Brought to you by Penguin Wrath of Poseidon by Clive Cussler and Robin Burceau Redview by Scott Brick Prologue 1 Sardis Persian Empire 546 BC The steep acropolis of Sardis loomed against the night sky while far below at the city's edge Flames consumed the reed-thanched buildings General Mazaris, dispatched by King Cyrus II of Persia, the moment he'd learned of the revolt, had ridden through the night, leading a unit of armed heavy cavalry. According to the Imperial Messenger, the Ionian mercenaries were sent to spark the revolt and dawn. Apparently, they'd gotten an early start. Fools, Artaban, his lieutenant, called out over the sound of hooves as the horses neared the gates. A wooden building exploded near the gold-refining works. Do they not realize that Cyrus will crush them? There is nothing left to crush, Mazaris shouted. I'm surprised that there's anything left to burn. It was the second time they'd marched upon Sardis. The first was when King Cyrus's army had broken the siege of the wealthy Lydian capital, captured its king, Cresus, then plundered his vast treasury. If not for this revolt, Mazaris would be accompanying the bulk of Cresus' treasure back to Acbatana. The quicker we quell this rebellion, the sooner we get home. He eyed the flames swirling from several structures just outside the gates. As they neared the inferno, Mazaris realized the purpose of the fires. He and his horsemen were almost blinded. Waiting for them, the insurgents, with their backs to the blaze, had the advantage. Within moments, Cyrus's cavalry was attacked by a shadow army of soldiers armed with spears, axes, and swords. Dividing his men into two flanks, Mazaris led the left, Artaban the right. The deafening clash of metal rang through the night as his horsemen, blinded by the flames, battled the unseen enemy. Mazaris thrust at an armed silhouette. His blade struck something solid, the rebels' shield. Shouting, Mazaris ordered his left flank to close in, while Artaban did the same with the right, sweeping in behind the rebels, who suddenly found themselves sandwiched between both flanks. Spurring his horse to rear, Mazaris blocked the thrust of a spear, and drove his blade into his opponent's chest, piercing through the man's inadequate armor. Pulling his sword free, he wheeled his mount to the right, then swung at the next man, felling him as well. Within minutes, it was over. The insurrectionists fled. The flames of the wooden structures, no longer being fed, began to die as a smoky dawn in the eastern sky burned, along with the embers of the failed revolt. Mazaris surveyed the scattering of bodies, none of them his men. The speed with which they put down the insurrection troubled him, as he met up with his second in command. Tell me, Artaban, does it not seem suspiciously convenient that the fire was confined to the outer wall, and that the skirmishers dissipated almost the moment we rode in? And why wouldn't they? Artaban nodded back at their troops, who were awaiting further orders. If you were a group of outnumbered mercenaries and you beheld Cyrus's immortal cavalry charging, immortal they were not, but the ease with which they'd won this so-called battle would certainly add to their legend. It did not, however, lessen Mazaris's concern. It was something more than the desertion of the city gates. His unease grew as he led a contingent of horsemen into the city. They trapped, asked Artaban. I fear something else entirely. He raised his hand, his men halted in the Agra, looking down the empty streets on all sides. Before his departure from Sardis, King Cyrus had appointed Tabulus to govern the newly conquered city in his stead. Tabulus's guards could easily have crushed the insurrection as small as it was. So why have we not seen any of his guards on the streets? Perhaps their governor is part of it. Let us hope not. Magus, take charge. If there is any evidence that the rebels are regrouping, end it. Artaban, bring back one of those rebels, alive. And where will you be? Artaban asked. I intend to find out- Sample complete. Ready to continue?


Super cheap, lots of benefits. Hey, what's up guys, it's Josh here. I have a short and sweet video today on my acupressure mat that I found on Amazon. Like I have maybe six, seven, eight back relief tools, things to kind of stretch my muscles out, stretch my back out after a long day. And the acupressure mat is probably the one that I use the absolute most and it's actually very affordable. Now I did see it being advertised on Instagram at one point. I think I said this in my original video. I clicked on the link and it was a pretty expensive, $60, $70 for like an acupressure set, decided to go on Amazon and I found pretty much the same thing for like $15, $20. And then it came with like a mat and a pillow so I pretty much only used the mat but I did get the pillow as well. And I gotta say this is just one of my favorite things to hop onto if I'm stressed out, if my back's a little bit tight, I just need a big refresh. It's one of my absolute favorites. Some of the benefits they have online is that it eases muscle tension, promotes relaxation, relieves chronic neck back and foot pain, reduces menstrual pain, alleviates chronic headaches, helps depression, anxiety and stress and boosts your energy levels. It's really just getting more blood flow into more areas and that's just all good all around which makes sense why it's gonna help relieve pain because a lot of times you have pain when a muscle's overused or it's underused and you have a lot of old built-up blood in the unused muscle. So it does hurt getting it out but once you've kind of flushed that old blood out and got your healthy new blood in there, it's more circulating and it's taking all the old toxins and things that were out of it. So if I'm having a bad day, just lay on it. Sometimes I'll even try to go to sleep on it if I can just cause it makes my back feel great. So I'll try to put a picture on screen of the kind that I got. I use it for all the main stuff like from sore, if I'm not really feeling so good, it does probably boost your mood a little bit. Most of the time, if your mood is a little bit lower, you're probably not stretching out, you're probably not moving. And it does have this kind of like pain that ends up feeling good afterwards cause it starts off hurting but like once you get the flush of the blood and your muscles can stretch out, it feels a lot better. So it is this initial pain but you get over and it feels a lot better. Kind of similar to like working out or like a really tough massage. But anyways guys, I can't recommend this enough. It's so cheap. The only thing I could say negative is that it does hurt a little bit, especially when you first get on it.  But my back, I've gotten used to it to where it did really hurt a lot in the beginning. I don't think maybe I was unable to lay on it permanently but now I can pretty much lay on it as long as I need to. The only time I get rid of it is like if I wanna turn on my side I don't only want it on my side so I'll take it out for money for that. But super cheap, lots of benefits. And if you can get over that kind of pain hump it's not really a bad pain hump because it's something that's eventually gonna help you for most people.  I love it. So anyways guys, we're on the road of 50,000 subscribers and I couldn't do without any of you guys to help you guys are the best. Having a great day out here. Hopefully having a great day at home. See you all in the next video. Peace.


All right, thank you very much It's a huge honor to be here. It's really cool to see talks Like those given by the banks Because as many of you here know my life currently is heavily involved with open source But maybe sort of lesser known is that I was really only able to start getting involved and learning to code because of open source So as a quick sort of story leading into this talk I started to learn to code around 11 or 12 years old and we had one Dial-up line and I was not allowed to connect to the internet during the day because my dad needed that line So I really only had two ways to learn to code and one way is super not open source at all my dad would drop me off at Borders bookstore, which I think is now out of business, but Will drop me off at the bookstore. I would run to the corner where the computer books were I would like crack open the book to not crease it because I couldn't afford to buy the books I would I would crack it open read and read and read and I would read the same like five pages Over and over because I had to memorize them because I couldn't take this book home And then I would run home and then try to sort of apply What I learned and I would repeat that sort of weekly And then the other thing was when I discovered that open source existed And you know github doesn't get github didn't exist Google was relatively new So I tried all these different search engines, but eventually found out you know people uploaded Tar balls and stuff of code and so I would download a bunch of tar balls at night and then during the day when I couldn't Dial up. I would just read the source code and learn sort of how to Apply these abstract concepts books are really great at abstract concepts open source is really great at applying those concepts and showing me Sort of how it could work. So that's how I really got started and and so it's very fitting And I'm very happy that I'm able to sort of build my career around open source software And around eight years ago. I started a project called vagrant and As the bank said in the talk it wasn't my first open source project There was many before that that no one knows or cares about But vagrant ended up being something a little bit more special Became a lot more popular than I could have imagined But also taught me a lot about you know community the non coding side of open source I had to learn about the challenges of building a community the challenges of bringing on contributors Governance those sorts of things and the success of vagrant and and getting into DevOps is also what led me to found Hashi Corp, which is a company aimed at building Striving to build you know the best infrastructure automation tools and I Started this company with my best friend Armand and together We've built seven free and open source licensed products that we've built our company on which Do development environments infrastructure automation cluster scheduling? Security software there's a pretty wide array of things and and When we started Hashi Corp We knew for sure we were building around four or five of these things and we started that process And when we started it was just the two of us and when there's two of you And especially when yours as as close as we were It's it's easy like you have a mutual vision a mutual understanding You know how you want these things to work how you want them to feel and you can be really really productive But we knew that to really you know reach the goals and ambitions that we had for a company Like Hashi Corp, which is to make a much broader industry impact That it was going to be more than the two of us and very likely much much more than the two of us So our biggest fear was how are we going to scale? our Not not company vision, but product Development and design vision like how how we're going to build an engineering Organization that really understands why we do some of the things the way we do some of the things we do not necessarily just what the problems are solving, but how we solve those problems and so Very early on in the company and to most very surprisingly early We wrote a document called the Tower of Hashi Corp. I think we wrote it when we were still around five Employees so it was really early. We published it about a year later. I think we published it in 2014 But I'm not a hundred percent sure if that's right But what the Tao is is really sort of the principles that guide sort of our vision our roadmap and product design of our products and When you know, I've worked in companies where when I hear the word principles I think it's some HR checklist that the company just has and with something like the Tao I think it's it's important to know that we wrote it when we're five people. It's important to know That we've applied it really rigorously sort of to our tools And so what I want to do in this talk is go over the principles of the Tao Explain what they are why they matter to us and how we've applied them to our tools and my hoping with my hope with That is that you could take this and it's more broadly applicable to not just building tools But also a design decision and in when you're adopting tools like is adopting this tool Is it are certain elements of the Tao important to you? And so an architecture? You know issue with this tool is not going to work for you and Yes, so let's get started going through the Tao There's eight of them a few of them are really similar so they'll be very fast and some I'll spend more time on So the first thing in the Tao is workflows not technologies and it's not a mistake that we put this one first It's really really important to us. And so the idea behind this is that technologies both software and hardware Continue to shift like continuously will shift if you expect that anything is going to remain constant you're in for a rude surprise and so with this idea The the core thing is that product design should start with an envisioned Workflow to solve a problem. And so I use a few words here I say product design and I'm probably going to say that a lot when we start a new tool Whether it's a new open-source tool or a new internal sort of back-end service We don't start with directly with algorithms or architecture things like that We ask ourselves. What's the problem? We're trying to solve and then we start designing and and I use the word design not in the in the meaning of visual design, but actually holistic design of how is it going to feel but also how is it going to work and so the examples I have here from our industry are Sort of looking at you know mainframe to server server to VM VM to container or if you look at physical data centers to cloud All these are huge paradigm shifts that have happened sort of over 10 to 15 year periods and in all of them lots and lots of software was replaced and If you look though the core problems that people are trying to solve just didn't really change And so the idea behind this is that most problems don't disappear some do but most problems don't disappear Some problems are created of course, but there's a big group of problems which stay the same And so if you work if you aim at work close first You could often create software that could span Multiple paradigm shifts, and that's really important to ease the adoption and transition into these different Different stages, and so the best examples for our software here are terraform and vagrant Terraform is a tool for creating infrastructure With terraform you could create VMs you could create Physical servers you could create containers you can manage sass applications, and we have people from hobbyists all the way up to You know the world's largest on any given day company I'm using terraform to manage Vastly different things, but using the exact same workflow And so the last thing I'll say about this against really important So spending a lot of time on this slide, but is is this is how we also start our software terraform and vagrant actually both started as Bash scripts and when I say that people usually think I built like a minimum viable product in bash I didn't I just put a bunch of echo statements and pretended something was happening when nothing it did absolutely nothing But the reason I do that is I as I write it in bash, and then I use it for a day I use it given it does nothing But I use it and try to see if it feels right like is this how I actually want to work on a day-to-day basis Because I'm gonna commit to sort of like building something that I'm gonna you know work on and use for years Is this how I want it to work? That's really the focus on workflows The next thing much simpler much more obvious I think is is the concept of simple modular composable This could kind of be described as the Unix philosophy in a way, but I think that's That the Unix philosophy is always interpreted to the speakers like you know The most beneficial definition so instead of saying that the goal of this is really just to prefer Smaller components with well-defined scope That are functional on their own but can be integrated and composed with other tools to do new and interesting things And I think this is really obvious in our and in our in the ethos of our company We are sort of one of the few startup companies that Built so many pieces of software that do vastly different things instead of bolting features on to different products Vault is a good example We built vault because people were putting secret information in console Which is a key value store and we didn't feel console the console security story It's just not built for that and so we felt sort of guilty in a way that people were putting secret information in there And we didn't have a better option to give them We could have bolted on encryption to console and certainly from a pragmatic standpoint That is the way we started the design But as we started thinking through the turtle tower of security problems You know beginning with where does the key stored? How do you rotate the key who has access to the key? How is security bootstrapped? You know as we went through these turtle problems. We realized that it would be a humongous problem Solution to just bolt on to console and would be way out of scope for what that project was trying to do And so that's an example of building vault The next thing is communicating sequential processes So if you're building these modular components is built on modular They have to be able to communicate with each other and for that We just follow this model where all our software should be a standalone process that runs well on its own with a well-defined Sort of communication interface An API that the use that they use to compose each other. I think that's fairly straightforward The next thing is immutability and so immutability is of course the idea that things should not be able to be changed And the most popular and and very appropriate for this conference Example is version control right with things like get we we get the Benefits of immutability which is sort of the idea of atomic You know it's changeable with some surgery But these atomic units that represent a version and in a history behind that and who did you know a sort of blame trail And these are only possible if you can't just change things whenever you want in the middle And so immutability is really important to us because these concepts are really beneficial to other things And a good example is infrastructure if you consider sort of each stage of your infrastructure as an atomic unit That's driven by a specific change then you could see the history of your infrastructure You could see who Impacted that history when the change was created But you could also reason about desired states Current state and sort of how to get between them if you know that what you should look like is exactly this and not This with these little changes It's exactly this and you're in this other worldview you could transition to that and that's really important for infrastructures Next thing the next two are really the idea of codification so Codification is the idea of encoding knowledge as code and so with infrastructure Historically knowledge is passed on via oral tradition You join a company in their you know system in or it or ops or whatever group You look at the infrastructure and you ask, you know, why are things networked this way? How does you know from from a web user to this service? How does how to what what applications that go through how do things work and the most common response has historically been Oh for that you should talk to this person and for this you should talk to that person because they're the subject matter experts on that And the problem with oral tradition is is it's slow It's also just fickle because human memory is fickle and so you don't quite get you know the truth usually And so we ascribe very heavily to the idea that knowledge should be represented in code and use that as a source of truth so with tools such as Begurant for example, how do you create your development environment the answer before used to be read this? Humongously large read me and follow this series of steps and the answer today can be We have this code that is also executable like the document and description of what is your dev environment is also the configuration for the creation of that environment and So that is the second thing which is automation through codification The idea that manual processes should be encoded as code and automated And this in our industry is just necessary and I heard talks yesterday talking about just growing scale When you're this this sort of scale of Infrastructure growth and the scale of application change in the data center is has already exceeded the point of human capability And so we need automation as a tool to leverage ourselves to be able to manage that and From the talks I saw yesterday, you know if you imagine a world if you believe in a world where there's only gonna be You know an order or more orders of magnitude more devices whether they're Whether they're my refrigerator or I really don't want my refrigerator to ever be connected to the internet But whether it's my refrigerator or you know a car or a watch or whatever it is This is just gonna put more and more burden And so we're gonna need automation to be able to scale these processes properly. So automation is Absolutely critical and the vehicle for automation that we choose is codification Which is different than simple scripting, which is the type of code It's this idea that sort of the knowledge and processes is the same document that is automated And the next thing is the idea of a resilient system. So we're deploying more and more things We're moving more and more to Cheaper less reliable whether it's hardware or just heart, you know infrastructure providers and Distributed systems smaller devices things like that and so in building this systems have to be resilient It's not a matter of You know if there's gonna be a failure It's a matter of when that's always been the case, but the when happens to just happen all the time Nowadays in large enough infrastructures and so you have to consider resiliency and there's various approaches to resiliency But the approach that we like to take is this desired state model It's the idea that to build a resilient system the system needs to be capable of understanding a desired state it needs to be capable of Determining its current state and it needs to be capable of creating a plan or a path to get from its current state To the desired state and if you consider that abstractly that describes leader election systems that describes infrastructure Configuration and desired state that describes configuration management This is sort of how we build resilient systems and if you could build this property into your software The idea is that they all should Strive and automate to being in the correct state all the time And so you could see examples of this with terraform which is tear when you run terraform There's a plan sub-command and when you run plan what it actually does is inspects the current state inspects your Desired state and shows you what it would do to reach that because changing infrastructure is scary So it shows you that before you could apply it With things like console leader election like I said Could move to his desired state of having a server a leader And then the last element of the towel which is really my favorite But also the one that has actually been the most challenging for for some people that have worked at our company We've actually had some people leave the company which which have left because we're just we're just too pragmatic but it is pragmatism and Pragmatism is the understanding that Everything I said before is a set of ideals and there are ideals that we should aspire to and ideals that we should fight for and build into our tools, but if it's just simply not the right pragmatic solution then we need to be able to be You know understanding enough to re-evaluate those ideals for the given problem set at hand and so a good example of things like this is That not everything we do is immutable Immutability introduces a number of challenges and immutability is not always practical And so the biggest Offender of this is console is our KV store console doesn't actually have Versioning sort of built-in doesn't have this concept of immutable Chainsets it has idea of transactions and things like that But but to some people that's not enough and so they'll reference our towel and say you know You're violating this principle you had Which is fair, but at the same time. It's it's a practical. It was a pragmatic decision at the time and we certainly You know got built features around that more and more as time has gone on So pragmatism is absolutely important when building these building against ideals and so that is really the towel of Hashi Corp and and I hope that if you've used our software You could see it every day and what you're using All the tech leads or projects all the engineers of our projects continually sort of apply this on a on a RFC by RFC basis when new features are being developed But this is also the principles that we use when evaluating software that we don't write when there's five different choices out there We tend to choose the choices that align Closest to our ideals and by having them written down and having them clear You know our opinion isn't isn't changing and isn't Isn't just what's best for the moment. It's actually really clear of what we care about and what we're looking for So I hope this helps. Thank you


My project Micro Recovery is about using mycelium to recover metals from e-waste. Mycelium is the vegetative part of fungi, so when we think of fungi, we, including myself before this project, often think about the mushroom, which is just the fruit, but there's actually lots of stuff happening underground, so mycelium are these networks of thread-like cells. They connect, for example, all the trees in the forest or in the park, so trees exchange nutrients and communicate with each other using this mycelium. And nowadays, there's lots of fascinating collaborations between artists and scientists and designers who use mycelium in really innovative ways, so there are people who use it to make artificial meat, to make leather substitutes, to make packaging, even bricks. And I found out that mycelium has potential to recover metals from e-waste. Fungi are the circulatory and nervous systems of nature, and the providers of nutrition, remediation, healing, and spiritual awakening. Micro Recovery is a speculative future that aims to heal our planet using mycelium, the vegetative part of fungi consisting of networks of thread-like cells. Today, among other things, mycelium can restore damaged environments, absorb radiation, recover metals, grow buildings and furniture, produce vegan leather by degradable packaging, and artificial meat. This chapter of micro recovery deals with the fastest growing waste stream in the world, discarded electrical and electronic devices, known by the umbrella term of e-waste. Following the waste management hierarchy, the speculation offers a system for dealing with e-waste, using mycelium in recycling and landfill remediation. The narrative also covers the societal and economic changes that we need to go through to turn micro recovery from speculation into a probable future. To initiate a transition toward this vision, we share a library of relevant projects by various fungi enthusiasts, both connected to e-waste management and other fields. The goal is to spread the spores of wisdom, connect the creative threads and inspire more chapters of micro recovery. When I found out that mycelium has this potential to recover metals from ores, I thought why not use it not only on our natural reserves, but on e-waste, because e-waste is actually the fastest growing waste stream in the world. I was oblivious to it, but it turns out that we generate, you know, it was, I think, 59 tons of electronic waste last year, and this number is growing, and we have to do something about it. So my project is a speculation, so I've speculated the future in which we're using mycelium to heal our planet and to solve our problems, including the problem of e-waste. And I've also considered, like, the societal and economic changes that we have to undergo in order for this project to become viable. The preparation for the exhibition has been a little bit stressful this time, to be honest, mainly because of all the uncertainty around COVID. We didn't know until basically the last moment whether it's going to happen, but it's happening. So even though we're planning to have everything online, so all of our projects are in the shape of a website, but it's a collaboration between, the exhibition is going to be a collaboration between future design and fine arts, so the fine arts guys are going to have all the, you know, physical projects, such as sculptures and paintings and whatnot, whereas we will have a projection, each of us has their own individual video, so you get kind of a glimpse into everyone's project, and we're going to try to make it atmospheric, so add some sounds and visual and see what works in the space. The challenging part really over the last 12 months has been that we haven't really met in person. We have a really amazing group, we're five students, and right now two of us are not even in Prague, so it's, yeah, it's been difficult to, you know, to connect and to stay on top of things in a way, but luckily everyone is responsible, so we're figuring it out. Well, each of us has prepared a video for the exhibition, and we called our exhibition Letters from the Future, so each of us got assigned a planet, so to say, that reflects our project, and you will just have to come and find out what it's going to look like. And what I would like to say to the audience? Well, I think it's really exciting that finally things are opening up, you know, we can go to restaurants, to gyms, to cinemas, and to arches, so I've really missed it myself personally, so I think for sure, if you get a chance, you should come and get inspired and meet some people, have some wine, and enjoy our work, we've been working a lot on it, so yeah, I hope you come and have fun.


So I've created a Slack channel where you can post questions. And also, I can copy questions. So I get questions from emails. And I will anonymously post those questions onto Slack with the answers. And so on that Slack channel, I'll be constantly trying to answer those questions in real time. And also, I'll be updating the lecture notes, fixing typos as we go along, and also perhaps elucidating points which were unclear that either in the lectures and the notes to help verify some issues. OK, so today the goal will be to essentially set up the correct effective field theory for electrodynamics in order to calculate the radiation from a binary in-spiral, electromagnetic binary in-spiral. And then tomorrow, we'll start off with a real problem going into GR. But once we've established all this machinery, that will be relatively simple. So let's remind ourselves about where we started off. So we're considering some charge distributions that are bound. And we want to calculate radiation at infinity from such a system. And we said one way to do that is to calculate the vacuum persistence amplitude. So here, I'm just writing it down for, say, a general scalar field like that. We would actually call that usually the partition function in a quantum field theory class. So in our case, for the electromagnetic field, can you hear me? Yes. Oh, OK. I'm lucky things went out. OK. Then that would be the vacuum persistence amplitude. And we said that we could write this as e to the minus e of j t, at least for static sources. And here, we're replacing j will be our, will actually be j will be basically our world lines. So j will be replaced by axonist derivatives, where we said we have some world lines, x i of lambda i, where each particle is parameterized by some affine parameter. OK, I see. Sorry. So people are joining the Slack channel in real time, which is fantastic. But I can hear it each time someone joins. I have to figure out how to turn it off, because it keeps beeping on my computer. Must be some way to turn off the notifications. All right, well, maybe, yeah, if you could hold off joining the Slack channel, at least till the end of this lecture, since if you have questions, now you can just put it in the chat. And then you can join the Slack channel afterwards, because every time somebody joins, it beeps on my computer. So it's hard for me to hear if anything's going on. So if you could just hold off on that just for a little bit. I'm sure there's some way to turn it off, but I don't want to take the time to figure it out at the moment. OK, so we're coupling our electromagnetic field to these world lines, which we're going to treat as sources. So the logic will be, well, first we'll treat them as sources, and eventually we'll elevate them to dynamical degrees of freedom by doing the path integral over them. So the action for the electromagnetic field, and here I'm working in the Feynman gauge, because it's convenient. That's our gauge fixing term. And now because the electrodynamics problem is Gaussian, the action is quadratic in the fields by completing the square in the exponent, we can actually do the integral exactly. So z of j just becomes e to the minus i of 2, where interestingly enough, so gf is the Feynman propagator. So gf in momentum space is going to be minus, sorry, g mu nu over k squared plus i epsilon. So at this point, you should be puzzled already, because this is a complex valued function. When k squared goes to 0, it picks up a complex contribution. And clearly, we're doing a classical problem. So normally, the propagator, the two-point function, should be real. If we're really interested in real propagation. But let's follow our noses. As particle physicists, this is what we do. And certainly, if we include quantum corrections, that's not a problem. We should be able to take the classical limit and still get the answer. So we're going to just plow ahead and not worry about that. And then we will worry about that. We'll come back to this issue of the i epsilon later. Now it's important to note that I didn't have a choice in my i epsilon prescription. Because to make the path integral convergent, I needed to add an i epsilon a mu squared, where the choice of this sign is fixed by the convergence of the integral. So I had no choice. I had not done anything by hand here. It's not ad hoc, OK? Good. OK, so what are our sources? Well, j mu of x, I'm going to write as v mu of lambda d lambda d4 x minus x of lambda, OK? So let me label since we have more than one particle. Now, notice that I've already worked in the point particle approximation, right? So I've taken my blobs. I've reduced them to point particles. Now we know, as we discussed last time, that we can include finite size effects by adding more terms to the action. And we'll come back to that in a little bit. OK, so now let's plug this in. So this z of j, we define as e to the i omega of j, OK? So omega of j will be equal to minus i over 2.  Several terms. I'll have d4k over k squared plus i epsilon. And I should change this sign here. So this sign actually was point in the notes that sign was wrong. That was pointed out to me by a diligent student, who I appreciate very much. And so this is what our result's going to look like, OK? And I still have, sorry, and I have e to the ik dot x minus y. And then I still have my two delta functions, d4 x minus x1, number 1, d4 y minus, sorry, yeah, y minus x2, number 2, OK? Good, all right. So let's explore out what this means, what this object looks like in the non-relativistic approximation, OK? So let's consider the non-relativistic approximation, because that's what we're going to be interested in, as that's the, so in electrodynamics, we could solve this problem relativistically. It's linear, it's simple, it's not a problem. But non-relativistically, it won't be so. So to warm up for GR, let's do the non-relativistic expansion in the electrodynamics case, OK? So the first thing we're going to do is we're going to choose lambda to be the time. And we're certainly free to do that. And let's look at the leading order result. Actually, you know what? Let me take it back. Before we do the non-relativistic approximation, let's explore this a little bit, what the meaning of everything is here, OK? So first of all, we see that there's going to be a real part and then there's going to be an imaginary part. And remember, we said this thing's going to go like e to the minus i ej of t. And this is, by definition, e to the i omega of j. So we're going to try to extract the energy. So the real part of the energy, so the real part of w, will be the energy or the potentials. And the imaginary part of w will be related to the decay rate or the power loss, OK? So how can we see that? Well, if we write k squared plus i epsilon, we can write this as minus i times the imaginary part, sorry, times delta k squared plus the principal part of 1 over k squared, OK? Thought of as a distribution. So the first thing we can see is that the imaginary part puts the photon on shell, right? It imposes k squared is equal to 0, OK? That means it's a solution to the equation's emotion. It's a propagating photon degree of freedom. So this corresponds to radiation, OK? Now, if we're interested in the piece, independent of i epsilon, this is going to be the real part. And this is going to be the potential, OK? So in fact, so I'm not going to go through this in the notes. But if we look at the imaginary part, so the imaginary part, I'm sorry, I'm not going to go through some detail because it's in the notes. And it would take a little bit too much time. So this we can see this is trivially true, OK? And this follows from just coming all the way back here. And taking the imaginary part, we get a, if I go to momentum space, this gives me a delta function of k. And then the fact, so this, remember that the famous identity, sorry, that the delta function on shell condition, OK? Which is just a phase space integral. So this on shell condition just converts it into a phase space integral, OK? So I don't know. Yep, there we go. Sorry. Good. So what I show in the notes, which I encourage you to look at, is so how are we going to relate it? We can relate the imaginary part to the power because we can see that the probability to emit nothing, so this is the vacuum persistence amplitude, is just the square of z. This is the probability for no emission. It's just going to be e, sorry, e to the minus d3k over 2k j squared, OK? And that just comes from squaring this vacuum to vacuum in the presence of the source. That's nothing like this, OK? So what I show in the notes is that if you want to calculate the power loss, then all you have to do to get the power loss is to weight the probability. So we know that from quantum mechanics that the real part or the imaginary part of the energy or the real part of w of j is just minus gamma t over 2, right? Because any probability, any decay probability just goes, sorry, any decay amplitude. And so the square that gives you the decay rate, so gamma is the width. Remember, this is the imaginary part of the energy, right? So what I show in the notes is that the power loss is just given by 1 over t e to the k over 2k times j of k squared. So I haven't shown you that here. Here, all I've told, the only thing we actually calculated was a vacuum persistence amplitude. What we really want, if we want to calculate the power loss, is the probability to emit n-quanta weighted by the energy of each quanta. And as I show in the notes, to get that, it's exactly given by this result, OK? All right, good. So now let's consider some examples. So let's work out an example. So we're going to start with, this is the exact result. Haven't made any approximations yet. That's our exact result. And we've claimed that once we calculate the real part, that will be the energy or the potential of the two sources. And once we calculate the imaginary part, that will give us the power loss, OK? So that's almost what exactly, so what we're really interested in as an observer infinity is the power loss, because we're measuring the power. We would also like to be able to measure the phase. And at this point, we don't have the mechanisms, the power to calculate that. And we'll have to do that. But for the moment, actually, even at LIGO, the first thing they really calculate is, or the first thing that they really are interested in is the power loss. From the power loss, you can get all the information. I'm not all the information, but mostly information that you want. OK, so let's do a simple example. And we're going to do it in the non-nulltavistic limit, again, to prepare ourselves for the real deal, GR, OK? And we're going to do the simple example in two ways. First, we're going to do it by taking the full theory result, the exact result, and expanding it. So in QED, we have that ability. In GR, we will not. Then we'll use that intuition to build an effective theory and re-derive that same result directly from the effective field theory. And the reason we want to do that is because we need to understand how to build the effective theory. Once we understand it in QED, doing it for GR will be trivial. So for our sources, again, we're going to take, so j mu of k is going to be the Fourier transform. So we're going to use the Fourier. So we're going to use our delta function. So there is our source. And so now we're going to make a non-ralthavistic approximation. And we're going to choose lambda to be t, right? So now, excuse me, w is 1 half dt prime. Those are the two times for the two different world lines. There is our propagator. And then we have e1 squared, 1 minus v1 squared, e to the ik, x1 of t minus x1 of t prime. So we have to make sure that we use different names for our affine parameters. I'm calling 1t and 1t prime. Then we have the same thing with 1 going to 2. And then we have e1, e2, 1 minus v1. Let me put this in explicitly. Plus, again, 1 goes to 2, which in this case gives an identical contribution. So I've used the d4x integrals. And that's why we end up with the positions in the exponents. OK? OK. So I should put it, we're making the non-ralthavistic approximation, and we're going to keep up the order v squared. So the first nontrivial correction. Now, how do we know that there's no linear piece in v? And the answer is that the symmetries forbid such a term, because we know the system is time reversal invariant. So the first correction has to go like v squared. If we were looking at some other effects, which did not preserve time reversal invariance, like a dissipative effect, friction, something like that, then that will no longer be true. OK. So the first thing we do is we know that in the static limit, let's consider the static limit. So in the static limit, we're going to take the b to 0. We're going to add two static charges. And then if we look at the exponents and we look at the k0 piece, we're going to have t minus t prime in the exponents. And so this is going to lead to delta k0 once we integrate over t. So we can see that the static limit is associated with vanishing energy. And I should say to gain just a little bit of intuition, what we're essentially doing here is we're considering the exchange of one photon. So electrodynamics is linear. So that's all you get is just one photon exchange, and you get nothing else. When we get to gr, this will get dressed by nonlinearities and so forth. But for the moment, that's all we have. So what this is telling us is that in the static limit, there's no energy transfer between the photon carries no energy so that we know that in general, away from the static limit, the energy transfer is going to scale like some power of the velocity. And we further borne know that k, just by dimensional analysis, the only scale around is r. So we can see that in the non-relativistic limit, k0 over k is over order v to the n, where we haven't fixed n, but we will in a moment. So that means that we can treat k as being a small quantity that with which we can expand. Now let's consider the terms we have in our expansion. We have these terms that go like e1 squared and e2 squared, and then we have the cross terms. Let's get some physical insight into what the meaning of those terms are. So let's consider the e1 squared term. So we have something like dt dt prime 1 over k0 squared minus k squared. And then we have e to the i k0 t minus t prime e to the i. Now we're going to take the small velocity limit. So if we drop that term at leading order, then we can see that this thing is going to be proportional to, sorry, and then I have a dk0 d3k. I do the time integrals. So now I'm going to have d3k over k squared, delta of k0, actually, OK. Actually, I'm sorry, let me do the k0 integral first just to clarify things. So then we have 1 over k squared, and then we have delta of t minus t prime dt dt prime e to the i k dot x1 of t minus x2 of t prime, OK. So we can see t is equal to t prime. This goes to 0.  That's infinite, OK. That shouldn't surprise us. So this is just the self-energy of the charge, OK. So that's the usual self-energy in classical electrodynamics. And from the point of view of quantum field theory, this is a divergence, which is a power law divergence. And we can just absorb that divergence into the charge, OK, or into the mass, in this case. So from the point of view of quantum field theory, of course, we could do this integral, say we have to regulate it. We could do it in dim reg, dimensional regularization. But in dimensional regularization, that gives 0, right? So in dimensional regularization, every scaleless integral vanishes. So that's a little bit weird. So is it infinite or is it 0? And the answer is from the point of view of effective field theory, it doesn't matter. Because any time you have a power law divergence, it's telling you that it can be absorbed into a local counterterm. And there is no logarithms involved. So I don't really have time to go into this. But basically, whenever you have a power divergence, you can just toss it. It has no physical effects. It can always be absorbed into a counterterm. And physically, or I should say technically, of course, it had to be true. Because if I used a cutoff, I would get one answer. And if I used dimensional regularization, I get 0. And it shouldn't matter how I regulate my theory, I need to get the same physics, right? So it's clear any time you get a result, which depends upon the regulator, assuming the regulator is not breaking any symmetries so you don't have to worry about anomalies, then you should get the same answer with any regulator. So if you're getting 0 and 1 and thinning the other, that means it just can't be physical, which means it has to fall out of any prediction for any measurable quantity, which is another way of saying that it can be absorbed into a counterterm. So that means we can come back here and we can toss these guys. So we can toss this guy and we can toss that guy. And let's just look at what we're left over with. So we're left with the cross terms. So what do the cross terms look like? Well, let's look at the leading order piece. So we're going to set v to 0. And we're just going to look, so this is a purely static limit. And in that case, we have minus 1 half dt dt prime d 3 k k squared. So this is w. And then we have a factor of 2 because we have two cross terms, e1, e2, e to the i k t minus t prime, e to the minus ik, sorry. And then we have, yeah, so this is k0. So I still haven't done the k0 integral. So sorry, I realized I just introduced some notation. So in my lectures and in my notes, ddk with a bracket means ddk with a 2 pi to the d. All right, so now I can again do the dk0 integral. That gives me a delta function and kills one of the 2 pi's. And I have to be careful because this is a plus sign. Where did this plus sign came from? That comes from this minus sign, e1, e2, delta t minus t prime, e to the i k dot x1 minus x2, OK? Hopefully I've tracked the signs right. There's a possibility that I have not. Now I could just do this. So this is do this integral over time. And then I just get the Fourier transform. There's one time integral left over that's just going to give me a t. And this is equal to minus e times t. So e is d3k, or it's just a Fourier transform. OK, so that's a sanity check. We've just re-derived the Coulomb force law. So you might want to do, if you're not familiar with this integral, and the easiest way to do this integral is to first put in a mass and then take the limit when the mass goes to 0. So you might want to do that. Just take you a few minutes in your spare time. Now let's go to order v squared, or next to leading order. And there are two sources of v squared corrections. So let's come back and to see where those two sources of v squared corrections are. OK, so we're ignoring now the self-energies. So there's a v squared correction here. And there's also a v squared correction hidden in here. OK, so there are two sources of v squared. The first one is just the explicit v1 dot v2. And the second one is from the fact that k squared. I know you have a question. Uh-huh, OK. Matt? Yeah, hi. So my question is about this order v squared corrections. So you put lambda equals g, right? But it seems to me that from there, you also should get a v v squared correction because it is proportional to tau times this gamma factor. So you should get v squared from that as well. Yeah, so in this, so in general, you write there are v square corrections from the affine parameter, writing it in terms of t. But in this case, there isn't. So in the case of, so certainly the m, this, from the kinetic energy, you will get that. But in this case, you don't, right? Because I'm taking lambda just to be the time, not the proper time here, right? But you're right, in the kinetic piece, you will get that. OK, I see. Thank you. Yeah, OK, good. So here, so this guy is a v squared correction as well. If we take v, k, sorry, to be of order v times k, OK? So remember before we said that k0 is of order v to the n. And we didn't exactly specify what n is. But now I claim that n is equal to 1. And you will see that quite clearly because, remember, k in x space is just d by dt, right? k0 is a d by dt, right? In other words, I can replace in my any integral I have here, starting at this point, the very beginning, any k0 I would have say here, I can just replace by a d by dt. And then by an integration by parts, that's going to give me a v when it hits the x's, OK? So k0 will scale like v. So this term will be order v squared, which is of the same order of the term that we're keeping, OK? So the first term will just be proportional to the Coulomb potential, right? We don't have to redo all that work because if we come back here, we could see that it's just the whole thing goes like 1 minus v1 dot v2, right? So there's no effect on the calculation. It's no this different than it was for the Coulomb potential. But for the second term, we're now going to get something like dk0, d3k, k to the fourth, k0 squared, e to the i, k0t minus t prime, minus ik dot, OK? So we can rewrite this as plus d by dt, d by dt prime. Let me write now this is delta x just to save some time. And now there's still a dt prime integral here. So now this becomes, and now these guys are evaluated at the same time, OK? So I can then just perform this integral and take the derivatives. And the result looks something like, so you've got something like v Coulomb times 1 minus v1 dot v2 minus e1 e2 over 8 pi r v1 dot delta x over r v2 dot delta x. And this is the first, so these are the first, these are the order v square corrections to the Coulomb potential. So just in general, so this is to do the k to the, 1 over k to the fourth integral, I'm going to give you a magic formula that you can prove for yourself as a fun exercise for a general power alpha in d dimensions. I'm doing it here in d dimensions. The reason I'm doing in d dimensions is because eventually, when we get to more complicated integrals or complicated situations, these integrals will diverge logarithmically. And as opposed to power divergences, logarithmic divergences are physical. So in fact, one of the reasons I actually originally got involved in this whole business was because when people were using standard calculation methods of GR to calculate these post-nuttonian corrections, they kept running into divergences. And this was a bit of a mystery. And they introduced ambiguity parameters. And it got rather complicated. And Walter Goldberger and I, my collaborator, not realize that, wait a second, from the point of view of quantum field theory, we see divergences all the time. It's not a problem. We don't have to introduce the ambiguity parameters. You just have to introduce counter terms. And you can introduce counter terms once you've built up the correct effect of field theory. So it's kind of the motivating factor behind trying to come up with this formalism. And you get logarithms in a classical calculation. And you get 1 over epsilon. And you get a renormalization group. And you can sum logarithms just like you do in quantum field theory, except these logarithms are not due to quantum fluctuations. So it allows you to use all the tricks of renormalization group in classical field theory to make predictions for logs at higher orders in all the power that comes from doing the RG. OK, so maybe this is a good time to take a break and to see if there are any questions that we want to talk about before we go on. No? OK, so I think this is a good. So when we come back, we'll talk about how to do the radiation. And then we'll talk about how to build up the effect of field theory. Perfect, so I'm back in five minutes, OK? Yeah, all right. But there is a question if you want to take. Yes, I have a question. So since we are in classical field theory, I'm wondering where does this correction to the collop to come from? So is this due to some reaction for us coming from the photon, or what is it? Yeah, I think it's just a purely relativistic correction. Ah, OK. In fact, you can think of, so the k0 squared correction is a correction to instantanity. It's the first correction due to the finite propagation speed of the light that's being exchanged. OK. It's the best way to think about it. OK, I went whenever you want to. OK. OK, so sorry. So let's talk now about the radiation. So the radiation comes from the imaginary part of the energy, or the imaginary part of w. So let's go back to our master formula for w. But now let me, instead, let me not multiply it out. Let me just write it like this. And then I, sorry, and I still have my exponentials. So why do I want to write it like this? Because it'll clarify the physics a little bit. So now we're interested in the imaginary part. So the imaginary part will be proportional to delta of k squared. Excuse me for one second. Today is the last day of school, and my children are home.  I'll be right back. OK, sorry about that. OK, good. So we said that k0 is of order v times k, which is order v over r. OK? So k0, which is now of order k in this, sorry, so hold on. Sorry, I need to come back. I need to take a step back for a second. So in the case of potential modes, or in the case of the potential, we found that there was a hierarchy that k0 was of order vk. OK? But now we can see that k0 is exactly equal to k because it's on-shell radiation, right? But the wavelength of the radiation, we know, is going to be k0, is going to be of order omega, right? Which is of order v over r. OK? So we can see that for radiation, we have k mu is of order v over r v over r. And for potential, we have k mu is of order v over r and 1 over r. OK? So at the very beginning of the first lecture, we talked about the fact that we're going to break up the field into long distance and short distances. And one thing we can see immediately is that the radiation is a long-distance piece and the potential is a short-distance piece, OK? So in a few minutes, when we do the EFT, that will be very important. So part of the reason I wanted to go through the full theory calculations through the exact calculation was to teach us what are the short-distance modes and what are the long-distance modes. And so especially for those of you who are familiar with effective field theories, you need to integrate out all the short-distance modes to get the long-distance physics, and that's what we're going to do. But again, we haven't got there yet. Let's take one step at a time. So what we do notice, however, is that k, which is border 1 over r, is order v. And it's therefore much less than 1, OK? So this is just a statement that the wavelength of the radiation is parametrically longer than the distance scale set by the distance between the two objects, OK? So the radiation with some lambda is much greater than r. But this is nothing but the multiple expansion. This allows us to perform the multiple expansion. So we're going to take k dot x is much less than 1, OK? So now let's look at our w in the multiple expansion at leading order. What happens? Well, let's come back up here. So at leading order, we're going to take, so dx mu vt is just going to be 1, 0. And then we're going to end up with d4k k squared plus i epsilon. So at leading order, I'm going to take, remember, e to the ik dot x at leading order will be 1. So I can drop all those guys. And then I'm just going to end up with e to the ik0 t minus t prime. And then the products of the ease, again, all coming from our master formula here, OK? So now if I take the imaginary part at leading order, this is going to give me something like d4k delta of k squared, where I've dropped everything which is sub-leading. Now this thing, so this is going to give me d3k over 2k as usual. And then I still have the dt prime and the dt. There's no scales involved. So this is some ill-defined divergent quantity, which of course, we can adjust as well just set to 0. And this is just a statement, as we'll see in a second, that there is no monopole radiation. Well, at the moment, that may not be clear. But physically, it's due to charge conservation. Any time a field couples to a conserved quantity, it's not going to radiate, right? The charge can't change. So essentially, all we've done here is asked whether or not actually, yeah. I mean, you can actually show this as explicitly 0, even without regulating it. And that's just a statement that if my source has no time dependent configuration, if the charge is constant, the charge can't change. The charge can't change. It can't radiate, right? Radiation comes from time dependence of the charge distribution. So we need to get radiation. We need to go to next leading order, OK? So that means we need to keep the first piece in our expansion of the plane wave. So this should be familiar to you from quantum mechanics, where this is how you calculate radiation from an atom, right? And you do it in the multiple expansion by taking the fact that the radiation has wavelength, which is parametrically for non-relativistic systems, which is parametrically larger than the size of the system. So now if I keep this term, then the imaginary part of W will look something like, now there's a sum over the particles implied, OK? But so this is the decay rate. And if I take this integral and I weight it by the energy, that will give me the power loss. So weight by the energy. If I weight this thing by k, so I add to this a k, right? Then this thing is equal to, and you can do this as an exercise for yourself, as the inner product or the dot, the square of the second derivative of the dipole moment of the system where the dipole moment is. So this is called dipole radiation. And I have skipped steps in going from here to here because of time limitations. But this is in the notes. And the crucial point to understand is we've done a multiple expansion. The leading term in the power loss, we get from the imaginary part and weighting it by the energy of each emission. And then we just get back the standard result for dipole radiation of electromagnetic system, OK? So if we were incredibly naive and just said, OK, let's do the simplest thing we could possibly do, we've calculated the corrections to the potential, and we calculated the power loss. So then in order to calculate the signal that we'd be expected to see the detector, we would solve the equations of motion, take those solutions to the equations of motion, plug them in here, and read off the power loss. OK. So let me pause for a sec because now the next logical step is how do we derive from an effective field theory? So how do we build an EFT to reproduce this result, which seems pedantic, but will allow us, when it comes to GR, just to jump in immediately and solve the problem, OK? So are there any questions? OK, yeah, we have a half hour to go over the questions, so let's say that for now because I'm kind of running out of time. So how do we reproduce this result in the effective field theory? OK, so the basic rules are, one, every term in the action should scale homogeneously in your power counting parameter. This is the prime directive in effective field theory, right? So why is that? Because your goal is to calculate systematically. And in order to calculate systematically, you have to keep track of every source of a correction. And by having each term in the action scale in a definite way in V, allows you to figure out what you need to do in order to include all corrections up to whatever order in V you're interested in calculating, OK? So in the full theory, you have nothing scaled homogeneously. Everything is just glob together. It's just everything you need. But we want to break it up into the pieces that we want, OK? So that is the prime directive. And how do we go about doing that? Well, we have to separate out all the pieces which contribute in different ways. So the first thing we need to do is we need to take our electromagnetic field, and we need to split it up into the two pieces which scale differently. Now, our intuition from going through this problem tells us exactly how to do that. There are two different types of modes in the theory, and they scale differently. So the full theory electromagnetic field, we can write as a mu potential plus a mu radiation, OK? So the momenta of this field scale as V over R, 1 over R, and of this field scale as V over R, V over R, OK? Now, mathematically, how do we do this in a sensible way mathematically? Well, we have to make sure, for instance, that we don't double count in our path integral. And we will discuss that later on. But more importantly, we have to figure out how to do this in a gauge invariant way. So the way we do that is we work in the background field gauge, which hopefully you're all familiar with from your quantum field theory classes, where we treat the radiation as our background field, OK? So we can choose a particular gauge fixing condition such that we can fix the gauge for the potential independently of the background field gauge, all right? So we'll fix this with a fine with a fine engage, as we did before. And this guy, by working with the background field method, we can choose any gauge we want in independent gauge. We could choose a fine engage again, OK? This is basically the Born-Oppenheimer type of approximation where you have fast moving degrees of freedom and slow moving degrees of freedom. The potential of the fast moving degrees of freedom and the radiation of the slow moving degrees of freedom. So we freeze the slow moving degrees of freedom and treat them as a background field. And then afterwards, after we integrate out the fast degrees of freedom, then the slow degrees of freedom are allowed to evolve dynamically. It's much like what we did when we took the point particles to be sources and then allow them eventually to be dynamical, OK? So now our partition functions, z of j, we can write as d potential deradiation. And then we're going to do the path integral over the fast degrees of freedom to get an effective action, OK? And the idea is we want to reproduce what is this theory. This will be our effective field theory. And we must choose as effective to reproduce all the long distance physics to the order of v of interest. Good, OK, so let's take, so now what we're going to do is we're going to take this decomposition and we're going to plug it into our action. Then we're going to derive what this action is. Then the next step will be to do the path integral by calculating Feynman diagrams to get rid of this guy and to generate that guy, OK? So the first thing we're going to do is we're going to take the potential and we're going to Fourier transform it, but only in space, not in time. So we're going to do a partial Fourier transform, OK? And then we're going to plug it into the action, OK? So that's our interaction between our source and this is j.a. And then we have the bulk action. First we have it for the potential field, OK? That is just f squared plus del dot a squared. And then we still have the action for the radiation field plus whatever gauge fixing term we're going to choose for the radiation, OK? So this is the first stage of the effective theory. And we want to make sure that in this stage of the effective theory, all our terms scale homogeniously. So we need to check the scaling of each term, OK? So how do we determine the scaling? So well, first of all, the field scales.  Once we know how each one of these scales, then we'll be able to read off how each term in the action scales, OK? So first, some of them are easy by dimensional analysis. So we know that k for a potential, right? So these k's, so remember this k here, everything which is k, is acting on a potential field. So remember the potential is the guy that is, you could put p's here just to remind yourself, but then it gets a little confusing between p and k. So let me not do that. But at least here, the only thing we Fourier transformed are the potentials. And we already know how the potential momenta scale. They scale like 1 over r, so v to the 0. They don't scale at all. We know how d by dt acting on a potential field here scales. So d by dt acting on a k, we know goes like v over r, a k, OK? So therefore, we learned that d by dt is down by v, as we expected, over r. Now what about dt and dx? Well, dx basically is just conjugate to 1 over k. So we have to worry about, so here, we have to distinguish between how the measure scales in radiation versus in potentials, OK? So notice that we don't have a d4x for potentials, because we Fourier transformed. The only place we have a d4x is for radiation, right? So the dx here will scale like 1 over the k of the radiation, which scales as v as r, this thing goes like 1 over k radiation, which goes like r. And then we know this is a scaling of v, because remember that k0 scales like k, this is for radiation, scales like v, right? So this guy, so dx is going to scale like 1 over v, OK? What about dt? Well, for the energy, dt scales like the inverse of the energy, because it's conjugate to the energy, or you can think about it. The only time scale is the orbital frequency. So dt is going to scale like 1 over omega, which goes like 1 over k0, right? So it's going to go like 1 over v, OK? I'm not worried about the units. The units are already always made up by r's. We're only interested in how things scale in v. So now we're all set to figure out what is the scaling of the field. To determine the field scaling, we make sure that the leading order piece scales as 1, OK? So this is how we read it off. So let's figure out first how ar scales, OK? So we have ar squared, and then we have a box squared, and then we have a d4x. And this thing should scale as v to the 0, OK? So we have ar squared, and the box scales as v squared, and the d4x scales as 1 over v to the fourth, right? Because the scales is 1 over k to the fourth, each k scales with a v. Box squared scales as k squared, each k scales with a v. So this has to be v to the 0. So therefore, ar has to scale as v, OK? And if we wanted to get the units right, we would know that it goes like v over r, OK? How does ap scale? Well, let's come back and look at the leading order piece for av. Now we know this guy is the leading order piece, because we know this dt is going to give us a v, and this guy is going to go like v to the 0. Remember, k, for potential, scales as v to the 0. It's a short distance guy. So now we have, now remember, we're looking at ap in momentum space. So we want to know how apk scales. So we have dt, d3k, then we have a k squared, and then k squared, OK? And this guy should scale as 1. So dt scales like what we just said, 1 over v, because it scales like 1 over omega, which scales like k, and k for potential, k0 for potential, remember, goes like v. Now the k, the 3 vector k for potential, scales like no powers of v, right? So we said that there are no powers of v for the 3 vector piece. So that gives us a v to the 0. This gives us a v to the 0. This gives us a v to the 0. So ak potential scales like v to the 1 half, OK? So that's the first step in the algorithm is to figure out how everything scales and make sure, and you figure out the field scalings by taking the leading piece and making it scale as 1. So now we could see, for instance, that this piece here in the potential is order v squared, OK? And sorry, I'm running two minutes late. So now that means this has to be treated as a perturbation to the leading order piece. So this tells us, let's see how we can just get back what we already know. If we look at the propagator for the potential mode, this leads to the propagator, which is 1 over k squared. And this leads to a correction, which goes like k squared, k0 squared, right? So if we look at the propagator, what does it look like? Well, it looks like k 1 over k squared. So I'm going to draw a line like that for my propagator. And then I'm going to correct the propagator by inserting one insertion of this operator and so forth and so on. That's rederived the geometric series for the propagator. But we've done so in a systematic way in the sense that each term in the action now scales uniformly in v. So we should really treat k0 as a perturbation, right? So now if we come back and we want to calculate systematically the potential, well, we have the leading order exchange of a Coulomb. Let me draw Coulomb propagators now to distinguish them from the paths in the world lines. Well, this diagram, this is a 1 over k squared propagator. That just gives us the Coulomb potential. But now we need to correct that propagator by inserting a k0 squared there.  And then finally, the v1 dot v2 term is just going to come from back here, here, where we pick out the v piece that couples to this potential here. So that's just going to be we can draw that as this, one other diagram. So we have two Feynman diagrams, two non-trivial Feynman diagrams at next to leading order, v1, v2. So this is an insertion of v1, this is an insertion of v2, and this is going to give us v1 dot v2 times vCoulomb. So what we've done here is we've parsed the full theory into homogeneously scaling pieces and use that to calculate the v squared piece. So this seems like we're just, we are literally reinventing the wheel, but for a very express purpose. OK, so the next thing we'll do next time is we'll reproduce the power law from the effective theory, and then we will move on to GR. OK, so let's stop there for questions. I think I apologize for going a few minutes over. No problem. Thank you very much, Ida. Nice lecture, and now we can start the Q&A.


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Hey everybody, welcome to this week's video update. Today is Friday, June 9th. Hope everybody had a great week of trading. Let's jump in to the trade. So first trade we made was an opening trade in EWZ. Implied volatility was still really nice and high in there, so we've put on and taken off a couple of positions in there over the last couple of weeks. Ever since they came out with the corruption going on with the president, the IV spiked up, we already had a strangle on which we had to manage our way out of. But we've continued to, we actually put on a strangle here, had that huge drop, had to roll down our calls, manage our way out of that one, roll it, and then we've put on and taken off a couple other strangles in this period here, giving us some nice opportunity. So the current position that we have on, that we put on on Monday is this one here. So it's still very centered, not quite enough profit to take off yet. The implied volatility did have a little bit of an uptick today with the down move. So we, so we'll wait to take that off for a little bit more profit. Next trade was a closing trade in wheat. So we closed out our put verticals. So in a previous, a couple of weeks ago, we took off our call side because the put side got breached and then price in wheat made a nice reversal, nice move up. And so we booked that profit in wheat, giving us a really nice profit on the overall iron condor. And then in EWZ, I already mentioned that we, this is one that we closed. So this was our 35, 36, our inverted strangle that we had rolled from June. So we took that off, took, got back a lot of that original loss that we took on that huge move in the EWZ ETF. And then we'll continue to add on positions if implied volatility stays high. Also had an opening trade in corn. So again, remember guys, when we, when we have these periods of low implied volatility, like we've had, I'm going to turn to some of these vehicles like corn and wheat and soybeans that, that where you get a really good credit for the amount of risk that you take. And I know some members don't have that futures opportunity to trade yet, but you want to get that as soon as possible because they are really good trading vehicles. So let's take a look at corn. And I try to, I try to spread these out.  Some people only trade these futures. So I like to sprinkle those in some only trade equities. So I, I use those too, but the concepts are the same. Okay. And this is really about learning how to trade these different strategies. So here's the iron condor and corn. You can see it's still well within our range. Nothing to do there yet. Next, next trade was a, was a closing adjusting trade in wheat. So we closed out the put side of our wheat iron condor price breached to the upside. So we, we, we took off the untested side just like we teach in the course. So here's what that looks like now.  And if it moves down some more, then we'll take this off and bank a profit on the overall trade. And then when I make that adjustment, what I also like to do simultaneously is add another centered iron condor. Okay. Collect some more credit, give ourselves more time to be right and widen those break even. So I did that and this, this position is in July. So I added the new one out in August, which currently has 42 days to expiration. So right in our wheelhouse and as you can see, you know, still, still well within our range here, nothing to do. What you'll notice if when you're putting on these grain trades right now because of the, the little spikes that we've had and the option premium increase, the options are pricing the risk to the upside. I mean, look at when I put this on, the price was about right here, but look at the difference in how much we have from our downside break even versus our upside. Look how much more room we have on our upside than we do on our downside. So just interesting the skew that's in those grain option premiums right now. I still like to put it on based on the deltas and let that, let that skew just play out, but that kind of interesting to see. We usually don't see that much skew in those next trade and last trade was a, what did I just mention the iron condor that we added in wheat in the August cycle. So those were, those were the trades for the week. Let's take a look at some of our other current positions that we still have on, still have this strangle on in oil prices hovering near our downside down here, but nothing to do yet. We'll continue to wait and monitor oil. We've, we've seen a little bit of a decline in the implied volatility of oil. As you can see here, but, but we need a little bit of a move up to profit on that trade at this time. Next trade, I mentioned corn. I mentioned soybean. Did I mention soybeans? I don't think I did. So in soybeans, we've got this. We've still got the put side of our iron condor. So prices moved back up into our range. So looking for a little bit more of a move up before we take that off and that's in the July cycle, which only has 14 days left. So we'll probably make take that off late next week. If we don't get a move in our favor or potentially early the next week. And then same thing here. We added another iron condor in soybeans. So that's still very centered. Nothing to do here. DIA. We've got this iron condor on in DIA. What's interesting today in the market's not closed yet. Look at the Nasdaq. Nasdaq is down over 2%. Yet the Dow is up 60 points. It's up over a quarter percent. I mean, you just don't see that usually. Typically these indices are much more correlated. But interesting how these large cap stocks are holding up. Whereas, you know, some of the big name high flyers that we've been that we've been seeing just climb and climb and climb like apples and Facebook's and Netflix. They're just getting absolutely crushed today. So it's actually been good for our portfolio because we had a QQQ vertical on which I'll just go to that now actually which looked like he was going to be a full loser. But look at this. It's moved all the way back up here. And, you know, part of the reason if you remember why we're holding this is just to add to keep some of that short delta in our portfolio because as premium sellers, we want to have that protection to the downside. And I mean, look at it now. It's really helping our overall portfolio out had to make it up about $300 just on this trade today. And something something's going on out there. You know, something spooking the spooking the techs. Let's see if we'll see if we get any follow through in the in the other stock indices and specifically the large caps. S&P's basically flat just down a couple points and the Dow still holding strong up 65 kind of crazy. Let's see. So I mentioned the Dow. So we need a little bit of a down move in Dow. If it continues to move higher next week, we will adjust that by taking off our untested side and then potentially adding another centered iron condor around that. There's only oh, no, we're in we're in 42 days there. So we got plenty of time to adjust that to adjust that that trade EFA looking for a little bit of a down move in EFA before we bank that profit. We've got that adjusted strangle in EFA in EWZ. I mentioned that Microsoft big move down today, which has helped us. It was hanging around our break even wasn't quite to the point of adjusting. But, you know, is going to do it either today or early next week. If it continue to move higher, we've got a huge move down right back and centered in our range. So that's that's excellent for Microsoft. I mentioned the Q's SPY. We've got this double calendar on just looking for a little bit more profit before I take this one off. So we've only got let's see seven days left in the front month of the of our SPY calendar. So we'll be aggressively taking this off early next week regardless of what it does. But hoping for a little bit more of a down move potentially get a little bit of follow through on what's going on in the Nasdaq before I take that off. And same thing with TLT. I just just need a little bit more of an up move before we book that profit in TLT. As you can see, we've had a little bit down move there implied volatility still around zero as far as the IV percentile and IV rate go. So nothing else to do in TLT at this time. So hope that was helpful. Everybody have a great weekend. Look forward to next week where we will be live Monday morning and we'll be we'll jump back into that. So hope to see you then have a great weekend.


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Hello, everyone, and welcome. This is Melissa Armour with the stocks, which are viewing Nike. I called two really nice, strong trades in this. And again, conviction helps a lot. I know some people killed it. And this is working. So I called the 90 calls and the 92 calls back when the earnings occurred, not even a week ago, but last week it was. And I knew this was higher. And this is on its own without the market. So ideally, 95, 94 could happen here today. You could be out of this right now. This was a trade that was down before it went, but it worked. And sometimes this is what happens when you're doing overnights, you kind of give them a chance to work. But the conviction helps. The rating system helps. How do I call trades like this? How did I know Nike was higher on this day when I called it? Because I rated the gap. And per the Golden Gap rating system, it rated that the stock was higher. Even though that particular day, it didn't make the move. It took a couple of days to work itself out. And this just looks higher now. So 93.90 was the high here. Actually, it's at it right now. Yeah, this is basically at 94 now. 247. 247. Yeah, Nike is on its own. Nike has nothing to do with the market. And that's the best way, yeah, 94. So again, everyone, you can watch. The people that are still in it, this is a nice move, a nice trade. It couldn't continue tomorrow, but today is a good solid exit. And really nice move. Really nice move. And congratulations to everyone that kept it. A bunch of people in the room said they did. So again, conviction helps. Reading the gap. Reading the gap analysis. Some trades work immediately. Some take a little time to go. And that's why you have to assess your risk every trade. Good luck, everyone. If you're interested in the Gap Options newsletter, email me and Melissa at thestackswush.com. Have a great day, everyone.