 All right, so it's time if you, it's okay, we can start. And so I welcome everybody. It's my pleasure to welcome Ira Rothstein from Carnegie Mellon University, who will be giving a lecture on effective field theory for gravitational waves, please. Thank you. It's very nice to be here. I look forward to interacting with many of you as much as we can given the situation. So hopefully you've been made aware of the fact that there are, sorry, there are lecture notes available and hopefully, so as I mentioned in the, or as I noted on the front of them, that you should be careful because there could still be typos in there and I apologize for that, but I would appreciate if you do find typos that please send me an email. And in general, feel free to email me in between classes. If you have questions that I'd be happy to answer in between classes. And I hope that you'll consult those notes for time sake, I'm gonna have to skip some steps calculationally, but most of the details of all the calculations of many of the things that I'll be talking about in class will be are in the notes. So for the steps that I skip during lectures, during lectures, please consult the notes. Also good, there are exercises in the notes that I would recommend doing if you have time and we can maybe in the question and answer section on the following day, I realize today's not too late but I think on my next lecture on Wednesday if you've tried any of the problems and you wanna go over them, we could go over them in the discussion section. Also I should warn you, so I'm lecturing in the morning and so about this time is when my children go to school and so you will probably hear noises and I apologize, it will only last for 10 or 15 minutes before they leave. So, okay, so I apologize for that ahead of time. Okay, so hopefully everyone can see my iPad. My iPad is visible, right? Okay, great. Yes, yes, sir. Great, so we're gonna talk about gravitational waves from the point of view of an effective field theorist or someone who has a training in particle physics. So I'll have to teach you both about gravitational waves. I'll have Valerie will be teaching the same subject from a more canonical point of view. So this will be complimentary to her lectures and but I'll be also teaching effective field theory at the same time. So I'm trying to teach sort of two subjects at the same time I've only got four lectures. So I'll do my best. At some points, I will have to ask you to just take things for granted and then consult the notes because there simply won't be enough time for me to teach you both effective field theory and gravitational waves. So I'm gonna assume a minimal amount of knowledge of GR. So just if you know, I'm assuming you know what a metric is and I'm assuming you know Einstein's equations and so forth, but not know nothing at the level of differential geometry or any of the technical issues that are often dealt with in GR. And I'm assuming that you've had a basic class in quantum field theory. But I think this my lectures will be self-contained to that extent. So hopefully you'll have seen or be familiar with most of the concepts that or technical tools that we'll be using in these lectures. Okay. And also please feel free to ask questions during the lectures. And if you have any questions if it's technical and we take a lot of time we'll wait till the discussion session, but hopefully, you know and if it's not just needs clarification, please interrupt. Okay. So before we go into this, I would just wanna say a little bit about why this is interesting. So if we're gonna learn all this technical machinery and do these calculations, we should have at least some motivation for putting in the hard work. Okay. So gravitational wave physics, why should they be, why is it interesting to say a particle physics or someone who's interested in fundamental physics? So let me just give you a few of the reasons why I personally am interested. So in particular, I believe that in terms of fundamental physics the great potential is for new sources or new astrophysical objects, something like say a bow star or axion halos surrounding binary systems. And I think the fact that we still don't know what the dark matter is gives us hope that we will find signals that we have not searched, we have not thought of in the past or something beyond the standard model. That is my personal opinion as to our best hope for finding new physics and gravitational waves. You know, the other possibility of course, is that we find deviations from GR. I personally think that that's probably highly, highly unlikely given the fact that GR is a very delicate theory in the sense that if you mess with it it starts to fall apart very quickly. There are theories of massive gravity but they all suffer some sort of, I don't wanna say pathology but they're very difficult to make work and be consistent with all known tests of general relativity. Finally, from a fundamental physics point of view the question of the neutron star equation of state is still an open question. So is neutron star in a hydronic phase or is in a cork phase, cork blue on plasma? These are unknown questions. That in principle, we will be able to address or learn about from gravitational waves in the future. And we'll talk a little bit about how one could go about trying to learn something about the nature of the equation of state of neutron stars. So this is really a question of QCD and is related to heavy ion collisions and so forth. So from a particle physics point of view that's really an interesting question. Now there's also the issue of what can we learn about black holes? Now it's very unlikely, very, very unlikely that we'll be able to learn anything about say hawking radiation or the information problem from gravitational wave physics. And then if we have time in the last lecture I'll, we'll talk a little bit about maybe about how hawking radiation, why hawking radiation is so negligible, even though at first mathematically it may seem to the contrary and there are a remarkable number of claims in literature that hawking radiation could have an effect on that signal but I'll explain to you why I think those claims are false. But one really interesting aspects about black holes which I hope we get to by the final lecture is that they have a very unique property in the sense that a classical property that hasn't been tested before. And in sense that they have vanishing polarizability. This is a prediction of GR which is to say that if you put a black hole in a background field it will not generate a quadruple moment. So think about the analog of an electromagnetic case if you have an atom and I put it in an electric field it generates say if I put it in a constant electric field in this direction it will generate a dipole moment in that direction. But black holes if you put it in a background field then it will not generate any multiple moments. This is a really strange result which is sorry which is a prediction of GR and we will be able to test that in gravitational wave experiments. From a point of view of a particle physicist this again is rather enticing because it leads credence to the idea that black holes are fundamental objects. In a sense if I had a point particle if I put it in a background field it doesn't distort because it has no size. So if we think about black holes as almost as fundamental objects even though they have finite size then that's consistent with that interpretation as a rather remarkable prediction of general relativity. So those are the fundamental from the point of view of fundamental physics why gravitational waves are interesting. Astrophysically of course has its own motivations for studying gravitational waves. So in particular there's a question of black hole formation out of black holes form. Okay so as you all know what we're looking for in a gravitational wave experiment are in spiraling binaries. So you've got compact objects that are caught in an orbit and they radiate as they radiate the in spiral and they form this famous chirp signal that probably you're all familiar with by now but we don't really know how those binaries form. So there are various mechanisms for how two black holes can capture each other. So actually so the three known mechanisms one is the so-called in the field and in this mechanism you have two stars that are in a binary. One of the stars during its lifetime grows to be a giant and envelops the other one. They have a cloud that surrounds them then eventually they both go supernova, they explode and you end up with two black holes after it blows ejects off all the material and these black holes typically are predicted to nearly have the same mass and the spins are aligned with the orbital plane. Then there are dynamical binaries and these are binary. So typically these in the fields are born in low density regions and a dynamical binary is born in a high density region usually in what's known as a globular cluster and when you have a very high density region it's just there's some finite probability that the two guys will be in a dynamic range of each other and they'll just get captured and in this case there's the masses are not expected to be necessarily the same order the spins are random and so in principle if we could measure the spins and the masses then that would give us some clue as to whether or not it was a dynamical binary or an infield binary and then finally there's primordial black holes which could have fallen under the category of fundamental physics and these are black hole that were formed in the early universe prior to say star formation even where they were due to fluctuations in say the Inflaton field whatever cause structure to be formed in the early universe and this is of course more speculative we have some ideas of how they formed and it's sort of sensitive to the power distribution and the nature of your inflationary model so this is more speculative however in order by making these measurements we can distinguish between how many black holes we have we expect are from infield binaries versus dynamical binaries and so you can make a prediction for the black hole population and the mass distribution and there's a prediction I'm not gonna go through the details that there should be a mass gap so there should be a region between 50 and 150 solar masses where there are no black holes this is called the mass gap this is what the standard predictions are for the black hole populations and already we know that in fact that this is not true so there are that mass gap has been populated there are events in there and so we've already learned something I wouldn't perhaps say new physics is probably too strong of a word in the sense of a particle physicist it's nothing beyond the standard model but there seems to be some tension with the theory of black hole formation and the data at present so already even though LIGO has only really been running for a few years we're already learning new things about black holes that we hadn't expected so as the number of events grows and the number of detectors grow I think I would be surprised if there weren't many more surprises in store for us and hopefully we'll see signals that don't fit any of our predictions whatsoever so our job as theorists is to make predictions for some sources that we imagine but if we want to distinguish between a black hole say and a magnetar and a Bose star or an axion source or any dark matter sources that may be compact then go through in spirals then we have to know what the background is so much like at the LHC if we want to look for supersymmetry we need to be able to eliminate the background so from the point of view of a particle physicist thinking about gravitational waves just as we would make a prediction for the LHC if you're going to build the Susie model or a Technicolor model whatever your favorite model is that goes beyond the standard model before you tell the experiments what to look for you have to tell them what the background is so whether you're interested in gravitational waves because you're looking for new physics or if you're interested in gravitational waves because you're interested in astrophysics in either case we need to be able to make precise predictions for the gravitational wave signal so my lecture is what we're gonna do is discuss how we use effective field theory techniques to make very precise predictions for gravitational wave signals and along the way of course we'll have to learn about effective field theory a little bit in order to be able to develop the proper formalism so let me pause for a second and see if there are any questions Yes, there is one from Anna, please Professor, I just wanted to ask if you could please tell us a little bit more about the theoretical origins of the mass gap claim because I've seen that you know I sort of disproved the mass gap claim but I'm not really sure where the claim came from in the first place. Right, so I can say a few things about it I don't wanna get too far afield for two reasons first of all it's not my area of expertise but second of all it could take a little bit of time but basically if I remember correctly so if you have let's take certain limits okay, so for instance how large of a black hole the mass of the progenitor will determine the size of the black hole essentially so without going into details depending on the size of the progenitor you may blow off a different amount of stuff when it goes supernova, right? So in order to predict the mass of the remnants you have to have some theory about how much stuff gets blown off. So if you're too massive what happens is basically everything gets blown off and all you're left off is the core so if you knew the mass of the core then you would know the mass of the hole but if you have intermediate masses then different amounts of stuff could get blown off and so there's some region where whatever there's some region of the progenitor masses where whatever's left over will never fit in that mass gap. Now to be completely honest how they get the exact numbers for the mass gap I think they have some models so I mean is it a huge surprise that the mass gap is actually populated? I mean personally as a non-expert in the field I wouldn't say probably not is because in the old days in fact the computer simulations couldn't even get the supernovas to blow up. I don't know if the status is now but not too long ago the computer simulations of supernova explosions were really struggling to even to catch on fire, right? So as an outsider and non-expert in those calculations those are extremely difficult calculations to do so to calculate the mass of the remnant is really hard but in short the answer to your question is is that for different ranges of initial progenitors the mechanisms for collapse can be different, right? So there are certain ranges where you just can fill the gap where those between 100 between 50 and 150 solar masses. So as I said I'm not really, I'm far from an expert on that subject so I'm afraid I can't give you a better answer than that. There's another question by Anirut. Hi. Hi. Regarding this mass gap as far as I know like you can have some inflationary models in which you can still produce these black holes from 50 to 150 solar masses during the radiation dominated era. So is it different because I don't see, I mean at least from what the consensus is right now like there should be no problem to have an inflationary model which produces these black holes in this mass gap. Yeah, so the mass gap, the mass gap is a prediction the only source of black holes are stellar. Okay, okay. Okay, so let's talk about how a particle physicist goes about thinking about predictions for gravitational waves. So we have to talk first thing we should discuss the word of the observable. Okay, now if we think about collider physics what are the observables in collider physics? Okay, so in collider physics of course we have some scattering, we have some in and we have out and we calculate cross sections, right? And these are proportional to some matrix elements, squared. Right, and so we measure at you have our detectors we consider our detectors like at being at infinity and we calculate some transition rates. So how do we apply the idea of the particle physics to gravitational waves? Well, if we think about gravitational waves what are the observables? Well, you've got some sources. Here's some in inspiring binary and here's our detector infinity and all we see is a wave. So basically there are two observables one is the amplitude and one is the phase as compared to say a collider where you can make up an infinite number of observables, right? You have inclusive cross sections, exclusive cross sections, weighted cross sections whereas here things look much simpler, right? All we have is an amplitude and a phase. Now of course a crucial distinction in the sense of how much information is in the signal lies in this time dependence, right? So if you consider the time evolution of the system as time goes on the system starts to shrink and the velocity increases. So this is V1, V2, V3, okay? So there's a lot of information in the time dependence of the signal. So probably you've all seen this if you look at it as a function of frequency so if you look at the peak frequency as a function of time it looks something like this, right? And you hear that whoop from that's called the chirp which comes at the end here, okay? So our goal will be given some time initial conditions and some set of parameters what we'd like to be able to do is to make a prediction for A of T and phi of T, okay? It seems simple enough but why is it quite to the contrary? So let's think about this all the physical effects that are going on here. So we've got some matter distribution let's call this 1 and we'll call this 2 and they're trapped so during there's some from the point of view of particle physics we would call this there's some exchange of gravitons here we'll call this that generate a potential, okay? And that potential is the binding due to gravity and these guys are caught into an orbit and what happens? What's the dynamics responsible for the input? Well, there's a huge amount of physics going on here so these guys of course will radiate as they accelerate and as they radiate they'll lose energy and then that in course increases the velocity as they start as the orbit starts to shrink but there's a lot more to it than that in particular will deform each other, right? There'll be a tidal force which will elongate them and this tidal force changes their shape the shape will then change the potential will be new due to the change in the shape, right? Which in turn will change the rate at which the in spiral occurs on top of that because gravity is a nonlinear theory really interesting things have to happen so you can imagine this guy emits a graviton that graviton scatters off the gravitational background of this guy and comes back and hits itself that will change the motion right? Or you could have this guy radiated a graviton this guy radiated a graviton scatters and come back and hits itself that will change the motion this is an incredibly complex problem and to calculate the phase and amplitude is extremely complicated and difficult so in order to calculate the radiation we've got to sort of solve all these problems together at the same time now of course there's one the in some sense most straightforward and I use that term in a remarkably glib and insensitive way to those who work on this incredibly hard subject that is numerical relativity so you just plug it into a computer and calculate so indeed there's a whole field of numerical relativity and numerical relativity in principle is the solution to this problem in practice it's extremely computationally expensive in that running a simulation takes months and the way the experiment works is they use what's known as a template bank so I'm not going to go into details about this but I think it's important to understand so basically what they do is how do they search for a signal they do they use something called match filtering basically what that is is you have your prediction and then you take your signal and then you you convolve them you integrate the product of the two and you see how well they overlap so but in order to do that you have to remember that the signal let's call it s will be a function of m1 m2 s1 s2 as well as other parameters which I'll discuss but every basically every binary has it has is described is characterized by the masses and the spins there's also some other parameters here which we will discuss which are finite size effects which to me are the most interesting effects so these include things like the polarizability these are sometimes called love numbers which we'll discuss so for instance these these love numbers would distinguish the black hole from a dark matter star or whatever whatever exotic object you're hoping to find in the signal and so the way they build a template bank basically is for every value m1 m2 s1 s2 they have to they have to have a signal right they have to have a prediction so what they do is they have a bank of these predictions for all these masses and obviously that's a continuum number of these and you can't possibly hope to build a bank numerically it's just it's impractical because as I said each simulation takes months so to build the banks they actually use a combination of analytic results which is what we'll be discussing and then and they put them into some models so it's not quite completely systematic from the point of view of the errors are 100% under control but the part where the errors are under control the parts we'll be talking about so as particle physicists we've been spoiled to always calculate in a systematic way where you can make a prediction and then tell the experimentalists this is our prediction and this is how big the errors are and as a theorist if you make a prediction without bounding the errors in some way then you can't claim to make a discovery so any prediction at the LHC when they say there's a 5 sigma which is the criteria for discovery a 5 sigma signal that includes some sigma from theory without a sigma from theory then that discovery is of questionable worth so which isn't to say that every prediction at the LHC is completely under systematic control itself so it would be disingenuous to say particle physicists have some sort of higher standard because there's always modeling of hazardization and fragmentation we believe we understand those things but it's often just based upon a model so I'm going to distinguish between a model and a first principle prediction that first principle prediction must have an expansion parameter that bounds the errors so we will be discussing analytic methods based on first principles with well bounded errors in the context of effective field theory okay so we have to have an expansion parameter in order to make a systematic prediction see my screen again I know it went off okay okay so we need an expansion parameter so our expansion we have to have some small dimensionless quantity so I'm going to use particle physics units okay and a natural expansion parameter arises always in certain some limits you want to take some limits to have total control of the situation so one such limit is when the velocity is much much less than one so I'm going to drop C from now on but um um that's the limit we're going to be interested that's going to be our expansion parameter and essentially in GR this is this is the only expansion parameter under which we have systematic control this is sometimes called the post Newtonian approximation and basically it's an approximation when so when V is much much less than one the curvature is small that is it's nearly flat space so the way to see that this is that this is nearly flat space so if we look at the metric of a of a spherical symmetric object right so the metric is the short shield metric oh and I'm also going to be working in a plus minus minus minus um so this is what the short shield metric looks like angular piece it's an omega so if we have a binary then the centripetal force should balance the uh or the centripetal force needed to sorry to to keep this system intact should be equal to the Newtonian force the gravitational force and so this tells us that so V squared so if we were to set this to zero we'd be back at flat space and you can see that it's V goes to zero we approach flat space okay so um the reason we can get calculational control in the small velocity limit is because we're approaching flat space we know how to calculate everything okay um good so um that's our expansion parameter and um the next step in um in approaching this problem is to parse this problem and to simplify it is to understand let's understand all the scales involved so what are the relevant scales so let's let's see what's going on here so here are uh our objects so let's let's assume that the objects have the size order the same size which is typically true almost all the cases will be interested in so that's we'll call that big r and we're going to make the assumption that r is much much greater than r and that's a reasonable assumption to make if we're interested in the small velocity limit right so they'll be moving slowly when they're far apart as they radiate they'll start to uh the orbital distance will get smaller they'll speed up um and uh so this hierarchy is consistent with our small velocity approximation now we have one other scale in the problem which is the wavelength of the radiation so the wavelength of the radiation goes like 1 over omega which goes like r over v okay so this is we have another scale in the problem when v goes to 0 so those are the three scales in the problem um and what we would like to do is to um simplify this problem by treating one scale at a time so the essence of effective field theory is divide and conquer treat one scale at a time because when you treat one scale at a time all the calculations simplify um so okay look what we're really doing what our job is to solve Einstein's equations but we will never even write down Einstein's equations okay we're going to solve the problem um by using methods of quantum field theory despite the fact this is a completely classical problem so uh even if you're not interested in gravitational waves per se the methodology will be discussing will be applicable uh to um to many other systems and in fact the methodology that we developed was based on ideas um used to study uh heavy quark bound states so the physics of heavy quark bound states um is remarkably similar to the physics of an inspiring inspiring binary right so um when you have a heavy quark bound state um uh the uh the typical radius is very small compared to the qcd scale so it's weakly coupled so those of you who are familiar with qcd know that qcd is asymptotically free which means at short distances it's weakly coupled so a bound state of heavy quarks is a weakly coupled bound state in a non-linear theory but what is gravity it's a non-linear theory and it's small v it's a weakly coupled bound state now the huge distinction of course is that quantum effects are order one in corconium and quantum effects here will be suppressed by uh h bar over the orbital angular momentum of a binary system so you can imagine how small quantum quantum effects are so you can do a little back of the envelope calculation and take h bar and divide it by the typical angular momentum for say solar mass black holes that's a pretty small number okay so I think we're 48 minutes in so why don't we take a little break and get up and stretch a little bit and come back in five minutes great so we'll resume at 54 right? okay great thanks here we are there may be questions I don't see any alright whatever you want okay good so how do we attack this problem so as usual you first start with a simpler problem and build up your intuition and your technical chops on that problem before you attack the harder problem so the natural analog or the toy model of this problem would be electrodynamics okay so first we're going to look at electrodynamics and this problem is considerably simpler because the theory is linear in the sense that the photon doesn't interact with itself like the graviton does so it's a much simpler problem to solve but to be completely honest it's also intractable right for the same reasons that we discussed before you know even if I have a system of charges that are interacting with each other and I want to calculate the radiation that problem in itself is not solvable in closed form obviously for the same reasons we talked about before these guys deform and as they inspire all the deformation changes and as they deform the forces change and as the forces change the radiation changes and the the radiation leads to new forces some so called radiation reaction forces which hopefully we'll have time to discuss and so this problem is again unsolvable but we again can attack it in a small velocity limit so let's talk about how we would solve this problem so basically if we think about it from the point of view of a field theorist we have some partition function over the field where phi now is a generic set of fields now the idea in effect of field theory is to break up the system into different sets of fields which live at different scales so before what the scales are we have this hierarchy so we're going to break the fields up into phi, let's call it phi r by r in phi r over v okay so this field only has momenta of order 1 over r and this guy only has momenta 1 over r and this guy has momenta 1 over v over r okay that's incredibly imprecise at this moment we'll have to sharpen that slightly but for the moment just so we can get a little bit of intuition so then we can write our path into grow in this way okay now the idea is well at long distances what's happening at short distances starts to become irrelevant in a very precise way so we do things in steps first we do the path integral over the shortest distance modes and then we end up with a new effective action so this new action will have encompass all the physics of the short distance modes here okay and then the next age okay and this is what we're really interested in so as an observant infinity the only fields we see are these okay so this procedure is sometimes called integrating out so we integrate out the short distance modes so these are the shortest distance modes these are the next shortest distance modes and as an observer and infinity all we care about are the radiation modes which are the longest distance modes so if we knew this as effective we could calculate everything we wanted the phase and the amplitude and then we would have solved the problem of course we can't do these integrals exactly but as long as it's systematic we can still have a prediction which is meaningful okay so what are these modes so let me just tell you the answer and then we'll spend the rest of today and pick maybe part of tomorrow depending on how much time we have justifying it so there are basically the three types of modes in this problem the phi r modes are the modes responsible for the internal dynamics of the object so if this were a neutron star would be the normal modes of the neutron star or whatever the object is in the case of a black hole they're called quasi normal modes they're the ringing modes of a black hole if you strike it it will ring and really to me this is the most interesting piece of physics because it's the short distance physics that distinguishes between normal matter and say dark matter but of course we don't have direct access to those modes we only have indirect access through how those modes affect this action okay now all of you actually know how to do this first path integral okay let's look at the electrodynamics case if I have a collection of charges and I ask what is the potential at some point r we know at least if they're static okay so that is what the potential would be at some distance now we know that if this guy has some region and we support r and we're looking at a distance r and r is much greater than r then we can expand this integral and we end up with q times 1 over r plus p dot e times r cubed plus dot dot dot dot this is called the multiple expansion you're all familiar with that and this is the simplest example of an effective field theory it's a completely trivial example of effective field theory nonetheless it captures a lot of the ideas so notice that all the short distance physics is encapsulated in these coefficients so q if I knew row of r exactly q would be this and p would be given in this way we will call these short distance coefficients and they're also sometimes called Wilson coefficients and to think about this more clearly we could label short distance objects by red and long distance objects by green okay so the short distance physics it would be incorrect to say the short distance physics is irrelevant for the long distance physics the more precise statement is that the short distance physics is encapsulated by a finite number of coefficients which are just pure numbers okay so we can describe this long distance physics the fields at infinity by a finite set of numbers q p and then I can have a magnetic dipole moment and then I would have a quadrupole moment q i j octopole so forth and so on and so these are just a collection of u v coefficients u v meaning short distance okay so if I don't know what this distribution of particles is like in the case those these particles would be the analogs of whatever the degrees of freedom are that make up the objects that are in spiraling so we don't know what they are we're trying to figure out what they are these are a set of unknown coefficients and we're going to use the data to measure them so that's exactly what we're doing in a in a gravitational wave experiment okay so for the sake of simplicity let's assume that our system is symmetric right so we're going to assume that our in spiraling objects are how the shapes is fierce if they didn't we would know what to do we would have to include multiple moments for them so we're going to assume spherical symmetry and of course for astrophysical objects that's an excellent approximation if they're not spinning at least so non-spinning black holes are are are described by the Schwarzschild solution which is a spherically symmetric solution to Einstein's equation okay so once we do the multiple expansion then this leads to the point part is basically a point particle description okay so the first step in our approach is to reduce this to a point particle and reduce this to a point particle okay now we have to make sure we capture all the finite size physics because that's what we're interested in when we're done right we're interested in knowing you know how did this thing deform how did it you know all the stuff that is all the physics has captured in the internal dynamics of the object so what we want to do is we want to figure out what is as effective what is the effective action for point particles okay and the logic behind this the process of determining as effective is we write down an action which is consistent with all the symmetries right so the first of all you have to figure out what the symmetries are but before we do that you know we started off with an action for a bunch of point particles and now we have an action for one point or two point particles and we have to describe the position of two point particles so we're going to describe them by world lines so i is equal to one or two lambda is some affine parameter affine parameter meaning it's just a parameter that has no preferred origin so there's no such notion of lambda equals zero being some preferred point so this is our x mu of lambda and I've left things relativistic here so x zero is a time coordinate which may or may not be the same as lambda which we'll see in a second so here's our x mu i of tau x one here's x two we'll give it a different parameter lambda prime and now we want to write down a Lagrangian which is a function of x i say and perhaps x i dot for these point particles right so this is going to be our effective action so we can write s so let's write it as a sum over i one and two m i so this is the kinetic term for a relativistic point particle so 8 mu nu for the rest of these lectures is the flat metric okay I'm not going to derive this for you hopefully it's familiar to you there are lots of different ways of deriving this but for the moment let's just take this for for granted it may be during the discussion section if there are questions about where that comes from we can discuss I prefer in general not to just say just ask you to accept a result but for the sake of time for the moment let's in this case just assume that that's true and now we need to couple the point particle to the electromagnetic field right so a mu of x is going to be our gauge potential and now we want to couple it in a way which is consistent with all the symmetries so what are the symmetries that's the first step in writing down any effective theory you have to know what the symmetries are okay so certainly in QED we know that it better be gauge invariant and if we're writing down a point particle description it had better be independent of how I parameterize the world line right I could parameterize the world line in any way I wish and the physics shouldn't change right I could if I have some parameterization x mu of lambda I should be able to parameterize it by lambda prime where lambda prime is some function of lambda and leave the action invariant this is called reparameterization variance so what can I write down okay so the first term I can write down in addition to to this kinetic piece here is I can write down d lambda v mu of lambda x mu okay and I'm going to put the charge in front of it and I'm going to sum over I so so let's see if this is invariant well under gauge transformation our first symmetry so delta s look like d lambda dx mu d lambda um d chi dx mu so I have my lambda's look like chi's and this thing is a total derivative so I can just set it to zero and it has no effect on the physics so this is indeed at least from gauge invariance we found that it's invariant under gauge invariance is it invariant under reparameterization variance well d lambda goes to d lambda prime sorry lambda over d lambda prime but v it should not be a partial sorry the variation of this guy exactly cancels the variation of v and indeed that isn't allowed term in the action okay now of course we also have to include the action for the electromagnetic field which does not live on the world line this is just qed so we're going to call this the notation I should really write this in red this is a short distance coefficient and corresponds to the total charge of the ice blob right now from the point of view of effective field theory I have to write down every possible term so of course there are an infinite number of terms that I could write down so let's start to write down the next order higher order polynomials in the fields that are consistent with gauge invariance and reparamperization that live on the world line these will account for the finite size effects so let's write down s finite size well we have our integral over our affine parameter for each guy and now let's write down everything we can which is gauge invariant well I can write down the world line f mu nu f mu nu that is still of course gauge invariant and I have to include some short distance parameter let's call it c1 into the action this is unknown but in principle calculable if you know what the underlying system is okay I can write down also at quadratic order in the fields I can write down another term with another short distance parameter which is gauge invariant I should say I realized I glossed over something which is that we're also assuming Lorentz invariance sorry, apologize for that hopefully that was obvious to you guys in the sense that we want to write down Lorentz invariant theory because we're going to take a non-relativistic limit but for the moment we want to be we haven't made any approximations about small velocities everything now still is completely relativistic now we're not quite done because this system we haven't imposed reparameterization variance but I can fix that by putting in a square root of v squared here so you can check for yourself v transforms in the opposite way as lambda so these guys they cancel each other and I've got two v's upstairs so I've got to put a v down here and that leads to the Jacobian the transformation cancelling okay so there are of course infinite number of of terms that one could write down but we've truncated so we have to be truncating in some expansion so let's do a little bit of dimensional analysis these operators have um sorry these uh these coefficients just on dimensional grounds scale as r cubed okay so we can look at by just counting derivatives and dimensions of the fields if there are questions about this we can go through a detailed example in the discussion um and so r remember is the parameter which we're taking to be small so as long as we're studying scales r little r much greater than big r then terms which say we'll scale like r to the fourth and then r to the fifth lead to our small corrections which in principle could be included so as an exercise you can start to write down the next set of polynomials the next sorry the next set of operators at higher order in the fields so so so far we're doing the point particle approximation we're doing expansion in little in big r over little r um and we've essentially accomplished our first stage of the effective field theory we've removed all the short distance physics the short distance physics is completely encapsulated in our red Wilson coefficients okay okay so let me pause any questions okay well if there are no questions um so what's the next stage well the next stage is um removing the next set of degrees of freedom and uh in order to do that we have to know what they so what are the what are the next set short distance next collection of short distance degrees of freedom now um that first is the easy step right that's the the multiple expansion is something you learn uh in in undergraduate physics the next step is more more difficult um so to give you the answer before proving it um there'll be modes which are responsible for the binding those are called potential modes so here's the radiation modes the potential modes will have typical wavelengths of 1 over r and the potential modes will have typical length of v over r okay so um the next step of um of modes will be uh short distance modes will be the potential modes so we're going to want to integrate out those guys um these are just words let's show that this is true um and the way we'll show this is true is by using the path integral formalism and understanding the details of the problem so let's consider the path integral of the partition function for some system where uh electromagnetic system where x will be the sources so these are charge charge source notice that I've assumed that um the action only depends on x and x dot um and the reasons for that is that if you had x double dot you could always use the equations of motion to replace x double dot by x and x dot now um the obvious question there is well equations of motion are classical and if I'm working the partition function what gives me the right to use the classical equations of motion because quantum fluctuations don't obey classical equations of motion um we can talk about that for us of course it's not important because um we're doing a classical problem so we're certainly allowed to do that we don't need to worry about quantum fluctuations um if people are interested we can have a discussion about why even quantum mechanically that's an allowed uh that's an allowed manipulation okay so quantum mechanically even though we're doing a classical problem um the partition function we know is known as the vacuum persistence amplitude okay so this is the probability to go from minus infinity in the vacuum to plus infinity at the vacuum in the presence of these sources of these charges and this is just normalized to um the the transition probability without the sources okay so in our case in terms of the path integral we can write this so it's a vacuum vacuum transition amplitude in the presence of the sources okay so what can we learn from this well if we look at this object then um we know how the vacuum will evolve in time it's just e to the i h t acting on the vacuum so we can write z as just a phase times t and this is where t goes to okay time some overall normalization coming from the denominator right so if you're in the vacuum and you're asking uh what's the probability be in the vacuum time the system the system evolves we're assuming some notion of adiabaticity so you've got some energy at the ground state and as the system evolves it accumulates a phase when that phase oh sorry there's an integral here sorry because there's time dependence and it just develops a phase over time that's given by just the energy of course if you're in the presence of sort of sources which are dynamical they will radiate and so we'll see in a second that this energy has an imaginary part and that imaginary part will correspond to the rate at which things are radiating the real part will correspond to the potentials and the imaginary part will correspond to the radiation okay so let's see if we can calculate this object in inelectronomics and in fact the reason we're doing uh E&M is because we can solve this problem right E&M is um E&M is uh Gaussian so let's we can do this exactly without without even making the multiple expansion or we can do it for arbitrary sources so remember what is the action for QED so here this is the gauge fixing term and I'm going to always work in the fine engage just to make our lives easier so this integral the path integral is doable this is quadratic in the fields Gaussian so z of j okay so notice here I haven't made any approximations this is a general j right eventually we'll replace it by point particles okay so now this guy this object here is essentially IE times T okay and what I claim is is that the the real part of this object will tell us something about the power loss in radiation and the imaginary part will be the potentials which gives us the forces between the objects um okay so let me since we're running out of time let me just quickly say so what we're going to do next is we're going to take j mu as point particles so this they will have the following form since j mu of x and we're going to plug that into here and from that we're going to extract the power loss and the potentials okay so that's the next order of business we'll have to say a little bit about what this g is here so for the hopefully that's familiar to you from your field theory classes so that's going to be a propagator but we're going to have to figure out what are the correct boundary conditions on the propagator in order to get the physically correct solution so I think I am out of time for today so we can stop and have a discussion all right thank you very much thank you very much Eira and we have already a question from Max