The Biggest Ideas in the Universe | 9. Fields

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today's big idea is something that builds on the last couple of videos where we talked about quantum mechanics and then entanglement today we're going to talk about fields in particular quantum fields right quantum field theory as it's typically known and what's going on here is we've already mentioned the idea of fields we talked about the electric field the magnetic field the gravitational field very briefly we didn't go into any details about them now we're going to be applying the rules of quantum mechanics to this idea of fields so far we've been talking about quantum mechanics it's more or less been in the context of a particle or a couple of particles or a spin or something like that but the big news is that according to modern physics the world is made of quantum fields and that should be a little bit puzzling at first glance because we already started with a particle and then we quantize it and found out that it had wave-like properties right the electron is a particle you treat it quantum mechanically it's a wave function but we think that in experiments in nature there are also particle-like properties right the electron leaves a track in a bubble chamber or a cloud chamber it looks like it has a trajectory as if it were a particle but if we start with a field and then we quantize it so we promote it to having a wave function how in the world are we going to get particles out of that that is going to be the major idea that we try to pursue today so first let's talk about what we mean when we just say fields a field we already said is something that has a value everywhere in space a value of some sort or another at each point in space and when I say a value that's gonna be a lot of the tricky part going forward so weak it's easy enough to draw space okay and then a field is basically at every point in space there's some mathematical object representing the value of that field so in the simplest case here's a point X we'll put a little loops let me draw it a little bit more effectively their ex a little vector sign over it means the three-dimensional vector that locates us in space X and associated with that point there is a value of the field Phi of X so the Greek letter Phi is often used for the simplest kind of fields in quantum field theory but it could be something more complicated so what it means in this case is that Phi is just a number so if Phi if the field is just a number at each point then we say we have a scalar field we talked about scalars before as numbers you could multiply things by the example of a scalar field the classic example the only known example in nature of a fundamental scalar field is the Higgs boson took us a while to find it 2012 but we've done it there it is there are scalar fields in nature the number could be a complex number or it could be a collection of complex numbers but it's nothing that changes as you change space-time itself as you rotate or look at different ways whereas if the field equals a vector then guess what you have a vector field look at that I wrote scalar nature because I have still not developed the ability to write and talk and I said the word nature so you know that there are vector fields in nature because you heard of the electric field the magnetic field at every point there is not just a number there's a little arrow representing the electric field vector or the magnetic field vector in fact there's both this is the crucial thing that separates classical fields right now we're just being classical we haven't quantized anything yet the reason why classical field theory is different from quantum mechanics where there's still a wave function right is because when you have many particles you don't have many wave functions overlapping you have one wave function for everything when you have a set of classical fields at every point there are a bunch of different classical fields so literally at every point in space the modern field theorist says in the classical limit at least there is a value for the Higgs field there's a for the electric feel the value for the magnetic field value for the gravitational field and also as we will see there are fields for all the matter particles like electrons and neutrinos and quarks and all those things they all have fields with values at every point so we will get into the different kinds of fields right the vector fields there's there's more than just scalars and vectors but those are probably the ones you're most interested in in when we talked about the potential wave physics could have gone in the early 20th century if quantum académie mechanics hadn't been the reality we could have imagined a view where particles were matter and fields were forces okay so classically there is that dichotomy that particles and fields were playing different roles quantum mechanically fields are everything and that's going to be part of what we try to understand here so let's think about how to quantize this idea right this idea of a classical field we want to quantize it I'm using the word quantize very loosely here if you read quantum mechanics books they will try to give you a cookbook recipe to go from a classical theory to a quantum mechanical theory it's a little bit misleading when they do that in fact because there's no one-to-one map between classical theories and quantum theories there are classical theories that have more than one quantum counter part in other words there's their classical theories that are the limit of more than one very different-looking quantum mechanical theory and vice-versa you can have a quantum mechanical theory that has more than one kind of classical limit so there's no immediate correspondence between classical theories and quantum theories nevertheless because we human beings are so sort of intuitive in terms of classical physics but not so much in quantum physics physicists usually start with taking a classical theory and turning it into a quantum theory in some way that's what we're gonna do with quantum field theory but we're not doing any systematic way I'm sort of going to tell you the answer is in various ways try to motivate them along the way so classical particles just remind us what we did have a position let's say X right remember that and then we invented quantum mechanical versions of particles and they had a wavefunction so quantum particles have a wavefunction you may notice that I'm being a little bit inconsistent because I already said that there is only one wave function of the universe so I really shouldn't say that a quantum particle has a wave function but in terms of the entanglement that we talked about last time there can be situations and there very often are situations where a single particle is not entangled with anything else in the rest of the world in that case it makes perfect sense to talk about the wave function of that particle so as long as we keep our wits about us there's nothing wrong with talking about the wave function of a single particle or even two different wave functions for two different particles as long as they are not entangled which can often be the case so the wave function looks like capital sine as a function of X in other words for each possible position that the particle could be observed to have we assign a complex number that complex number is capital psy of X and we call it the amplitude so this is yet another way in which we are being a little sloppy when I write psy as a function of X usually what you have in mind is the whole function for all the different X's if you pick out a point sometimes you will call that you know X star or X nought or something like that and writes I of X star that's fine so mathematically we think of the wave function psy as a map from positions all the possible positions you could observe the particle to have two amplitudes and then the amplitude be a complex number you square that complex number to get the probability of actually observing the particle to be at that position were you to measure it and to be super Matthew about it what is the set of all positions that a particle could well we're not being super physic see about it so don't think about you know string theory and extra dimensions in everyday life the set of positions is three dimensional so this is our three where our means the real numbers so this is the three dimensional version of the real numbers or you have three dimensions telling you which direction you're moving in so this is a three dimensional vector space the set of all positions in flat Euclidean three-dimensional space and then they get mapped to the complex numbers a single copy of the complex numbers so you give me three real numbers that tells me where we are in space the position X and then I get this is X and then I get the complex number sigh of X single complex number and I will square it if I observe where the particle is the probability of seeing the particle at X is equal to the magnitude of sigh of x squared and if you don't remember I think I did mention this very briefly if a complex number sign equals a plus I B where I equals the square root of minus 1 then the absolute value of size squared which is the thing that is the probability is just a squared plus B squared so it's always a non-negative number because it's the sum of two squares and a and B are both real numbers themselves as the I that is the complex unit that makes the combination a plus IB be a complex number okay so this story as a whole is what we want to generalize to this set of quantum fields to the situation where we have quantum field instead of having a position that we can observe but we'll have for a field is an entire field configuration that we can observe okay so let's just tell the story again but Ella for fields start with classical fields and a classical field has a configuration which is to say a value of the field at every point in space so Phi for example of X would be the set of all values of vital X's that's the configuration of the field that's one particular configuration the field could be in okay so I have to say this looks like a wave function right I write 5x here I write psy of X up here they look like similar kinds of things but they're really conceptually very very different sigh the wave function is the thing that you use to get the probabilities if you added more particles in you would not add more size you just add more X's into the arguments for sy whereas this is a classical field which means that it has equations of motion it would have a Hamiltonian if you like they told you what energy it had it could have an action and you could minimize the action to find what the equations of motion were but you could observe it in principle as accurately as you like there's no sense in which Phi of X is sort of secretly encoding some observational outcomes it just is its value and it can be observed as a classical field okay sadly the world is not run by classical field theory it's run by quantum field theory but this is the idea you have in mind when you talk about classical fields they look notationally like quantum wave functions but conceptually they're a very different thing okay so what happens if you try to turn it into a quantum field well quantum fields are going to be things where there is a wave function but the wave function is not a function of position in space it's a function of field configurations all throughout space so they have a wave function capital sigh which will write with square brackets as a function of some particular field configuration 5x so what we imagine here never actually happens in real life what we imagine here is the thing that we're measuring is not the value of Phi at some point X but simultaneously measure all throughout space what the value of Phi is this number psy is a complex number attached to every specific configuration if I can have throughout the world okay and then we attach a complex number to that square it to get the probability so this 5x is a field configuration just SuperDuper explicit about this throughout all of space and this sigh is the complex amplitude associated with that particular field configuration so in other words in quantum field theory you have a wave function of a wave a wave function of a field that is stretching out all throughout the universe rather than just the wave function of a single particle and then psy squared versailles is a function of v x is the probability of observing or measuring the field to have the configuration 5x it's beginning to seem hard my field theory right because we've already had to imagine doing this observation of a field all throughout space which we can't actually imagine doing in the real world but conceptually there it is now in a real working quantum field theories would imagine in fact observing the field point by point at different locations or in different regions of space and you can use this wave function of the large can field configuration to calculate the probability of getting different outcomes at different points in space but again conceptually this is the thing that is the the center of the universe the actual description of the universe in quantum field theory so to say that in Mathew words because we like to say it both in English and in more mathy words to be math you about it let let's invent some notation here let capital F of R 3 R 3 is just space three-dimensional space the real line times itself 3 times let F of R 3 be the set of maps from R 3 to R that's suppose to be an arrow there that's supposed to be our okay so we're imagining in particular to make our lives easy that 5x our field is a real scalar field so at every point it's just a number it's just a real number there are also fields that are complex or multi valued or vectors or whatever we're just imagining it's a real scalar field so this R 3 represents space and this R represents the value of the field so this is X and this is 5 X so f of R 3 is the set of all field configurations the set of all maps from R 3 to R and then sigh the wave function is basically a map from F of R 3 to the complex numbers it's a map that says you give me a particular field configurational throughout space I'll give you a complex amplitude okay now sometimes notationally you will hear this sigh as a function of a field configuration referred to as a wave functional because sometimes when you go from finite dimensional vector spaces like r3 right like the positions X to infinite dimensional vector spaces like f of r3 this by the way is an infinite dimensional vector space sometimes they think of elements of infinite dimensional vector spaces as functions and functions of functions as functionals this is the word that is used so sometimes we use we use the word wave functional for this but you know what usually in quantum mechanics we do not make big distinctions between finite dimensional and infinite dimensional vector spaces we just say it's the wave function so I'm just gonna say it's the wave function this is the wave function for set of quantum fields ok so again this seems hoof seems a little bit complicated well more particularly it's not the complication that worries us what worries us of what might be worrying you is we take this field which is already not very part of and make a wavefunction of it which is not particle II either how in the world are we supposed to say that this represents our universe which seems to be full of particles at least when we observe them that's gonna be the tricky part okay how do we get particles out of this it's easy enough to say that we go around assigning complex amplitudes to every possible field configuration but let's see it in action let's actually get some work out of it okay so consider the simplest possible field theory not just a real scalar field but remember these fields have dynamics they have equations of motion and so when you tell me I have a real valued scalar field or I have a real valued vector field or a complex field or whatever you're not done telling me what the field is you have to give me its equation right you have to give me the way it evolves does it interact with other fields just interact with itself etc so the simplest possible thing is to consider a non interacting field a field that doesn't interact with other fields or even with itself in a way that'll make precise in just a second and the technical term we use here not very technical is free so this is not free in the in the sense of free beer this is free in the sense of it's free to do whatever it once it's not interacting it's not being bumped into okay it's like you know free from slavery you're not free from paying for your dinner so this is a non interacting free field and it's a scalar field a real scalar field etc okay so let me in fact let's mention let's imagine it just in one dimension okay so here is let me do that again cuz I'm gonna want to do it twice here is space X and here's the value of the field we're thinking classically for a moment here so let's you can do it another color to make it stand out the field is zero most places but there's a little bump in it here and zero there's a little bump in it there okay and this bump is moving to the right this bump is moving to the left and we will label this bump bump a and bump B maybe even though I'm not gonna be able to draw it very effectively maybe bump a has particular Wiggles and bumpy has some particular Wiggles for a free field you can imagine that those Wiggles can maintain their shape to some extent and then what's gonna happen over time is that as a moves this way B moves that way it will evolve into a new configuration and the point of being a free field is they will just go right through each other these two bumps these two perturbations these two excitation Zoar whatever you want to call them in the free scalar field do not interact with each other so after some time what you're gonna see is a field configuration it looks like this that's B and that's a so B is still moving that way a still moving that way and despite my poor drawing skills they're supposed to be more or less unaffected by having gone through each other okay it may be that they're both individually waving up and down but they're not affecting each other so that's a simple example that we can think about it's a sensible starting place for our investigation and so we're gonna consider the quantum theory of a free scalar field that's that's our job but we're gonna show how it gets us to particles somehow so remember that any field configuration and the field Commissioner is the function of space a real valued function of space okay so any function remember when we talked about Fourier transforms or Fourier analysis any function can be written as a sum of sine waves essentially sine waves with different amplitudes and different wavelengths I use the word amplitude there it means something plea different different heights and different wave lengths so any function can be written as a sum of sine waves and what I mean by sine waves is sort of the most general notion of a sine wave just a wave that goes up and down with a constant period or a constant frequency if you want to call it it could be a cosine just as well but sine waves is that is the general term that we use here and this is the Fourier transform and I'm just stating it I'm not gonna prove it I'm not gonna give examples you just need to remember this fact okay you don't understand this fact that's fine but it's a fact to be able to say back to me can any function be written as the sum of sine waves yes it can and uniquely we can do you give me a function you give me some field configuration I can figure out exactly the combination of sine waves that would reproduce it so I start with something that is some arbitrary function right doing something like that that is my 5x as a function of X and I can write it as a sum of things that look like some long wavelength mode plus some things that look like a little shorter wavelength plus some things that look like an even shorter wavelengths plus dot dot if I pick correctly and there's a mathematically correct way of doing this like pick correctly the different wavelengths and the different heights for these sine waves I can add them together to make any function I want and it's absolutely unique and it is reversible I can go backwards and go from the sine waves to the function I go from the sign from the function to the sine waves we call each one of these a mode of the field okay a Fourier mode one of the contributions in this way of thinking to how to describe that particular function now just as a note just because you're gonna say well I've drawn these pictures in one dimension what about three dimensions so there's a three-dimensional version of this okay three spatial dimension illusion it's hard to draw a sine wave and oscillating wave but you can visualize that you can visualize for example a sound wave right you've ever visualized a sound wave moving through space there's a plane in which it's more dense air and then a plane flips right next to that in which the air is less dense and then more dense and then less dense and these vibrations and density are what make a sound wave so we will sort of conceptualize a plane wave as they're called in three dimensions as exactly that sort of a set of planes moving in some direction okay so this is all three dimensions of space X Y Z and there's a vector k k is called cleverly the wave vector and the wave vector says what direction the things going in k also says let me see if i can get this right yeah k also tells us the wavelength so the wavelength here is the difference between one peak and one other peak and we call it lambda traditionally so these planes that i've drawn are supposed to be where the function in three dimensions has a peak remember it's a sinusoidal function going up and down and in between the planes it gets to be a trough or then a peak again and so the wavelength lambda is just the distance the number of centimeters or whatever in between two peaks and k is related to the wave vector in the following way or sorry lambda the wavelength is weight related to the wave vector by lambda equals 2pi divided by the length or the magnitude of k okay one of these ubiquitous two pies that appears in physics so what this means all you need to know forget about the two pies I don't remember where they go all the time what do you need to know is a big wave vector corresponds to a small wavelength and vice versa so the wave vector contains almost all the information you need to know about to reconstruct the plane wave this is called a plane wave because of exactly what it looks like bunch of planes traveling waving and the other thing is of course you need the height of the wave so the if we just draw a cross section and we draw a sine wave then lambda the wavelength is the distance between two peaks or two troughs and let's call it h to be the height and I will warn you ahead of time I'm gonna have trouble remembering that because the actual word that scientists use to be to refer to the height of a sine wave is the amplitude of the wave but that's a use of the word amplitude that has nothing to do with quantum mechanics it's just a synonym for height and so since I've already told you that amplitude the complex number that you square to get the probability of observing something ie a certain value of the wavefunction I don't want to reuse the word amplitude for the height of a sine wave so let's just call it H the height okay and I'll try to you all try to remember that this this will be my job to remember that so just like in one dimension every single function can be written as a sum of sine waves appropriately chosen in three dimensions every single function can be written as a sum of plane waves and the plane wave will be specified by the wave vector K pointing and giving you the wavelength and also the height the size of the wave so given any field configuration three dimensions I can write it as a sum of plane waves and I call these once again molds the different modes are each mode has a wavelength and it has a direction given by the wave vector okay now also true is that each mode has an energy there's an energy for each mode for a plane wave and there's different way that's different contributions just like a particle moving in a potential has a kinetic energy and potential energy a wave also has a kinetic energy and it can be thought of as one-half the rate of change of the field over time squared so if I took the derivative with respect to time of Phi at any one point and then I squared it multiplied by 1/2 integrated that over all the points in space I would have the total kinetic energy of the field now space is big space is infinitely big r3 right the set of all points ok so this is really the kinetic energy at a point the kinetic energy density the thing that we would integrate over all space to get the total kinetic energy but that's infinite this is the finite thing so this is what we care about there's also a new thing we had kinetic energy for particles - there's a new thing called the gradient energy because there's a something that a field can do in a particle can't which is it can vary from point to point in space not just for moment to moment in time but the formula the sort of pattern is exactly the same you calculate the rate of change of the field over space and you square that so in other words if I have a field that is very languorous ly changing from point to point in space it has a low gradient energy if it's vibrating really really quickly in spin space forget about what it's doing in time but it's just changing very rapidly from point to point it will have a lot of gradient energy the gradient is the gratings just yet another word for the slope of the field how fast it's changing over space and finally the field can also have its own potential energy and in principle that can be something very complicated it can be a complicated landscape so for every value of Phi the potential energy is just the energy that doesn't depend on how the field is changing in either space or time it just depends on the value of Phi but we said we're looking at a very simple case of non interacting free fields and in that case the potential energy takes a very simple form it is one-half times a parameter which we'll call M Squared for right now don't ask me why well I'll tell you why in a second times the field value squared so the potential energy unlike the kinetic ingredient if you potential energy is something that we choose when we invent our theory the static energy and gradient energy are basically fixed for a scalar field theory potential energy can be more complicated and when we quantize it so we're choosing the simplest possible thing one-half M squared Phi squared basically right and then what we're trying to get to is from quantum fields to why we observe particles and guess what this parameter M that we choose when we invent our theory will be the mass of the particles so there is a parameter in how we define our field theory that will eventually tell us what the mass of the particles is and also by the way k the wave vector of a certain mode will be the momentum I'm going to try once again I'm not necessarily promising to be very good at this but I'm going to try to reserve K for the wave vector of a plane wave that we're gonna use as a mode that decompose into which field is decomposed and what we're gonna do is analyze the field mode by mode 1 mode at a time that's a good thing to do but then when we make it into particles particles will have their own momentum the particles associated with a plane wave of wave vector K will have momentum K but usually what's going to happen very quickly is that we're gonna use all this machinery of plane waves modes and all that stuff to let us start talking about particles even though we're doing quantum field theory and then once we start talking about particles I'm just gonna use the letter P for their momentum just like we always do when we talk about particles but it's the same thing it's the same concept ok so these are the different contributions to the energy of a certain mode a certain plane wave of our field ok again this is just a simple way we can analyze the field by breaking it up into these modes it works especially well for non interacting fields but it's a good starting point even when we look at interacting fields and then what we're gonna do is we're gonna notice something namely every single one of these expressions we wrote down has the field squared in some form right either the field itself squared for the potential energy or the rate of change of the field for the kinetic energy or the rate of the change of field over space for the gradient energy so if we have two plane waves that we consider and they're the same except for different heights so same wave vectors okay just the height is different then the kinetic energy the gradient energy and the potential energy will all change proportional to the height squared if you change the height of the field then its kinetic energy will have to change because it has further to go when it's changing in time if you change the height of the field the gradient energy will change because it is now moving more from point to point because you increase the height of the sine wave okay so what we notice is that if you if we that all the energies are proportional loops to H squared where H is the height of the sine wave the amplitude when you call it that of the plane wave of the mode that we're looking at so let's plot this okay here is the height of a certain mode here is the energy and it doesn't matter whether we're thinking of the kinetic the gradient or the potential energy they all transform the same way when we let the height go up and it looks like this h squared okay there's some proportionality constant there but that doesn't matter to us what matters is how the energy changes as we change the height okay so I forget actually whether we talked about this concept but the idea of a parameter that you can change and when you change it the energy of the system just changes as H squared on both sides positive H and negative H this has a name that's a famous name for a physical system it is the simple harmonic oscillator it is the one of the simplest physical systems you can imagine thinking about it's the spherical cow of quantum mechanics simple harmonic oscillator because for the simple reason we're gonna be able to solve all the equations exactly for a simple harmonic oscillator a more physical example of this of a simple harmonic oscillator is a wall with a spring and on the spring there's a box I think we talked about this particular example and then the spring has a strength K sorry that it's K again that's what it is and the energy the potential energy V of X if this is the position X is 1/2 sorry 1/2 K squared times X minus X naught squared where X naught is the equilibrium point so the box moves slides along frictionless surface it wants to be as equilibrium if you stretch the spring or compress the spring the energy goes up as the displacement squared that's an example of a simple harmonic oscillator but what we're saying here I mean this is pretty awesome stuff I don't want to you know undersell this because this is like crucially crucially important what we've done is taking a lot of simplifications we've imagined a field that is non interacting just a scalar field free field and we've decomposed it into modes right into plane waves with given wave vectors and different heights and we've said as the height changes the energy of the mode changes as that height squared a negative height is just we're measuring it down in the sine wave rather than up it's the same thing so it's gonna be symmetric around zero and in fact there's this good physical argument that we gauge that the energy is proportional to H squared so the system of a single mode can be thought of as a simple harmonic oscillator that is very good news physics wise because man people have analyzed simple harmonic oscillators to death in fact you could do this box that is attached to a spring there are plenty of systems in quantum mechanics when you first are in quantum mechanics that act like simple harmonic oscillators to a very good approximation a hydrogen molecule so hydrogen atom is a proton with one electron around it a hydrogen molecule is two protons with two electrons that keep them together and if you imagine perturbing the distance between those two protons it acts just like a simple harmonic oscillator so there's many many systems in physics that had this simple harmonic oscillator form and what we want to do is quantize this harmonic oscillator okay so remember what we're doing here we have the wave function for our field but we're decomposing is that the right word to use yet decomposing our field into a sum of modes we're expressing it as a sum of the particular modes and so we can look at the wave function of each mode separately and then sort of combine them at the end of the day so let's look at the wave function of a mode single mode that means a single plane wave with a fixed wave vector and a fixed height and all that so this is saw of Phi and this is a single modify so it is parametrized by H the height of the mode and K the wave vector that's all you need to know to tell me what exactly mode I am talking about and a wave function is gonna tell me the probability of observing that mode to have a different height different wave vector okay in fact actually this is probably I'm realizing this is probably not the best way to say it the mode is fixed by the value of K so let's write the mode itself as Phi sub K okay so Phi sub K is a mode in other words a plane wave with a certain wave vector in a certain size of the wave vector and direction in which it's pointing and that field configuration corresponding to that mode only depends on the value of H I think that's the better way of saying given the wave vector we know what the plane wave is doing we just don't know how big its deviations are up and down how much the field is actually oscillating that is what is fixed by this height H H is the maximum value of oscillation of the field up and down so even though we have this complicated field Phi is function of X everywhere by looking at it mode by mode and we realize that if you tell me what the wave vector is the only thing that varies to tell me what the mode is doing is its height I have turned this complicated wave function of the whole set of field configurations into a wave function of a single number H right and furthermore I know what the energy of H looks like as H changes the energy goes like H squared it's a simple harmonic oscillator so this is a quantum mechanical system that I know how to deal with and I say quantum mechanical because this actually I should have said this a long time ago there are people who think that there's quantum mechanics and that quantum field theory is somehow another theory passed quantum mechanics ok that is not the way of thinking about it you nonrelativistic quantum mechanics and you also have quantum mechanics of point particles you also have quantum mechanics of fields and the most popular quantum field theories are relativistic quantum fields we're not gonna worry about the fact that they're relativistic but these quantum fields are compatible with the rules of special relativity which we talked about in the spacetime video okay so my point is that when we do quantum field theory we're not replacing quantum mechanics we're doing a version of quantum mechanics nevertheless some people including myself will sometimes lapse into a lingo which uses the phrase quantum mechanics to mean the quantum mechanics of a particle or a finite number of degrees of freedom as opposed to a field that has a degrees of freedom at every point an infant number of degrees of freedom okay by taking our field writing it as a sum of modes and thinking about only the height of each mode we've taken this complicated quantum field theory problem and turn it into a set of very very simple problems because every mode just a function of one variable H and we can do quantum mechanics of that single variable so let's do that although I do that I mean I'm going to tell you what the result is I'm not really gonna like solve the equation all of its glory let me just redraw the plot of the energy okay so this is a value of H and this is the energy of the mode so this is the energy of Phi sub K as a function of H and it looks like this it is a parabola it goes like h squared so remember when we talked about the electron in the hydrogen atom we said that even though the electron had a wave function the wave function was perfectly smooth you want for physics reasons I think I explained this best in the Q&A video for the quantum mechanics lecture rather than in the actual quantum mechanics video itself look at the Q&A video you want the wave function to go to zero at infinity far away from the nucleus of the atom there's a competition where the electron wants to be close to the nucleus but also the wave function of the electron if it varies too quickly if it's gradient is too large then that costs a lot of energy so the minimum energy state is one where the way function of the electron is bunched up near the nucleus and it goes to zero there is a next highest energy where the wave function electron varies a little bit wiggles once but nevertheless it still goes to zero far away from the nucleus in all directions and this is a very common phenomenon it is exactly like as we keep saying a string where you tie down the two ends of the string and then you pluck it right there is a lowest note that you can get the fundamental of that and there are overtones or harmonics from that which have more than one wavelength in them okay so that is the origin of discreteness in quantum mechanics is not because you start with discrete ingredients you start with this field and you quantize it and get a wave function of it but when you solve the equations you find that the solutions come in a discrete set that's also true for the simple harmonic oscillator in fact it's easier to see in the case of the simple harmonic oscillator you can solve this and guess what because the energy is going towards infinity up here if you go up high enough there's never a point where the energy stops growing okay the energy it cost more and more energy for the height of the wave to be bigger and bigger right so if you want to look at modes of this toy problem this thing up whimper this is just one little part of our big problem our big problems quantizing the field we have chopped that big problem down into a set of little problems this problem is quantizing a single variable the height of the sine wave it's energy goes as H squared and if you want finite energy solutions to the quantum mechanical problem of just this one mode those wave functions for H better go to zero at infinity because as you go to higher and higher Heights it costs more and more energy potentially an infinite amount if you never come down it will cost you anything amount of energy so there are a discrete set of solutions to this equation there is a lowest energy solution a next higher energy solution a higher energy solution after that and in fact one of the reasons why the simple harmonic oscillator is nice is because the energies are exactly evenly spaced between each other the difference between lowest energy and next highest is the same so if you plot the energy levels here's energy level n here is N equals 0 n equals 1 and equals 2 N equals 3 and it's the same distance in energy ok so to go from one energy solution to the next highest is an equal change every single time and what I mean by this I visualize this I should have drawn this here is the form of the wave function of lowest energy call it size 0 of H this is really sigh of Phi of K of H but just keep in mind that we're fixing what K is and we can just write it as Phi of H ok and so this is the lowest energy mode it has an energy which I'll tell you later but right now it's the lowest energy mode and then there is one that is next higher in energy that looks like this sigh one of H and then there is one next higher energy after that this green good yeah green is good like this this is sy 2 of H etc the point is I said it before go and keep saying it because it's just crucially important we start with like the smoothest thing in the world this field that is stretching out through all space and we have a wave function of that field but then when we look at the lowest energy state the field can be in it's a sum of modes and every mode will be in its lowest energy state we can also have states of the field of slightly higher energy by taking one of those modes and bumping it up into its next highest energy state ok and there's a discrete difference in energy between those two possibilities and the number the amount of energy that you change by is you increase what mode you look what what energy state you're looking at is the same every time ok so there is an interpretation of I should let me just say one other fact about this the pinning down here of the mo of the modes I keep saying modes depending on the wave functions these all go to 0 at minus infinity and plus infinity ok it's a little bit different than the electron the electron in the nucleus in the in atom was had a wave function that was nonzero near the nucleus and went to 0 a large physical distance away from the nucleus this is a more conceptual thing so I'm asking you you know this is not an easy thing I get I'm asking you to bear with me for a little bit of a conceptual leap here this is the wave function is going to 0 as H increases so this is a wave function of a mode parametrized by its height what it's saying is we don't know when we measure this mode how high it's going to be what the height of that particular mode is going to be because quantum mechanics right because all we can do is predict the probability these colorful lines that we've drawn are the wave functions for different states the mode could be in and we would square them to get the probability of observing the mode to have that height and what we're saying is because we want minimum energy States because we just start with zero energy and then build from there all of the modes will have the property that there's very very very little probability of ever measuring the mode to have too much height because that would cost too much energy so they're pinned down at large values of the height of the mode not at large values of distance from the nucleus but the math is the same now ok now is the big reveal there is basically an interpretation of this and I wish I could do a better job at explaining why this is a good interpretation but it is a good interpretation and maybe I've justified it enough the real thing is that you know there's this constant difference in energy between these different energy levels the interpretation is size 0 the lowest energy state of the mode that we're looking at can be thought of as a collection of 0 particles it is the lowest energy state it's the vacuum state it's something that is just there there's nothing happening there's nothing moving around it's the lowest energy state you can be in there is a state that has slightly more energy and that we interpret as a state containing a single particle there is a state that has slightly more energy than that side too and that can be interpreted as a state that has two particles in it and this goes on there's an infinite Tower of states cyan which you can think of as having n particles why does that make sense well there's a cheap and easy way of having it make sense I mean think about the these are all for a single mode for it so for a single wave vector so for a single momentum of the particle if I give you one particle in a state just classically one particle with a certain momentum it'll have a certain energy if I give you two particles with exactly the same momentum the same kinds of particles it will have twice that energy and if I give you three particles with the same momentum the same kind of particle you know three electrons moving the same direction three times the energy so these equally spaced energy levels are at least consistent with the idea that we can interpret them as describing different numbers of particles now it goes way beyond that you can actually do you can dig into this and you can do an exact correlation an exact map between the wavefunction not just of the height but also of the position in space you can make way packets by superimposing different modes on top of each other and find that if you have like a bunch of different modes all of them are in their N equals 1 state all of them are supposed to be you know one particle that's just a complicated wave function for a single particle it acts exactly like that okay this is why quantum field theory is a theory of particles because you can think of the theory you can think of the individual fields as being described by a set of different modes and the modes because of this harmonic oscillator kind of behavior that it costs energy for the height of the mode to be big have discrete sets of solutions to the Schrodinger equation with discrete energies that are equally spaced and therefore can be thought of as describing different particles now you want to be more than that obviously you want to do more than that you want to describe why these modes when they hit a cloud chamber will leave a trajectory okay and that comes down to a different feature you know we can't answer that question at this level because we assumed from the start that we had a non interacting feel this is not an electron the electron has to interact with the cloud chamber to leave a track behind so to answer the question of why particles leave tracks is a bit trickier and the answer is of course because the answer relies on the fact that the interactions between the electron and the rest of the world are local in space the electron only interacts with things that are right next to it that's why it leaves a track we talked a little bit about that back in the time when we were talking about force and energy in action and we talked about momentum versus position and velocities and the Hamiltonian all that stuff it's locality that makes things look particle like this story just justifies why you can start with something as wavy as you can get right the wave function of a set of fields and it can be interpreted as a set of discrete particles so in fact that is the punch line and I'll just write it again because it's worth saying one quantum field it can be thought of in the right circumstances and these circumstances are right so far but they'll be more complicated ones but can be thought of as a superposition of different numbers of particles in fact this is one way to invent quantum field theory we invented quantum field theory by saying let's say we have a field that's quantize it you can also invent quantum field theory by saying let's say we have many particles and try to come up with a convenient mathematical way of characterizing all the particles at once and what you do is end up inventing a quantum field it's the same thing so by superposition I mean you know we decomposed our wavefunction of quantum fields into wave individual wave functions of different modes but the real whole shabang the big thing sigh of phi of x is a superposition of all those different modes and every mode can be a superposition of these different energy states ok so not only does this particular quantum field it not only could it describe one particle or two particles or three particles it can describe a set of particles you know how many there are when you observe it there's a probability that you'll see three particles or twelve particles or whatever that's the beauty of quantum field theory and in fact you can observe it at different times and the number of particles can change I'm not going to justify this but there is a there's a folk theorem which means it's not really a theorem but there are good arguments in favor of it that says that if you want quantum mechanics and special relativity and you want to be able to change the number of particles change number of particles the only way to do that is in a quantum field theory QFT quantum field theory so there's a justification for why things need to be quantum field theories that's the only way to obey all these cherrish principles all at once ok did I get that right yes I think I got that right so that's why we have we start with quantum fields we get particles it's it's at least again I didn't justify it all the way but I tried to make it seem like a plausible thing now we're not done we're not done with quantum field theory we have a long to-do list okay we're not gonna do it all today we'll we'll do some of it later on we'll do a lot of it but you know quantum field theory is hard like come on field theory really is the combination of hardest and most important subject in physics is what I would say you know you can argue that you know some specific very technical field of high-temperature superconductivity or string theory or whatever is harder than quantum field theory because we don't know what the rules are etc but those are only interesting to some people who are specialists in those fields quantum field theory is central to modern physics and it's really hard in part it's really hard because there's a lot of infinite quantities you know the dimensionality of various Hilbert spaces is infinite as we'll see there's a famous problem in quantum field theory with an infinite answer to certain well posed questions what is the probability that these two electrons bounce off of each other if you calculate it sloppily you'll get an infinite answer and that's bad this is the problem of renormalizing the infinities okay so there's a lot of subtleties that go into quantum field theory but it's crucially important it's absolutely central right now quantum field theory is the best way we having we have of describing nature at its deepest level so the hardness is worth it we should we should do it so what we'll have to do and not necessarily disorder is imagine that there are different types of fields so we mentioned that already so there are not only scalar fields there are vector fields there are tensor fields like the gravitational field I'll explain what that means and there are even spinner fields which are not vectors or tensors things like the electron neutrino etc fermions as they are called the matter particles are made of spinner fields well we'll get into that don't worry so different types involve the scalar oops scalar fields like the Higgs vector fields like electricity and magnetism now the cognoscenti in the audience will know that secretly the electric and magnetic fields are not vector fields after all they're both part of a single tensor field we'll get to that and it's worse than that because there is a vector fee from which they both get derived the vector potential field all I'm just you know shouting out to the people who know what is going on here the electric and magnetic fields come from an honest vector field even though they're not honest vector fields themselves sometimes they're usually in particle physics these come from gauge fields and that has to do with a certain symmetry that the fields have and we'll try to explain where that symmetry comes from and why it's important in the forces of nature and then you also have Fermi on fields spinners spinners spelled in a weird way o RS not it's one n ORS not two N's ers not a set of things that spin but spinners is the mathematical formalism and these are matter particles right electrons quarks etc so there's a enormous amount of richness in quantum field theory because there can be many different types of fields and then the richness continues because you have to let them interact right and the way that you do that is the simplest way this sort of first step in learning how to let them interact is by thinking about Fineman diagrams so what I did here all of this stuff is to sort of justify why under the right circumstances you can think of a single quantum field as a superposition of a collection of particles and then finally diagrams sort of take that license and really put it to good use they say good let's just do particle physics let's imagine we have a certain set of particles to start and it does quantum mechanics the particles interact and they evolve into a superposition of different numbers of particles outgoing so you might have for example two electrons II - II - and the Feynman diagram will say that they scatter off of each other time is going this way by exchanging a photon photons are usually denoted gamma the Greek letter gamma in particle physics okay so we'll have to understand what's going on here how you can use this cute little picture to actually calculate the probability of this kind of interaction happening and then we have to talk about you know that messy set of issues that I mentioned infinities renormalization some theories can be renormalized some can't it's an issue etc ok so there's a lot of conceptual issues that we have to get on with there is one thing I should probably ease on is an hour of my computer telling me it's been an hour we're not gonna talk about any of these things today I'm going to end the video but there's one thing I really do want to talk about because it is not talked about in most discussions of quantum field theory but cosmologists care about it a lot and that is the issue of the energy of the vacuum so in ordinary quantum mechanics you have a certain number of particles and that number of particles remains fixed right in quantum field theory when you really have a field you can interpret that as particles but the number of particles comes and goes it turns out that in quantum field theory even the zero energy state even the state of lowest possible energy is still an interesting place so the vacuum in quantum field theory is the lowest energy state and the reason why it's interesting is because you still have at every point in space there's still you know the quantum mechanical version of Phi of X right there's still some quantum state for the field that is filling all of space and indeed the lowest energy state in quantum field theory is a superposition of the lowest energy States Oh all those modes right so I can take every single mode of my quantum field which is purely hypothetical right like remember classical field has a configuration a quantum field has a wave function over the set of all configurations so even if we're in the lowest energy state we don't just say the field is zero we say the field has a wave function if you observe the quantum field even in the vacuum you can get any answer there will be a nonzero possibility of you getting any particular value for the field if you observe it because the wave function doesn't ever go exactly to zero okay it goes to pretty close to zero oh it eventually goes to zero when you get to infinity but you never get to infinity so at any one point you have actually that was that was mistaken infinity in energy at any one point in space you have a value that of the field Phi of X that you could always get in principle you can't get Phi equals infinity but you get the same answer at every point in space okay so even the vacuum is an interesting place in quantum field theory and you say well ok it's the lowest energy state this is state with zero particles right in quantum field theory you have the vacuum the lowest energy state you have States with one particle you have States with two particles etc the set of all those things is the set of states that the system can be in that the thing can be in why would the vacuum be interesting well you see look at this yellow line here conceptualize that maybe I can just I always am slightly nervous when I try to cut and paste in this program this is not the program I usually use copy good what are the chances I can paste it on I think I did this once before successfully paste huh it worked I could do that 20 more times in the course of these videos I will always be pleasantly surprised okay the point is for a classical particle in a potential like this let me see if I can find yet another color to use here's blue okay so here's a classical particle I'm comparing a classical particle in a potential to a quantum wave function in exactly the same potential so I'm not sure how visible this blue is but the classical particle now it's not visible at all is it I'll use white again minimum energy of the classical particle in a potential V of X equals let's call it for future reference sake 1/2 Omega squared x squared okay I know that this picture I wrote H but we're just doing the general idea of a quadratic potential of an x squared potential that's the simple harmonic oscillator at its essence so what is the minimum energy state well kinetic energy of the particle can be 0 the particle can just sit there doesn't have to move and the potential energy of the particle can also be 0 it can be at the minimum at x equals 0 so total energy can also be 0 the minimum energy of a classical particle in this potential is 0 I'm belaboring a point that you are not surprised at all to hear but look at that yellow curve there right that yellow curve is changing that's the minimum energy quantum wave function in the same potential but it's not just a point sitting at the origin right it has some support everywhere sort of spreads out a little bit so we can calculate the energy of that profile you know we I didn't over here you know when we're talking about those I said that they're equally spaced but I didn't tell you by how much they're spaced I didn't tell you what the first one was so that's math you just do it you plug into the Schrodinger equation it's still a Schrodinger equation even in quantum field theory there's a version of the Schrodinger equation don't believe people who tell you the Schrodinger equation applies only to nonrelativistic particles the equation Schrodinger himself wrote down does in fact apply only to nonrelativistic particles but it's easy to generalize the Schrodinger equation is H sy equals I D by D psi the Hamiltonian acting on the wave function is I there's an H bar in there that is I set equal to 1 times the time derivative of sy and that's true for any theory that you have you just have to give me what H is what the Hamiltonian is it works equally well for quantum field theory as for anything else ok so what is the energy of this quantum thing the quantum thing is in yellow here so the energy of the lowest energy state in quantum theory quantum minimum energy yeah try to squeeze things in so my handwriting is becoming less good well you work it out it equals 1/2 let's call it anot 1/2 H bar Omega where Omega is this parameter that appeared in the potential right there so it's not zero it's greater than zero what make is a positive number ok there there's a minimum energy in the quantum theory that is different than the classical theory and in fact it's a little bit it goes beyond that I can actually tell you what the answer is for the other energy states the energy for n particles equals n plus one-half times H bar Omega so what is it that's the energy of all these different wave functions sy 0 sy 1 sy - they're all sy n where n plus 1/2 times H bar Omega is the energy so the zero energy state has 1/2 H bar Omega the next highest state has an energy three-halves h-bar Omega 2 particle has 5 halves H bar Omega etc okay it almost never matters what the energy of the lowest energy state is what matters is the difference in energy between one excited state as we call them there's the lowest energy state the ground state and then any other state is called an excited state it's excited to be having more energy than the ground state the difference is in energy U which in this case is just some multiple of h-bar Omega those matter because you can see them if you have a particle or some system that is an excited state and it undergoes a transition to a lower energy state let's say by emitting a photon much like an electron in an atom goes to a lower energy state than emits a photon then you can measure the change in energy the difference in energy between two states so usually in physics the ground state energy itself doesn't matter after all we just took this potential and let's say we added a constant to it plus C we raise the whole potential up or down it doesn't change anything at all anything measurable except and there's one thing that it changes I'll erase that here just because one thing that it changes is the energy of empty space remember this is supposed to be the real world and this is saying there are fields in empty space those fields could have energy so let's imagine you did the following thing let's imagine you set the classical energy of empty space to zero why well because we said so that's that's the reality of justification we're imagining that's what you chose to do the field the quantum field is a superposition of an infinite number of harmonic oscillators the modes of the quantum field the plane waves right that's what we said the quantum field this is just a general fact the quantum field can be thought of there's a superposition of all these different modes you can put every single one of those modes into its lowest energy state but all those lowest energy states still have energy even though they're the lowest energy this is the extra quantum bit over and above the lowest classical energy so quantum mechanics QM contributes an infinite energy to empty space that sounds bad it is bad this is known as the cosmological constant problem Einstein remember Einstein maybe not remembered maybe never heard the story but back in 1917 Einstein already had general relativity he figured that out in late 1915 1916 and he was applying it to the universe as a whole and he was looking for a solution to his equations that had you know stars distributed equally throughout the universe which is kind of prescient it wasn't obvious that that's how things were but now we know that distributing galaxies equally throughout the universe is pretty good approximation and what he found is that the universe would either expand or contract those are the two possible solutions there was no solution where the universe just sat there static and he asked his astronomer friends is the universe static and they said yes because that's what they thought in 1917 it was 1920s where Hubble discovered that the universe is actually expanding so Einstein figured out a way to add a new term to his equation for general relativity which he called the cosmological constant and it sort of pushed things apart and it created balance the matter was trying to pull things together because much class was pushing things apart so he could find a solution to his equations it which in which the universe is neither expanding or contracting the so-called Einstein static universe now we now later reinterpreted Einsteins cosmos with constant as a measure of the energy density of empty space and I assigned once he realized universe was expanding he was like ah that was a bad idea get rid of the cosmos what constant nowadays we've discovered it we discovered the universe is accelerating in 1998 strong numbers discovered that and we attributed to the cosmological constant the problem is the cosmological concept on the one hand is just a free parameter you can pick it you can set it to be whatever you want if you're God and setting up the rules of the universe you can make it whatever you want but it seems like the difference between your classical theory and quantum theory is infinitely big when it comes to the cosmological constant so whatever the observed cause Marvell constant is here's the logic that people go through and we don't know this logic is right in fact somehow it has to be wrong but here's the logic the logic is wherever you get the cosmos will constant from you can always think of it as a classical piece plus a quantum piece and naively if you figure out the quantum piece it's the sum of the zero point energies the vacuum energies of an infinite number of modes an infinite number of simple harmonic oscillators so literally at every point in space you should have an infinite energy density just from the quantum contribution now you can make the classical contribution equal to minus infinity you could exactly cancel it right but there's no reason to there's nothing special or good about zero cosmological constant and now things have gotten worse because in 1998 we discovered we don't have zero cosmological constant we have a small but nonzero value that's crazy we have no reason to think that we should have a small but nonzero value of the cosmos bullet constant if it were zero like when I was a graduate student everyone thought it was zero and we said well we don't know why but there must be some symmetry that makes the cosmological constant exactly zero and people say well let's look for that symmetry or some dynamical mechanism or something like that when we discovered it in 1998 that there was a nonzero cosmological constant but there was nowhere close to the quantum mechanical contribution by itself everyone is now more puzzled than they ever were so the solutions the search for solutions to the cosmological constant problem are ongoing not to get too far out of what we're supposed to be talking about here but the here's how bad it is the best solution on the market into why the cosmological constant is small is the anthropic principle if you imagine there are different parts of the universe where the cosmological constant takes on different values than if it were really big infinitely big or even finite but really really big you wouldn't be able to form a galaxy right it would just blow everything apart the cost marshal constant either if it's large and positive pushes matter apart at a tremendous rate or if it's large and negative causes the universe tree collapse very very quickly in order to get a universe that lasts long enough for galaxies to form but doesn't blow them apart with a super fast accelerated expansion the cosmological constant has to be small so if you live in a multiverse where the cosmological constant takes on different values in different places we living complicated organic beings who rely on a long-lived universe with galaxies in it to exist would only ever find ourselves in regions where the cosmos are constant is small I'm not saying that I agree with that answer or that I like it but it's on the table as a possible answer and in fact I can say in my judgment it's better than any of the alternative answers as of right now so really what that means is not that the anthropic principle in the multiverse are right but then we don't know why the cosmological constant is small and it's a very interesting way to get at the kinds of problems that come up when you try to marry quantum field theory with gravity of course we're talking about the biggest ideas in the universe so we're gonna get to some of the ways you can try to marry quantum field theory with gravity eventually first we have to do a little bit or understanding and quantum field theory itself our journey is not yet over

Channel: Sean Carroll

Views: 155,115

Rating: 4.9222341 out of 5

Keywords:

Id: Dy1LNk_B6IE

Length: 76min 30sec (4590 seconds)

Published: Tue May 19 2020

Reddit Comments

[u/BlazeOrangeDeer]

Sean Carroll has been making a series of lectures to give approachable introductions/summaries of physics concepts. He recently did episodes on special relativity and quantum mechanics, leading into this one about the combination of the two: quantum field theory.

Playlist of the lectures and Q&As

[u/JonJonFTW]

I love how much depth he went into about the basis of QFT. The furthest into physics I got was part of my Nuclear Engineering degree. I had to take some "qualitative" QM courses where fundamental concepts were discussed but they were light on the math, but thankfully I was able to take some QM courses in the physics department. Didn't get much further than solving the Hydrogen atom in one course, and flying by the seat of my pants with relativistic QM in the second course (no one understood anything, we all got curved massively). So thankfully I could follow most of this, but I agree this was definitely too in-depth in comparison to all of his other episodes. I still appreciated it, though!

You'd think I would have understood the importance of the SHO but I gotta say, I was shocked when Sean put it all together at the end and showed how it comes out and gives you particles. My favorite episode yet!

[u/AshleyvanderBeck]

Sean always does such a good job explaining the concepts simply.

[u/AddemF]

This is great for someone who knows high school physics. But I think he may still be talking over the heads of the widest possible audience. Maybe he knows that and is fine with it, not sure.

But anyway, cool vids.

Yet I still watch these and think "Ok, I get position in the classical picture. I get the idea of being able to talk about the probability of observing a particle at a point. Nothing forbids making that the magnitude of a complex number, but ... this seems so random. What motivates this choice?"

Eh. Just gotta learn QM one day.