I was trying, I've been trying for a while to figure out how to work nunchucks into a presentation and I think the thing about nunchucks is I told somebody like it's not that they're dangerous.    He's crazy. Thank you. Thank you for being here, for waking up on day three. Thank you for coming back to in person on behalf of myself and on Mars. It's been, you know, certainly a challenging three years. Thank you if you stuck with us through virtual. If you've been with us in the past, welcome if you're new. We really, really appreciate it. I keep, I keep trying to reflect on the past three years and I can't, you know, I can't summarize what we've all been through. But I was up here Sunday night. I don't know if you know the speakers come in for our walkthrough on Sunday and we all get up here and make sure everything works and the slides don't look weird and the videos run and all the PowerPoint things.    And it's good to be home and it's good to have you here and thank you for being with us. You know, as Rob said, I've worked on the Google Algo project. I followed the Algo. But I want to talk about a very specific story today about how I think Google is pushing searchers down the sales funnel with inserts and what that means for us. And hopefully what we can learn from that, I think we all get a little freaked out about it. I'm going to talk about some of that. It wouldn't be one of my talks if I didn't have those scary stuff first. But I think there's something we can learn about what Google is doing, why they're doing it and how we can be a better part of that funnel and move people towards action. But my job involves a lot of falling into rabbit holes and sometimes leaping into rabbit holes, sometimes dynamiting the bottom of rabbit holes. Following the paths between the rabbit holes, I took my 11-year-old out to do this ill-advised photo shoot.  She's 11 going on 105 and I'm going to murder her soon. No, I love her. But she does like two photos. She's a pretty good photographer, so I brought her out. She does like two photos. She shakes her head and it's like, I'm out. Like, this is embarrassing. So I go in to get my nine-year-old. He's not a very good photographer, but he's in for anything. He's like, I'm here. Let's go. And what I don't realize is I'm out there for 30 minutes and he becomes like this director bully.  And by the time I'm done, he's like, man, I think we're done and he's got like 60 shots. So I'm not that pretty. So sadly, this is about the best we ended up with. But this is my life, like falling into those rabbit holes, trying to figure out where they lead. But I think as SEOs, a lot of times when we look at SERPs, we look at them in isolation. We look at, here's where I'm ranking, and you know, we're getting smarter and we get into our click-through and we get into things like that. But we don't look at this journey that can happen across searches. And so that's what I really want to talk about today. So this slide and the next couple of slides, if you saw Andy speak on Monday, are going to be very familiar.  Are we talking about exactly the same thing? He's like, no, no, no. It's okay. It's all good. We got this. I think actually what he said was, Dr. Pete, I'm in Spain and, you know, I'm having a good time. So I'm sure it's fine. But we got it covered. I love Andy. But as he said, you know, we've all seen this journey over time. This is a search for Tornado in, I believe, 2002 from the Wayback Machine. Thank you, Wayback Machine. And what do we see? Blue links, right? We know this. We see one indented result, which is interesting because those were with us years ago when the way it came back. Now the same search in 2022 and I'm in Chicago. I'm adjacent to Tornado Country. And what do we have? We have this huge top stories pack, right? We have this knowledge panel on the right. We scroll down. We have also in the news, which is the local extension of top stories. We have videos. So we have more knowledge panel. We have people also ask and then we have that organic result, right? And we know that we've seen this. I've talked about this for quite a while. We've seen this evolution. We've seen how things are being added to SERPs. But I want to scroll up and you may also have seen this in that knowledge panel. You see these sections like cities with most tornadoes. Hooray. We're in that list. And here's Chicago and you click on that. And it takes you to do a search for Chicago, right? And it takes you to do a search for Chicago with this context of cities with most tornadoes. So it has this awareness of your journey, but it's taking you to another SERP. Amanda was just up here. You may have read the study by Spark Toro in 2020. Two-thirds of Google search has ended without a click. So we started talking about these zero-click SERPs. I think this is a really important concept, but I think we see it in isolation and we go, oh, that's bad. That's not for us. And it is scary and it is evolving and it is something we need to be aware of. But I want to talk to you about what I think Google is trying to do with this and why it's important. And I'm going to focus on one feature and it's maybe something you overlooked because it's usually at the bottom of the SERP and it's this refinement feature. And so here's an example for loose leaf tea. I'm a tea drinker where you can go in and start to refine by type, by flavor, by brand, et cetera. And we're starting to see this for more and more things. I don't think this is something that's getting a ton of clicks right now, but this will tell us the story of where Google is headed. And I think this is really critical. So in a minute I'm going to talk about the messy middle. Jean-Lucas actually wrote a great paper recently about this topic. A paper Google wrote a couple of years ago. But first I want to talk about the classic funnel. Like we've probably all seen this. I simplified it here, the ADA funnel. I like to just simplify to awareness and dressed in action because that tends to map to informational searches, investigational and commercial. I did not know until creating the stack that this was created in 1898. The core ADA concept is almost 125 years old. I don't usually do this and I don't want to put anyone on the spot.  I'm guessing there's a couple of people. Thank you for being here.  It would be the same people. Yeah, thank you. I love you. I'm with you. I'm there. That's a long time. That's something that maybe needs to be rethought and reinvented somewhat. I think classically as SEOs originally, and we've all hopefully moved on a bit, we focused at that type of funnel. Those broad keywords, those head keywords, those vanity keywords, whatever you want to call them. And why? Because they got all the traffic, right? They're the high-volume keywords. Great, that's what I want. I want that traffic. And then over time we started to recognize that, well, on the other hand, those people don't know what they want and that doesn't convert very well. So we moved down the funnel. And great, high conversion. I know we've talked about that this week a bit. But on the flip side, there's not many people down there. And so our choices sometimes seem like a tiny percentage of a very big number or a big percentage of a very small number. And I want to be clear, you know, I reached cover this on Monday. That whole idea, and I think we'll run until we get into this, this is a big topic for Will usually, that whole idea of prioritization and knowing where people are taking action, where they're landing is critical, really, really important. But I think the other problem with this is because we simplify the funnel, we miss that this bottom and top, they're a very, very small segment of this whole process. And there's so much in between. And I think Google is seeing that too. And seeing that we're missing that journey along the way, and we're also kind of artificially what is top and bottom. When you draw a picture of a funnel, it's really concrete. In real life, it's messy. And that brings us to this. Google wrote a paper called Decoding Decisions, Making Sense of the Messy Middle. It's a good read. They did some experiments on how people do brand switching, how you build brand affinity, not a real long paper. But I'm going to focus on one aspect, and it's this model. This is Google's reimagining of the funnel. And I'm going to tell you the important pieces. And again, it's an abstraction, right? This isn't based on real data. This is based on how they think things are changing in search. So triggers can be anything. You know, I stub my toe and it hurts. I watch TV commercial for something. My friend has electronics that look cool. I'm hungry. Anything that sets you off on a decision process. And what Google is saying is essentially there's two sides of this process. One is exploration. And exploration is a process of widening my options. You know, I know I have a need, but I don't really know what I want. I don't know what's out there. I don't know what my options are. I'm going to explore. And then once I've explored, I need to narrow those options back down to a choice. Google is also recognizing, though, that we go back and forth through this loop. It is not a simple, straight down to a decision process.  You know, I remember shopping for strollers and they're so expensive and they're so complicated now. And I'd get this review and it's like two stars. This strollers great. It was, you know, it was a good price point. It comes in these great colors. It was durable. My kid loves it. It's safe.  What's the two stars? The cup holder wasn't quite big enough for my super big golf, you know, Lightning Mountain Dew.  You know, that's not going to go.   You start to make weird emotional decisions because we all do that. And so we go through this cycle and we go back and forth. We go through these marketers. We hate this because it's not, oh, that funnel is so nice. We just, you just go down, right? That's it. You buy the thing. But we know this, like we do this. We don't make these linear rational decisions. We loop through these things, and especially if it has any significant price point or we're going to have it for a few years. So how do we get through this? How do we cope with this? I shouldn't even say that because this next section is worse. This is the, we're not going to get to the coping yet. We're going to get to the wake-up scary part. I want to show you the journeys Google is creating. I'm going to start with a simple example, and I'm going to move on to a more weird one. So let's say I do a search for loose leaf teas. I'm kind of up near the top of the funnel. Like, I have some interest. First of all, Google's already kind of presenting us options, which is interesting. I'm a teacher here, so it's funny to me that the options are green tea, white tea, and Earl Grey. English breakfast, forget it. Tajiling. My hypothesis is that someone at Google was sitting around saying, you know what, it's good enough for Captain Picard. It's good enough for you. So the things, they're not out there. But we're going to skip past that. That's nothing to do with anything. And at the bottom of this, we have this refinement option. I'm like, okay, you know, I think I'm leaning towards green tea. I'm going to click on that. It takes me to loose leaf green tea. And we've had a SERP refinement. We're back on another SERP. Right away, I get into a giant local pack for tea stores. This is interesting. Google's been, like, categorizing the packs. I think Miriam's going to write about this soon. We're doing a little collaboration. And so immediately, I can go look for stores, but let's say, well, I'm not quite ready. So I go back down to that refinement. And now I can do more. And I'm like, oh, Jasmine green. That sounds nice. It takes me into another search. Jasmine loose leaf green tea. In this case, department stores, which is weird because two of these aren't department stores. I'm not a local SEO, so I'm not going to worry about that. And then I get to this popular carousel, which is a list of products. And I can navigate within that popular carousel. And NuMe is a brand. They do a lot of organic teas. And so I click on NuMe. And I'm like, oh, that's, you know, I just want a couple of boxes that I want to think crazy. That's a nice price point. That sounds good. I click on that. And I get this thing that kind of looks like a knowledge panel, but is products and is paid, except I said that the other day. And John Mueller got a little upset with me because now those aren't all paid listings. They're a mix of paid and free shopping listings. So Google costs that organic because it's not all paid, which in Google vernacular is correct. For us is a little weird because this feels paid to me. So it is not a knowledge panel in the typical sense. But the thing to reflect on here is that this entire journey down to the point of potentially making a purchase or almost making a purchase happened in the SERPs. I never left the SERP to take that entire journey through decision process. If that doesn't bother you, then let's amp it up a little bit and get into an even weirder one. So this query is interesting to me because by tablet, traditionally as a marketer, if you had a very strict view of the funnel, you would consider by tablet to be a bottom of the funnel phrase because it has the word buy in it. But you know what? No one piping by tablet knows what they want. This is a very broad, very unclear. They know they want to buy something, but they have no idea what it is. And I wrote in our recent, our professional guide SEO just came out. I was half joking about how we do search. If you walked into Best Buy and you just started saying, buy tablet. Hey, buy tablet.  Like nobody talks like that. This is how we search, but this is not how people do things. But Google's optimistic here. Google's like, hey, here's some products. They're shiny. They're nice. $2,000, $1,500. I'm like, no, no, no, Google. I'm not ready. And then they hit me with top 24 tablets and they're still like, you need a tablet son. I'm like, I'm not there yet. I keep scrolling down in store availability. So now we have a local pack, not only a local pack, but a local pack that's completely geared towards telling me which stores carry this right now and whether they have it, which became much more important during the pandemic. Interesting trend because no one had toilet paper and hand sanitizer and these things. So whether they had it in stock. And then I go down to this refinement.  I heard that's kind of cool. So I'm gonna click on Apple. This is a total aside, but that was interesting. Verizon in the shopping listing lists a $0 price. Okay, technically true. A little strange. And I go down also nicely refined by color. I'm like, wait, hold on. I'm not ready to pick a color. That's a big decision. I'm not ready for that. I'm gonna go back. That's not enough for me. And so I go back to buy tablet. Back to another shirt. I go down. I see network connectivity. Oh, okay. What's that all about? Click on that. Oh, 5G. I've heard that's good. I'm an important person. I gotta do stuff fast. 5G's fast, right? You know, I know Bill Gates is there's something going on there, but I'm not gonna worry about that. I don't believe that. But the 5G is good. So I click on that and I get 5G tablet. And again, Google is trying to sell me a tablet. I'm like, all right, Google, chill. And I see operating system. I'm like, okay, okay, let's see. Well, no IOS. That's weird. Don't know why. Oh, you know what? My dad's dog walker. They got Android. I heard that's good. Click on that. Google trying to sell me again. Notice in the shopping, though, we see these as ads. You click on that arrow and you get an entire shopping experience. So even that is a side journey Google could take me on. Popular 5G. I see something here. I see 12.4 inch screen. I think, ah, I'm over 50. I say it's not what it used to be. That big screen. I'm gonna need that. And so I add that to my query. Android tablet, 5G, 12.4 inch. Now still trying to sell me. I see videos. Well, I'm at a point now where I'm thinking, you know what? Actually seeing some reviews might be nice. And I look at this 10 key moments. What's that all about? I click that. Oh, processor. I need it to be fast, right? And so I'm gonna watch this video. And now Google, if you haven't seen this, it's been around for a little while now, is taking me to the exact point in the video where they talk about the processor. So now this is a very specific product review of a very specific product. I'm moving through the decision process and they're gonna tell me exactly about the processor. Well, it turns out the part about the processor in this video isn't that useful or interesting. But I see this pen. And I'm like, damn, I need that. I don't know what that is. The S Pen. That sounds awesome. So I type that. And so now I'm like Android Tablet 5G 12.4-inch S Pen. I see the shopping ads up here. Samsung Galaxy Tab. I'm like, oh, this S Pen, that's not something you can get with anything. I gotta get the Samsung. Well, I gotta have that now. I know that. So I change the Android Tablet to Samsung Galaxy Tab 5G 12.4-inch S Pen. Now the ads are relevant to me. Now I'm at a point where I might be ready to make a decision. I expand that. I look, I'm like, hey, 670. That sounds pretty good. 64 gig Mystic Black. Come on. That's sweet. I'm gonna need one of them. Again, this entire journey happened within Google. Now I want you to understand, I don't think people are doing this right now. I don't think in 2022, people are going down to refinement and taking an entire journey within Google. But I think Google understands a couple things. I think they understand that this journey is essential. And they understand that their bottom line is in what? Paid search. Their bottom line is in shopping and ads. And if they hit them with all these shopping and ads like they did to me at the very top of that decision process, I'm not gonna bite. And if they can move me through the process to a decision, their ads are going to be that much more effective. The other scary thing is, if you look at our missing middle diagram, all of those queries basically happened on the right. Half of it we haven't even gotten to. Now, this is a little gray. This is not a distinction that is as black and white as we might like. But if you look at what I did, I did a process of refinement. I did a process of contraction, like 5G tablet, Android tablet 5G, Android tablet 5G 12.4 inch, right? I'm refining what I want and I'm narrowing down my options. That's happening mostly on the right. So I want to talk about what happens on the left, because I think that's where we can shine the search marketers, and that's where we can really tap into a process and help to move people down the funnel. Not just be at the top or bottom, but be a part of the journey that takes them to a decision and why that's important. Google says a great thing in this paper. Sometimes all it takes to make a big impact is to show up at the right time. But the right time is not always when someone's finally there to make that final decision. The right time is when they don't know what they want. The right time is when they're confused about a feature. The right time is when they're trying to make a decision between two or three difficult things or a compromise or a trade-off. And if you can be there when they need you, this is critical as marketers. Alright, so let me give you another example. What's interesting is all the keywords I list on these next couple slides are ranked for by one single site in the home improvement space. And so these are real keywords that that site ranks well for. So over on the right in this evaluation, this narrowing process, we have things like kitchen designers, kitchen remodeler, kitchen and bathroom remodeling, Houston kitchen remodeling, mid-century custom kitchen cabinets Toronto, custom kitchen cabinet services in Manhattan, New York. We don't see the word buy there or anything like that, but we see these action words, remodeler, remodeling, right? These are actions. These are people I might hire. Houston, Toronto, Manhattan, location implies that I'm going to take action. I don't just type in mid-century custom kitchen cabinets Toronto because I'm curious about what's up with custom kitchen cabinets in Toronto from Chicago. I'm like, people in Toronto are nice. I got no beef with Toronto, but I don't just do this for fun. I'm looking to take an action. And just the fact that I said mid-century custom kitchen cabinets means I know something, right? I've gone through some of this other process. So what's over there on the left? Well, stuff like this, kitchen design ideas, modern kitchen ideas, kitchen inspiration, mid-century modern kitchen, mid-century kitchen cabinets, 75 modern pictures, kitchen pictures and ideas. People looking to explore and discover what's out there, all they really know is that they want to remodel their kitchen. They don't like how it looks. It's getting worn down. They're bored. They don't really know what this whole space of options is. And so I think this is often where we shine, is in helping people explore the space so that they can start to go through that evaluation phase and contract. But we're over there too, right? Like these queries. I'm not showing you the organic results, but there are organic results for kitchen designer, remodeler for all these locations or local results. We can move them along this journey. And I want to show you something interesting. I think we can start to plant the ideas. This is where I don't want you to get so focused on the bottom of the funnel that you miss the journey. So let's say I'm exploring what I want. I don't really know what the options are. I don't know the language of kitchen cabinets, you know, the complex language of kitchen cabinets. And so I finally am going through this process. And on this site, I see a page for mid-century kitchen cabinets. And on this particular site, they're showing photos. They're showing what that involves. They're showing what goes into that. But they also are a service provider site. They're helping you find service providers. So they're moving you from mid-century kitchen cabinets where they rank to mid-century custom kitchen cabinets, Toronto. Now they bypass Google completely. They've taken you on that journey themselves because they intercepted you at a time when you had a need. And the need was, I just know I want a new kitchen. I don't know exactly what I want. I don't know what it's called. I don't know. But once they serve that need, they've planted the idea of what I want. And they've moved me to the bottom. And Google understands that. And I think Google is trying to intercept that.  Classically, we think of phrases like buy a refrigerator, compare refrigerators, refrigerators on sale. Well, two things. We've already talked about how buy a refrigerator isn't really a very actionable kind of phrase, right? We know we want to buy something, but we don't know what it is. But these are also highly competitive. These are highly commercial. These are expensive. But what about phrases like, should I replace my refrigerator? Why isn't my fridge getting cold? Why is my refrigerator so loud? We think of those as informational top of funnel queries. But you know what? A whole lot of these people are going to need a new refrigerator. A whole lot of these people are moving into the decision process. And I think as Lily Ray said this morning, the funnel itself isn't that clear, right? Like, if you serve them at this point, if you serve them that informational content, that authoritative content, that trust-building content, you don't have to push the hard sell. And Google may even be saying, hey, you know what? These people don't push that hard sell. These people are there when they're needed to show you what your options are. And you know what? If some of those people don't end up needing a new fridge, they can fix it themselves. If you give them good advice, that's fine. Because there's a whole lot of people starting this journey who will need you. And if you're there for them at that moment, that will mean something to them. I want to give you a way to kind of explore this a little bit and play with it in Mods Tools. If you use Keyword Explorer, there's a couple things that are a little hidden. And so I'm going to do a search for kitchen cabinets. And this is Keyword Suggestions, which just returns other keywords that might be related. But in that pull down on the left, buried down way too far, because I love it, is the are questions option that will just return questions. And we also have exclude and include filters. And I'm going to exclude the word buy. I'm going to try and get away from kind of the boring bottom of funnel kind of queries. And I get things back like, how much should I budget for cabinets? How many cabinets in the 10 by 10 kitchen? Is it worth to reface kitchen cabinets? Again, to what Lily Ray was saying, some of those people are going to reface their own kitchen cabinet. Maybe they're going to buy some stain. They're going to do that basic job. And that's fine. But other people here are going to see that you gave them good information and they are going to look for a service. And if you provide that, you're in the right place. But look at a phrase like, how many cabinets in the 10 by 10 kitchen? I don't type this because I love geometry. I don't type this because I want to know how much can fit. I would actually. I do love geometry. That's a bad example. But these are people who have a 10 by 10 kitchen, right? They're trying to put cabinets in it. They're looking to take action. And so what if we can intercept them at this point? Is this informational? No, I think this is a post trigger signal that they want to make a purchase. But they're just not that far along the way. So if we lose them and say, top of funnel, not interested, low conversion, we're missing that this is a person on the start of the journey that will take them to the bottom of the funnel where the conversion is. And so we can't ignore that process. This is very simplistic. But my other argument is that, let's imagine that you're there. You're there at the bottom when they search for that action keyword. And you're feeling good about yourself. And your competitors there, too. But you rank number one. They rank number six. You're crushing that. You feel good about yourself. What if they're everywhere along the way? And you're not. What happens to the person who went through that journey when they see you and them? What's your cook-through rate look like compared to them by position? What's your conversion rate look like compared to them? You've already built that trust in that relationship and been on that journey with them. And I think that's really important. So I want to give an example. I was, I had to thank Areege because there's something about conferences and she said something that just sparked I don't even know what she said. And I'm, I apologize because I'd love to remember what it was exactly. But sparked some idea with me because I was struggling with, I know this is a lot to process. And I know it's a lot to, it goes against some things we say. And again, I don't, I am not saying do not focus on where the money is. I am not saying do not prioritize. I am saying that the journey is more complex than we'd like. And that if we're not on it with the buyer, then when they go to take action it may already be too late. And so I want to give you an anecdote and we've all been through I think probably something like this I've multiple times. You're making a pretty big decision. So let's go back to that buy tablets. You want to buy a tablet. You don't know much. And you do a couple of hours of research on the internet and you're kind of feeling good. But for some reason you want to go to the store. Like you want to hold it. You want to kind of see what the options are. You still feel a little more comfortable with that. It's a decent price point. So you go to the electronic store. You go through this process again, right? Like you have to find the aisle. You have to kind of find the category. You have to see what the things are that they have. You have to compare that to your other research because they're not going to have everything.  You're starting to make weird, you know, this antenna is sexier than the other antenna. This one has a racing stripe. You know, whatever they have. And you're going through all this and you finally kind of narrowed in. Okay, I think this is the one. The box is in your hand. And along you've been here like half an hour now and you've done two, three hours of research at home, right? And along comes the sales associate. Hey, can I help you? Like, bro, I've been here for half an hour. I'm a Midwestern guy, so we can't just say that. You've got to make conversation, right? So it's like, oh, yeah, you know, I'm looking for this and looking for this. I talked to my uncle and I don't know if you know him. He's up north a bit. And you go through this whole thing like, I think I've settled on this and five minutes later, he's like, oh, yeah, yeah, yeah, that's a good one. That's got all the megabits, man. That's called the mega pickles.  I'm not even, I've totally tuned out at this point. I'm like, okay, cool, cool, cool, you know, have a good one, see you, man. Go through the 17 step Midwestern goodbye. And I'm walking away.  If your name's Chad, no offense. This is a completely fictional Chad. Hate him. Like Chad.  I was home for three hours working on this. I've been here for half an hour.  And I think sometimes we can make customers feel this way. And I don't think it's that extreme.  We're not that bad. But when they get to that bottom and we're only there then, where do they go back? You know, we're no important, more important at that point than the credit card they used to make the transaction. I would actually say they're more loyal to that credit card, knowing how I feel about my own credit cards. And so what happens when they go to make another purchase? What happens when they go back? You know, I like tea. I shop at Adagio, for example. And I go back there. I skip Google completely because they have a good search engine. They have good in-store browsing. I can find new things. Now if you're sitting there at that bottom of the funnel, I never even got to you. You know, I've gone straight to them because they've been with me on that journey before. And so, again, I don't want to oversimplify this. I don't want to scare you. But what I want, the point I want to make, and I'm apparently running a minute over, is be there on that journey. Prioritize, focus absolutely. Know where your money comes from. Absolutely essential. And please listen to people like Ariesian Well, who talk about that. But don't oversimplify it so much that you're not there for the rest of the process because that funnel and that process is so much more complex. And it's essential to be part of that search journey. Google knows that. Google knows that there's billions of dollars for them in that. And we need to know that, too. That's what I got. Thank you.


Now you're on. Okay. Yeah. Thank you. I'm going to first review the Landau's Fermi liquid theory and then describe, reformulate Landau's Fermi liquid theory as a dynamic of shapes, using the method of the coaxial in orbit. Based on the idea that the state of a Fermi surface can be described as a point on the coaxial in orbit of a group of canonical transformation and that would lead to a reformulation of Landau's Fermi liquid theory. The work that I will describe to you in this talk is based on a paper written this March by Lucca de la Cretas postdoc in Chicago. Yixian Du human meta to students in Chicago and myself. So let's first remind ourselves that Landau's Fermi liquid theory was first formulated by Landau in the 1950s. And the idea that Landau has is that even if we start our description, if our system consists of fermions which may interact strongly with each other, near the Fermi surface, the notion of the quasi particle becomes more and more well defined. And the notion of the Fermi surface survive even when the interaction between the original fermions may be strong. Landau's Fermi liquid theory has been successfully applied to many systems like helium three electrons in metals, neutrons in neutron star and possibly many other examples. The Landau's Fermi liquid theory is sometimes described in the popular press as a theory of free theory theory of free particles. But as we will see the structure of Landau's theory is very different from the example of free field theory that one usually learned from textbooks like free scalar field or free fermion field theory. It is instructive then to first go through the formulation of Landau's Fermi liquid theory as it was originally described by Landau. So Landau tells us that the state of a Fermi liquid at low energies can be characterized by phase space distribution function, f as a function of t, x and p. From the point of view of quantum mechanics this might be and might see that is contradictory to specify the both the position and the momentum of the particles. But once I have to think about this distribution function and describing particles that can be localized only in a certain volume in space. And when this volume is the uncertainty in the definition of x is large enough then we can describe the phase space distribution function both in terms of x and p. I read in a recent interview by Cillian that Cillian claimed that it was he who solved Landau the idea that a Fermi liquid can be described by a phase space distribution function or at least convince Landau that this notion exists. So now let's consider the case of free fermion. Of course for free fermion if we put that free Fermi gas in external electromagnetic fields then one can write down the new wheels equations or one can think about that also as the Vlasov equation or collision less Boltzmann equations. So the f over dt the time derivative of the distribution function is given by the evolution depends on how fast the particle move in phase space. So this v of p is the velocity of the particle in configuration space that is the epsilon over dp the group velocity this particle and here e cross v cross b e plus v cross b is the Lorentz force acting on the system p dot. So this equation completely determine how the phase space distribution function evolved as a function of time. Given external field we can determine the response of a Fermi liquid to such an external perturbation. Moreover one can notice the following if the distribution function is is the zero or one like it is in the ground state of a Fermi liquid then the property that this function is always zero and one preserved under the new wheels evolution and that's because the new wheel equation preserves the phase space and so if we follow each volume of phase space that it move in phase that it move the distribution function remains either one or zero depending on whether it was one or zero in the beginning. And so the new wheel equation preserved the sharpness of the Fermi surface and then one can think about this time evolution as the evolution of the shape of the Fermi surface. So here what happens so I'm drawing here the original Fermi surface say at t equal minus infinity we have a spherical Fermi surface in a Fermi liquid with rotational symmetry that would be the ground state and then we turn on some electromagnetic field and what happened is that this shape deforms so the figure on the right one have to think about that as the particular shape of the Fermi surface at a particular point in space and time at every point in space and time you will have a different shape and our task is try to formulate the time evolution of the shape in a mathematically convenient way. If there is the connectedness of the Fermi surface can it split in several order? I think mathematically this equation doesn't allow it but if the equation doesn't apply itself then the Fermi surface can split but if one can apply this if this equation applies then I think there is the surface cannot split. Given some properties if the property of the function for example epsilon of p is regular function with some nice properties etc the land of Fermi liquid theory is also predictive in the sense that one can use it to compute physical quantities for example the linear response of a system to external say electric field. So here is an example of the calculation of the density-density correlation function called pi zero zero how the density of the system changes under external scalar potential. One should just solve the equation the Liouville equation in an external electric field which is gradient of the potential a zero and then solve it into linear approximation one find the response function to be this integral. Now this integral turned out to be exactly the same as the one compute one would find using the usual Green's function method in the limit of small q. If q is small then this formula would be valid. So that illustrates the validity of the Landau's theory although it doesn't know when we write down the Landau theory it's written not in terms of the individual electrons in terms of some grossly, grossly defined distribution function but it reproduces the correct response function. Now Landau's contribution is actually one on three we'll learn that he figured out how to include interactions between the electron between the original fermion into the effective Liouville description. He showed that the only thing that one need to do is to include the so-called Landau parameter which is a function of two point p and p prime in momentum space or rather to the extent that one needs it only p will be prime near the Fermi surface. And then the modification in the Liouville equation becomes the modification of the energy of the quasi particle with momentum p and at point x. The energy is the bare energy plus the contribution that comes from the distribution of particles at other values of momentum but at the same point in space. So this integral is a linear correction to the energy coming from the deviation of the distribution function at point x and p at the same point x but at different p I forget to write a prime so delta f at x and p prime. And the kinetic equation then written down in a very similar way one has to also include into the time evolution of coordinate is momentum the deviation of the energy from its bare values so here the true velocity and also there is another contribution to the force that depends on derivative of this function with respect to x and that's all that one needs to be ordered to compute reproduce the linear response of an interacting now Fermi system to external electromagnetic field. Now the Landau's Fermi liquid theory as Landau has written down is not a field theory if you can see it's formulated completely in terms of the some equation of motion not in terms of an action and that has some consequences. First of all we lose access to a large two box of field theory effective field theory we don't lose the Wilsonian point of view although Landau's Fermi liquid theory looks like a effective low energy field theory but it's not something that one can understand from a Wilsonian perspective. There is some advantages if one can reformulate Landau's Fermi liquid theory as a field theory and one of the advantage maybe that one can better solve the problem of non Fermi liquid. So these non Fermi liquids presumably exist in some places in nature. One place is two by two plus one dimensional electrons in the high magnetic fields the quantum hole effect. The quantum hole effect the half field Landau level is supposed to be dual to a theory of composite fermions at finite density coupled to a gauge field and that problem of fermions at finite density with a Fermi surface coupled to a gauge field is one of the place where presumably a non Fermi liquid would appear. So to understand the Fermi liquid non Fermi liquid one needs to first try to at least form my perspective one it's useful to reformulate Landau's Fermi liquid theory as a effective field theory. Previous approaches have been there have been some previous approaches towards this goal. So earlier in the 1990s they would work by Benfato Galavoti and Gendruchin and Schanker trying to reinterpret Landau's theory as an effective theory of the Fermi liquid particle and so one write down some Lagrangian or some action that depends on the fields of the fermions and try to include the Landau's interaction as a four Fermi interaction between this quasi particle fermion. So that Lagrangian you will see is not the most simple Lagrangian so one of the the drawback of this formulation is that it has to be done in momentum space and so in momentum space it's very difficult to couple the theory for example to external gauge field. Also another thing that makes this formalism that say maybe something else perhaps to be done is that in this formalism there is a unnatural cancellation between diagrams so if one compute not two point function for example but the three point function one see that there are three diagrams that almost cancel each other exactly in the infrared and that that non-natural cancellation survive to four point five point function and become more and more severe. And so something else probably is here L1 is a momentum in K1 is momentum okay so here actually I lost yeah I'm not I'm not sure yeah it's this L I think it's from Konchinsky and this L may be P maybe K plus L I forget his notation. I'm not saying that I this is important for the the thought I'm just trying to illustrate that it looks like an original Hamiltonian but one has to to interpret this V as some effective interaction so the idea is pretty simple that there is some effective interaction that can be renormalized as one goes into infrared. Another set of idea comes from one plus one dimensions where one knows that a fermion can be bosonized into a boson and during this bosonization the density and the current becomes a derivative of some bosonic field. But to transfer this idea to higher dimension the situation becomes very complicated that's because now instead of one boson one in one plus one dimensions the Fermi surface are just point but in higher dimension the Fermi surface now has an internal extension like it is a line or a surface and so one would have now one chiral boson per point on the Fermi surface and that makes the theory very complicated and there is no general rule to write down such an action one try to one would try to go from the microscopic theory and derive such an effective action but but doing doing that is it technically complicated and doesn't exhibit the general properties. Recently Elz, Tolkien and Senzel try to think about this problem in a perspective of anomalies and that's one thing that they notice is that there is mixing of mode with different theta and that mixing also is very make this bosonization very non-trivial because suppose suppose we put this electron in a magnetic field then suddenly one fermion from one point can move to a point fermion at another point so somehow different bosonic modes of different theta need to couple to each other and how they couple to each other is a very non-trivial question. Now in this top I want to answer a simple classical problem and the the the question is can one formulate reformulate Landau's equations as an equation that comes out from some action s so action depending on some few phi so that delta s over delta phi equal to zero is identical to Landau's kinetic equation and the answer my answer is to this question is yes one can do that and the method that one need to use is the method of co-watching orbit so let's let I'm going to flash to you just a few mathematical uh trans transparencies and then we go to the more physical uh application of this formalism so consider a group that we call g and corresponding to that is the algebra that I also call g but it's with a different font that small g is the algebra and large capital g is the group and for each elements of the algebra one can exponentiate it into a element of the group now we know from the theory of a group and the algebra that there is something called the adjoint representation so for the algebra the adjoint representation is simply realized by the by the by the commutator that is if I have a an element of the algebra g of f then an action of the element g in the algebra on f is simply the um given by the commutator of g with f so the adjoint representation is realized on the algebra and one can exponentiate that to a representation of the lead group uh so for unitary lead group usually we write that f go to u f u minus one u is the exponent of that g now there's something that usually we do not learn in uh as a physicist in the course of of lead representation of lead group uh it's the co adjoint representation because in a lot of time this co adjoint representation is very similar identical to the adjoint representation so mathematicians like to make this distinction between the space and the dual space so the lead algebra we call little g and the dual space g star are the space of all the linear map from g to to to the number real number so for each uh uh elements of the dual space and uh elements of g each of such pair one can put a scalar product that belong to that is a real number and and this dual space also realize a representation of the lead group and the lead algebra uh we define this adjoint co adjoint representation so that if we have an f and a capital f and apply a co adjoint rotation on capital f and co adjoint rotation on this f that scalar product is is preserved so that is how one can define the co adjoint representation okay and then what is the co adjoint orbit the co adjoint orbit is defined as the orbit of a given point in co in in in in the dual space under uh under the action of all possible uh elements of the lead group so if I take one point and then use this co adjoint representation and then rotate this point and one one can define this co adjoint orbit and this co adjoint orbit actually is a coset of the lead group and the subgroup that that keep this reference point fixed in space okay so all this might look a bit abstract but I'm going to go to a physical uh case of a Fermi a Fermi liquid that's relevant for Fermi liquid theory so the idea is the following the idea is that uh all the uh different shapes of the Fermi surface uh that uh evolve that is the result of evolution of original Fermi surface and the Liouville equation can be thought of as the result of a canonical transformation on the original Fermi surface so all the state with a sharp Fermi surface and connected Fermi surface can be obtained from the original spherical Fermi surface by applying a canonical transformation and this canonical transformation acts in the face space of a single particle the canonical transformations um acts as follows uh so first the the the infinitesimal canonical transformation or the elements of the algebra are all functions of a face space x and p so x go to x prime equal to x minus epsilon gradient over p of f and p go to p prime equal p plus epsilon gradient of x you can see here that this is basically a a Hamiltonian evolution under a time epsilon with the Hamiltonian given by f and one can see that this preserved the Poisson brackets like all Hamiltonian evolution uh the Li algebra uh is given by the Poisson brackets that is the commutator of two of such infinitesimal canonical transformation is another another canonical transformation whose function is the Poisson bracket of these two function f and g so it is a Li algebra and exponentiating this Li algebra one get the Li group that is the group of all finite canonical transformation into the power of f now the dual space what is the dual space f so i'm going to define this dual space as also function of face space x and p and the scalar product between f small f and capital f i'm going to define a simply integral over the face space of the product of small f and big f and this equation has a interesting physical interpretation if if one interpret capital f as an observable for example p square over 2 m one possible function capital f then this small f can be interpreted as the distribution function it's a state and the scalar product is the average value of this observable capital f in the state given by f it's very similar to the trace of the product of rho and f in quantum mechanics instead of taking the trace of the product of the density matrix and the operator here which is take the classical integral over the face space of the product of two functions so the adjoin and co-adjoin action of the group in this case are the same i turn out that action of group element exponent of g on is realized in the same ways in the Li algebra in the in the dual space by the same formulas one can see that if we do this adjoin rotation then the scalar product of these two functions remains the same by some properties of the Poisson bracket one can do integrating integration by part and show that any question okay so now let's let's go back to the problem of the Fermi liquid so let's take the our let's try to construct the coadjoin orbit the coadjoin orbit we take the reference state and try to apply different canonical transformation on that reference state and our reference state is simply the state with spherical Fermi surface and let's try to work in perturbation theory that is for small perturbations so let me take the one element of the Li group of the canonical transformation hq now it's exponents of minus phi which minus is just for convention and phi is a function of face space x and p so now using this this rule that we have before a new distribution function is the action of the coadjoin the coadjoin action on this f zero and it's just f zero minus two leading order in phi minus the commutator of phi with f zero by commutator i mean the Poisson bracket so this this can be calculated and rewritten in term of another theta function you see here that the theta function is not different from the original theta function the position of the Fermi surface has now moved this p the this here p hat is the direction of p in momentum space p hat cross dot with gradient of phi is how much the Fermi surface has moved in the direction of p the gradient because gradient of p naught of what pd phi dx because the Poisson bracket so the Poisson bracket between phi and f zero is like px by the p of f zero yeah f zero has p dependent yeah but but but there is um but um all the turnout that all the uh if the the the the coadjoin orbit does not depend on the choice of the reference point you can choose different reference point get the same uh orbit now there is an ambiguity in the um in the uh parameterization of the point of the coadjoin orbit as the elements of the group two different elements of the group can correspond to the same point on coadjoin orbit and that's because if we i multiply this u by u times v but v is the thing that preserve f zero then acting this u v on f zero give me the same as acting u and f zero because v doesn't do anything and this procedure allows us to do a gauge fixing in our phi in particular one can uh think about phi as not a function of x and p but only a function of x in the and in the direction of p we can assume that phi is in but it is the same at all point along the radial uh direction in p depends only on the angle uh parameter of p and so phi is now a function of only x and theta and that's that's how a scalar field at each point in the Fermi surface appears in this formalism okay so now let me write down the actions that turn out to give rise to Landau's equation so the action is written in the following way so there is a very phased term or a wetsumina written term that i'm written in this way so u is remember u is a point parameterizes point in the coadjoin orbit so here is the first derivative how this point moved in time and then this is the Hamiltonian that depends on the distribution function the first term can also be written alternatively as the integral of a two form which in mathematics is called the Kirilov-Gostan-Suryo-Symplectic two form but it turned out to be also possible with some uh um and with some some ignoring some term that is totally derivative as the just integral over time for three for me on this this this omega i can write it but probably i can write it privately for you this this omega i have a slide i can i can show you this uh actually it might be too cryptic for this slide but this omega can be defined as follow if i take a point and then you find two uh tension vector they'll uh i i i i can this decipher this uh for you a bit later uh but i want to uh go to more physical um um um calculation it is it has the properties that it is uh closed uh two form so it can be used as a as a definition of Poisson bracket on the as a symplectic form on the coadjoin orbit so this um this this this is an interesting um place to um to stop and think about the this first term the first term basically is the bary phase term which tells us how much the bary phase a system acute when the Fermi surface changes its shape so we have a Fermi surface that move uh in shape changes and then we have a bary phase and apparently Landau's theory specify what is this bary phase in a precise way the bary phase does not depends on the Landau parameter or the interaction at all it is properties of the phase space itself how did you choose this action why is this one this one is this this Hamiltonian yeah we can check that the equation of motion are all so if we make this choice and we can verify that all the correlation function from Landau's theory the equation of motion the Liouville's equation appears from this action as i i i try to illustrate later in particular the coefficient in front of this term has to be chosen to be one for example then everything should the this fine derivative time derivative term is pretty unique uh yeah i can't see it can you uh unitary perturbation theory and time i think blue but um if i'm not wrong yeah i i was i wasn't aware of this so yeah thank you okay so let's uh now write this action and expand it quadratic order in phi so that's very simple and we get the quadratic action that basically uh chiral boson at each value in phi and the two-point function which in the original theory of free fermion is a one loop graph in this effective theory becomes a three-level diagram with a propagator of phi and the all the formula the calculation can be done and one can check that the two-point function is reproduced but one can also go to higher point function so here the free fermion theory map to an interacting theory of the scalar field the interaction comes from various places it's because for example in this uh wesomino-witton term there are all terms of higher order in phi and we expand the exponent of the boson bracket uh and also in the energy there is also non non non linear terms and when we write it down the result is complicated but very unique the prescription is given we just follow the calculus the the formula and then we can write down all the interaction terms it interestingly free fermion become interacting uh bosons in this approach and bosons a different value of theta coupled to each other in uh unlike in the linear approximation quadratic approximation this is the Hamiltonian uh that is the minus the integral of dt of the Hamiltonian so that Hamiltonian is just the energy associated with the a Fermi surface that is perturbed integral of p square over 2m for example and this swzw is the first derivative term in phi so here we see um the result is complicated but one can do the following check which is quite non trivial that is one can compute three-point diagram three-point functions row row row correlation function and check that it is exactly equal to the sum of all the three graphs that arises from the interaction between the phi fields so uh loop diagrams in fermion is exactly equal to uh the sum of the three graphs and moreover here we see the cancellation between two different Feynman diagram in the case of fermions uh become natural in the bosonic language that is if individually each of these fermion diagram is one over q and cancel each other to be q to the zero of power each of the diagram in the bosonic theory is already automatically of order q to the zero it's a non trivial check thank you and including the deland out parameter in this theory can be done uh uh by just saying that instead of uh the Hamiltonian being the integral of uh energy over distribution function we also include for example the lambda parameter terms so here i write the classic lambda parameter term but there in principle one can also write down all kind of lambda parameters that depends on say derivative of the distribution function or three-point uh the lambda parameter etc so here in principle one can envision a program similar to effective field theory that is one write down effective action based on certain symmetry and then expand it in derivative okay so i have a few minutes left let me mention some a possible extension of this formalism so one thing that one can notice is that the fermion bilinear in a theory of fermion former closed algebra and in the long wavelength limit that closed algebra is actually the algebra of canonical transformation and that means that we can uh extend the theory to include for example spin full permissurface we need to extend the algebra to include not only psi vector psi but psi vector sigma a psi where sigma is a spin matrix pauli matrix and also we can extend the theory to a bcs to include bcs uh other parameter by extending the algebra of canonical transformation to algebra that contain psi psi operator on psi decar psi decar operator this would still be a closed algebra okay so let me conclude the method of co-echoing orbit provide a natural way to write down an effective theory of a fermi liquid uh the the outcome of this formalism is a non-linear posonized version of landau's fermi liquid theory it reproduces uh non-linear and linear both linear and non-linear um fermi liquid responses and perhaps it's a suitable starting point for the study of non fermi liquid thank you very much maybe this is a very naive question but i'm still having trouble understanding where the advantage here is on what people do all the time in the sense of trying to write down an effective ginsburg landau by decomposing exactly the fermi surface into its various shape fluctuations and then taking a four fermi on term doing harvester tonovitch and integrating out is it is it in control is it in the way that you can extend it where exactly is the advantage of using the co-echoing orbit this this formalism give rise to a very unique construction that is all the interaction for example at the three three point three point function uh interactions are all well defined when people do this let me let me just say that this formalism allows one to not think too much and just follow the mathematics um i actually don't know if there is a way more a way to actually arrive to this uh interaction between the bow zones in a way that people have been trying to do so far in the formulation of the Boltzmann equation there is also an approach that you introduce an inner product that looks just remotely similar of course um so the question is could i uh write down the field theory for the Boltzmann equation and um or reversely write down the hydrodynamic version of a fermi liquid uh just as i can write down the hydrodynamic theory of and that follows from a from a Boltzmann equation by the Boltzmann equation you mean just the equation this this uh with the collision term with the collision term well for example because then you have i mean you can write down the Boltzmann equation also in terms of inner products that look very very similar to the ones that you have written down there but you always stay of course at the level of the equation of motion you try to show a type form and so forth but i could now conceive that you write using this action principle to write down the hydrodynamic field theory that would follow from the Boltzmann equation or vice versa that would apply to fermi liquid theories this yeah the collision term has break time reversal so i'm actually not sure if if uh if the simple action uh would be sufficient it might be an action on a double contour but i haven't thought about that uh very interesting uh so if it is a true quantum theory so you you also need to tell us probably what is a measure of integration what's independent fields so is it some kind of hard measure of of your capital use i think so yeah it would be a hard measure of in capital U invariant under the group action yeah and in terms of your phi field it's just supposedly flat that's i'm not actually sure near presumably yeah presumably yeah there is a question in the chat from Andrei Chubukov does this formalism allow one to study non-analytic corrections to fermi liquid example the t-squared terminal specific heat in two dimensions i would hope so that would comes from a loop correction in the bosonic field theory but we have not analyzed it there is one more question in the chat from Ahmed Salah are all elements in coad joint orbital regular values i'm not sure regular values yeah so we are here concentrating on the path of the coad joint orbit near f zero so that makes everything simple but we have not tried to think about the global properties of the coad joint orbit maybe maybe the last question actually this this is probably an extension of your plus Andrei's question so you showed nicely you can very quickly get the polarization so presumably you can get the zero sound collective modes now in helium three it's also known very simply calculation and experiment that you increase temperature and it goes from zero sound to to regular sound you just get this crossover inside can you do this with this theory do you also get it's just increased temperature the crossover probably not exactly this theory because this theory is is zero temperature because f zero we start the coad joint orbit or have sharp for me surface so to extend that to non-zero temperature we have to extend presumably the coad joint orbit to include non-sharp for me surfaces but you could do it with disorder as well probably so it stays your temperature but have a disorder type lifetime but that's also not included right the lifetime of the zero sound in a zero temperature or yeah i give us give the fermions a lifetime by adding disorder but i guess that's also not in this theory and not too in the in the outcast not in the three-level version of the theory but presumably also if it's included in the in the loop some loop diagram