 Okay well first of all it's a pleasure to be back in Trieste. I want to thank the organizers for the invitation to speak and also for not scheduling my talk first thing in the morning. So the subject of what I'm going to talk about today is easy to describe in words. It consists of two compact stars, black holes or neutron stars, still far apart in orbit around each other interacting gravitationally in the process emitting gravitational radiation. In Newtonian gravity this is a freshman physics problem these things just go around and round forever and ever as long as they're sufficiently far apart. But in GR all hell breaks loose and this is a nonlinear problem and it's in fact a research problem. So my goal is to describe the dynamics of the system at least in a certain regime using methods borrowed from particle physics namely in statistical mechanics as well namely the concept of an effective field theory which I will review in this lecture or I will introduce in this lecture. So first of all let me start by introducing the scales in the problem so I'm going to be talking about compact stars or compact astrophysical objects and what I mean by that is that the size of the star if it has a well-defined size let's call it the physical radius is of order or only slightly larger than the gravitational radius of that object. So if this is a system this is an object with some mass M the gravitational radius I'm gonna call it gravitational radius but it's also just the short child radius of the system and so that's two times Newton's constant times the mass. So by compact I mean an object whose actual radius is of order that size and the units are c equals 1. So certainly a neutron star for example with a mass close to the mass of the Sun in a radius of order 10 20 kilometers or so that is a compact object because because in in these units and gravitational units this is something like a kilometer or so so the short child radius of the Sun is a kilometer so we're talking about an object whose radius is close to the short child radius and certainly for a black hole these two scales coincide. What I mean by the size of a black hole is just its short child radius its gravitational radius. So these are the objects that I'm going to consider and what we're interested in is in the gravitational wave emission from so first of all a system of one object isolated object is stable it just stays like that forever so so what I'm really going to be interested in is in dynamics so I have to have at least two and two is hard enough so let me just do two and not do the three-body problem etc so what I'm interested in is in describing gravitational waves from binaries consisting of compact objects. So this is in a cartoon the problem that I want to solve let's focus on the case most of my lectures will be about black holes because they're cleaner it's a purely gravity problem as opposed to neutron stars that have all sorts of other physics in the nuclear physics etc so let's just focus on the black hole case and so what we want to solve is we have a black hole here black hole one it's tidally distorted by the presence of a second black hole here the whole system is emitting gravitational radiation with some frequency spectrum omega and I have a gravitational wave detector way out here on earth and so what I want to know is what the spectrum of gravitational waves looks like way out here so near the system this is some solution of the Einstein equations and let's imagine that we know the metric then to know the experimental observables all that we have to do at least in the black hole case is solve this equation Ricci curvature equal to zero subject to some initial conditions and then if we want to know the experimental observables all we have to do is take that solution and take the limit where we're really far away from the system of emitting black holes and if we pick smart enough coordinates the metric out here looks flat plus a small piece and then this small piece is what gravitational wave detectors measure way out at infinity so what we want to calculate is that and then there's going to be some some spectrum of gravitational waves so we could also do this in frequency space just by taking the Fourier transform in some suitable coordinates that go to flat space coordinates out at infinity that part is important so that's the goal so it's a pretty simple goal if you think about it but of course the question is how do we do this some non horrible nonlinear partial differential equations and so the question is how do we solve it well the first thing that we notice is that these equations are actually scale invariant at least in the case where we're talking about black holes and we're just solving the vacuum equations because they're scaling variant because if I rescale the metric by a constant so if you take the metric as a function of x and you rescale them by some constant these equations stay the same so the only thing that this problem can depend on is this scale the the gravitational radius and the frequency of the gravitational waves so what determines the form of the spectrum is the following parameter just by this sort of scale invariance some parameter that I'm going to call epsilon which is a dimensionless parameter and so if you take the short shell radius what I'm calling rs is also when I'm gonna call Rg so I use both interchangeably so this is the parameter that describes the physics so this is I don't know that this is what determines the shape of the spectrum of waves the distances you mean to the source or between them so in the nonlinear regime where these things are very close to each other that's not a very meaningful quantum we'll talk about the distance soon but in the nonlinear regime it that's not a meaningful scale right because these horizons could be merging or something like that but in some regime that certainly is a meaningful parameter so we'll talk about that in more detail anyway so this is the parameter that determines the qualitative features of the spectrum so we can talk about the different limits we can talk about the limit where the where this parameter is of order one and therefore the spectrum of waves sorry the wavelength of the waves is of order one over the short it's of order the short shell radius and that's what's called a nonlinear regime and that's a hard problem so the nonlinear regime is where this epsilon is of order one if that's true then that means that the typical wavelength of the gravitational waves that I'm looking at divided by the gravitational radius is of order one or maybe a little bit larger and so in this limit these two black holes are close together so this is exactly the limit where it doesn't make sense to talk about a separation between them it's some horrible nonlinear problem and in that and in that limit in that limit we're kind of stuck because the way I know how to solve physics problems is either to use symmetries there are none in this problem or to have some small expansion parameter that I can use to do perturbation theory and there isn't one here so this is not tractable by pen and paper I'm a pen and paper kind of guy so instead you need to use a supercomputer and I don't have one of those so I'm not going to talk about that limit yes it's because it's because these gravitational waves have a radius which is a word of the gravitational radius so the gradient energy in them is of order the energy of the system so if you like every you can't expand Einstein's equations in a small you can't expand them around flat space that limit doesn't exist in this case so it's not it's just not a useful limit or it's not a tractable limit so anyway so that's the super computer regime for most of the time that the system spends in this parameter range for most of the time that it spends in that parameter range meaning that I can look at this nonlinear system so here's once again the two black holes very close to each other tidally distorting each other's horizons etc emitting waves and this regime is called the merger because these things eventually well they're unstable to emitting gravitational waves so they eventually collapse into each other so they merge and then they merge into some object and if what we understand about gr is true what they merge into is a black hole because it is even though not proven mathematically rigorously it is widely believed that the endpoint of gravitational collapse is a black hole and it's not going to be a black hole that's going to be in its ground state if you like it's going to be highly perturbed so it's still going to be emitting away most of its multiple moments and that face is called the ring down the process of doing that is called the ring down and eventually if again if everything we know about gr is a guide what it settles down into is a black hole so the endpoint of this collapse is a very simple object it's just described by two numbers the mass and the spin and it's stable so the nonlinear regime is when the two things are colliding super computer time they merge this part actually you don't need a super computer for this part you can actually solve using the methods of work what is called a black hole perturbation theory black hole perturbation theory is just the I don't know the the enterprise of expanding the Einstein equations around the black hole solution so the the ring down is tractable semi analytically meaning that it's a you can turn it into first-order differential equations that you can then solve on the computer so the ring down just consists of expanding the metric around the metric of an isolated black hole plus the fluctuations the thing that it gets radiated out to infinity and then the fluctuations obey a wave equation and that wave equation in cartoon form looks like that it's some sort of hyperbolic linear wave equation for the perturbations in the black hole background and excuse my Italian it's called the Reggie Wheeler equation I apologize for my Reggie so that that's tractable so that's an interesting limit and the system certainly lives a fraction of its time in this limit but it's not the limit I'm going to talk about today because as I said I'm a pen and paper guy and even this problem is not a pen and paper problem you need Mathematica so that's beyond my abilities so the problem that I'm going to focus on for the rest of today and tomorrow is the linear regime where the parameter epsilon is small so in this limit now it starts to make sense to introduce a separation between the objects and you can draw a different picture of what the system looks like it looks a lot cleaner than this thing because now the objects are really far apart and almost point like and so we can introduce a new scale which is a separation between them that was a question earlier earlier in the lecture and so these guys are now in some sort of orbit around each other which is nearly Kaplerian they're still that orbit is still unstable is still there's still gravitational wave emission but the wavelength of these gravitational waves is pretty large compared to the other scales in the problem so now we have multiple scales we have the size of the object if there are black holes there it's the gravitational radius the separation between them and then there's the wavelength of the radiation that's being emitted and there's a hierarchy of length scales in the problem so we have the gravitational radius that's the smallest scale you can have in classical gr and then we're assuming that there is a there is a well-defined orbital scale and it's a lot larger than this thing and then because epsilon which was just the gravitational radius times the frequency is it means that the wavelength of the gravitational waves is much bigger than every other scale of this problem and in fact in this problem because the gravitational radius is so much smaller than this thing we recover Newtonian physics to a good approximation so for example we can define a typical velocity and the velocity squared is what you would get from Newton's loss that is the gravitational radius divided by this the orbital separation so this is of order r gravity over r and it's a lot and it's and it's a small scale in this problem so now we have well this is the expansion parameter but this is also can be regarded of the expansion parameter it's a small scale so what we have here is a nearly Newtonian system and so we can now hope to do perturbation theory around Newtonian physics around Newtonian gravity that we understand pretty well so it's nearly Newtonian orbits and then if you want to understand the gravitational waves being emitted you can just do linearized gravity around that that limit so the answer to a really good approximation in this regime the the the dynamics in this regime is described by two simple equations yes it is that's a good question I should have said that half an hour ago so Holst and Taylor won a Nobel Prize for studying exactly this system so so gravitational waves emit in their case it was pairs of neutron stars separated actually by a much larger scale than the one that's relevant for LIGO I forgot the exact number but it's exactly a problem of this kind and if you look at how the orbit the case is a function of time it's bang on with the GR answer what else can I say about that and it's a system that has been has been tracked for so long it was discovered in the 70s that you actually need to go to second order in the in the expansion parameter to understand to get the to fit to get the right fit so that's really one motivation for studying for my whole talk which is this is a system that's out there right apart from the fact that it's going to be sort of the bread and butter type signal for gravity wave detector so you'll get more about that in the afternoon lectures thank you for that question so anyway it's a nearly Newtonian system so we can write down equations to describe its evolution almost immediately their textbook equations yes excuse me these are astrophysical objects so zero zero charge yeah we're not doing quantum gravity here they're not extreme all supers metric black holes just plain old short child objects or cur objects I guess yeah so nearly Newtonian system so here are the equations that describe it a Lagrangian for they're basically point particles particle one or black hole one with mass m1 particle two with mass m2 and position x interacting through purely Newtonian gravity Newton's constant m1 m2 over r so that describes the orbital evolution to a good approximation but there are corrections to that limit and I'm gonna discuss those corrections in great detail and then the instability the fact of the system emits waves it is encoded in a formula that Einstein derived probably a hundred years ago I guess I think it was in the original paper and it's the famous quadrupole radiation formula which says that the rate of change of mechanical energy of the system at least if you average over many orbits so there's a minus sign is equal to the power emitted in gravitational waves and the power emitted in gravitational waves is related to the moment of inertia of this orbiting system of point masses so the actual formula is that it's Newton's constant divided by five five is because it's a quadrupole to l plus one and then it involves the third time derivative of the quadrupole moment squared and the brackets here just denote a time average so I'm averaging over many orbits and this Q i j is just a moment of inertia but it's just a traceless part of the inertia matrix so Q i j is the sum over the point masses times the coordinate so 1 and 2 times the coordinate of each point mass subtract out the trace and that's what it is and then we can get some feel for the for the dynamics just by plugging in a simple orbit so I'll leave that algebra so these are all textbook formulas I got them out of MTW but they're in every textbook well this one I didn't need to look up in MTW so let's just do for simplicity circular orbits and just plug in and then the binding energy of the system the mechanical energy of the system is just a half mv squared minus because it's gravitational energy and the quadrupole formula in that case turns out to be equal to 32 divided by 5 and then it goes like velocity of the system to the 10 divided by Newton's constant kind of weird but okay and then the reduced mass divided by the total mass so that's what the system looks like or what the relevant equations are like for a circular orbit they simplify and we can solve these equations just by setting the rate of change of the mechanical energy to the to the gravitational wave power spectrum or power it's not a spectrum yet to understand the dynamics of the system at least crudely yes yeah it's just the conserved energy associated with this Lagrangian so a half mv squared for each guy minus gm over r it's not really conserved right but yeah so we are we are saying that the total energy is changing because energy is being radiated away so yeah so so so it's a bit of a cheat really right it's called the adiabatic approximation so you you and it's actually consistent and it can be derived rigorously in the sense that using this where was it just setting this equal to this okay it's not really rigorous but the error is some power of velocity so it's small in this limit thank you okay so with that so that's called the adiabatic approximation when you do that anyway let's use the adiabatic approximation let's just assume that it's true and then let's solve let's set d by dt of one guy equal to the other guy and see what we get and when I plug in this formula I'm going to solve it in terms of the frequency the frequency of the orbit is related simply to the gravitational wave frequency by a factor of two actually so the velocity if this is a Newtonian system is related to the frequency by this formula that's called Kepler's law so the orbital radius and the velocity are fixed in terms of the gravitational radius and the frequency and if you plug this into these formulas at the time derivative of this equal to this you get an equation and I'm going to omit order one number so I'm going to assume for simplicity that mu is of order m so these are roughly the same mass objects so then setting e dot equal to minus the power in gravitational waves gives me an equation for how the frequency evolves in time so this is the formula that Halston Taylor had to fit to back in the 70s and 80s and 90s and I think they ran out of money but so this is the equation that we have to solve if we want to know the orbital evolution and it's a pretty simple equation to solve d omega dt goes like a power of omega so from this equation we can understand for instance how long does it take for the orbit to decay the orbital the time that it takes to decay is just an integral over the frequencies involved so if it starts out with some initial orbital frequency and then you stop observing it at some final frequency the amount of time that it takes to evolve from this initial orbital frequency to this final one is an integral d omega of dt over d omega we know dt d omega in terms of omega so we can just calculate that integral the only scale in this problem is the gravitational radius so time in units of c equals one goes like the radius and then the non-trivial dynamics is the power law because this goes like some difference of these powers of the frequency so that's the result of doing that integral and if we plug in some numbers for LIGO that operates very roughly speaking you'll get more details about LIGO in the afternoon it operates in a frequency band of order so omega over 2 pi is the initial it's the lower cutoff on the frequency is something like 10 Hertz and so it goes between that and something like let's say 10 to the 4 Hertz maybe closer to a thousand something like that so that's the final frequency and if you plug in numbers into that formula you know how long it takes for the system to live inside of the LIGO band so it's a signal in the LIGO band and it scans that frequency band in a period of time that if you plug in numbers you get something like this and I'm being very schematic here because I'm ignoring all sorts of numerical constants when I do this but if you actually do it you get something of order a few minutes let's call it 10 minutes and it depends on the mass of the system so in units of the mass of the sun it goes like this so the system lives for a long time in the in the in the in the LIGO band for example and then you can also calculate the number of orbital cycles that it spans in that band and that's just the total phase so it's just the frequency integrated over the time spent and then you get something that goes like well it's dimensionless and it goes like omega to the minus five-thirds and if you plug in numbers for LIGO you get a huge number of orbital cycles minus five-thirds as well so we already get just from this very crude approximation a lot of information about what the system is doing for typical numbers of experimental relevance so if we start out I don't know for instance with neutron star neutron star pairs at a frequency of 10 Hertz they're separated by a distance of about a hundred kilometers that follows from Kepler's law a velocity of order 10 to the minus 2 in c equals 1 units and then it speeds up because it's emitting gravitational weight wave so it's falling into a gravitational potential and then by the time it's at the upper end of the band the velocity is closer to like a tenth the speed of light the radius is right about the actual radius of the object so that's when they're getting ready to collide but the point is that for a for a for basically the entire band except a small region at the endpoint the system really is nearly Newtonian and so if you're gonna set out to discover the sources this is really all the formalism you need basically I mean you have to clean it up a little but that's about all you need and that's good enough for discovery so most of the most of the most of the dynamics is in this Newtonian limit most so that's good enough for discovery but once you discovered it and you want to learn about gravitational physics from these systems you want to extract the masses of the actual black holes their spins etc etc you you have to look at things in more detail you have to look at corrections to this formula the experiments actually are sensitive to those corrections because the system is coherent over the band of the detector for many cycles so any tiny deviation from from the actual theoretical value is detectable I guess what I'm trying to say is that if you really want to extract all the parameters you have to calculate the dynamics to pre-high order in the velocity and it was determined in the 90s how far can you go and that LIGO is sensitive to so the fact that the system goes for like 10,000 orbits in the LIGO detector gives you gives you an estimate of how far do you have to go in perturbative GR corrections away from the Newtonian limit the answer has worked out in 1994 turns out to be that beyond this leading order Newtonian result over here you have to go to to the following order again this is not for discovery what we did oh it's already good enough for discovery but for parameter extraction it turns out that you need corrections that are order velocity to the six beyond Newtonian gravity so you actually have to solve the Einstein equations as a perturbative expansion in powers of velocity to a rather high order so for example we would calculate the energy of the system and in the Newtonian limit we get something like this but then there's GR corrections from the fact that the systems have well from nonlinear GR or from just retardation effects that Newton's loss that the interaction is not instantaneous for example etc etc so there's an order v squared correction to that and then there's an order v to the fourth in an order v to the six so etc so for the energy you have to go at least to that order for the power you have to go well it starts at v to the 10 that's the the leading order Einstein result and then the next term is order velocity squared the next term turn out to be velocity cubed I'll I think I will explain that next next lecture velocity to the fourth velocity to the fifth velocity to the six and it turns out that a velocity to the six you start also finding logarithm so the velocity in the expansion I will also explain that next lecture so this estimate that you have to go to this order was worked out by Thorne and his collaborators Finn Larson who else sorry I didn't write down the reference but it's Thorne from 1994 or so that was the original motivation for going out to this high order there's a more modern motivation for doing that which is that numerical GR has really I don't know come into its own in the last decade now they can finally really evolve these systems numerically over many orbits and it's the and the new question is how do you match these the perturbative limit to the to the numerical results and that also motivates going out to pretty high order in the expansion so I'm going to assume that that's well motivated and what I'm going to try to do is construct or tell you how you compute these corrections but again if all you care about is discovery we're done and then you can go to lunch but if you want to hear more stuff stick around by the way the the activity of computing these sort of corrections is called the post Newtonian expansion in GR it has its roots with the work of Einstein himself who worked out some of these corrections in his original paper and later on as well and then it's really took off in the 80s and 90s motivated really by gravitational wave detection and what I'm going to talk about in the in these lectures is basically the post Newtonian expansion but rephrased in a sort of modern particle physics language because I would like to convince you that that's a useful thing to do the reason why that's a useful thing to do why use these sort of fancy particle physics methods to address this problem which is just basically the problem of solving Einstein's equations order by order and perturbation theory that's all we're doing really but there is a good reason why one should think about this in a different light and it's because the problem involves a hierarchy of length scales so at a given order in velocity we have effects coming in from three different different scales and they're completely different physical effects once again the physical scales are the actual radius of the object so the physics associated with whatever sets that radius for a neutron star for example that comes into play here so that's a border of the gravitational radius there is in this limit a well defined orbital scale and there's physics associated with the orbital dynamics and finally there is the wavelength of the gravitational radiation itself these are all scales that play a role in this problem and they're all correlated by the kinematics the kinematics tells us for example that if you take the ratio of the gravitational radius over the orbital radius by Kepler's law that's v squared so the expansion parameter v much less than one is related to a ratio of scales but it's also related by the multiple expansion to the ratio of the gravitational wave scale over the radius that's of order v again much less than one so if we want to compute at a given order in velocity we have effects coming from all these different scales in the problem and the question is how do we organize that how do we think about how to do that efficiently and now I come to the subject of well the main subject of these talks and it's how to actually do this how to organize the stuff in a in a in an efficient way and there is a set of methods for doing that and it's called effective field theory so what I would like to do for the remaining time I have is set up this problem which I just described using this effective field theory language yeah yes thank you so yeah so correlated scales means that it's useful to use the language of effective field theory and so now I'm gonna spend some time giving you a sort of schematic introduction to the ideas of effective field theory and then I'm gonna fill in sort of my sketchy introduction here with examples drawn from this system as we go on so I will just assert a bunch of things that are true and either you already know them in which case you don't need this or this is new to you in which case you'll have to take my word but just so that we're on the same page because there's people with many different backgrounds here I'd like to tell you what the ideas involved are and then we can apply them so whenever I say effective field theory I'm gonna abbreviate it by EFT and I'm gonna be schematic here and before I get started for my own sanity just because it's what I'm used to I'm gonna use units in which even though there's no Planck's constant anywhere in what I do I'm gonna set it to 1 and I'm gonna set the speed of light to 1 and I'm not gonna go all out and set Newton's constant to 1 I'm not doing quantum gravity here that means that everything has units of mass to a power so anything goes like mass to some power in particular Newton's constant goes like 1 over mass squared and in my system of units it is equal to 1 over 32 pi m Planck's squared so those are the units anything goes like mass or energy to a power and if there is a I don't know a mantra to the effective field theory idea is the concept of scale separation the reason what you do effective field theory is because it allows you to separate scales explicitly so I'm gonna explain why that works so it's called scale separation but it's also called decoupling so to illustrate what the coupling means I'm gonna consider a schematic field theory classical or quantum even though I'm gonna be using quantum language everything I do applies to the classical case as well as we will see and so to illustrate the idea I'm gonna consider a model where I have some fields like the gravitational field for example or the electromagnetic field and some of them are what I'm gonna call heavy and some of them are gonna are gonna be what I call light so I have some sort of field theory it's described by a Lagrangian containing two types of fields which I'm gonna call Phi and Phi for clarity little Phi and Big Phi it involves some sort of Lagrangian let's call this thing the full theory so this is our universe it describes everything that's why it's called the full theory but I'm gonna imagine that these fields are such that Big Phi is very massive for example so it describes in a quantum field theory particles of mass Big M and little Phi is light so little Phi of X is what I'm gonna call the light fields meaning that the mass of Phi is small in fact it's probably gonna be even zero so like a photon or a graviton and then the second set of objects Big Phi are very heavy so this M Phi has a mass which is certainly much bigger than the mass of the light guys and sometimes I'm gonna call this mass lambda Big Lambda as units of mass or energy and we're gonna call this quantity the ultraviolet cutoff this lambda or the mass of the Big Phi is the biggest energy scale in this problem or the shortest distance scale in this problem so it's called the UV scale UV ultraviolet short distances so this is the system it's described by some Lagrangian it has two types of fields in it and in this toy universe we have an LHC which has enough energy to make the light particles like the standard model for example in the Higgs but it has not enough energy to make any new any heavy particles we haven't seen any we think we hope there's some but I don't know it's not looking so great these days but anyway let's pretend they're still there and ask what happens so we're gonna be doing some sort of experiment like a collider experiment in the energy scales of that experiment let's call that energy scale omega it's gonna be of order or less actually or greater than the mass of the light particles but it's certainly gonna be much less than the mass of the big heavy particles so that's the kinematic situation we're in we have the theory of everything so we can calculate whatever we want using this theory but it seems kind of natural because we don't have direct kinematic access to the heavy fields we can't make them to sort to try to remove them from the physics because experimentalists that are doing physics at this energy scale don't know anything about these heavy particles the theorists in this true in this toy world know everything there's some string theorists and they use anomaly cancellation to figure out the Lagrangian something like that but the experimentalists are not so sure that the string theorists are right so so they still have to build an LHC so if you want to make predictions for this experiment it doesn't make a lot of sense to carry this big fire around since you don't make it directly so instead we're gonna be doing something which is called in the language of effective field theory integrating out really what that means is that in this field theory there is a quantity which generates all the observables that symbolically I can write as a path integral over the lot over the heavy fields and over the light fields so you use your favorite method for computing this path integral if the theory is weekly coupled you can use Feynman diagrams otherwise you have to put it on a computer whichever way we do the calculation it makes sense to do the integral sequentially first we do it over the heavy fields and then we do it over the light fields and the reason why this is useful is because of this decoupling concept so let me try to explain that so we'll do the integral sequentially meaning that we'll first do the integral over the heavy stuff waited by the action and then since we're integrating over the big five all that's left is some function of the little five let's call that quantity as effective of little five and effective means that this is what we call an effective action okay and here's where the coupling comes in the reason why it's useful to do this is because this effective action and it's associated effective Lagrangian can be written in a very simple form so here's what the coupling means so there's some effective Lagrangian for the light fields and what the coupling means is that this effective Lagrangian can be written in a local way meaning that it has the following form so it's a local function local meaning that it's in other words just a function of phi at x so this is of the Lagrangian itself is a function of x because we're integrating over x to get the action and what the coupling means is that you can always write this Lagrangian as a local function of the light fields in their derivatives so it is a sum over many terms in general an infinite number of terms indexed by some index and then it involves a bunch of what are called coupling constants and then a bunch of as I said local terms so these are just functions of phi at x and the derivative of phi at x that's what the coupling is the first person to make a big deal out of this observation is Wilson and he did this in the 1970s trying to understand both the strong interactions QCD well pre QCD trying to understand the strong interactions and also trying to understand the critical phenomena in statistical mechanics he found that this language this decoupling language this idea of integrating out heavy degrees of freedom was very useful because of this and so I'm going to explain to you why this is a useful observation these coupling constants I'm going to assume that it was not Wilson who gave him this name but you never know are called Wilson coefficients they're just coupling constants the beauty is that let's ignore the mass of the little phi so it's zero the beauty is that they're complete work not completely but they're largely determined by dimensional analysis that's one of the nice things that the coupling tells you because there's two scales and well there's really yeah there's two scales in the problem the energy omega and the mass of the of the phi particle the energy omega shows up in the derivatives of the five fields the heavy scale though has been removed right the heavy field is no longer there so the I don't know the remnant of that scale is in the coupling constants of this low energy theory of this effective theory and we can just do dimensional analysis to figure out how they scale yes you mind speaking a little louder I can't hear you know it in the general Lagrangian I assume they can interact in any way I like so that's the magic of the coupling that I have to make very few assumptions about the form of the full theory and whatever happens at low energies is of the form that I write down that that's I'm not proving that to you it's that's that but that's the the main conceptual thing and we'll see examples in the context of the binary system not till tomorrow I guess alright so yeah Wilson coefficients we can do dimensional analysis everything has units of energy to some power the action is dimensionless as units of h bar as units of angular momentum so it goes like mass to the zero power therefore the Lagrangian because position has units of one over momentum so one over energy therefore the Lagrangian has to have in four space-time dimensions units of mass to the four so any term in this Lagrangian has to have units of mass of the four now let's assume that this term this oh which is something that you build out of the fives and their derivatives and so on has dimensions to some power let's call it delta so this is some dimensionless number so now we know the units of the coupling constant so this is mass of the delta this is mass of the fourth so C has to have units of mass to the four minus delta has units of mass there's only one scale in this problem which is the heavy mass one explicit scale the the light scale is in the five field so whatever this coefficient is has to scale like some dimensionless let's call it equal some dimensionless parameter divided by the scale delta to the minus four divided by this UV scale delta to the minus four so that's what the coupling means that whatever happens in the UV whatever the full theory looks like the effective field theory for justify is very simple in other words the effects of short distance physics on light stuff it's pretty mild there's only two possibilities it either and it depends on how big this delta is the critical number is four so delta is either bigger or less than four or equal to four terms with delta less than or equal to four are what are called relevant terms terms with delta bigger than four are what are called irrelevant they're not irrelevant in the sense that you can just throw them away they're irrelevant in the sense that they become small at low energies because we can just do dimensional analysis to try to understand the effects of one of these terms so let's call this delta I so the effects of oh I will therefore scale like omega over land over the scale lambda to some power and the power is the one that's given by dimensional analysis delta to the minus four so if delta is bigger than four then this is tiny and you can ignore it if if delta is bigger than or equal to four you cannot ignore it so it's relevant but it's a term that was probably already there in the original Lagrangian maybe modified by the interactions so the effects of the UV of the short distance physics are as I said very mild either they just generate terms that were already there so they just redefine the low energy parameters of the five field like it's mass or it's kinetic energy or something like that or they just generate these new terms these terms that are very small and that's the concept of the coupling it's useful because it tells us that we can organize calculations in in terms of the dimensions of the terms in the Lagrangian so the terms in the Lagrangian get more and more complicated as you include higher and higher dimensional terms they have more derivatives on the fields there's more of them etc but at a given order in this expansion they get smaller and smaller so you can hope to truncate them this is especially true given the fact that every experiment has some sort of experimental error has some sort of a resolution let's call that epsilon and it's hopefully less than one the experiment is good and so therefore we only have to keep terms up to a given order in this dimension expansion up to some maximum dimension so even though the expansion in operators or terms is an infinite expansion you only have to keep a finite number of terms so not not delta less than four let delta less than this delta max because if you keep more there's no point unless you build a better experiment so it's all really follows from dimensional analysis together with this highly non-trivial fact which won this guy in Nobel prize what else can I say about this so so so these theories have predictive power because we know that we always have experimental resolution that is finite so predictive power is there because really only a finite number of terms are needed the number of terms being dependent by the experiment so that's the basic fact and I hope to give you examples in the context of the binary system I guess probably not till tomorrow but now the question is how do you use this fact and I can think of two ways of using this information depending on whether you're a string theorist and you know the full theory or you're just like myself a lowly particle phenomenologist and you're more agnostic so how do we use this information well let's first of all one way of using it is to say that we actually know the full theory then you can certainly you're free to calculate using the full Lagrangian but that might be complicated and painful and it might just be simpler to remove the large scales in the problem in this way that I mentioned first and then do calculations in the low energy theory so what this what this concept what this decoupling idea or what this effective field theory idea in this case it's just a way of organizing an expansion a systematic expansion in powers of ratios of scales so it's it's a bookkeeping device if you like but it's a useful bookkeeping device and it organizes an expansion in powers of the observable energy scale the scale of the experiment divided by the short distance scale lambda sorry say that again so so it's it's the case where it's like the LHC before the Higgs so we kind of know that it's going to be there but we haven't made it so why should we keep it in our theory we should remove it in principle you can get them by integrating out but in practice you do something else so that's actually gonna be more my second example so I'll say a few words about that you're consistently a little ahead of me in these like so it's a bookkeeping device it's a really useful bookkeeping device because it doesn't just tell you about the powers of omega over lambda it also tells you about non-analytic dependence on omega over lambda that's because what I haven't told you so far and maybe it's a slightly more technical than it need to be these coefficients obey what are called the renormalization group equations they depend on a renormalization scale that you need whenever you start doing radiative corrections in any field theory so the bookkeeping that I mentioned coupled with something called the renormalization group which means that the coefficients here actually depend on an arbitrary renormalization scale that you need in order to make sense of loop diagrams Feynman diagrams it's a function of mu it's a function of some scale renormalization scale it's not a physical scale it's a theorist scale that you used to do calculations and so these parameters evolved together according to renormalization group equations that you can calculate in the effective theory and they look just like I don't know equations for a particle moving on some space so if you solve these equations they depend on initial conditions a nice place to set initial conditions is at the large scale and that's where you set initial conditions where these parameters are typically of order one order one not one and then evolve using these equations down to a scale of order the physical scale that you're interested in and when you do that you get logarithms so if you know the effective theory if you know the full theory it's still useful to do effective field theory because by solving some pretty simple renormalization group equations I hope to give an example of one again next tomorrow you can get logarithmic dependence on the ratio of scales as well so a typical observable using this glorified dimensional analysis this decoupling together with the renormalization group the prediction is that a typical observable is going to scale like powers of this ratio times powers of logarithms of the same ratio and the renormalization group tells you exactly what those logarithms are and exactly what those coefficients are so doing that is a lot easier than doing calculations in the full theory where you have all sorts of scales in the perturbative expansion and it's a big mess so another way of describing what I just said is that this bookkeeping procedure views the full theory in as a sequence of effective theories which are easier to deal with so the picture is that we have a full theory for example let's imagine a full theory where you have two heavy fields five one and five two and a light field and then what you do instead of calculating with that big mess with all sorts of particles with different masses is you just integrate out each one at a time so you first integrate out the heaviest one let's call that five one and then you just do renormalization group evolution in an effective theory that just contains the light fields which relative to five one are big five two in little five and you do that until you hit another threshold which is the mass of the second heavy guy so from the point of view of this theory this guy is light but once you get to low enough energies it's no longer light so why keep it around so it's a good idea to just remove it by hand removing it by hand this what I just did before this thing of integrating out it's also called matching so that's a buzzword that I will use from time to time and then you run down again using the renormalization group equations in an effective field theory that now no longer contains this just contains the light stuff and then you keep going all the way through you get to the scale of the problem you're interested in so this cartoon is how effective field theory handles calculations the prototypical example of that is is the weak interactions in particle physics the weak interactions as as as Salam and Weinberg and Glashow pointed out are mediated by a charged heavy spin one particle called the W boson and a typical weak interaction process a leading order involves the exchange of the W boson so here on the external lines are some fermions doesn't matter for my purposes there's some order one coupling constant here associated with the weak interactions but now let me imagine that I have been doing experiments at momentum transfers Q which are much smaller than the mass of the W in the mass of the W it's about 80 gv it's known to like four digits five digits but let's just say it's 80 so this is the full theory it contains light particles like fermions and it contains heavy particles like the W boson but but at low energies it makes a lot more sense to just remove that particle because its effects look just like a local interaction between the light particles with an effective coupling constant known as G Fermi which is of order G squared divided by the mass of the W squared Fermi didn't know about the W boson nevertheless he actually knew about the scale because what he did is he didn't know the full theory he knew the effective theory and he just used it to compute things and computed this parameter and found it to be roughly one over 250 gv squared so he could have figured out the standard model not really anyway so that's that's the sort of prototypical example of an effective field theory and you might say so what this diagram is not that complicated so what do I need to go through all this trouble for the answer is that if you want to do radiative corrections it's a lot easier to do them in the effective theory than it is in the full theory at least that energies smaller than the W mass so if you were going to do the radiative corrections in the full theory you would have to compute Feynman diagrams involving W boson exchange and then exchange of let's say these are quarks you would have the exchange of what's called a gluon a massless particle and so these are the sort of diagrams you would have to calculate but as was actually first pointed out by Whitten in the 1970s it's stupid to calculate these box diagrams because you can just do them in the effective theory and get answers same answers you have to get the same answers but a lot easier so this this calculation is a calculation that involves all sorts of stuff it involves the masses of the light quarks it involves the masses the mass of the W etc and it's a you have to calculate some box diagram it's not the end of the world but it's not something that I particularly like doing however in the effective theory where you remove the W boson it's a lot cleaner all you have to do is calculate diagrams that look like this where you have gluon exchange and it is an easier thing to do so that's sort of the canonical example and certainly the kind of effective field theory that I'm going to use for the binary problem it's going to be it's going to use effective field theory in this sense it's also going to use effective field theory in the following sense so the second way of using effective field theory is what happens if you do not know the full theory so we don't know what quantum gravity is for example nevertheless we can still make progress because the effects of quantum gravity decouple if what if the full theory is not known and you only know the light particles because or those are the only particles that you made at an accelerator for example you can still use this concept to try to understand the effects of unknown heavy physics short-distance physics because the coupling guarantees that whatever this effects of the UV scale they have to show up as local terms in some effective Lagrangian involving just the light fields so the Lagrangian would just look like the stuff that we wrote down earlier today just involving these local terms so it's a way of parameterizing ignorance order by order in a ratio of scales that's why it's so useful given the experimental resolution is finite I only need to keep a finite number of unknown terms and fit them to data and this is particularly useful when there are symmetries in the problem that constrained the form of the Lagrangian so it's useful because it's a way of parameterizing ignorance and so I'll write that in a systematic order by order in the expansion parameter way and it's most useful and it's most of the time used if the low energy dynamics has some known symmetries associated with it because then you can use the symmetries to constrain the parameters not the not the coefficients themselves but the the operators the terms in the Lagrangian I will also use effective field theory in this language I guess I'm almost out of time today so I'll just give you examples of this second way of doing effective field theory I'm pretty hungry so after that I will stop and go eat I don't know what you guys are gonna do so I guess one typical exam one good example is classical gr I do not know what quantum gravity is but I know that whatever new degrees of freedom it has probably live at the plank scale so if I'm interested in I don't know doing graviton graviton scattering at low energies meaning much smaller than the scale and plank 10 to the 19 gv I can do so in an effective field theory that just includes gravity it just includes the graviton so in this case what I call the light degree of freedom is just the metric g minu of x so it corresponds to a massless particle mass equal to zero called the graviton and then you can without knowing everything there is to know about quantum gravity write down an effective Lagrangian for that field by itself the guide to doing so are the symmetries of the problem because what Einstein told us a hundred years ago is that there's something called the equivalence principle and it can be interpreted as something called diffeomorphism and variance which okay if you're gonna be a stickler it's not really a symmetry but let's view it as a symmetry and so this low energy dynamics has the following symmetry I can take the coordinates and change them at will any way I like this metric transforms in some prescribed way and so then you can start writing down the most general Lagrangian for the effective field theory based on the symmetries so the simplest term that you would write down would have some sort of would have no derivatives on the field that term is the cosmological constant term then the next simplest term you would write down the symmetries tell you that it's got to go like the curvature linearly and it's got some coefficient let's call it the plank scale itself because that's what it has to be by dimensional analysis and then you could keep going and add our squared terms and now these involve actually these guys don't have a scale but if you go to our cubed you get a 1 over m plank squared etc etc and so a very low energy is what survives are just these terms and indeed pretty much every lecture I think every lecture that you will have will involve these terms in some way so that's one way in which effective field theory is used if you don't know the full theory you can constrain the low energy physics in a model independent way and systematically parameterized deviations the second example is the standard model itself because we know all the particles in the standard model by now and we know their symmetries but we hope that there's some sort of UV physics near near the TV scale I still hold out hope keep the hope alive but what constraints that theory is a symmetry called the SU 3 times SU 2 times U 1 and so the Lagrangian has a bunch of terms with dimension 4 which are in every textbook like pescan and shorter involving quarks leptons gluons Zs Ws photons etc and then it has terms with a dimension 6 for example etc etc and then this scale experiments constrained it to be by now something of order 10 TV so even if I don't know what's happening at 10 TV I can put this into the Lagrangian and put constraints on it based on low energy data that's called precision electroweak physics and I guess since I'm hungry I'll do one last example I was gonna talk about QCD but I'm too hungry so the last example I will give is of condensed of condensed matter systems they also use effective field theory precisely in this way at short distances they might have some complicated crystal or a fluid or sorry yeah complicated crystal or a system of interacting many body atoms or some many atoms or something like that but at low at low frequencies you can write down an effective Lagrangian for the low energy degrees of freedom an example so this is a condensed matter and an example that I like of effective field theories of this sort are the effective field theory for fluids that were written down by Nicholas and his collaborators starting in 2005 I think it was and they've been actively is that right 2005 6 and till now they've been working out the consequences so that's a very beautiful example of an effective field theory so I think I'm out of time is that right yep I'm starving so I think that's all I will say for now I can answer some questions if there are any if not I'll continue next tomorrow I guess Walter wants to go to lunch so it's possible to have lunch either at a cafeteria here or the bar upstairs or even the cafeteria down at Adriatico which is also open okay let's thank you Walter for this beautiful lecture