Raman Sundrum Lecture 1 on SUSY and BSM Physics in 2026

Ajouté : 2026-06-05

[00:00:00]Even if you don't notice it the day you get home or something, it it'll it'll show up. Okay, thank you. Um, okay. So, I'm going to talk about uh where am I holding this? Uh, BSM and be beyond the standard model physics and uh and in particular super symmetry. Um, these are substantially similar technically to the lectures that I gave four years ago uh here uh in 2022. So you can I' I've uploaded it into the Google Drive or whatever, but you can look up my Tassy lectures from 2022 and you will see that there are equations and things written down there and hopefully the order one factors there are at least correct and it has the useful references to many of the things that I talk about where you want to learn a little bit more. Um and and so that frees me up a little bit uh to say things slightly differently than I did back then. Um but so try and listen take your own notes here and and and also refer back to those. Um there's sort of two parts to the or two tastes to my lectures. one is sort of going to be there's going to be a lot of narrative um uh storytelling if you like where in some sense I'm trying to convey the beauty of the subject um and give you context and inspiration for uh the the sort of physics beyond the standard model which is a mystery of course we don't know what it is um but uh so there's that side of it and then also some part of it is going to be the technical tools right like I'll say something about super symmetry and and so on and um and that part I'm hoping to give you the tools that I think are modular meaning I'm not telling you a story that's set in stone and then you just absorb it and that's the end but I want to give you tools that I have found modular meaning you can modify them and use them in your own research when it comes up and uh so for example in recent years the same tools that I'm teaching you I have used probably mult multiple times in the last five years. Um, for example, even sticking to super symmetry, uh, I've written papers in the last few years where we were thinking about the seemingly small deviations but noticeable deviations in the G minus2, the magnetic moment of the muon experiments from the standard model and trying to say, well, what would be a way in which new physics could solve that anomaly or explain that anomaly while not doing damage all over the all over the place for other things that we've measured extremely well and and it's this kind of collective model building that's very difficult and requires having some perspective and that's what I'm trying to give you or similarly it came up when I was thinking about dark matter modeling so I'm hoping that what I'm talking about here is not some sort of the physics of or theoretical physics of 1990 um and and and it's set in stone. I I I want it to be usable in that sense. The other thing that I'm trying to say under the radar of your consciousness is simply that quantum field theory matters and uh and that my guess is that deep quantum field theory underlies the deep mysteries and I hope that that's part of what comes out. Um so I will be talking about classic paradigms like super symmetry and I'll mention you know string theory and so on. In some way I'm telling you about paradigms that are classic and they're also giving you kind of a world view.

[00:03:35]But again it's not my intention to give you a world view with that you have to stick to. You can having absorbed this way of looking at the world. You can go in a completely orthogonal direction.

[00:03:45]Okay? It's it's just meant to be a springboard. So you have some place to stand and if you want to carry out a revolution against everything I'm saying, great. At least you know at least you know what is sort of the classic background to this field. Um so uh let's see um the other part that I hope comes out is uh as a kind of naturally born BSM person I rebelled especially back in those tassy days I rebelled at learning about the standard model. Um it was like okay it's a standard model the name itself tells you it's boring and and so that's it.

[00:04:21]However, I found that in the course of learning about going beyond the standard model, it forced me to reappreciate what is so glorious about the standard model, right? Where that name just is the wrong name to call it. But um and and so it'll come up. You'll see that I'm going to have to review little things about the standard model. And sometimes if that's unfamiliar to you then or or you've forgotten it, then you know relearn it um so that you can go down the BSM path as well. Um, understanding the standard model will both be something that is beautiful in its own right. It'll also constrain anything any attempts to go beyond the standard model and it'll explain a lot about the way data works because so far data and the standard model seem to be going hand in hand and uh and it'll also tell us about the shortcomings of the standard model which is explaining why we would even think to go beyond the standard model. Okay. Um so then the last thing I want to say is just sort of a quantitative thing as far as this quick introduction is concerned.

[00:05:24]Um and that is sort of on the experimental frontier. You may have seen this drawn in terms of our ability to do experiments. Um there's obviously the energy direction and you may have seen this where you sort of plot there are two frontiers going to higher and higher energies and going to weaker and weaker couplings. And of course there's some sort of a frontier for how you do that. Depending on experiments you can go and looking for very weakly coupled particles. Maybe they are particles related to dark matter that don't couple to us very strongly and so you have to go in this direction. Or maybe you go to some, you know, super symmetry, going to higher energies at colliders and looking for it. And there's an interesting terrain where you might have special sensitivity in little islands that seem above ordinary reach uh or extremely weak coupling. Something interesting happens that actually makes a signal more distinctive. Um the a classic example of this, I'll mention it briefly and you probably hear lectures from Austin Joyce on it. uh cosmological collider physics where cosmology suddenly kicks in as a tool or long live particles where colliders these these particles that are very weakly coupled are often longived and that makes them more distinctive. So there's a terrain like this in in in the phenomenology and mostly I am sort of living in this my lectures are going to be here hopefully there are orthogonal lectures or from past tassies um where they focus on all of this all things dark matter um or or dark forces or hidden sectors but but I'm mostly here thinking about high energies um at collider energies or aspirational collider energies uh and and couplings that are relatively strong order one standard modelish. Okay. Okay. So that's uh a broad introduction. Um I wanted to sort of frame the the mystery or or the category of mysteries um that that we're tackling in these lectures um which I would broadly call hierarchy puzzles. So everybody's heard of the hierarchy problem as a kind of technical problem and in a sense I'll get to that quite late in the lectures um because it is often debated, misunderstood, illphrased, taken too seriously, taken not seriously and and we need to have some balance and context to really appreciate what that is. So I want to come at it from much more intuitive like everybody should feel it in their gut sense of hierarchy puzzles. Okay. And uh and so for that I want to um hope I'm yeah I think I'm writing the right space. Um I just want to point out that we of course have um particles sort of organized according to masses or energies if you like or if you want in terms of one over distance.

[00:08:28]Okay. So basically the energy scale again just like here and um and I'm the first thing to note is that in a sense the ultraviolet and infrared bookends of particle physics uh come from just looking at the uh lrangeian for general relativity. Okay.

[00:08:52]So, here it is. Uh, whatever.

[00:09:00]Again, I make an honest attempt at factors, but hopefully a better attempt in my printed lectures from 2022 and in the references that actually were done by professionals. Okay. Um so this is the lrangeian of general relativity. This so-called cosmological constant. So this is the Einstein term curvature. This is the so-called vac for a particle physics physicist we'd call it the vacuum energy. The energy of all the sort of vacuum expectation values of all the scalers and composite scalers and the theory. Um and uh and you might have thought that it should be of order some particle physics scale like the weak scale or something like that. But but we know that this thing gravitates.

[00:09:53]So we can actually see what it does. And and and and roughly it it tells us according to Einstein's equations how much the universe is curving. And in particular today where we're sort of vacuum dominated in our cosmology, the vacuum energy is pretty much the dominant energy in the universe right now. Um this Hubble scale, today's Hubble scale is kind of the size, you know, the sets the horizon of our universe and the time scales uh you know the age of our universe. And so somewhere down here is this Hubble scale and it's kind of the infrared book end of particle physics. I can't look outside that horizon by any known mechanism. Um and and we know that that this vacuum energy to fit this number that we see today is is mill electron volt to the 4th.

[00:10:47]So >> just to remind you, I know the cosmological constant is traditionally hard to observe, but things hard to Yes. Yes.

[00:10:59]>> And please do keep interrupting me, any of you, if you see me making one of these presentational errors, but don't point out my substantive errors.

[00:11:18]Okay.

[00:11:24]And and and so this thing if I if I did it right would be something what is it 10 to the 10 the minus 33 electron volts.

[00:11:34]Okay. So so that's sort of the lowest energy scale or or distance scale if you want in the universe. And and the other scale that you see here is well it's hidden in G Newton which has dimensions minus two mass dimensions minus two but of course its significance from a particle physics perspective or a quantum field theorist perspective is if this thing was sitting in the in some sort of a path integral right if I was doing some sort of a path integral then nominally that's where the h bar sits okay so if you G Newton is the actual face of H bar when you're talking about quantum gravity. And so this G Newton since it has dimensions, the dimensionless strength of H bar, so if I put it like this, the dimensionless strength of the H bar effects quantum effects is some sort of energy scale squared time don. That's the dimensionless strength. Um which tells you that when you get to energies where this is of order one then then so which which is yeah so when you get to energies of order one which is known as the plank scale right um and this is the 10 the 18 dB that quantum gravity has become a big thing the whole idea of the spaceime having a kind of classical fabric like we're experiencing right now is wrong okay the entire idea of spacetime has sort of effectively broken down. So, so this is sort of where quantum gravity lives. Um, erase my path integral at least.

[00:13:18]Okay, this is different different conventions. Put some four pies or two pies here and there, but but roughly it's that. So, this is where quantum gravity lives. If you go to even higher energies, you're basically starting to, you know, if you're starting to do collider physics at even higher energies than this, then you're in the land of creating black holes.

[00:13:41]Um, okay. Um, down here, effectively we have things like the photon or the graviton. They could be massless, but effectively we can't test their mass to anything smaller than the infrared cutoff here.

[00:13:57]Um and of course everything that particle physics can do is is in between these two ends and and so quantum gravity having a so the the dimension of G Newton as I said the scaling the scale dimensions minus2 that's the coupling so this is a non-renormalizable theory uh if I take it as a quantum field theory it's a non-normalizable theory um and and so it requires if you go up to those energies it requires requires replacing this effective field theory of quantum gravity based on this um by some fully complete theory. I'll say more about but but but but somewhere up there we expect therefore something like something like string theory again string theory is the theory best known that can act as a theory of quantum gravity um but could be something else okay so but let me just stick to strings just so I can draw a picture and there there's evidence that I will go through quickly with you to suggest there are other interesting physics way up there like grand unified theories um unifying the non-gravitational forces um but we live sort of halfway in between here roughly at order the weak scale or the TV scale or the 100 GV scale right so so that's where we are and an amazing feature of the standard model is that the standard model particles are kind of spread out many orders of magnitude in masses below the weak scale.

[00:15:41]Theoretically, we think of the weak scale as sort of the defining scale of the standard model.

[00:15:48]But indeed, the particles and before you diagonalize mass matrices, the mass matrix elements are extremely hierarchical in the standard model. Um, and so this is my cartoon to say that's that's an interesting fact. Um and uh and then maybe one more thing I'll try to squeeze in on this side is simply the mystery of dark matter where doesn't go that far down but dark matter could could be some composite object heavier than the plank scale could be black holes it could be things that are very light could be fuzzy dark matter um we don't know it's it's it's one of the grand mysteries which I will have relatively little to say just a little bit in the context of um some of the BSM models. Um okay did I cover that?

[00:16:43]>> Yes.

[00:16:44]So the main thing is fundamental physics question.

[00:16:51]>> Sure. Sure.

[00:17:07]Aren't there crazy lucky?

[00:17:16]>> Um, >> could you question?

[00:17:17]>> Yeah, sorry. So the question is can one do interesting things where you can measure at least even if you can't measure absolute masses of these super light uh bzons could we perhaps measure mass differences okay um so when you're down at sort of so possibly if if we are talking about masses which are slightly somewhat bigger than the Hubble scale the absolute Hubble scale of today um because we can look at they're interesting lever arms like looking at how fast these things travel from very far away. So even if there's a tiny difference in their speed because one is slightly got a different mass, their arrival times on Earth might be big enough yet. But so and and so nutrinos, you know, I I didn't put nutrinos down here. We know that, you know, they're somewhere with actual masses we can measure more like these mill electron type scales. So So there are lots of cool experiments one can do at that level. when you're really down at super light masses like Hubble, right? Then I don't know any practical experiment that can do it. I don't even know how to give meaning to mass if it is softer than the softest scale physically possible according to this theory. Okay. Um great. So, so one, so one puzzle, what I'm calling the hierarchy puzzle, is simply if all of this is fundamental physics, it sure is distributed in an incredibly diverse range of scale. And and I don't want to turn it into a technical question yet, but I want to say what would we like? What would be satisfaction walking out of the course right by the end of the week? and and and so so all these large hierarchies we you know we'd like to understand well what would understanding mean I'm going to pose what I think is the way it works in in in the field which is that you're looking for explanations where there is a mechanism where these things arise ex as exponentially large hierarchies, meaning they're exponentials in terms of some moderate um input parameters, right? So really some fundamental theory just has sort of order one numbers here and there. Why?

[00:19:42]We don't know. We don't try to push that far in our understanding. There's a sort of order one or order a few. But there's a mechanism by which they get stretched exponentiated and all of a sudden you're seeing huge hierarchies right and I want to say if we could understand those mechanisms we would uh say okay that's it that's that's sort of one grand goal of particle physics of course there are others like what is dark matter right that's another question but I want to pose this as an organizing principle in our thinking okay um okay so I did want to sort of compare um sort of the status of string theory which is not my focus in these lectures but I just want to at least give you again some some mental picture if you haven't studied it again there are references that allow you to read a little bit about it yourself even in an introductory way in my that I refer to in my 2022 tassy lectures um but so the fact that the coupling constant of general relativity is negative and so it's a non-reormalizable theory. Well, I'm just reminding you of things that you may want to learn more of that non-renormalizable in the old days used to mean not a theory bad. Now we understand that there's such there's a subject called non-reormalizable effective field theory and it's completely usable and it's a tool worth learning and understanding.

[00:21:17]um and in particular this aspect of general relativity uh was I mean I don't think there was something in incredibly novel in the philosophy but by John Donahue uh again I think I'm going to have to leave you to look him up um treating general relativity as a non-normalizable effective field theory and how to still make sense of it even at the quantum level. A kind of analogy is um the fermy theory. So you remember the fermy theory of weak interactions before the development of the standard model used four firm interactions. I guess they would have been something like G Fermy Sidebar uh left-handed uh currents.

[00:22:09]Okay, these kind of poor firm interactions were Ferm's attempt to to model the weak interactions and they were non-reormalizable. They also had minus2 and this is also a kind of non-reormalizable effective field theory. Um and there are others theory of pions is another famous non-renormalizable effective field theory called the chyro lranjim and a lot of this sort of non-renormalizable effective field theory a gentle introduction to non-renormalizable effective field theory is the book by George I that I strongly recommend in my tassy lectures from last time um which are aimed at the standard model but they're aimed at your understanding the standard model with the tools that allow ow you to go beyond the standard model like effective field theory and all of that. Um and it's beautifully done in a fairly it's a very thin book. That's why I like it. Okay. Um uh good. So this sort of methodology is something I will refer to but I'm not going to you know spend a lecture going through what is effective field theory.

[00:23:16]There are tassy lectures on effect I think like Tim Cohen has tassy lectures on effective field theory of non in including non-normalizable effective field. Okay. So then if we sort of compare >> Yeah.

[00:23:37]>> So if we sort of compare um the standard model electroeak theory which is reormalizable we can sort of compare it. We can compare it in this analogy I'm making with string theory sort of Okay, that that this is a sort of a UVMPlete theory. It's not a non-reormalizable effective field theory. This at least perturbatively is not a non-reormalizable effective field theory. It's an it's a reormalizable effective field theory. So these are sort of the classic complete theories of the of their type. Particle physics, quantum gravity, they're both perturbative. So they're this is perturbative.

[00:24:25]It's the only perturbative quantum gravity that we have thought of. And I vaguely conjecture. It might be the only one in existence. Quantum gravity which is perturbative. But um and this is perturbative.

[00:24:41]Okay. At high energies. Um let me tell you what you mean when you say it's perturbative. I mean it is believed that many nonperturb.

[00:24:50]>> Yes. Yes. Exactly. So, so the standard model has that that many phenomena can be captured perturbatively. It's not it's not like nothing is perturbative, right? It like like many things can be done perturbatively and here many things can be done perturbatively on top of which there are um beautiful non-perturbative phenomena here and beautiful non-perturbative phenomena here some of which we'll talk about. Um but it can at least in other words as you're just sort of trying to crawl towards doing quantum gravity and you say well can I use fineman diagrams well as soon as you do that you're led in a certain path and string theory has those fineman diagrams whereas there are other attempts to deal with quantum gravity where I can't even start with a fineman diagram right it's I've got to do something radical and and and those have typically not been advanced as far est the beauty of this is perturbation theory I you when I was young I used to tell everybody I hate perturbation theory because it's for total losers anybody can do it now now I wanted to only do non-perturbbit but but but that is a mistake perturbative detective work is the pathway that humans can take that can finally lead them past their own confines of perturbativity to new phenomena okay that are truly non-perturbative. So even if you love non-perturbative phenomena, you must master perturbation theory in all its in all its guises. Um okay. So uh so of course at the effective field theory level there are things like the fermy theory here.

[00:26:35]>> Yes.

[00:26:36]>> Aren't all physical theories like Yeah, indeed.

[00:26:50]>> Indeed. So for the purposes here, so repeat your question. Isn't everything got some sort of perturbation analysis?

[00:26:56]Um indeed anytime you solve a problem non-perturbatively by some brilliant stroke, you can then perturb around it.

[00:27:04]So for this purpose, let me just say Fineman diagrams. Yeah, that's good enough. Yeah. Um, and in fact, I try to turn every question at first pass into a question about Fineman diagrams. It cuts short so many long conversations where you just say, "Just show me your diagram and shut up." Right? And so, so again, I'm a big believer in the power of perturbation theory to at least conceive of what the hell you're talking about, right? Um, okay. So, this is a Fermy theory. There it is. There's a final diagram here. Uh, there is GR, right?

[00:27:37]and and and and yet there are things that that the full theory has that the effective field theory doesn't have. For example, here there are the new particles like W's and Z's and you know everybody else. Um and here there are string exitations right that in string theory instead of having particle interactions that look like this in finement diagrams you have string whirl sheets in spaceime that look more like this. And these strings can can be excited or they can be very excited.

[00:28:06]>> Okay. So these are the extra kind of states that exist in the full theory but not in in GR extra particles that exist in the full theory but not in the firmmy effective field theory. Um and just like here G Fermy defines what we call the weak scale. If I do one over square root GF I get the weak scale. But that doesn't mean that there's a particle living at the weak scale.

[00:28:33]It's just a coupling. I've turned it into something with mass dimensions by doing one over square root of it. But that doesn't mean that there's a particle living there. Of course, we know the standard model. So we know that this is really G ^2 over MW^2 up to factors of two.

[00:28:49]Okay, the electroe coupling that that's what it really is. So the particle actually lives here. So if G is small, then this is going to live below the weak scale, which it is. Like I think if you do this, you get 246 GV. If you do this you get like 80 whatever it is GB right depends on this coupling and similarly here G Newton which defines the plank scale right that I wrote the plank scale doesn't have to be where particles associated with this theory live they are again given by some sort of gring squared over so this is a bit of a cartoon but mr squared and so these strings if the coupling is weak so that you have a fineman diagram analysis, then then the strings would live somewhat below the plank scale. As I've shown in my picture here, the strings may not live right up here. They may live here. If the string coupling is incredibly weak, maybe they live down here, right? But but I want to offer so so I want to say just because we say there's a plank scale and I've drawn this picture I've already introduce which has been questioned in the past and you should question it again which is maybe there's nothing up here maybe when I point up here there's nothing there because string theory or whatever is really the physical scale associated with quantum gravity may maybe it's far lower okay but that's where I want to mention granification that that I want to later point to some circumst circumstantial evidence that suggests that these incredibly high scales getting close to this are actually scales of physical particles. Okay, I want to at least pre present that evidence, but that's why I bother to make this trivial point. Um, >> question.

[00:30:36]>> Yes.

[00:30:37]>> So, if you're kind of defining an upper limit, is there a lower limit?

[00:30:42]>> Yes, in in terms of mass scales. That was my Hubble. Sorry. The so the question was is there a lower limit on mass scales and operationally stuck in this universe which again I'm not convinced we are but anyway we're stuck in this universe. Um it's the Hubble scale the size of our horizon if you like or the age of the universe that's sort of the softest we can get.

[00:31:01]>> What about the strings though?

[00:31:02]>> Um the strings are also uh in in this kind of setting would also be confined into that box. So you can imagine a string with an extremely low mass scale maybe below the Hubble scale but it wouldn't describe it wouldn't fit with any of the things we're trying to do here right because some of these strings the least excited strings should be identified with the standard model particles and so on which we know have significant masses.

[00:31:36]Okay. So then maybe the last thing I wanted to say about this part of the table is simply um there's anomaly cancellation cancelling.

[00:31:53]So for example uh gauge anomalies again subject you haven't learned it learn the elements of it. Um after the discovery of the first particle of the third generation, the tow mezons, um one of the motivations, not the only one, but one of the strong motivations to think that there would be a top and bottom quark is that in the absence of the top and bottom quark, uh if you only had the tow, um you would have an anomalous standard model gauge symmetry.

[00:32:23]Okay, it would suffer from and which is a form of not breaking gauge in. So the theoretical consistency of the theory can imply particles that you haven't seen like the top and the bottom. Okay.

[00:32:36]Um so this is a subtle and interesting aspect of of of the standard model quantum field theory uh and everything that you'll ever do beyond that. And oddly in string theory, there's something very similar that um that happens that as you attempt to do string theory for the real world, you discover that anomaly cancellation comes to bite you and you find that you need new degrees of freedom. But here they are.

[00:33:01]So I'm going to use this thing that my former student told me to use so I don't have to keep writing extra dimensions again and again. This is it. I'm telling you once uh see everybody can see it.

[00:33:14]Take notes. extra dimensions are in some sense the simplest way of accounting for anomaly cancellation in the gauge symmetries that control string theory.

[00:33:24]Okay, so they're predicted in that sense and and in a in a subtler version of that uh so is Suzie.

[00:33:33]Okay, so there are some symmetries which are perturbative gauge anomalies. There are some which are non-perturbative gauge anomalies. All of these have analoges in sort of this kind of field theory and and they have here but they anomaly cancellation the simplest way to cancel these anomalies that afflict this theory are by introducing some of the ingredients that we're going to be talking about in the course. Um and then maybe the only other contrast I want to make here is this has been fabulously well tested by experiment.

[00:34:02]This we have not seen any direct evidence for whatsoever. we just see something that is compelling, beautiful and very much in keeping with the kinds of physics that we see here. But the best you can get here is it's quasi quasi realistic. Okay, we have quasi realistic string theory constructions um with gauge fields and chyal firm on matter and Higsfield like things. So, so one could easily imagine this being the real world, but but we haven't got direct evidence of that.

[00:34:37]Okay. So, this thing reduces below the string scale to quantum field theories of this type, if not exactly this, maybe exactly this, but of this type. Okay. Um, the last thing I wanted to say about strings for the moment, um, ex except to say that there's a certain plausibility to these exotic ingredients by the fact that they are sort of being they they came out unexpectedly in a way by just following your nose with with string theory. Okay. Um, good. So, so the last thing I wanted to say is that >> can I suggest a pretty So, I'm using >> Yes. Yes.

[00:35:24]>> No, I >> um wait now.

[00:35:30]>> Now I have screwed up >> all the way. Okay. Sorry. And there's a question somewhere. Yeah.

[00:35:48]>> Yeah. So, so one of the question.

[00:35:51]>> Yeah, sorry. Yeah, to repeat the question, you could there is bzonic string theory which doesn't have super symmetry in it. There are some anomalies that need to be cancelled uh and and they can be cancelled with having you know famously 26 spacetime dimensions, right? Um super symmetry is more related to um first dealing with what's called modular invariance in string theory and to solving one of the problems that's related here which is that it this this theory is technically speaking unstable.

[00:36:24]It has a tachionic scalar famously. Um so writing a theory that removes that tachionic scalar uh requires a certain so there's a property of super symmetry that we'll touch on very soon um which is important in terms of cancelling the bad behavior of the purely bzonic theory that in turn sort of gets you into a new game of anomalies that super symmetry solves.

[00:36:53]Um, good. So, the last thing I want to say about this is that the sort of interplay between these ideas, this deep thing that I'm not going into um, for the most part is that gauge field theory.

[00:37:10]So, we're all used to in perturbation theory gauge fixing, gauge field theory and uh, sorry uh, so let me take this question first.

[00:37:27]Yes. So, so indeed for energies below, so the question was, is there a way to go from here to here?

[00:37:35]Okay. And I'm saying that for energies below or much below M string the characteristic defining scale there um string theories of this type will have quantum field theory or quantum field theory or effective field theory um uh well well they'll be approximated by that right so the stringiness will not be obvious and it'll just look like point particle physics.

[00:38:06]Now, different string constructions will look like different quantum field theories.

[00:38:12]Maybe one of them is the standard model, right? Um there was a question somewhere. Yeah.

[00:38:23]>> Yeah. So, there's a very beautiful thing. Suppose I throw away hypercharge in the standard model. Um I still have SU3 and SU2. And if you count the number of SU2 doulets in the affirmionic doulets in the standard model, you see there in every generation three uh colored or cork uh dlets and one leptton d four. So that means there's a total of 12, right? Suppose I had just introduced the tow the the left-handed towel leptton and its neutrino which once upon a time that was all that was discovered of the third generation. That would be nine, right? because because they wouldn't have seen the top and bottom giving the other three. So that's an odd number of SU2 chyro firm doulets that is anomalous but it cannot be seen in a perturbative fineman diagram and it took the genius of Ed Whitten to to discover it and so they're often called Whitten anomalies. Um so there's an aspect of that that's also comes there's some there's some an analog of that those are called uh not gauge invariant under large gauge transformations as opposed to perturbative yeah it's the SU2 yeah and there's an analog of that in sort of the story of Susie here um okay so gauge field theory I just want to say we use usually perturbatively gauge fix and think of it as a bunch of point particles like luons and so on.

[00:39:51]But actually the fundamental variables of gauge field theory the the gauge invariant the gauge invariant.

[00:39:58]So the invariant degrees of freedom are really Wilson loops and Wilson lines like uh um trace of e to the i over some contour in space time. It's this kind of thing, okay, that a lattice gauge theorist like one of our organizers Zora would would know that's the way lattice gauge theorists think about what they're calculating that these are the degrees of freedom that they would think about and these things depend on contour or sort of contours in spaceime and if you think in terms of canonical theory um these contours in space are really a kind of loop are kind of or maybe I should just call it what it is they are kind of string.

[00:40:53]They are a kind of string. They're a kind of relativistic string. So, gauge theories are really also theories of a kind of relativistic string, right? Um, but they sometimes don't behave that way when you study them quantum mechanically, but fundamentally that's what they are. And this is clearest.

[00:41:12]This this is evident. uh the the the nicest way of seeing that is is in the famous ADSCFT dualities and ads CFT duality um uh between gauge theories and string theories. So sometimes sometimes these kind of differences are are not it gets it gets confusing because there's aspects of this which are very stringy. Okay. Um, if you want to read again there references telling you where you can read reviews of that kind of stuff.

[00:41:50]Okay. Um, let me just actually do something.

[00:41:55]>> Sorry.

[00:42:05]>> Yes. Yeah. No. No. So, so in the canonical theory, so here I'm saying if you ask a lattice gauge theorist who thinks in spaceime, they're often having a contour in spaceime. So then your question is reasonable, which just to repeat is that this looks like a string. This looks like a loop in spaceime whereas there I'm drawing sheets in spaceime.

[00:42:28]Actually, if I break it down to like canonical quantum mechanics where I say, well, what's going on in space? What are the degrees of freedom in space? The canonical structure of gauge theory is that these are contours in space.

[00:42:43]They're they're loops in space. And indeed, if you think about the dynamics, these loops in space can trace out uh sheets in spaceime.

[00:42:55]So, so no, no, it's it's it's quite close in parallel. I agree that in the quick way that lattice gauge theorists talk about these objects that aspect the canonical structure what what are they doing when they calculate something is not so obvious. Um but we can talk about it later. It's a very beautiful seeing the stringiness of gauge theory come out is one of the beautiful miracles. Um okay so there uh let's let's push this up. Um so of all of these I've spent a long time sort of just pointing to some rather exotic ingredients extra dimensions uh super symmetry certainly strong quantum effects are going to be part of quantum effects can make magic. They can make things that look like on paper they should be stringy degrees of freedom behave like particles when you put in quantum mechanics or things that look very particle-like on paper behave very stringy and and and all sorts of other things, right? Uh they can create extra dimensions out of just interactions, strong interactions. Um so all of these things are very exotic ingredients that I want to say are nevertheless plausible. They are both magical in terms of the power for understanding the hierarchical structure of our universe, but they are also um uh not done by just sort of some minor tinkering. There's there's full-on blown magic and they are also um uh plausible if you buy my story. Right?

[00:44:41]So, I've said it very quickly. you have to sort of see whether you think it's plausible or like like okay that's all very exotic I don't think it's part of our world our world is just too boring to ever have any of these ingredients right that that you have to decide um by the way clearly I'm around for a week and I would love to have all these convers I would love to debate you over lunch whatever whatever you want right figure out how to have these conversations okay so let me just start with extra dimensions because I would just want to give a snapshot of each of these um well extra dimensions and sus in particular. Um, so we we can sort of just the quickest thing I want to say in terms of showing the sort of power of extra dimensions is is to start by just motivating it as sort of parallel universes.

[00:45:27]Right? So let me draw two parallel universes here. each of which so these are these are all 3 + 1 dimensional universes and there are some particles that live on this universe right so universes in this I'm drawing planes like this and they are offset by being at two ends of an extradimensional interval okay um and so there's some particle that propagates along here on this slice and there's some particles that propagate on a different species of particles that propagates on this slice.

[00:46:11]And if they're in parallel universes, then they would not talk at all. Right?

[00:46:15]But as long as there's spaceime, then at least there should be general relativity because general relativity is the dynamics of spaceime. And and and to keep it simple, let's just say that there's some other um I'll just draw it like this. there's some other instead of writing GR all over the place, let me just put in some scalar field, there's some scalar field that is my toy model of whatever this gravitational field is and it connects these two parallel universes, right? Maybe this is dark matter on this side and matter on this side and they only talk gravitationally, right? It could be this. This could be the answer. Um, so how does how does something how do we like what's so magical about this, right? Um and this is meant to be microscopic. So microscopic we haven't got the energies to resolve. We don't have the wavelengths to resolve it. Right? So then it looks we don't notice that there's an extra dimension. But technically what is the punch of extra dimension? So we start with this field which is a function of the usual 3+ one dimensional spaceime as well as this extra coordinate x5.

[00:47:23]Okay let's just take one extra dimension.

[00:47:26]And uh let me give this field noman boundary conditions just so I choose some boundary conditions on those two ends.

[00:47:34]Right? And so if it's noman boundary condition then to make sense of it in the standard language of four-dimensional people or 3 plus one dimensional people we can just write a a frier series.

[00:47:52]So we can look at all the things that satisfy the noman all the harmonics that satisfy the noman boundary condition with some coefficients which are then functions of x. Okay. So I've just fer transformed effectively I fer transformed away the fifth dimension from the quantum field and it's now just sitting in these trig functions. Um great. So, so instead of talking about this, I just have an infinite number of four-dimensional fields. And what's there to notice about these fields?

[00:48:24]Well, let's look at sort of the Klein Gordon equation that description of uh a free field phi. Okay, approximately free field phi. Start. We always start perturbation theory with free fields. Okay, so the Klein Gordon equations has the dal bersian that you know and love, but it also has this fifth component, right? So this is the five-dimensional version of a dal bersian and acting on phi um plus possibly plus possibly some mass term.

[00:48:57]I put m5 to say it's a fivedimensional mass term, meaning that's the mass parameter that appears in the five-dimensional Klein Gordon equation.

[00:49:05]And this should equal zero.

[00:49:08]Now I'm just going to simplify life by just taking the massless case just to simplify the math. And so this thing is just saying that this term is for each of these component fields. Each of these is like n pi over l.

[00:49:29]So this would look like the four-dimensional wave equation.

[00:49:34]This extra term is just adding for every n for every component field fn every one of these effective 4D fields it's just acting like a mass term from the four-dimensional viewpoint. Okay. So the usual picture of this already an interesting kind of particle physics is to write what are called the kulutline masses which means the effective the kutline masses after the pioneers of extradimensional theory in the relativistic era. Uh the cluticline masses are the effective four-dimensional masses that an experimentalist would call mass namely four-dimensional masses. Um if they were unaware like as humans they're unaware unable to see this distance but they have high energy colliders then they would see a spectrum of four-dimensional particles which would sort of look like n= 0 n= 1 n= 2 um where these splittings would be whatever um pi over l right so those effective mass passes would show up as this kind of splitting. If their collider has enough reach, it would start to see these recurrences, right?

[00:51:01]So-called kitline exitations. Um, and if if the energies in this regime, suppose you didn't have enough energy to get there, you couldn't even create the first kutle exitation. You would just think that this was the only particle.

[00:51:18]it would just be one four-dimensional particle and it would be described by a purely four-dimensional effective field theory.

[00:51:26]Okay, so that's the approximation. I can only I can do this where you take off your glasses and it looks like that place and that place are the same place, right? Um you don't you're not able to resolve the fact that they're separated. So there's a kind of four-dimensional effective field theory, but if you had higher energies, you'd start to see look something exciting is happening there, right?

[00:51:48]And and so what I want to consider just to give you one data point is the scattering of this guy and this guy on their parallel universes via my scalar analog of a gravitational field. Right?

[00:52:03]This this this thing here um and so this exercise was done by Arani I think I don't have this reference in Arani Hamemed Gman and Schmaltz in this paper.

[00:52:19]So they they sort of did the experiment at low energies and high energies. So if I think of time as going up vertically, then this is kind of a T- channelannel exchange, right? So they just looked at what is the amplitude for scattering.

[00:52:36]And so the amplitude for scattering is basically so the amplitude for that scattering that you can see there is basically going like that 5D propagator that is straddling that you can see the curly line that's straddling between the two parallel universes. and and and in fact if t is much smaller than uh m kk uh squared by which I mean again pi over l okay so so if the energies are down here right why do I say it's like a 4D effective field theory well if you actually do the calculation fully in the fivedimensional theory then you find that this thing goes like 1 / t plus something which is order one uh uh L^2 um or 1 over MK^ squ. So it looks very much like a four-dimensional scattering process where in when you have a T channel exchange of a massless particle you get one over the propagator which is 1 / T or you might have a contact interaction. Now actually there is no contact directly between here and here but the effect of all of the all of the exchanges of all of these guys is to effectively create a constant. So if I only saw this experimentally I would not think ah I've discovered extra dimensions. I wouldn't I would just think I had a conventional fourdimensional theory with a T channel massless particle being exchanged and some contact interaction. However, the surprise is when you go to t much bigger than mkk^ squ, right? Um, in that region it looks like this.

[00:54:22]Okay? Meaning suddenly you get something exponentially and you might say why would that be happening? You can see that there is some sort of cancelling going on because on this side or or at x5 equals 0 at x5 equals 0 this thing just looks like one regardless of n but at x5= l that's l the size of the extra dimension. Sorry I should have said that that's the size of the extra dimension. Um at x5= l this thing is flip-flopping between plus and minus one. So as you start having the energies where these exchanges at higher end matter, they've got some kind of cancellation and the net effect from the infinity of them is to give you this kind of exponential kill which is really the evidence that these parallel universes are parallel. They're not in the same place. Um this kind of form factor this is very reminiscent of form factors in say composite systems like nuclear physics. Okay. In the scattering you will often see these kind of form factors which at low energies you don't see any sign of things look like a pion looks a point particle but at high energy you start to see evidence that it is not a point particle something it's in fact dramatically different from a point particle and they're captured in in in these kind of form factors that we normally associate with strongly interacting or composite systems and it turns out that it's it's not a complete coincidence, right? It it it it comes back to the fact that higher dimensional theories as in the famous ADSCualities, higher dimensional theories can capture some of the subtlety that we normally only see in strongly interacting composite four-dimensional field theories. Now, here's the thing. This kind of behavior where the ultraviolet changes its behavior dramatically from what you infer in the infrared is key in understanding some of the uh or can be key in understanding some of the issues that come up with hierarchy puzzles and hierarchy problems.

[00:56:46]This kind of form factor clearly affects the ultraviolet dramatically.

[00:56:51]So it's the kind of ingredient we'd like to play with. Humans do not understand strongly interact very being heroic and well that means they're down here and the subject is here. Okay.

[00:57:06]So the the beauty of extra dimensions is they give you a perturbative back see that was a fineman diagramraph they give you a perturbative backd dooror to the kind of that would normally take you sort of a higher IQ to be able to capture in your heads. Right. Um, and so this is a great tool. Having expions as a kind of tool for you is is is is modular and and easier on the brain for many parts of the modeling. Um the one catch with extra dimensions is that extra dimensional quantum field theory is uh non-normalizable and and we can see that uh say by writing say a fivedimensional action for that scaler there. Um let's do something simple.

[00:58:01]D5. There's various derivative terms either higher derivatives or the usual mu derivatives and and then let's just have it be self-interacting in some way because this tells you sort of the the general feature.

[00:58:16]So you can easily do your dimensional analysis and you can see that in five dimensions in order for this to be an action which is dimensionless and fundamental the dimension of the scalar field has to be three halves. Okay. So that this is dimension five and this is dimension minus five.

[00:58:35]That's true. Then from here this is dimension six and therefore the dimension of this coupling ph to the 4th which in in standard field theory is dimen is dimensionless in four dimensions here it has dimension minus one. Okay. So as you go to higher dimensions couplings that you are familiar with that are dimensionless or reormalizable even uh become non-reormalizable. They have negative mass dimension. Okay. So this is a useful tool but it is restricted in in what we can uh do with it. Um okay.

[00:59:14]>> Yeah.

[00:59:38]Yeah. So to repeat the question, the question is here I did the simplest case of an extra dimension. It's an interval.

[00:59:43]If you have a higher number of dimensions, they may have interesting topologies and shapes. And therefore, when you look at the kulitzline spectrum, uh would it look different or could you choose those shapes so that you get a different spectrum? Maybe one that you prefer. And the answer is yes, they are correlated. It goes back to this ancient problem for mathematicians.

[01:00:03]Can you hear the shape of a drum? And so different drums will have different sort of notes, harmonics that they hit. and and yes so so there is something like that typically it'll be some discrete thing um but that discrete spectrum would be the thumbrint in a fer transformed language of the shape of the extra dimensions >> yeah great >> yeah exactly so the question is what if I had just done five cub and and actually fi cubes in the perturbative regime is the exa exception, but it's a it's not really an exception because normally if I have only fi cubed, I actually have an unstable potential. So of all the sensible theories you'd ever play with, you can have this and it would be okay.

[01:00:57]But it wouldn't exist in isolation.

[01:00:59]You'd have to have something like this as well. And then you're back to what I was saying. Yeah.

[01:01:08]>> Um Yeah. So it would ch yeah so these things would vary the kind of form factors that you would get depending on which boundary conditions you would choose um yeah so it would change even the boundary conditions as well as the shape of the extra dimensions will modify the clut spectrum for example if I had chosen durishlay norm on boundary conditions I would have ended up here in particular I wouldn't have this massless degree of freedom so the collider primitive collider would not see even one particle of this.

[01:01:41]Yeah.

[01:01:47]>> Yes. Yeah. Yeah. So, so of course it depends a lot on these questions about the shape and this the boundary conditions etc etc. But let's do the crudest thing which is a perfectly ballpark answer which is if this if this is of order one over L right that's that's that and I haven't seen these right then the answer would be 14TV colliders typically probing of order well let me do it in round numbers 10TV 10TV energies if I could have produced a 10TV TV particle, right? Um, more like 5TV, but doesn't matter. 10TV and and so L has to be smaller than an inverse of 10TV, whatever that is. Okay.

[01:02:37]Um, yeah.

[01:02:43]kind of getting structuring like not looking at it like that or something like from the physics I can actually go down without >> yeah so my interpretation of the question is suppose I don't have the energy to literally get these particles on shell is there stuff that I could see with the you know the poor person's budget of energy could I see some circumstantial evidence for this kind of amazing structure and and yes so with specific incarnations of theories of of that sort. There can be we'll talk about some of them in the lectures uh signs hints that maybe something like this is going on that there is sort of a extra dimension doing interesting things.

[01:03:52]We'll give examples of that type um for sure. Yeah. So, so it can be done but of course it's you will have to put some input in by pure thought you will not like it will depend on the exact scenario.

[01:04:08]So if you're lucky Yes.

[01:04:13]>> Yeah.

[01:04:14]>> Yeah.

[01:04:21]>> Yeah. Yeah. So um impressionistically this looks like some kind of QCD like form factors for elastic scattering. You know normally when you hit two hydrons together at high energies you get many hydrons but suppose you only look at the ones where you go two to two like two pie and scattered to two pines then that there can be things which are impressionistically like this. Um but this kind of extra dimension the particular shape etc etc would not be a perfect match for anything in in real world. Right? So if you want the real world analog of extra dimensions, if you want the ads CFD of QCD, right, is a very very messy theory involving an extra dimension. In fact, perhaps more um so this is at best impressionistic.

[01:05:18]Uh when you go beyond the standard model, this may be a better model of the kind of thing we're after than QCD, right? Um so we as I said we are poor in these deepest of deep mechanisms our understanding and we go where we can with the lights that we have but yeah >> yeah there are theories that are higher dimensional which can be reormalizable in a special sense. For example, they may be strongly coupled where the usual power counting and analysis of when something is well behaved or not well behaved can differ and some of those are known. I'm giving you just the diagram way of that. I think there was a question there.

[01:06:13]>> So we're getting Yeah. So if if for example, let's even not do something as violent as intersect.

[01:06:33]Let's say it did that, right? Whereas this one went straight on. Okay. So your question is what if something like this happened? Um well remember this, let's say this is time, right? This would say that the laws of physics as observed by scattering experiments, right? which clearly the size of the extra dimension is observable at high energies would mean that as or or even at low energies.

[01:07:00]Right. Right. So if the size of the if the distance L is varying from the early universe to the late universe or from yesterday till today then you would see scattering amplitudes or scattering cross-sections which were time dependent. I know I did this experiment yesterday. I know I did it right. But now my grad student says you know I was a factor of two off and and right so that's that's what would happen. So to the extent that we have evidence that the universe in our directions is very uniform which we have very strong evidence of that right then then we would need this to be sort of fairly constant as a function of the x muse.

[01:07:47]>> Got it. Very good. Um good. So the other thing I do and I think that might fit in the 10 minutes is the the quick impressionistic not this is completely super symmetry. Okay. So I want to introduce super symmetry in a different way than I did in those lectures years ago. Um because I I somehow think it's more insightful. I mean in a lot of this stuff um question is what to leave you to learn if you get excited by it and what to tell you in this valuable time and so let me go at it by talking about super symmetry starting from quantum mechanics. Um so all of you know the DAC equation and DAK faced this problem in his kind of you know they had done first quantized quantum mechanics uh when he first started and they were trying to make sense of relativistic quantum mechanics because they hadn't quite invented second quantized quantum field theory and so his way of coming at it was to sort of say I want some sort of first order version of the Klein Gordon equation. So the Klein Gordon equation makes use of the damburian uh D mu^2 and he wanted a kind of first order version of it by sort of like asking for some sort of square root of the denominion and um you might say oh you know just get the igen values of the denominion and take the square root of all of those and that implicitly defines some square root but that is not a local he wants some local differential equation so he wants a local version of this which is almost like a zen cone in the sense that like this can't be done. Okay, but he breaks out of it and invents the durac operator.

[01:09:35]The trick is he's introduced spin. Okay, these are matrices. They are given by the gamma matrices in spinner spinner space times this. Okay, so the square of this turns into the identity matrix times the denominion. This you're familiar with. It's famous. And you can wonder whether that kind of magic which ultimately led to second quantized field theory. You can you can wonder whether there's some kind of square root like thing that you can do in seemingly impossible situations. Um already at the second quantized level. So for example in quantum field theory like in all quantum theories there's a Hamiltonian in principle and it's the thing that does time evolution and you can ask whether at the full level of quantum field theory can I take a square root can I take the square root right um crazy but let's just just let's posit that it's true so that means that it is equal to some local quantum field theory operator squared. Okay, local quantum field theory. So this is local in the sense of quantum field theory squared.

[01:10:48]Um and if we're going to get there, the easiest way to do it is to start from the toy models of quantum field theory.

[01:10:58]And as you all know, you learn quantum field theory by analogy with the quantum harmonic oscillator. Right? If you take all the modes or the momentum modes of a quantum field, you get each quantum each momentum mode like a harmonicator.

[01:11:13]Let's lower the question and ask what about the quantum harmonic oscillator?

[01:11:18]Can I take its square root? Right? And um so the quantum harmonic o oscillator of course this thing is p ^2 over 2mm plus and I presume all of you know all of the motivation for going from harmonic oscillator to quantum fields. Um but let's just take that Hamiltonian and I want to take the square root of this thing. Right? So the brilliant answer because you can write it down completely explicitly is is this zero. So again you introduce a kind of spin by writing a matrix and and so you guys completely know you know the quantum harmonic oscillator you know the destruction and creation operators and omega right? So so this is this is it. This is the matrix and uh and you can square it yourself or you can square it in your head as I'm doing and and you find it's so it seems to live in this extra expanded spinner or spin-like or two degrees up and down whatever you want to call it matrix space. But then in each sector in each sector it's just a harmonic oscillator with some particular zero point energy or in this case the harmonic oscillator with no zero point energy. So since we normally don't worry about zero point energies you'd say yeah you've managed you've you've taken the square root in this sense right again in all of these there's sort of an impossible question and with some tiny fine print you can do it right. So, so okay. So, so this is it and and this is really the driving equation for full-on super symmetry. But it needs a little um change of notation to sort of see how we're going to get to how is this dumb trick going to get us to quantum field theory and super symmetry. Um okay so we can write we can write Q the trouble is quantum field theory has many oscillators one for every momentum node here I have one oscillator and I've already invoked this 2x2 matrix and it's basically if you try to go this way you will get very very large matrices to capture the full subtlety of the problem. So we need a little slightly fancier notation and and that's very simple. We can we can actually just define we can define matrices B dagger which is just 0 1 0 0 and B which is the con the hermission conjugate of that and we can define the the basis in spinner space you know down or but we can we can call this in quotes we can we can think of it is a state of affirmion. So I'm putting f to say firm.

[01:14:36]So forget about bzonic harmonic oscillator. This is just saying empty. I could have just said spin down. I'm calling it empty. And and and this I can just call it full. Okay. These are the only two states in a two in a two component theory. Um, and you'll notice that the B's satisfy the analog of the commutation relations of the A's. So two A's commute. And here you'll see that two B's anti-commute.

[01:15:13]Okay. So if I just square B, right?

[01:15:15]Anti-commutator of two B's is just 2 B squared. Square of that, that's zero.

[01:15:20]And similarly similarly if I do B dagger with B dagger it's trivially zero squaring that but if I do the anti-commutator of B dagger with a B that's like the analog of a commutator of an A dagger with an A and and and this is just one do that um so we can think of these matrices as what are called the firmy firmionic oscillator.

[01:15:52]Okay, where it has a a ground state, it has an excited state, but because of the fermy exclusion principle, that's it.

[01:16:00]It's like you can put one particle in or no particles, but that's all you've got.

[01:16:04]Okay, so it's a very very simple Hilbert space and you have creation and destruction operators, but they're not very fancy. They're just matrices. Um, and this is sort of the analog of the computation relations. So the full basis so the full theory with the with the original oscillator the full theory or the full uh Hilbert space is spanned by so if you want the full basis for the Hilbert space is spanned by states that look like some occupation in the Bzonic sense of the AA dagger harmonic oscillator tensor some either no particles in the firmionic side meaning this state or some states like this with one particle in the fionic states. Okay. So these states are sort of net bzons.

[01:17:05]They have an integer number of bzons and no firmians. If I'm calling this firm in the sense that it's an empty fionic state and here it's a net firmian.

[01:17:20]So how do we rewrite this Q and Q ^2 right? So this Q in this language you can check is just root omega and it's a dagger b plus dagger a by the way the a's and the b's just commute with each other that you can check yourself and and and the q ^2 is just omega a dagger a plus omega b dagger b okay so it looks much more symmetric there's a bzonic oscillator and a firmionic os oscillator that's the Hamiltonian and this thing is called a supercharge or if you want well it's a supercharge and it's conserved because you can easily check that Q commutes with Q ^2 because Q commutes with Q so so so in particular it's a symmetry because it commutes with the Hamiltonian right That's so it's conserved in that sense and it's related to a symmetry. Um and uh and and hence it defines well hence it's sus.

[01:18:33]So it's that that's what super symmetry is. Um maybe I think I'm running out of time. So let me just say this last thing and then I'll stop just to name things.

[01:18:42]You can see that what Q does is it creates or destroys a firmian and creates and and accompanies it by creating a destroying a a Bzon. So um here Q exchanges the action of Q on a on a net Bzon state is to turn it into a net firm state and the action of Q on a net firm state is to turn it into a net BZON state. This is a famous feature of super symmetry that it exchanges bzzons and firmians. Right? Um in this language once we turn once we switch from matrices to the language of operators the idea of going to quantum field theory where creation and destruction operators are labeled by spatial momenta as we do in standard quantum field theory allows us to generalize these firmionic oscillators to also have um momentum indices. And so this way of thinking where you switch to the language of bees allows you to take this very simple structure and elevate it to full-on quantum field theory which we'll do next time. Um and and indeed this is the analog this is sort of the toy model of all the bzonic fields like the photon field that you study and this is the analog of all the fionic fields you study. um which maybe I'll end with just Edwin's joke which was but he wasn't joking um which was we have firmians clearly in nature and this is sort of evidence that super symmetry is right um okay maybe I I should he I don't know if he would agree that's what he said I think it's what he said it seemed it seemed like it when I saw him as a grad student um good I will I will stop there. Next time we'll talk about, well, what about interactions? Let's do that next time. And then we'll get into the full-on MSSM and all of that. Uh, am I out of time? I'm good. So, [applause]