sohnius theorem which is traditionally invoked as a motivation for supersymmetry:

For Coleman-Mandula+Haag-Lopuszanski-Sohnius to be a motivation for supersymmetry, you first of all already need to believe that spacetime symmetries and internal symmetries ought to be unified. Even if you already believe this, then the theorem only tells you that supersymmetry is one possibility to achieve this unification, there might still be an infinitude of other possibilities that you haven’t considered yet.

For Deligne’s theorem the conclusion is much stronger:

First of all, the only thing we need to believe about physics, for it to give us information, is an utmost minimum: that particle species transform linearly under spacetime symmetry groups. For this to be wrong at some fundamental scale we would have to suppose non-linear modifications of quantum physics or other dramatic breakdowns of everything that is known about the foundations of fundamental physics.

Second, it says not just that local spacetime supersymmetry is one possibility to have sensible particle content under Wigner classification, but that the class of (algebraic) super-groups precisely exhausts the moduli space of possible consistent local spacetime symmetry groups.

This does not prove that fundamentally local spacetime symmetry is non-trivial supersymmetry. But it means that it is well motivated to expect that it might be one.

Tags:
54 replies
1. john baez says:

In short, Deligne's theorem applies perfectly well to the tensor category of representations of the Galilean group: it says this tensor category can be seen as consisting of representations of a group.  But that's not very surprising!

2. Urs Schreiber says:

[QUOTE="DrDu, post: 5590231, member: 210532"]What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.[/QUOTE]The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

3. Urs Schreiber says:

[QUOTE="DrDu, post: 5590231, member: 210532"]What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.[/QUOTE]The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

4. Urs Schreiber says:

[QUOTE="DrDu, post: 5590231, member: 210532"]What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.[/QUOTE]The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

5. Urs Schreiber says:

[QUOTE="DrDu, post: 5590231, member: 210532"]What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.[/QUOTE]The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

6. Urs Schreiber says:

[QUOTE="DrDu, post: 5590231, member: 210532"]What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.[/QUOTE]The theorem is about tensor categories, whose morphisms in physics translate to possible interactions between particle species. It doesn't say anything about potentials.

7. DrDu says:

What i wanted to say is: in case of the galilei group the interacting reps arent tensorial products.

8. Urs Schreiber says:

[QUOTE="DrDu, post: 5589993, member: 210532"]First, let me say that this is a very nice and interesting insights article![/QUOTE]Thanks. Glad you liked it.[QUOTE="DrDu, post: 5589993, member: 210532"] If I remember correctly, e.g. Galilean relativity fits well into the Klein schema but there are more general kinds of interactions possible than in the Poincare setting, namely the ones where particles interact via a potential. Are these representations compartible with the prerequisites of Delignes theorem?[/QUOTE]Where the theorem speaks about groups, the only condition is that these are affine algebraic. So all the usual matrix groups fit in.On the other hand, to make the theorem say something about physics, we are to think of these groups as spacetime symmetry groups at high energy, equivalently at small scales. That makes the Galilean group be an odd choice.

9. DrDu says:

[QUOTE="Urs Schreiber, post: 5548698, member: 567385"]Technically what I mean are the spaces of "intetwiners" between representations. In physics these are the possible spaces of interaction vertices.For instance the space of interaction vertices for two spinors merging to become a vector boson include the linear maps which in components are given by the Gamma-matrices, as familiar from QCD. But there is an arbitrary prefactor in front of the Gamma-matrix, the "coupling constant", and hence the space of interaction vertices is in fact a vector space.[/QUOTE]First, let me say that this is a very nice and interesting insights article!If I remember correctly, e.g. Galilean relativity fits well into the Klein schema but there are more general kinds of interactions possible than in the Poincare setting, namely the ones where particles interact via a potential. Are these representations compartible with the prerequisites of Delignes theorem?

10. Urs Schreiber says:

[QUOTE="Jimster41, post: 5571137, member: 517770"] Does it make sense to ask what Background(s) support the existence of regular groups – but don't start with any?[/QUOTE]Not entirely sure what you mean to ask here, but I'll highlight again that there is an utmost minimum of assumption that goes into the argument given in the entry above. All it needs is that locally the collection of particle species satisfies the most minimalistic conditions (such as that the space of interaction vertices is a linear space over a field of characteristic zero). No assumption on "backgrounds" enters. And crucially, no assumption on groups enters. The statement about the spacetime symmetry groups is all a consequence of the theorem (that appropriate groups exist at all, and that they span the space of algebraic supergroups).[QUOTE="Jimster41, post: 5571137, member: 517770"] Doesn't this bear on the question of whether or not space-time is likely to be discrete or continuous?[/QUOTE]No. First of all, the classical homotopy category is well known to be pathological in many ways, and the fact that it is not concrete is absolutely no cause of worry or concern, it only serves as a counterexample to concrete categories that people like to cite. If you are looking into actual geometry (discrete or continuous or whatsoever) then the classical homotopy category is not the place to look. It's not about geometry, but about abstract homotopy theory. More importantly however, apart from the word "category" appearing both in the above entry and in the blurb on the homotopy category that you cite, there is no relation between then two.

11. Jimster41 says:

"First of all, the only thing we need to believe about physics, for it to give us information, is an utmost minimum: that particle species transform linearly under spacetime symmetry groups." – Reference https://www.physicsforums.com/insights/supersymmetry-delignes-theorem/Does it make sense to ask what Background(s) support the existence of regular groups – but don't start with any?"The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable."https://en.wikipedia.org/wiki/Concrete_categoryDoesn't this bear on the question of whether or not space-time is likely to be discrete or continuous?

12. Urs Schreiber says:

There is a set theoretic size bound for the theorem to work (that regularity condition that I mentioned ), but it is very mild. I think it is hard to construct categories that violate this size bound,  and the example that do are contrived and won't show up in mathematical practice, much less in physics. This is a point that I should eventually expand on.

13. Haelfix says:

Out of curiosity, how many loopholes are there to this theorem?  For instance, the Coleman-Mandula theorem offered a rather large amount of interesting outs, and just glancing at the statement of the theorem, it looks like some of the same sorts of tricks can be used.For instance, Coleman-Mandula/Haag-Lopuszanski fails when you consider an infinite amount of particle species (like theories with an infinite tower of higher spin states)…

14. Urs Schreiber says:

The theorem itself speaks about super-groups in the general sense, which includes the super-group extensions of the Poincare group (physicist's supersymmetry) as well as more general super-groups. The theorem itself does not know the Poincare group, it only knows that spacetime symmetry needs to be some possibly-super-group. But given that we know that spacetime symmetry looks at least approximately (at low energy) like Poincare, together this means that all that can happen at higher energy is that some super-group extension of Poincare becomes visible. And the super-group extension of Poincare, that's what's conventionally called super-symmetry in physics.

15. ohwilleke says:

So, would I be correct in saying that the class of models consistent with this theorem is quite a bit broader than the class of models conventionally described as supersymmetric?

16. Urs Schreiber says:

There is a comment on that at the end of the article, where it says"Notice here that a super-group is understood to be a group that may contain odd-graded components. So also an ordinary group is a super-group in this sense. The statement does not say that spacetime symmetry groups need to have odd supergraded components (that would evidently be false). But it says that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of possibly-super groups. Not more. Not less."But let me expand on this point more:The theorem says that the most general admissible symmetry groups are algebraic super-groups (as opposed to more exotic things like quantum groups etc.) An algebraic group is equivalently a commutative Hopf algebra. An algebraic super-group is equivalently a super-commutative Hopf algebra, hence a Hopf algebra that may be non-commutative, but only in the mild sense that it carries a Z/2-grading and commuting two odd-graded elements past each other introduces a minus sign. What is excluded are Hopf algebras that are more non-commutative than this (e.g. quantum groups) as well as yet more exotic situations.It is the odd-graded elements in a super-commutative Hopf algebra that correspond to super-group elements that mix bosons and fermions. Now, as highlighted at the end of the article, ordinary commutative Hopf algebras, all whose elements are in even degree, are included within super-Hopf algebras. These do not mix bosons and fermions.The theorem does not say that the spacetime symmetry group necessarily needs to mix bosons and fermions, and how much. It only says that there is guaranteed to be a kind of spacetime symmtry group for every consistent collection of elementary particles, and the  kind of groups arising this way are precisely the super-groups (as opposed to more exotic things like quantum groups etc.)

17. ohwilleke says:

What precisely must a theory do to qualify as supersymmetric within the meaning of the theorem?For example, must there be symmetry between the Fermionic and Bosonic sector of fundamental particles generally, or must there actually be a one to one correspondence such that each fundamental fermion have precisely one bosonic counterpart and visa versa?

18. Urs Schreiber says:

Kodama, If you have any technical questions or comments, I'd be happy to discuss them. But please try to omit the believes and the agenda and the gossip. Try to stick to the objective facts that we are trying to discuss here.  There is some really interesting stuff here, and it would be sad to drown it in noise.The material you quote is subject to various misunderstandings. Since you are lazy and just copy-and-pasting stuff other people said elsewhere, I don't feel motivated to guess which questions you might actually have. Let me know which points you find unclear, and I'll try to help.

19. kodama says:

20. kodama says:

[QUOTE="Urs Schreiber, post: 5553950, member: 567385"]Possibly you are thinking of "Quantum gravity in the sky" (arXiv:1608.04228). [/QUOTE]yes[QUOTE="Urs Schreiber, post: 5553950, member: 567385"]This proposes two principles to fit LQC to PLANCK satellite data. Now LQC is, much like MOND, a formula without a theory.  .[/QUOTE]yeah that sounds about right /s

21. Urs Schreiber says:

[QUOTE="kodama, post: 5553846, member: 551353"]thanks for bringing this to my attention but LQC without SUSY also offers improved fits to data – if you want i can post papers from Ahsketar et al. and LQC does not make use of SUSY[/QUOTE]Possibly you are thinking of "Quantum gravity in the sky" (arXiv:1608.04228). This proposes two principles to fit LQC to PLANCK satellite data. Now LQC is, much like MOND, a formula without a theory. This may be of interest, but it does not seem to "offer improved fits to data".Moreover, there remains the issue with the Starobinsky model that I mentioned in #27: in plain gravity it needs an overly large initial homegenous patch to work in the first place.

22. kodama says:

[QUOTE="Urs Schreiber, post: 5553607, member: 567385"]Supergravity predicts improvements to fits of models of cosmic inflation to the latest data:The models of cosmic inflation that best fit the Planck satellite data are "plateau models" such as Starobinsky inflation aka ##R^2## inflation. See figure 12, and table 6 in

• Planck Collaboration, Planck 2015 results. XX. Constraints on inflation (arXiv:1502.02114)

As made explicit below table 6, there is preference for Starobinsky inflation in the data, even if not significant. However, in

• Alex Kehagias, Azadeh Moradinezhad Dizgah, Antonio Riotto, "Comments on the Starobinsky Model of Inflation and its Descendants", Phys. Rev. D 89, 043527 (2014) (arXiv:1312.1155)

it is argued that other models of Plateau inflation are actually physically equivalent to Starobinsky inflation.Starting with

• S. Cecotti, "Higher derivative supergravity Is equivalent to standard supergravity coupled to matter", Phys. Lett. B 190, 86 (1987).

it has been argued that Starobsinky inflation prefers being embedded into supergravity. This was reinforced with the Planck2013 data. On p. 17 of the above arXiv:1502.02114 it says:For review see

• Fotis Farakos, Alex Kehagias, A. Riotto, "On the Starobinsky Model of Inflation from Supergravity", Nucl. Phys. B 876, 187 (2013) (arXiv:1307.1137)

and

• John Ellis, "Planck-Compatible Inflationary Models", talk 2013 (pptx)

A particularly striking supergravity prediction has been reported by Dalianis and Farakos:While models of Plateau inflation fit the Planck satellite data best, to do so they need to start inflation from a relatively large initial homogeneous patch of spacetime of diameter the oder of a few thousand Planck lengths. This leaves the problem of why the universe before inflation was homogeneous on this scale. Now in

it is argued that when embedding the model into supergravity, that this problem goes away in that the required initial homogeneous patch shrinks to the order of one Planck length (see equations (4.11) and (4.13)).A review of this result appeared recently presented here

• Ioannis Dalianis, "Features and implications of the plateau inflationary potentials", Planck 2015 conference contribution (arXiv:1602.05026)

[/QUOTE]thanks for bringing this to my attention but LQC without SUSY also offers improved fits to data – if you want i can post papers from Ahsketar et al. and LQC does not make use of SUSY

23. Urs Schreiber says:

[QUOTE="kodama, post: 5553566, member: 551353"]so any specific and testable predictions if susy is well motivated due to Deligne theorem, even as lhc has ruled out low energy susy?[/QUOTE]

Supergravity predicts improvements to fits of models of cosmic inflation to the latest data:

The models of cosmic inflation that best fits the Planck satellite data are “plateau models” such as Starobinsky inflation aka ##R^2## inflation. See figure 12, and table 6 in

Planck Collaboration, “Planck 2015 results. XX. Constraints on inflation” (arXiv:1502.02114)

As made explicit below table 6, there is preference for Starobinsky inflation in the data, even if not significant. However, in

Alex Kehagias, Azadeh Moradinezhad Dizgah, Antonio Riotto, “Comments on the Starobinsky Model of Inflation and its Descendants”, Phys. Rev. D 89, 043527 (2014) (arXiv:1312.1155)

it is argued that other models of Plateau inflation are actually physically equivalent to Starobinsky inflation. Starting with

S. Cecotti, “Higher derivative supergravity Is equivalent to standard supergravity coupled to matter”, Phys. Lett. B 190, 86 (1987)

it has been argued that Starobsinky inflation prefers being embedded into supergravity. This was reinforced with the Planck2013 data. On p. 17 of the above arXiv:1502.02114 it says:

For review see

Fotis Farakos, Alex Kehagias, A. Riotto, “On the Starobinsky Model of Inflation from Supergravity”, Nucl. Phys. B 876, 187 (2013) (arXiv:1307.1137)

and

John Ellis, “Planck-Compatible Inflationary Models”, talk 2013 (pptx)

A particularly striking supergravity prediction has been reported by Dalianis and Farakos:

While models of Plateau inflation fit the Planck satellite data best, to do so they need to start inflation from a relatively large initial homogeneous patch of spacetime of diameter the oder of a few thousand Planck lengths. This leaves the problem of why the universe before inflation was homogeneous on this scale. Now

Ioannis Dalianis, Fotis Farakos, “On the initial conditions for inflation with plateau potentials: the ##R + R^2## (super)gravity case”, Journal of Cosmology and Astroparticle Physics, Volume 2015, July 2015 (arXiv:1502.01246)

it is argued that when embedding the model into supergravity, that this problem goes away in that the required initial homogeneous patch shrinks to the order of one Planck length (see equations (4.11) and (4.13)).

A review of this result appeared recently presented here:

Ioannis Dalianis, “Features and implications of the plateau inflationary potentials”, Planck 2015 conference contribution (arXiv:1602.05026)

24. Urs Schreiber says:

[QUOTE="kodama, post: 5553566, member: 551353"]so any specific and testable predictions if susy is well motivated due to Deligne theorem, even as lhc has ruled out low energy susy?[/QUOTE]Supergravity predicts improvements to fits of models of cosmic inflation to the latest data:The models of cosmic inflation that best fits the Planck satellite data are "plateau models" such as Starobinsky inflation aka ##R^2## inflation. See figure 12, and table 6 in

• Planck Collaboration, Planck 2015 results. XX. Constraints on inflation (arXiv:1502.02114)

As made explicit below table 6, there is preference for Starobinsky inflation in the data, even if not significant. However, in

• Alex Kehagias, Azadeh Moradinezhad Dizgah, Antonio Riotto, "Comments on the Starobinsky Model of Inflation and its Descendants", Phys. Rev. D 89, 043527 (2014) (arXiv:1312.1155)

it is argued that other models of Plateau inflation are actually physically equivalent to Starobinsky inflation.Starting with

• S. Cecotti, "Higher derivative supergravity Is equivalent to standard supergravity coupled to matter", Phys. Lett. B 190, 86 (1987).

it has been argued that Starobsinky inflation prefers being embedded into supergravity. This was reinforced with the Planck2013 data. On p. 17 of the above arXiv:1502.02114 it says:For review see

• Fotis Farakos, Alex Kehagias, A. Riotto, "On the Starobinsky Model of Inflation from Supergravity", Nucl. Phys. B 876, 187 (2013) (arXiv:1307.1137)

and

• John Ellis, "Planck-Compatible Inflationary Models", talk 2013 (pptx)

A particularly striking supergravity prediction has been reported by Dalianis and Farakos:While models of Plateau inflation fit the Planck satellite data best, to do so they need to start inflation from a relatively large initial homogeneous patch of spacetime of diameter the oder of a few thousand Planck lengths. This leaves the problem of why the universe before inflation was homogeneous on this scale. Now in

it is argued that when embedding the model into supergravity, that this problem goes away in that the required initial homogeneous patch shrinks to the order of one Planck length (see equations (4.11) and (4.13)).A review of this result appeared recently presented here

• Ioannis Dalianis, "Features and implications of the plateau inflationary potentials", Planck 2015 conference contribution (arXiv:1602.05026)
25. kodama says:

[QUOTE="Urs Schreiber, post: 5548729, member: 567385"]No. The article presents a fact that was discovered by somebody with no interest in supersymmetry, either way.Most of what you quote are standard arguments for unbroken low-energy susy. As explained in the article right at the beginning, this is not what it is about. Remains the Coleman-Mandula theorem, on which the article comments in some detail towards the end.Try to read it. Try to read it without ideology. It is an exposition of a mathematical theorem, which you may try to understand and accept, but which does not go away by becoming angry at it.[/QUOTE]so any specific and testable predictions if susy is well motivated due to Deligne theorem, even as lhc has ruled out low energy susy?

26. strangerep says:

[QUOTE="carlosdelamora, post: 5549438, member: 601942"]In what sense do you mean that G/H is a semi-group? the only way I can think you want to define the multiplication is by saying xHyH=xyH. […][/QUOTE] No, that's not what I have in mind. But to explain properly would require many pages (hence hijacking this thread) of unpublished work (hence not appropriate on PF).

27. carlosdelamora says:

[QUOTE="strangerep, post: 5548310, member: 70760"]Heh, well, in ancient history (when s.p.r. was a great place and you were a moderator) you once rejected a post of mine because I suggested that spacetime itself is not fundamental. I'm glad to see you've (apparently?) changed your mind. :biggrin:A small quibble about 1 point in your article:IIUC, the word "exactly" is not correct for massless particles — one must manually impose a constraint that the representation should be trivial wrt the continuous spin degrees of freedom (i.e., the 2 translation-like generators in the E(2) little group for massless representations). Weinberg vol 1 covers this.Separately, I have a question about Klein/Cartan geometry. Are there already any extensions of that framework for the case where G/H is a semigroup? I'm thinking here about how one might embed temporal casuality into physical theories in a more fundamental way, rather than being imposed by hand as is currently the case in GR and QFT.Cheers.[/QUOTE]In what sense do you mean that G/H is a semi-group? the only way I can think you want to define the multiplication is by saying xHyH=xyH. That would impose the condition of H being a normal subgroup and therefore G/H not only a semigroup but a group.

28. Urs Schreiber says:

Thanks for your efforts! But oh dear, what may have happened there? I don't know. What I see is that the following paragraph was added to the entry:"Wigner classified all irreducible unitary representations of the restricted Poincare group, including the unphysical ones. The latter cannot be used to define a free quantum field theory satisfying the Wightman axioms. Those that can are the physical ones and are characterized by a nonnegative real mass and a nonnegative half-integral spin; the zero component of the momentum has a nonnegative spectrum. Many of these are realized by paticles occurring in Nature, though not as ‘elementary particles’‘ but as bound states (in a suitable approximation, e.g., QCD). From the point of view of representation theory, the center of mass of a bound state behavs just like an elementary particle. Thus elementary is meant in this generalized sense."Did you add more than this? If so, maybe hitting "back" on your browser still recovers it?

29. dextercioby says:

For the record, to the extent that I know of, there is no formal treatment in the literature of the irreducible representations of the universal cover of the restricted Poincaré group in the language of rigged Hilbert spaces, therefore this part right here in blue could count as original/unpublished material.

30. dextercioby says:

For the record, to the extent that I know of, there is no formal treatment in the literature of the irreducible representations of the universal cover of the restricted Poincaré group in the language of rigged Hilbert spaces, therefore this part right here in blue could count as original/unpublished material. [ATTACH=full]104976[/ATTACH]

31. dextercioby says:

For the record, to the extent that I know of, there is no formal treatment in the literature of the irreducible representations of the universal cover of the restricted Poincaré group in the language of rigged Hilbert spaces, therefore this part right here in blue could count as original/unpublished[ATTACH=full]104976[/ATTACH] [ATTACH=full]104976[/ATTACH] material. [ATTACH=full]104976[/ATTACH]

32. dextercioby says:

For the record, to the extent that I know of, there is no formal treatment in the literature of the irreducible representations of the universal cover of the restricted Poincaré group in the language of rigged Hilbert spaces, therefore this part right here in blue could count as original/unpublished[ATTACH=full]104976[/ATTACH] material. [ATTACH=full]104976[/ATTACH]

33. dextercioby says:

For the record, to the extent that I know of, there is no formal treatment in the literature of the irreducible representations of the universal cover of the restricted Poincaré group in the language of rigged Hilbert spaces, therefore this part right here in blue could count as original/unpublished material. [ATTACH=full]104976[/ATTACH]

34. Urs Schreiber says:

Thanks, Arnold. That nLab entry is waiting for somebody to take a little care of it. It was started by John and/or people discussing with him long time back,but the editing was abandoned before a stable version was reached. Might you have 10 minutes to spare on this? It would be greatly appreciated! Just hit "edit" at the bottom of the entry. The syntax is simple and should be self-explanatory.

• Arnold Neumaier says:

I tried to edit the Poincae page but it is now in a state of limbo, without having accepted my edit or allowing me to edit them further. Maybe someone authorized to manage n-lab can recover my changes and place them there.

35. Arnold Neumaier says:

Nice article!You referred to https://ncatlab.org/nlab/show/unitary+representation+of+the+Poincar%C3%A9+group where you could delete the remark by John Baez. Indeed, Wigner had classified all irreducible unitary representations of the Poincare group, including the unphysical ones. The physical ones are almost characterized by causality requirements, but to exclude zero mass continuous spin (which is causal but apparently not realized in Nature) they should rather be characterized by the requirement that one can create from them a free Wightman field theory.

36. Urs Schreiber says:

[QUOTE="unknown1111, post: 5548567, member: 549492"] To me it reads like the usual SUSY propaganda: SUSY must be correct, because otherwise string theory is in deep trouble. Thus let's find some good sounding reasons why SUSY is inevitable. This article seems motivated by the current doomsday mood in the HEP community. [/QUOTE]

No. The article presents a fact that was discovered by somebody with no interest in supersymmetry, either way.

[QUOTE="unknown1111, post: 5548567, member: 549492"]
There are four main motivations for SUSY:
[/QUOTE]

Most of what you quote are standard arguments for unbroken low-energy susy. As explained in the article right at the beginning, this is not what it is about. Remains the Coleman-Mandula theorem, on which the article comments in some detail towards the end.

Try to read it. Try to read it without ideology. It is an exposition of a mathematical theorem, which you may try to understand and accept, but which does not go away by becoming angry at it.

37. john baez says:

[QUOTE="Demystifier, post: 5548488, member: 61953"]Suppose that the Standard Model, as we know it, is the final theory of "everything". Since it is not supersymmetric, it must violate some assumptions of the Deligne's theorem. [/QUOTE]No.  Deligne's theorem says, very very roughly, that under certain conditions particle must have some super-group of symmetries.  However, for the purposes of this theorem, an ordinary group counts as a special case of a super-group, namely one that has no transformations mixing fermions and bosons.  So what we see in the Standard Model is allowed.    Urs explained it this:[QUOTE]Notice here that a super-group is understood to be a group that may contain odd-graded components. So also an ordinary group is a super-group in this sense. The statement does not say that spacetime symmetry groups need to have odd supergraded components (that would evidently be false). But it says that the largest possible class of those groups that are sensible as local spacetime symmetry groups is precisely the class of possibly-super groups. Not more. Not less.[/QUOTE]

38. unknown1111 says:

I'm currently doing my PhD in theoretical particle physics. I understand SUSY, the Poincare Group and Wigner's Classification quite good. I've read the article twice. However I have no clue what the author is talking about.To me it reads like the usual SUSY propaganda: SUSY must be correct, because otherwise string theory is in deep trouble. Thus let's find some good sounding reasons why SUSY is inevitable.This article seems motivated by the current doomsday mood in the HEP community. Everyone was certain that SUSY shows up at the LHC, just as everyone was certain that SUSY shows up at LEP or the Tevatron. (And sure, the 100 TeV collider certainly will find SUSY.) Howecer, there is no experimental evidence for anything beyond the standard model and certainly no signal that hints towards SUSY particles. The fact that the LHC did not find any SUSY particles is a big problem for SUSY fans, because now one main motivation is no longer valid (SUSY as a solution of the naturalness problem).Therefore, SUSY isn't very attractive anymore. There are four main motivations for SUSY:    Solving the naturalness problem (Higgs mass problem)    Unfication of the three standard model forces. However this argument is rather weak, because any BSM theory with as many free paramters as SUSY can be easily fitted such that the couplings unify. In addition, it's quite unlikely that a big unified symmetry (SO(10), E6) breaks directly to SU(3)xSU(2)xU(1). Instead an intermediate symmetry group between the unifcation and the standard model group, like the Pati-Salam group possibly exists. If this ist the case the couplings ALWAYS unify with SUSY or without.    Solving the Dark Matter problem. This argument is rather weak, too. Any expansion of the standard model with additional particles contains a dark matter candidate if we impose an additional discrete symmtry to guarantee its stability, which is what SUSY does.    The Coleman-Mandula-Theorem and the argument that SUSY is the only possibility to mix spacetime and internal symmetries. I've also problems with this argument, but this comment is already too long. In short, I don't think that SUSY helps to understand why fermions and bosons behave so differently (which is one of the biggest mysteries in modern physics), because this difference is simply the assumption at the start of SUSY. Thus I don't see which theoretical problem SUSY solves or why the proposed "unification" of spacetime and internal symmetries helps in any way.Now the first one is no longer valid. The second and third are very weak arguments anyway. Thus it is not suprising that many SUSY researches are stopping to work on SUSY topics now. However there is a group of researches that can not stop and thus needs to find new motivation for SUSY: string theorists.This is how we end with an article like this. Lots of highbrow mathematics and complicated wording, which impresses students and laymans and leaves the impression that SUSY is inevitable.

39. MathematicalPhysicist says:

[QUOTE="Demystifier, post: 5548488, member: 61953"]Suppose that the Standard Model, as we know it, is the final theory of "everything". Since it is not supersymmetric, it must violate some assumptions of the Deligne's theorem. My question is: what these assumptions (violated by the Standard Model) are?[/QUOTE]How can this possibly be if it doesn't include gravity? (do you refer to SM of particle physics?).

40. Demystifier says:

Suppose that the Standard Model, as we know it, is the final theory of "everything". Since it is not supersymmetric, it must violate some assumptions of the Deligne's theorem. My question is: what these assumptions (violated by the Standard Model) are?

• Urs Schreiber says:

No, the entry comments on this point in the paragraph starting with the sentence:"Notice here that a super-group is understood to be a group that _may_ contain odd-graded components." But it need not.A super-group is a group in super-geometry. It's underlying space may have add-graded coordinates, but it need not. In this terminology, an ordinary group is also a super-group, just one where the super-odd piece happens to be trivial.It's all explained in the article, but since you missed it, I'll say it again: the theorem of course does not say that ordinary groups are ruled out, that would clearly be wrong. Instead the force of the theorem is to say that the largest class of admissible groups is that of super-groups (i.e. ordinary and possibly super groups), instead of, say, the even larger class of non-commutative groups or what not.

41. MathematicalPhysicist says:

[QUOTE="strangerep, post: 5548426, member: 70760"]sci.physics.research[/QUOTE]Are researchers in physics still using these usenet groups?I think that nowadays with stackexchange and PF that why would anyone still use those primitive forums.I know that they still exist.

42. strangerep says:

[QUOTE="MathematicalPhysicist, post: 5548403, member: 72"]s.p.r? is that a usenet group?[/QUOTE] sci.physics.research

43. MathematicalPhysicist says:

[QUOTE="strangerep, post: 5548310, member: 70760"]Heh, well, in ancient history (when s.p.r. was a great place and you were a moderator) you once rejected a post of mine because I suggested that spacetime itself is not fundamental. I'm glad to see you've (apparently?) changed your mind. :biggrin:A small quibble about 1 point in your article:IIUC, the word "exactly" is not correct for massless particles — one must manually impose a constraint that the representation should be trivial wrt the continuous spin degrees of freedom (i.e., the 2 translation-like generators in the E(2) little group for massless representations). Weinberg vol 1 covers this.Separately, I have a question about Klein/Cartan geometry. Are there already any extensions of that framework for the case where G/H is a semigroup? I'm thinking here about how one might embed temporal casuality into physical theories in a more fundamental way, rather than being imposed by hand as is currently the case in GR and QFT.Cheers.[/QUOTE]s.p.r? is that a usenet group?

• Urs Schreiber says:

The formatting here in the comment section tends to come out differently from what the people editing a comment expect. I think in the message by "Mathematical Physicist" above in fact everything except the last line is meant as a blockquote from a previous comment, the only line that "Mathematical Physicist" meant to add is"s.p.r? is that a usenet group?"to which the answer is: Yes. it is short for "sci.physics.research". Nowadays it exists as a GoogleGroup https://groups.google.com/forum/#!forum/sci.physics.research .

44. strangerep says:

[QUOTE="Urs Schreiber, post: 5548364, member: 567385"]Regarding the ancient history: I don't remember the contribution you are referring to, maybe you could remind me.[/quote] Oh, I didn't keep a copy. At the time, it was all too hard to convince anyone that symmetries are more fundamental than spacetime. Nowadays, it's a more widely respectable point of view.I have another question about your article. You talk about[quote=Urs Schreiber]The case of interest to us here is that of tensor categories which are ##{mathbb C}##-linear, hence where the spaces of particle interaction vertices are complex vector spaces.[/QUOTE] What precisely do you mean by "spaces of particle interaction vertices"? In the context of ordinary QFT, I imagine tensoring together the Fock spaces of the various elementary fields so that (e.g., in QED) one can express interaction Hamiltonian terms like ##bar psi A^mu psi##. But such Fock-like spaces are known to be incapable of accommodating nontrivial interacting QFTs (according to Haag's thm, etc).But perhaps you mean something else?

• Urs Schreiber says:

Technically what I mean are the spaces of "intetwiners" between representations. In physics these are the possible spaces of interaction vertices.For instance the space of interaction vertices for two spinors merging to become a vector boson include the linear maps which in components are given by the Gamma-matrices, as familiar from QCD. But there is an arbitrary prefactor in front of the Gamma-matrix, the "coupling constant", and hence the space of interaction vertices is in fact a vector space.

45. strangerep says:

[quote=Urs Schreiber]This is a powerful formulation of spacetime geometry that regards spacetime symmetry groups as more fundamental than spacetime itself. [/quote] Heh, well in ancient history (when s.p.r. was a great place and you were a moderator) you once rejected a post of mine in which I suggested that spacetime itself is not fundamental. I'm glad to see you've (apparently?) changed your mind. :biggrin:A small quibble about 1 point in your article:[quote]if one computes, […] the irreducible unitary representations of the Poincaré group, then one finds that these are labeled by exactly the quantum numbers of elementary particles seen in experiment, mass and spin, and helicity for massless particles.[/quote] IIUC, the word "exactly" is not correct for massless particles — one must manually impose a constraint that the representation should be trivial wrt the continuous spin degrees of freedom (i.e., the 2 translation-like generators in the E(2) little group for massless representations). Weinberg vol 1 covers this.Separately, I have a question about Klein/Cartan geometry. Are there already any extensions of that framework for the case where G/H is a semigroup? I'm thinking here about how one might embed temporal casuality into physical theories in a more fundamental way, rather than being imposed by hand as is currently the case in GR and QFT.Cheers.

• Urs Schreiber says:

Regarding the ancient history: I don't remember the contribution you are referring to, maybe you could remind me.Regarding the quibble: True, I have swept some technical fine print under the rug, in order to keep the discussion informal. Also, either way this fine print does not affect the point of the article.Regarding Cartan geometry for semigroups: I haven't seen this discussed anywhere. It seems plausible that one could generalize the definition to that case in a fairly straightforward way, but I haven't seen it considered.

46. Alex Klotz says:

Is Deligne's theorem here the same one, or related to, as the one that is used to help solve the twin prime conjecture?

• Urs Schreiber says:

What was used in the discussion of the twin prime conjecture is Deligne's theorem on extending the Weil bound on Kloosterman sums. This is unrelated to the theorem on Tannakian reconstruction of tensor categories that the above entry is referring to.Pierre Deligne proved many important theorems.