The Biggest Ideas in the Universe | 10. Interactions

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today we're going to be continuing on the theme that we already started in the last video we started in the last video we talked about the idea of fields quantum fields in particular and this is such a deep and rich topic it's gonna take more than one video to get through all of what's interesting about quantum field theory in fact in the last video really all we did was established that if you had free fields that just say fields that didn't interact with each other or even themselves then you could think of them as collections of particles today we're gonna take the next obvious step and let them interact with each other so today's big idea is interactions and the way to do that is of course like many of these videos I have the title then I have what I really want to talk about what I'm really talking about today are Fineman diagrams and how we use vitamin diagrams to calculate the interactions between quantum fields thought of as collections of particles so to remember where we are we have classical field configurations oops yeah let's try to make that prettier ok classical fields which is our starting point have a configuration and we just started with the simplest case of a scalar field scalar field meaning just that at every point in space there is a real number a value for that field more complicated examples would be vectors or tensors or spinners or whatever and then we quantize it okay so we get a quantum wave function and that is just an assignment to every possible field configuration of some complex number so this is not Phi at X but the configuration of five for all X's gets a complex number capital psy that is the amplitude for that particular field configuration so you might have in just a simple one-dimensional example if this is space X and this is the value of our field Phi you have some configuration and you call that Phi 1 of X and the wave function assigns a complex number called the sigh of y12 that particular configuration yeah some other configuration over here this is all when I say over here I mean over here on the pad that I'm writing on not there in space all these configurations are hypothetical possible configurations of the field in space so here's another configuration you could have call that Phi 2 and the quantum theory you would have an amplitude for that being the actual field configuration and then you measure it you observe the field everywhere somehow and the probability of observing some particular field configuration Phi is just the wave function for Phi absolute value squared that's the born rule for quantum field theory good so that's a big abstract mess and we needed to learn how to sort of make some sense of it to turn it into something useful and what we did was to show that in this simple spherical cow world where we had free fields not interacting with each other we could first decompose them into modes that is to say into collections of plane waves of field configurations that are just sine waves all throughout space with a certain direction they're moving in a certain wavelength and then we could treat each mode one by one and a wonderful spectacularly helpful thing happened each mode behaves like a simple harmonic oscillator we know how to deal with simple harmonic oscillators so in particular simple harmonic oscillators have a lowest energy state a next highest energy state a next highest energy state and so forth and it turns out that we can interpret this formalism in the case of the modes of a quantum field as corresponding to zero particle States one particle States two particle States and so forth so we have modes of a field and we could write those as Phi sub K K is the wave vector it tells you the wavelength in the direction that the field is moving in as a function of H the height so very roughly speaking you have some sine wave here this is X this is Phi K and the wavelength which is the distance let's say two troughs lambda is 2 PI over the size of K and the height is just this tiny value here H the amplitude of the highest peak in the sine wave and so we showed that modes of a field are like simple harmonic oscillators of course there are an infinite number of modes so that's an infinite number of simple harmonic oscillators vibrating all throughout space and we can quantize a simple harmonic oscillators solve the Schrodinger equation one by one and we find that there are energy levels equally spaced and then we interpret those energy levels as collections of particles so at the end of the day a quantum field is just an organizational tool I should say just is an organizational tool for talking about many collections of particles and in fact collections superpositions of different numbers of particles I should also say that this is in the spherical cow regime where the field is non interacting things become a little bit more complicated when they interact a little bit we're gonna ignore all those complications but when the fields interact by a lot or the fields do something dramatic this description in terms of particles might be completely off-base there are absolutely situations in which the field enos of the quantum field really does matter that it's not just the cheap substitute for thinking about particles hopefully we'll get a chance to talk about some of those situations later on okay so real fields I don't mean the real numbers I mean actual fields in the world are not free that is to say they do interact so we have to sort of turn on the interactions a little bit just like Galileo could drop a ball and say well let's ignore air resistance and put it in later we're starting with free fields and then we're putting in interactions later so in fact what happens is that particles you can still think of them as particles this is the the wonderful thing about field theory even when we turn on the interactions you can still think of them as part they scatter I'm not gonna write all this down they scattered dot they can scatter off each other they can annihilate they can decay write a neutron decays into a proton and electron and antineutrino they can create other particles and il-8 and so forth they can recombine so we want to invent some way of thinking about this we made a huge step by going from quantum fields to particles but we want to do the next step so what does that involve so here's what we should be thinking about the the nice thing as I as I said as I hinted at if you hear some scratching in the background I think that's Caliban he's a little bit excited this time of day the thing that we can do is not only think the free fields as particles but continue to think of them as particles even when we turn on interactions the difference is unlike ordinary non field theory quantum mechanics the number of particles is not a conserved quantity the wave function of the quantum field is a superposition like I said of different numbers of particles and the Schrodinger equation that tells you how that wave function evolves can change the number of particles through different kinds of interactions so we want a way of dealing with that but nevertheless we're gonna deal with it by thinking of it as particles oK we've sort of used the field NIST to justify that we can think of it as particles for the rest of this lecture it's particles all the way now I keep calling them lectures they're videos there are nice informal videos they're not lectures don't be intimidated okay so what we're thinking of is a collection of particles and we start by imagining the particles are unentangled so despite all the Cupra about saying that there's only one wave function of the universe we start by saying that there are two different particles let's say if we're doing the scattering experiment two different particles that have their own wave functions so here is particle one with its wave function it's traveling this way here's particle two it's wavefunction traveling that way and this under the short agur equation is going to evolve into I'm gonna draw it bigger because it's going to be more fun lots of little particles going out in all different directions right these little blobs I'm drawing you're supposed to be little wave packets a wave packet is just a little bit of wave function this sort of localized to one region of space and you can make these wave packets as we said even by superimposing plane ways that stretch all over space that's kind of the fun thing about the Fourier transform you can get a localized function by summing up a bunch of functions that are certainly not localized ok so this is the transition through time between let's say two particles coming in scattering out and this picture on the left is perfectly sensible we start with two different particles unentangled they have their own wave functions the picture on the right is entirely suggestive and not very sensible because after we're gonna have a situation where there's a lot of entanglement it's not gonna be that crucial but it's there but more importantly it will be a superposition of all sorts of different numbers of particles so will not look anything like this nice neat thing where you have just a certain collection of wave packets that are all separate from each other the individual components of the wave function will look that way so that's what we're gonna do we're gonna sort of do it chants fight chants right you have two particles coming in let's analyze the idea that they scatter in two particles scattering the three particles scatter the four particles etc but how do you actually calculate that and you know I'm gonna give you sort of a half-baked explanation of this by the end you will not actually be able to calculate anything at all but you will know how the calculations are done at least half of the ideas behind them so if you ever want to pick up a book on quantum field theory you'll be very prepared conceptually to follow what's what's going on and it was of course richard fineman who figured out the way that we all do this in the brilliant idea of Fineman diagrams and 5min diagrams you've all seen them right the little pictures a little cartoon stick figures of particles coming together but one thing that emphasized here is that they're not just pictures you might think if you just if you didn't go too deeply into what's going on here that the Fineman diagrams are just representations of things that can happen you know the two electrons can scatter off of each other by exchanging a photon but they're much more than that every Fineman diagram is a number it is a way of calculating a number and that number is a contribution to the amplitude to the quantum mechanical wave function for that set of part pickles that are going out of the Fineman diagram so the initial particles in the Fineman diagram represent a starting wave functions the outgoing particles represent a contribution to the final wave function environment gave us a recipe he didn't just say this was a good idea to think about he actually told us how to do it so he said that there are basis that we can think of some basic set of interactions so by interaction I mean let's say an electron can absorb a photon that's an example of interaction and then we combine them in all possible ways so if you start with some idea that you're going to ask well how can electron and a positron which is an anti-electron how can they scatter off of each other to form just another electron and positron not change their identity just you know scatter off of each other in some way well by starting with that fundamental interaction of photons and electrons you'll see you will see in a second there's a lot of different ways to combine them to get that final interaction and Fineman says do them all combine them all this is quantum mechanics everything that can happen is going to contribute to what's going on you know in the back of his mind he was inventing the path integral formulation of quantum mechanics where literally if you just remove point particles every trajectory the particle can take will contribute to the final answer that was sort of an inspiration for Fineman diagrams but you don't need to know anything about the path integral way of doing quantum mechanics to do Feynman diagrams it's much more straightforward than that and then you calculate the number using this combination the amplitude and you know what to do with the amplitude when someone hands you a quantum mechanical amplitude the probability of whatever final state you're considering is that amplitude for that final state square so I do this I write this over and over again so I really do want to emphasize this fact I already mentioned that quantum field theory is not a replacement for quantum mechanics it's a specific example of quantum mechanics all the rules of quantum mechanics are still here in quantum field theory it's just the thing we're quantizing our fields not particles okay so that here's an example the example I just mentioned let's think about an electron and a positron remember that Paul Dirac taught us that there's things called anti particles so an electron is a tiny particle with a negative charge and a positron is a tiny particle with a positive charge which is the anti particle of the electron but because the electron was discovered first and it's more important for our lives we write the electron as E - it is the electron with a negative charge the positron we don't write as P because P is for proton we write the positron as a plus it's the anti particle the electron we're gonna take an electron and positron and ask if they scatter into an electron and another positron so they don't change their identity they just maybe change their momentum they literally bump into each other now this is not the only thing that could happen you know you're not done okay you would also like to calculate the probability that this electron and positron do everything else that they can do and we will have a way of doing that but for right now we're focusing on this particular final state so we can find the probability of that and roughly you know schematically before we get into the details the picture is we have time running left to right so again different ways of thinking about physics conventionally choose different ways for time to run in relativity and space time time runs from bottom to top in particle physics sometimes it does that but more often it goes left to right and there are studies I mentioned Larry Bird it's key you know people who are Hebrew eaters or other people who read right-to-left sometimes make it go right to left so we will do left to right because that's my culture that I come from and we start with an electron and a positron and we draw lines representing these particles moving through time and then stuff happens okay so this is a blob that I'm going to label stuff happens and will be a little bit more specific about that in a second and then I'll emerge another electron and a positron when I say another one because they need not in principle be the same ones there's in sent some sense no such thing as the same ones you know these electrons and positrons are vibrations in the underlying electron and positron field so they do not have any personal identity over time okay but this is the the kind of interaction that we want to think about we want to know how do we actually calculate numbers for this okay well Flyman gave us the rule he says we start with some basic set of interactions for this process what we do the scattering we're gonna do it in the context of quantum electrodynamics okay QED as it is called quantum electrodynamics that is finally wrote a nice little book about it at a semi popular level that you can that you can pick up in stores it is the first really successful full-blown interacting quantum field theory it was the center of focus for people like Fineman and schwinger and tomonaga when they were thinking about renormalization the interactions and all that stuff and it's the theory of just electrons positrons and photons so it's electrons and positrons interacting with themselves and with each other but only through electromagnetism so we're not worried about the weak nuclear force or gravity or anything like that okay so in that context we have one fundamental interaction and sometimes because we're in Feynman diagram land we talk about interactions as vertices so the vertex in quantum electrodynamics is an electron comes in positron comes in and they hit each other and a photon comes out photons are for historical reasons and particle physics photons are labeled gamma okay so this is this is a photon so here's the interaction and there's one more little notational thing that happens which is that there is a feature of electrons and positrons that there is a certain amount of electron NISS which is conserved okay in quantum electrodynamics and other theories it's not but in QED the electrons are neither created or destroyed what they can do is travel backwards in time so this is the famous idea of Fineman that he actually he popularized if he got it from someone else whose name I'm forgetting right now but the idea that you can think of anti particles as particles traveling backwards in time and of course it's only a matter of our human prejudice that we label electrons particles and positrons anti particles the point is that they are anti particles of each other it's just as fair to say the electron is an anti particle the positron as it is the other way around but we are full of electrons so we you know give preference privileged to the electrons so we draw a little arrow on the line to represent the direction in which the electron is moving and because the positron can be thought of as an electron going backward in time we draw the arrow backward okay so you gotta get this straight the particles are moving from left to right the electron is for the electron is therefore moving left to right but because the positron can be thought of as an electron going backward in time the electron this is going right to left in the case of the positron okay so these arrows do not represent the direction of motion of the particles they represent the flow of this quantity called electron s which by the way no one ever uses the word electron s but it's the closest you'll get is something called Fermi on number and we'll talk about what that means but electrons are forming a number +1 positrons are Fermi on number minus 1 okay and this is the fundamental interaction from which we will build all of our quantum electrodynamics diagrams and the final piece of information need to know is like I said there is a number attached to this you use these pictures to calculate the amplitude and the number that is - this vertex is the square root of alpha alpha is the fine-structure constant and you'll see why it's the square root in a second the fine-structure constant is just a way of thinking about how strong the electromagnetic interaction is right it's a dimensionless number alpha is approximately 1 over 137 there was a time when people thought maybe it was exactly 1 over 137 and tried to explain that using numerology that didn't work this is not exactly 137 but the number 137 still holds a precious precious place in the minds of particle physicists so the fine-structure constant is a small number it's less than 1% right it tells us how strong the electromagnetic interaction is and every one of these vertices gets a factor of the square root of alpha so one thing to keep in mind here is that this is reassuring to us the fact that alpha is small because we're thinking of these interactions as tiny perturbations on top of the spherical cow of the free field theory if the interactions were really really strong you couldn't do that right putting on the interactions would change everything in a dramatic way the fact that alpha is small is what lets us get away with this way of thinking about interactions in quantum field theories ok and there is one rule you need to know the rule is you can flip particles lines of particles let's say these lines are called the propagators but we won't get into that detail the lines that we're drawing on the climate diagrams that's all you need to know we can flip the lines between it in and out so if a particle is coming in and this electron and then we have a certain diagram we can draw then we know there is another diagram where the particle is coming out and we just flipped it literally sort of moving it over the arrow now changes its direction at which it's moving so by interchanging particles with anti particles so what that means is we drew this diagram with an electron and positron coming in a photon coming out we said that's associated with the number alpha so particles go to anti particles means eehm eyeness gets interchanged with E Plus electrons become positrons and vice-versa the photon in this way of thinking is its own antiparticle so photons just remain photons there's no electric charge that would have to change assign photons or neutral particles so they're just their own anti particles in this way of thinking and therefore this diagram let me shrink you down a little bit here this diagram implies the existence of a couple of other diagrams it implies the existence of a diagram where an electron comes in and we take that positron line and move it over from coming in to going out and therefore we get a photon going out gamma and guess what another electron coming out so this is another diagram another interaction where rather than two particles positron electron coming in and annihilating into a photon we have an electron just spitting out a photon but it's the same Fineman diagram status so this diagram also comes with the square root of alpha associated with it and likewise we can flip both particles both electron and positron on the other side and make the photon come in and these go into a minus E Plus okay also associated with the factor numerical factor of square root of alpha and so forth and you can do other things as well once you get inside the middle of more complicated climate diagrams there's a lot of different things that you can imagine doing okay so I mean that's that's already kind of nice right it's kind of a little bit of a unification in some sense this is saying that there's something happening something that could happen called the annihilation of an electron and a positron into a photon there's a certain number the fine-structure constant that tells us how likely that is there's a completely different sounding thing which is the photon can be given off by an electron or by a positron equally well and that also is characterized by the same number and this is why at the Fineman diagram level because they're just the same climbing diagram flipping some particles from past to future because remember time is going this way and all these pictures okay so let's put it all together that we want to calculate an electron and a positron going into a photon no sorry that's not what we calculate we want to calculate these going into another electron and positron okay so how can we do that well Flyman says basic set of interactions that's what we got combine them in all possible ways okay so given that we want to start with e minus E Plus we have P minus E Plus they go into this what what can they possibly do all they can do is do that fundamental interaction that we drew right that's the only thing that we can start with in this picture I can keep it prettier than that and then they turn into a photon but the photon isn't what we want coming out we want an electron and a positron but guess what the photon can spit out an electron in positron so here's our arrows II - EEP Loess okay and that's a photon in the middle this vertex gets a factor of alpha this vertex gets a factor of alpha so there's a number associated with this diagram which is approximately alpha the fine-structure constant that's why it's a square root of alpha because historically things like this are what happened and so we use alpha to characterize this by the way when we square it to get a probability that'll be alpha squared anyway so this is a process that contributes and it gives a number we can calculate the probability that the electron and the positron will scatter off of each other there are other factors that matter right the fact that if you have an electron and positron far away they're less likely to scatter than electron and a positron very nearby that comes into other details that I'm just not worrying about right now so called kinematic factors what is the position and the momentum etc of the wave packets coming in but this is the fundamental quantum field theory nests of the calculation but it's actually not the only thing that can contribute right because I said the only thing the electron and positron can do is emit a photon but there is another I yeah I'm realizing that was a complete and total lie now that I say it out loud now that I think about it it is true that the electron has to sort of emit a photon the electron cannot sort of directly interact with the positron without emitting a photon but there is another way to draw that which is this e minus E Plus okay the electron comes in and it emits a photon the positron comes in and it absorbs a photon look at that I'm sure all the particle physicists the audience were laughing at me when I said there's only one way to draw the first one this is backwards this is another perfectly good diagram also of order alpha right the same size roughly speaking as a contribution to the final amplitude and we would add them together this is in fact an example of quantum interference these these diagrams have a typical size of order alpha the fine-structure constant but I didn't tell you what they are positive or negative or they can even be you know imaginary components they're complex numbers in general all of quantum mechanics is about complex amplitudes remember in the QED a video for quantum mechanics we talked about the double slit experiment and the fact that it's a crucial feature of quantum mechanics that the wave function is not just a probability distribution it can have positive and negative values and it can interfere and that sort of makes us think the wave function is a real physical thing this is still quantum mechanics we're doing these are contributions to the wave function we will add them together and then Square to get the probability of what's happening and these can interfere with each other and that interference has been observed in experiments all the Fineman diagrams give you separate contributions you have to add them up and they can cancel out to give zero there are situations where there's a certain kind of process that you would think happen if you just threw one vitamin diagram but it doesn't actually happen because all the diagrams just cancel out and you get zero at the end of the day so it's still quantum mechanics there can be interference the other thing to say is to go back I said that this is not the only thing that can happen right so there's another process there's also processes like this fun one where an electron and a positron come in and you go like this so you have two photons coming out and the electron this goes this way comes out that way okay so this is a different process this is not a - e+ goes to e - e plus this is e - e plus goes to two photons it's another thing that can happen you would not add this to the previous pieman diagrams because it's contributing to a different final state just like in the double slit experiment the interference happens at every point on the detector you add up what the contributions are there you don't add up contributions at different points of the detector you add up contributions to the same final state so these two diagrams contribute to the same final state electron and a positron this diagram and once like it contribute to different final states so they're there but there's different processes we don't have to go into them right now okay so in fact let me just get rid of these because they're there I didn't want you to think that you know we were done but let's let's continue on with this particular process we're not done even now drawing you might think are you like think about it all this looks like all the diagrams we can draw not true there are an infinite number of diagrams we can draw so in fact there's an infinite series and that kind of becomes important the infinite series is because let's draw it this way we have just very schematically we have this blob that represents stuff happening okay what we're saying is there's a contribution that looks like this once you get good at the Fineman diagrams you stop drawing the arrows and the labels with E - and E Plus you can also label these individual lines these propagators with their momentum and their energy these are properties that they have but you just keep those in mind implicitly and stop drawing them so as a diagram like this there's another diagram that we drew that looks like this okay but there's also diagrams let me see what I chose to draw here's a diagram with two photons going back and forth and of course once you realize that then you realized there's an arbitrary number of photons that's clearly going to be an infinite number of diagrams like I could put three photons for photons five photons it doesn't stop in any way but there's also different kinds of diagrams here's a diagram where the electron positron and I lead to a photon then the photon creates an electron and a positron but they reunite into a photon and then they go off right that that contributes to the final answer there's one where the electron and positron come in and I like to a photon but another photon gets exchanged between them before they annihilate and then they go off and then you know as you can imagine they become more and more complicated so this seems like bad news this seems like oh my goodness we're trying to do some simple calculation there's an infinite number of things I have to calculate and that sounds bad but in fact that's not bad remember alpha is approximately 1 over 137 so relatively tiny number and every one of these vertices comes with a square root of alpha so the more vertices you get like something here you have let's colorize it some like this diagram you have 1 2 3 4 vertices as opposed to over here you only have to write so you will in fact get more and more an infinite number of diagrams but only more more complicated diagrams we've drawn all the simple diagrams already and as the diagrams become more more complicated this diagram looks like alpha this one looks like alpha but this one looks like alpha squared as does this alpha squared alpha squared and higher order diagrams will look like alpha cubed alpha 4th etc because alpha is less than 1 as you raise it to higher and higher powers you get a smaller and smaller number so the hope with Fineman diagrams which will be partially real exactly realized but in a complicated non-trivial weighing the hope is that even though there's an infinite number of diagrams these diagrams become less and less important and in fact in practice you don't have to calculate an infinite number of diagrams you have to calculate the small number of important ones to get an answer that is really really good it becomes harder and harder to calculate the diagrams as they get bigger and bigger I'm leaving out a lot of details about what you actually have to do to calculate them but you get the basic scaling going on here so this is an example of perturbation theory which I mentioned before perturbation theory in action perturbation theory is the idea that you can start with something you understand perfectly like free fields or a ball falling without any air resistance and you can add in tiny effects and add them as perturbations so the way that the reason why that works is because they generally in perturbation theory more generally not just in this context you get an infinite power series by power series I mean so you get an infinite series an infinite series let's say a for the amplitude we're looking at can be written as a 0 plus a 1 plus a 2 plus dot dot where a n is proportional to alpha it's terrible that the proportionality sign looks like an alpha so I'm going to say some other number epsilon to the power n some small number epsilon epsilon is less than 1 so this general feature is that all of the terms in the series are smaller than the previous terms because they are proportional to epsilon to some power and epsilon is a small number you might worry you might say well proportional to covers a great deal of sins you know maybe the constant of proportionality is really big for some term later on and actually that's just a legit word you should worry about that I'm not going to poopoo that in practice if you're dealing with well-behaved systems perturbation Theory works very well ignoring powers of epsilon to the thousandth power R is a pretty good approximate but you have to be careful things can go wrong things can be subtle but this is that this is the thing that most particle physicists learn to do when they learn their quantum field theory of course the other thing that can happen is so this this feature here that you're a power series in a small number means that your interactions are weak and now what I mean is by weak I mean not very strong physicists love to use the same words in different contexts to mean different things so there is an interaction in particle physics that we'll talk about later called the weak nuclear force it's a dumb name for an important interaction so sometimes when you say weakly interacting particles like the dark matter candidates weakly interacting massive particles what you mean is particles that are interacting through the weak nuclear force other times like here all you mean is particles that are interacting through some interaction that is not that strong okay so this is just an interaction that is feeble and when interactions are feeble perturbation Theory works not always true in the so called strong nuclear force guess what the interactions are not weak they're not even feeble they are strong the thing that you might think is a parameter the thing that plays the role of alpha in the strong nuclear force can be greater than one and if this epsilon here in this power series were greater than one this power series would be useless every term would be bigger than the previous term and they would never add up to some finite number so it's not always true that perturbation Theory works but it's very is a very good approximation for a lot of very important problems certainly for QED through quantum electrodynamics it works very well okay so we let's do a little bit more digging into the details here so I said that this fundamental interaction in QED was proportional to the square root of alpha says who like who who invented that rule right well tell ya it was I mean the rules were invented by Fineman but they are ways of expressing rules that were already invented so Fineman diagrams are a calculational tool for dealing with quantum field theory so if I'm a did not invent quantum field theory but he figured out how to work with it in a plausible way in a way that everyone could do and you know there's the the joke that he tells like when he was first scribbling these diagrams he was immensely amused by the idea that if his idea panned out the physics journals would now be full of these slow scribbles and indeed you know the first time that he scribbled these things on a blackboard at a seminar he got made merciless fun of and now you open any physics journal it's full of Lyman diagrams so the rules where does it come from how do we assign numbers to vertices to interactions well you know even though we're doing quantum mechanics it's still physics so remember we did for classical mechanics we talked about the Newtonian paradigm with F equals MA but then we also talked about the Hamiltonian way of doing things where you have phase space and Hamilton's equations and we also talked about the principle of least action where you had some integral of some quantity over time and the minimum of that action that integral was the actual physical thing that you did the actual physical path that a real particle would take so the another thing that Fineman figured out is that you can this is this goes back to the path integral formalism is that you can do quantum mechanics with an action but rather than saying here's an action that is an integral over all time and the minimum value of the action is the real one in quantum mechanics they're all real but they don't all count the same the minimum value of the action is the one that will add up to the biggest contribution to the quantum amplitude but they're all there in some sense quantum mechanics takes advantage of all the different trajectories the system could take so it turns out for completely other reasons that if you're going to invent new laws of physics that are not new laws of physics but a model for physics that you don't yet understand it is way simpler more convenient to work with an action than it is to write down the equivalent of Newton's laws to write down the equations of motion for the fields to write down the equivalent of Maxwell's equations the reason I'll tell you the reason the reason is because these fields often have symmetries we'll get into that later but the symmetries matter a lot and it is way easier to implement the symmetries in the context of writing down in action than it is in the context of writing out individual equations of motion and that goes back to the feature that we mentioned earlier about these alternatives to Newtonian mechanics which is that they're often either in the action formalism or in the Hamiltonian formalism they're written down in terms of a single function the action the Hamiltonian you don't have to guess separately the equations of motion for every field you can write down one action and all the fields equations of motion come from that one object so we're gonna go back to the action in this particular way of doing things it will look classical but we will use it in a quantum mechanical way so the principle of least action says that we have an action s and there's an integral of kinetic energy of your particle or your system more generally - the potential energy and then you integrate that over time DT so remember integrals are sums so we're saying at every moment do you evaluate this quantity kinetic - potential you add them all up over all the different moments of time now the fact that you take this quantity and you integrate over time is always true so we often just forget the fact that we are integrating it over time again that's implicit in our background knowledge and we deal directly with this quantity which is called the Lagrangian the Lagrangian is just the kinetic energy minus potential energy is the thing you integrate over time to get the action so you can write down s equals something but it's much more common to write down L equals something so s equals the integral of L DT okay that's a nice way of saying what we want to do to invent our new theory is to invent the Lagrangian for that theory okay so saying this is true this is true in the context of particles moving in potentials and things like that but the same idea is true for fields but fields are a little bit more complicated right they're spread out all over space there could be different kinds of fields they could be vectors etc so you as you might expect the Lagrangian for fields is in principle a little bit more complicated but not that much so for fields Lagrangian L can be thought of as the kinetic energy minus something called the gradient energy we mentioned this before the kinetic energy is the field changing over time the gradient energy is the fields profile over space but we're not done yet there is still minus the potential energy so this is saying that in some theories of physics the value of the field itself contains energy when the field has zero value maybe at zero energy maybe not but maybe when the field is some large value that contains energy that's the potential the potential is not the derivative of anything that's what makes it the potential then also since we have more than one field in our game we have an interaction energy so if you have electrons and photons for example in quantum field theory they are both described by fields and there better be a term in your Lagrangian that tells how those fields interact with each other and that's what this term is so the game that is played by particle physicists the world over is to write down different lagrangian's for different theories of fields interacting in different ways now they do have one more big simplification locality locality in this sense means that two fields interact with each other two different fields or the same field in itself only when they are at exactly the same point in space so two fields far away might interact with each other indirectly like the earth can interact and the earth is made of fields in quantum field theory so is everything else so is the Sun made of quantum fields they can interact with each other through gravity but only because there's a gravitational field in between in some very strict sense the earth interacts with the gravitational field locally so does the Sun and that interacts back in all the way over in between so the the way that locality manifests itself in this language is that the Lagrangian for a quantum field is an integral over space of something which we call the Lagrangian s'ti and the Lagrangian city is a function of the field Phi and its derivatives oops I can't spell anymore I'll just say the derivative so I'm not gonna try to notational eyes that integrated over all of space so this curly L is the Lagrangian s'ti density is just something that we have at every point in space and then we integrated over everything to get a whole quantity right if you have energy density that exists at a point in space the energy in the volume is the integral of the energy density over that whole volume the LeGrande density exists at every point in space and we integrate all over all space to get the total value of the Lagrangian so this d3x is representing an infinitesimally tiny cube in space okay DX dy DZ that's what the notation d3 X means and this is just a yo an underappreciated fact about quantum field theory so it doesn't need to be this in fact let me be a little bit more clear here let's let me notice something here well it didn't quite work but close enough the fields are functions of X right so the field is Phi at the point X so when you're integrating over space both the fields and their derivatives have a value at every point and what this is saying is one field at point x interacts with other fields at point x naught with not at point y so it is not for example we could imagine taking you know Phi of X Phi 1 of X and a different field Phi 2 of Y and integrating d3 x d3 y so we integrate twice over space and have fields all over the place interacting with each other that would be spooky action at a distance right that would not be local that would be bad so let's just put a big X to that that's not what we do this way of writing the Lagrangian for a quantum field theory is just an enormous ly important feature of a reality the laws of physics didn't have to be that way they didn't have to be local now we talked in the video about space about the interesting fact that there is this thing called space in which interactions are local why is that no one knows why but we know that it's true and so in quantum field theory we take advantage of that in quantum gravity it might not be true this is why I keep emphasizing this because I'm trying to teach you ways that you know you eventually you're gonna grow up to be Nobel Prize winners who quantized gravity there's indications in quantum gravity that locality is not as important as it is in quantum field theory but right now we're doing quantum field theory work localities absolutely 100% central ok so because the action is always the integral over time of the Lagrangian and the Lagrangian is always the integral over space of Lagrangian s'ti the thing that particle physicists actually spend their time doing is inventing Lagrange densities ok and that's from the Lagrangian s'ti that we will get our rule for the Fineman diagrams so the QFT recipe quantum field theory what do you do if you want to be a grown-up quantum field theories first you invent a set of free fields and you write down their well kinetic energy gradient energy and potential energy right that's an easy enough thing to do and then you know you have the different kinds of fields maybe then you decide what symmetries the fields have can't go into this in detail right now need a whole another video to talk about symmetries but symmetries turn out to be SuperDuper important in quantum field theory whether or not it's things like gauge invariance which is a feature of electromagnetism and the nuclear forces or just position invariance time translation invariance all these things symmetries will restrict the different number of terms you can write down in your interaction LeGrande and then that's what you do you construct an interaction Lagrangian that is the thing that we indicated will exist oh by the way yet another abuse of grunge and yeah I got it right another abuse of language because the Lagrangian is always the integral of the grunge density not only field theorists particle physicists worry about the grunge densities they get tired of saying the word density over and over again so you will often refer to this curly L thing the thing that you integrate over all space to get the Lagrangian as the Lagrangian so I will do that too so curly L is the really LaGrant entity but we call the Lagrangian okay so this term here interaction energy is the interaction Lagrangian okay the thing that we want to great over space to get the actual Lagrangian so for example for an example we already have an example of one interaction an electron a positron and a photon can interact with each other in a certain way so we right L QED it's the interaction Lagrangian for quantum electrodynamics and I should polish to mention that in a real quantum field theory textbook they will spend pages and pages of introducing new notation and a lot of Greek letters and things like that I've already used all bunch of Greek letters myself but I'm trying to keep it as straightforward as possible so my doe tation is different but roughly speaking it is a number the square root of alpha again I'm hiding some four pies and things like that times the electron field times the positron field times the photon field this is the electron this is the positron this is the photon these E's have nothing to do with you know two-point-seven the Euler's constant gamma has nothing to do there's another constant the euler mesh roni constant I'm not going to pronounce it correctly but these are fields this is e minus of X is the electron field gamma in this very very simplified notation is the photon field and likewise for the positron field the reason why I'm sort of slightly smiling and saying that it's a simplified notation is the real photon field is a vector so it has indices it has directions and it's gauge invariant and it's a whole number of things you need to think about the real electron field and the positron fields are spinners and they have their anti particles and there are more indices yet to worry about but forget about all that we don't need to worry about that what matters is there's a term in the Lagrangian that has an electron field hitting a photon field and a positron field and together from them you get the interaction so this is the rule this is literally called the Fineman rules given this interaction lagrangian you get that this particular diagram II - he plus gamma gets the number square root of alpha the existence of a term in the Lagrangian with an electron field positron field and a photon field times a number means that there is a Fineman diagram with those three particles and the size of that Fineman diagram that interaction that vertex is the number whatever that number is in this case it happens to be the square root of the fine-structure constant and so this is the game that you play as a professional particle physicist you invent different Lagrange ten cities for different fields with different symmetries and so forth okay so you can see I hope you can see the basic outline I wonder how small I can actually shrink this yeah you can see the basic outline well look we've done a lot of stuff how awesome are we but there's a lot of details okay and what I want to do is rather than focusing on computational details right now I want to give you one little conceptual detail to close things off a Fineman diagram this is a point about what is actually physically going on when you look at a fireman diagram it's a story and it's not even an especially accurate story finding diagrams are a way of calculating something that is very rigorous and well-defined the amplitude for the wavefunction of the quantum fields to have a certain to be a certain number in a certain final state that you care about but the steps along the way have a different kind of status so by the way there's one thing I should have said I realized a while ago so when you when I wrote this diagram here you notice that for some of these diagrams or for this diagram well yeah for this diagram I wasn't very careful about whether the photon was going from left to right or right to left that's because you actually sort of it doesn't matter that really is what what matters because the photon is its own antiparticle so whether it's going forward in time where backward in time doesn't matter likewise when we draw so I erased that's why I can find excited it remember we drew let me draw it again you drew this picture so here's an electron going in with the positron and creating two photons gamma gamma I also didn't care too much about how to draw that intermediate line whether it's going forward or backward in time I just do it vertically and there you think it might matter because there's an electron or a positron or both well that's the thing it can be either one it doesn't matter because the electron and positron or the same things in some sense just one is going backward in time one's going forward in time but it's the same kind of field enos and I should also mention yet and yet another digression is many of you have heard of this idea called wheelers one electron universe John Wheeler who was a PhD thesis advisor for Richard Feynman as well as the PhD thesis adviser for Hugh Everett and also for Kip Thorne and a bunch of other famous physicists so the story goes that one day Fineman gets a phone call from wheeler and wheeler says I think I know why every electron in the universe has the same mass in charge and Fineman says it's 3:00 in the morning what are you doing and Wheeler says it's because they're all the same electron this is closely related to this idea that in a diagram like this there's only one line for the electron that is both representing the electron and the positron right the idea of Wheeler was that the electron goes forward in time then occasionally goes backward in time and that goes forward in time and if you drew the Fineman diagram for the whole universe there would only be one electron going back and forth and some complicated zigzag so many of you have heard that some of you have heard that theory probably few of you have heard it is wrong and that is not correct that theory so a lot of people sometimes ask me like what ever happened to that theory why don't people talk about that it's wrong and we know it's wrong it was a very clever idea not all clever ideas turn out to be right the reason why every electron has the same mass and charge as every other electron is because they're both they're all very vibrations perturbations in the same underlying quantum field okay we already knew the answer to that question why are all the electrons the same mass in charge they're the same quantum field vibrating in different ways furthermore we now know that electrons are not forever electrons can turn into other particles under the right circumstances like muons and stuff like that so it's just not true that there is only one electron all throughout the universe there are quantum fields but in the fiber diagram an electron going one way in time is the same as a positron going the other way in time therefore in these intermediate lines it doesn't matter okay that's what I wanted to say but that's not what I wanted to say that was a digression that got me away from what I wanted to say let me make this diagram bigger I can just reuse this diagram maybe I can make it bigger no I can't style nope can't make it bigger all right I need a bigger diagram so I'm gonna redraw it just one make everything clear here here's a Fineman diagram electron positron to photons by the way yeah nothing yet another thing it's life going on in Flyman diagrams so I keep forgetting things that I wanted to say why not this one why not just consider a minus e plus two a single photon as a physical process this is the certainly exists as a diagram right we it's the fundamental Lego building block that we use to make all of our other diagrams but we don't consider the process of an electron or positron dilating into just one Photon why not so I'll give you 10 seconds to see if you can think of the reason why not we can think of electron and positron annihilating to two photons but not one Photon the answer is that momentum is conserved okay so this electron and positron they're massive particles they're moving more slowly than the speed of light they have a total net momentum but for the two particle system and that total net momentum could be in its rest frame or it could be whatever but it's certainly moving slower than the speed of light right the center of mass of an electron and positron is always going to move slower than the speed of light whereas the photon can move only at the speed of light so you cannot have a situation where an electron and a positron annihilate each other into a single photon while conserving energy in momentum they can annihilate into two photons because two photons they're both moving the speed of light but in different directions so the center of mass of the two photon system can still be moving at zero speed or whatever speed the electron and positron removing and okay that's why we didn't do that and that way of thinking is very important for an about to say here because there is an implication of the fact that energy and momentum are conserved so what we can do is we can sort of complicate our diagrams a little bit by drawing little arrows for the momentum call it P 1 and energy e 1 this also has momentum P 2 e 2 for particle 2 this is particle 3 so it has momentum P 3 and age-3 particle 4 over here has momentum P 4 energy e 4 and so this is y 5 now you're in let's get complicated because you see the arrows representing momentum are separate from the arrows representing electron Nisour Fermi on number but okay the point is at every vertex the total momentum and total energy are conserved in a fireman diagram as part of fineman's rules that he made up so here you have a certain amount of energy and momentum which I could write in two different ways which are equivalent to each other so when p1 comes in it splits into the momentum coming down let me just write this as P v e5 and the point is that p1 equals P 3 plus p5 and likewise II 1 equals e 3 + e5 and that's because at this vertex right here p1 splits it goes partly into p3 and e3 partly into p5 and E 5 and then that P 585 recombine so we have p5 plus p2 equals E - nope equals e 4 what am i doing clearly missing my brain now p4 that's what happens at this diagram and likewise 'if I've + e 2 equals e 4 ok and then actually is enough to fix what he 5 has to be if you add these equations up and check you'll find that everything is consistent but energy momentum are related to each other right for a massive particle or for for any particle at all there's a relationship the energy squared equals the momentum squared plus the mass squared ok you can't separately decide what the energy is and what the momentum is this these quantities P 5 and E 5 will not obey in general this relation the energy is just whatever it has to be to satisfy energy conservation they will obey this so this these yes this no 4 e 5 is not true three-five okay so we have a way of saying that what we say is that when you get more colors sorry about that we say that these ingoing particles this whole bit here and the outgoing particles this whole bit here these are real these are real particles it means exactly what it means all the ingoing particles you can make right you can create some electrons and some positrons put them in the photons coming out you can detect right they're certainly real they're what you see in your detector but the intermediate stuff the stuff in between okay the particle that is neither coming in nor going out that has a name it is called a virtual particle in virtual particles here's the sort of big reveal virtual particles aren't really particles virtual particles are a way of talking about the vibrations in quantum fields what's really happening here is a vibrating electron field and a vibrating positron field two different wave packets are coming together and when they overlap all these other fields start vibrating along with them the photons electrons etc in different complicated ways and then something goes their merry way so this is why in practice quantum field theory is really hard to get right and to get rigorous because we understand what the particles are doing when they're far away and not interacting with each other both before we start and after but when they're on top of each other the math becomes very very difficult the Fineman diagram language says that we can think about what happens in between as if it were a set of virtual particles but these virtual particles do not move at the right speeds they do not have the right energies a virtual photon is not massless the energy of a virtual particle is not its momentum squared plus its mass squared in fact the energy can be anything depending on just making energy conservation come out right so they're not really particles but they're a convenient calculational device that's why I say apply my diagram is a story there is a true story which is what is the amplitude for sir reaction to happen but the intermediate steps are calculational device you kind of know that has to be true because the real process that you're looking at is the sum of all of these different diagrams and all the different diagrams have different sets of virtual particles in them they're not exactly the same so Fineman actually so anyway but that's that is the way of thinking about climate diagrams they're they're calculational devices for doing calculations in quantum field theory I just don't want you to take all of the steps along the way overly literally okay including the virtual particles in between now I'll read the reveal Fineman when he was inventing this he had an aspiration he was hoping that he was not coming up with a set of calculational tools to do quantum field theory better he was hoping that he was replacing quantum field theory with a good old fashioned particle theory the theory of particles that be created or destroyed following the rules of these Fineman diagrams turned out not to be right again many missteps along the way to any great idea and it turns out this is a way of thinking about quantum field theory not a replacement for it part of his motivation was the cosmological constant problem remember we talked last time about the fact that when you have fields pervading the whole universe these fields have energies and the energies can be added up and if you just do it naively the quantum mechanical contribution to the energy of empty space is infinitely big and this is one of those things that physicists have known for a long time but before around the 1980s they didn't worry about it that much but some people did so Flyman worried a little bit about it Philip Andersson worried about it a little bit for different reasons but they thought that this was a problem with quantum field theory right that it gave an infinite energy density to empty space if you could replace the fields with particles you could get rid of that infinite energy it didn't work so we are left with both this wonderfully accurate calculational device of quantum diamond diagrams and this somewhat unnatural formalism of quantum field theory we don't know what to do about that we didn't I'm not gonna reveal what to do about that we still don't know what to do about the energy density of empty space but we're thinking and it might be that we do in one way or the have to replace quantum field theory but in the meantime it is absolutely the best way we have of understanding nature currently available

Channel: Sean Carroll

Views: 75,761

Rating: 4.9202657 out of 5

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Id: PaRGj5Phpm0

Length: 65min 52sec (3952 seconds)

Published: Tue May 26 2020

Reddit Comments

[u/gillsh]

Sean’s output on these has been very impressive! I think I can speak for all of us when I say, thank you!!!

[u/NumberKillinger]

Haha these video backgrounds are getting wild