The Science of Everything Podcast - Episode 85: Introduction to Quantum Field Theory

Episode Date: July 29, 2017

A discussion of some of the major conceptual aspects of Quantum Field Theory, including the concept of a quantum field, classical field theory, harmonic oscillators, second quantisation, interacting theories, and the relationship of QFT to other branches of physics. Recommended pre-listening is Episodes 83 and 84, Advanced Quantum Mechanics parts I and II.

You're listening to The Science of Everything podcast, episode 85. Quantum Field Theory. I'm your host, James Fodor. So in this episode, we are going to delve into the strange world of quantum field theory, or QFT, for short. So this is a particularly advanced topic, and so once again, pre-warning to listeners who might be a bit queasy about advanced physics. But I'm going to try to keep this episode fairly consistent. I'm not really going to go into the details of quantum field theory because it's impossible to do that in an audio podcast. It's even more essentially mathematical than quantum mechanics is, but it does build on some similar concepts.

So I do recommend the previous two episodes, Advanced Quantum Theory Part 1 and 2, as recommended pre-listing for this show, even though it doesn't directly build upon those. it does refer to some of the same concepts, so it will be beneficial to have listened to those beforehand, and certainly to have some background in quantum theory. But really what I'm going to do today is just to go through some of the key concepts in quantum field theory, basically to explain what quantum field theory is and why we need it, and what the key sort of conceptual breakthroughs or conceptual innovations of quantum field theory are. so you can understand why it's such an important theory and sort of what it tells us about reality. So, in this episode, what I'm going to talk about is I'll begin with classical field theory,

a very brief outline of that, and then transition into quantum field theory, explaining what it is and what the key ideas are. I'll talk about second quantization, so essentially that's the relationship between quantum field theory and regular old quantum mechanics. I'll talk a bit about interacting theories of quantum fields, sort of where the meat of the subject is, about scattering and a bit about particle accelerators and how all that fits together. I'll talk about the levels of theory. So that's basically to explain how quantum field theory relates to quantum mechanics, special relativity, classical mechanics,

and how all of those sort of different branches of physics fit together. That's particularly useful if you're planning on further study of physics on your own or at university or wherever else, it's useful to understand how these fields fit together. Or even if you just hear people talking about these terms, it might be helpful. And we'll conclude with a look at some of the implications of QFT, sort of the take-home messages about what it tells us about the way the world is. So we've got a bit to get through, so let's make a start. And as I said, I'll begin by talking a bit about classical field theory. So classical field theory predates quantum mechanics. It dates back to the 18th century. And it is a mathematical and physical description of

fields. Now, I'm pretty sure I've talked about fields before in some of the episodes about electromagnetism. I forget exactly what the number is of that episode, but we'll just go over that again. A field is essentially a function, a mathematical function, defined over a region of space. In the case of quantum field theory, it's also defined over time. I mean, you can do that classically as well, but we'll talk about it defined over space time. So we can define some number at each point in space and each point in time, that number corresponds to the field at that particular point in time and space. So a field is an abstraction, but we also talk about fields as if they are real things, because as far as we can tell, well, they are real things. That is, fields seem to be part of the

way nature is constituted. So the electromagnetic field, for example, is one you probably would have heard of before. As far as we can tell, there's sort of some real thing that is the electromagnetic field that exerts forces, magnetic and electrical, on charged particles or moving charged particles. So, don't get confused. Fields are generally defined in terms of like a function over space and time, which sounds like it's a pure abstraction. But we think that that function describes something that exists in the world.

And I can't really say much more about that something other than that it behaves like a field, like the mathematical idealization of a field. but that's just sort of getting used to the different ontology, if you like, of a scientific worldview. Ontology meaning the study of things that exists. As far as we know, reality is made out of fields. So a classical field, to come back to specifically the topic at hand, is one that does not factor in or incorporate the effects of either quantum mechanics or special relativity. And we'll come back to that.

But the two main examples of classical fields are Newtonian gravitational fields, and classical electromagnetism, so the electromagnetic, well, electric and magnetic fields, which I already mentioned. So these can be studied using techniques called Lagrangian and Hamiltonian mechanics, and I'm not going to go into the details of those, but the field versions of these tools, essentially mathematical tools or descriptions of physics, allow us to describe the way objects or objects or particles move and interact with each other over time by deriving equations, so-called equations of motion,

beginning with a fairly abstract starting point. So to give you the sense of this, in, say, Hamiltonian mechanics, I mean, Hamiltonian and Lagrangian mechanics are both fairly similar in terms of the overall idea. You can convert from one to another. So don't worry too much about the names. I'll just talk about Hamiltonian for no particular reason.

The Hamiltonian, if you recall, from the previous episodes in advanced quantum theory, is essentially just an energy function. you define some sort of system, maybe it's a swinging pendulum, maybe it's a mass on a spring, maybe it's a particle moving through an electric field, it doesn't really matter. Whatever the physical system is, you define its energy function, so it's potential energy, and its kinetic energy in particular.

You add those together, that gives you your Hamiltonian, so it's a function of generally position, perhaps of time and other parameters you put in there. It gives you the energy of the system. Now, essentially, the principle of Hamiltonian mechanics is essentially you define some function over space and time, which is called the action, and you try to minimize that action with respect to small perturbations. This is called the principle of least action. Perhaps I'll do an episode on this specifically in the future. I'm not so worried if you don't understand exactly what this is, because I'm just sort of rushing through it. All I want really to get across here is you start with this Hamilton in this energy function, you carry out some mathematics, and you end up with the equations of motion of the the system in question. That is, equations that describe how your particle or how your spring, or how your pendulum, or how your whatever, behaves over time. And in particular, these equations will be equivalent to Newton's laws of motion, you know, F equals MA. So it gives you the same results as good old Newtonian mechanics, or Newton's three laws. It's just a more

mathematically refined way of getting those results. So this is essentially what classical field theory is. It is a generalization, well, it began as a generalization of Newton's mechanics to more abstract representations. And in particular, field theories allow us to describe systems that have a non-conservation of particle numbers. So systems where particles or other sort of entities, they don't have to be like electrons, but whatever we're treating as a particle, created and destroyed. So there's variation in the number of particles. That's very important in a lot of systems, and field theories are enable us to deal with that. Also, systems that have infinitely many degrees of freedom, so there can be variation, saying, in position or momentum along a continuous scale,

rather than the discrete numbers of positions. Field theory is particularly good with that. So that's the very quick version of classical field theory. It's a generalization of Newtonian mechanics that relies on this concept of fields, so functions defined over time and space. Each point gets a functional value defined at that point, and then you use, use the formalism of, say, Hamiltonian or Lagrangian mechanics to derive equations of motion that describe how those fields change over time and how particles interact with those fields and so on. I mean, that's the very basic idea. Now, how does that relate to quantum field theory? Well, quantum field theory is obviously a generalization of, or a continuation building upon

classical field theory, but it's designed to treat or incorporate quantum effect. So when we talk about quantum effects or quantum mechanics in this episode, essentially what I mean is phenomena that are only manifest at a very small, spatial or temporal or energy scales. So this is like a single atom, emitting and absorbing photons or electrons changing energy levels and a single atom, things like that. Very small scale. Billiard balls moving around is not going to be relevant to, quantum mechanics isn't going to be relevant to. that motion. So we would say that that system is a classical system, quantum mechanics isn't relevant. And sometimes it can be a bit of a bit unclear as to whether you need to consider quantum effects or not. But for our purposes, we'll just assume that it's generally clear as to whether quantum effects are relevant or not. So classical field theory does not incorporate the effects of quantum mechanics.

Quantum field theory attempts to do that. Quantum field theory, in the way it's generally described, also attempts to incorporate the effects of special relativity, although it doesn't intrinsically have to do that. special relativity, or just relativity, as used in this podcast, essentially refers to effects that occur at very high energies or very high velocities. So, you know, this is a rocket accelerating up to near speed of light, or neutrinos streaming in from the sun at almost the speed of light, electrons in a particle accelerator, stuff like that. So things that are moving really fast and have extremely high energies. So ordinary atoms, even if they're experiencing quantum effects or

emitting photons and so on, that does not. constitute a relativistic effect, or to put it another way, relativistic effects are usually not that important for systems like that, because they're not high enough, they're not high enough energies. So the basic dichotomy there is quantum effects, small systems, very small scale, relativity, high energy, high velocity systems. Now you can have cases where you have small systems that are at very high velocities, particle accelerators would be an example of that. In that case, you need quantum mechanics and special relativity to, together. And that's essentially where quantum field theory comes in because it allows you to incorporate both of those.

But more on the relationship between these different areas later on. I just wanted to point out that essentially that's what quantum field theory is trying to do. It's trying to use the ideas of classical field theory, but applying them to the quantum and the relativistic realms. So now I'm going to talk about essentially the conceptual underpinning of quantum field theory. By the way, I should say quantum field theory is a general term. There are many quantum field theories. Quantum field theory is not one theory, it's a type of theory. And one of the most successful examples of a quantum field theory is quantum electrodynamics,

which essentially is the generalization or the building upon of classical Maxwell's theories of electrodynamics, but now incorporating quantum and relativistic effects. So QED for short, quantum electrodynamics is an example of a quantum field theory, which is a particularly successful one, but there are many others as well. But anyways, here we're not going to get into the details of any particular quantum field theory. I'm just going to talk about the basic conceptual apparatus behind what you do when you're making a quantum field theory, or trying to understand a quantum field theory. What's the basic idea?

So, essentially, we start off with two fields. The basic idea is that each field represents a different, or each of the two fields, represents a different variable that you can measure or different property of the particle. So typically the two that we work with are position and momentary. but that can be others. They're called conjugate fields because they're directly related to each other, right? A particle will have a position and a momentum. So they're related to each other, but they're distinct. Now that's the same for classical field theory. We have essentially our position field and our momentum field, and there's some relationship between them. And, you know, we use our Lagrangian and Hamiltonian methods to derive equations of motion, which explain how those fields change over time and how interactions with particles occur in those fields and so on and so on. Similar idea in quantum field theory. with one important difference, is that in quantum mechanics, I mean it's in the name, quantum, there needs to be a smallest discrete energy scale. This is what quantization means. It means you can't just have any old value of energy.

You can only take on particular discrete values. This is the idea of the discrete energy levels on the atom, for example. An electron can't just orbit wherever it likes. It can only adopt particular orbitals in accordance with the quantization rules. Now, it's the same thing in quantum field theory. If you want to have an actual quantum, if you want to incorporate this quantization effect into our field theory, we're going to need to have our fields quantized as well.

So essentially, they can't just take on any old values. They can only take on discrete multiples of sort of the smallest unit, or the quantum. That quantum is very small. So when you scale up to the macro scale, it's tiny. And so at a classical level, it looks like you can have any value for energy. or position or whatever you like, but actually at the quantum level, not the case, there's

actually discreteness there. But the point is we need to incorporate this into our field theory, so how do we do it? Well, the basic idea is that our two fields, our position and a momentum field, our two conjugate fields that we use to build the field theory, what we will do is we will say that they don't commute with each other. We apply a commutation relation between them. And what does that mean? A commutation relation is essentially a way of saying that there's a difference doing something one way versus doing it in the other way around. So think about addition. 5 plus 3 is the same as 3 plus 5. It doesn't matter which order you add the numbers in. You get the same result. So addition is commutative. You might have even thought about this

before because it just seems obvious addition commutes, right? But not everything commutes. Subtraction is not commutative. 5 minus 2 is not the same as 2 minus 5. It matters the order you do it. You get a different result if you do it the other. way. Now that's a simple example, but there are other things that are not commutative as well. So if anyone has done linear algebra, a matrix of numbers, you can multiply a matrix by another matrix, so the matrix multiplication. Matrix multiplication is also not commutative. That is matrix A times matrix B is not necessarily the same as matrix B times matrix A. It matters what order you do it in. And the difference between doing something, doing an operation forwards versus doing it backwards or doing it the other way,

if you like, it's called a commutator. So, formally, this is defined as a times B minus B times A equals the commutator, but that's not so important for us. It's just the difference between doing it forwards versus doing it backwards is the simple way of thinking about it. Now, if something commutes, the commutator is zero. So for addition, the commutator

is always zero. Now, how is this, in any way, relevant to quantizing our conjugate field, our position and momentum fields? In classical field theory, position and momentum fields commute. So the commutator is zero. and simple. Quantum mechanically, they don't. That is, in quantum field theory, position and momentum fields do not commute with each other. There's a non-zero commutator when you try and multiply

position times momentum fields and then minus momentum times position fields. The multiplication of the fields together is non-commutative, so it matters the order you multiply them in. Now, how does this have anything to do with quantization? With having discrete energy levels that the system can occupy? Well, the best way I've been able to come up with to explain this, and there may be better ways, but this is the best way that I've been able to come up with, is to think about what it means for the fields not to commute with each other. One sort of loose way of thinking about that is that if you specify the position, then you can't just specify whatever momentum you like, and vice versa. If you first specify the momentum, you can't just then specify whatever position you like. there's an intrinsic dependence upon one on the other. So if you specify the position first, that restricts what you can say about the momentum. Or if you specify the momentum first,

that restricts what you can say about the position. Whichever you do first restricts what you can do about the second one. And that's where the sort of commutation relation comes in. The order you specify the matters, or the order that they are specified. Now, why would that be the case? Well, this relates to a mathematical concept called a Fourier transform pair, and position and momentum are a Fourier transform pair, but that's a bit hard to explain non-graphically. But I'll appeal to one of the previous episodes in quantum mechanics. It may be in episode 14,

where I talked about the uncertainty principle, because this is an instance of the uncertainty principle, the idea that you can't measure position and momentum to arbitrary accuracy at the same time. And essentially the reason for that is because these two form a furrier transform pair with each other, and the way to understand this is to think, in terms of waves, because of course we know that particles are waves, that's part of quantum theory as well, so it seems reasonable.

So imagine that we have an electron and we know exactly what the momentum of this electron is. Now, what does that mean in terms of the wave? Essentially it means the frequency of the wave is precisely known, because frequency relates directly to the momentum of a wave. Now, if we know exactly what the frequency of the wave is, it has a specific, So it has a specific defined frequency. That means it's what's called a plain wave. Just mathematically, if you write down a wave with exactly one frequency. It goes on for, it's a plane wave. It goes on forever. Well, in three dimensions, it's a plain wave. But in one dimension, you can think of it as a sine wave. The important point is it goes on forever, from minus infinity to infinity. There's no end to it. There's no way of having a curve like that with one single defined frequency, but having end points to it. You can't define that mathematically. If it has one defined frequency, it goes on forever to infinity. And if it goes on forever to infinity, that means that it's spatially non-localized. You can't say anything about where it's located.

So that is, if we know exactly what the momentum is, the position is not defined. We don't know anything about the position. And it goes in the exact opposite as well. It turns out the way that you spatially confine a wave like this, so that you can say the wave, the particle exists, or the wave is confined to this particular point, is by adding lots of frequencies together. and this is where, again, the Fourier transform part comes in,

this is the mathematical description of that process, adding in many different frequencies. It turns out that if you add in infinitely many frequencies, you can confine a wave to exactly one point. So then we can know exactly what the position is. It's exactly there. The particle has a precisely defined position. But then, of course, we have no idea what the frequency is

or what the momentum is, because we've added an infinite number of frequencies. That is an infinite number of momentum together. Now, those are the two extremes, a precisely defined point particle with completely undefined momentum or a precisely defined plane wave, essentially, with completely undefined position. But we can have intermediate cases where the position and momentum are each partially defined, so it's partially localized and we only partly know what the momentum is. This is the Fourier-transform peer relationship, essentially.

The way the variables are defined is such that there's a mathematical relationship between the spread of one and the spread of the other, and you can't precisely define each of them at the same time. this is the basis of the uncertainty principle. And as I said, I've talked about it in previous episode of quantum mechanics. So if you're confused, you can go back to those and relisten. But the basic idea here is that the fact that the two variables, position and momentum in this case, form a Fourier transform pair,

is related to the fact that we are considering the system in terms of standing wave patterns. This is another important principle of quantum mechanics. The electron, as it exists, orbits about the after, forms a standing wave pattern, so it's a wave that sort of constructively interferes with itself and is self-propagating. Again, go back to previous episodes in quantum mechanics if this idea is unclear. Also, the episode I did on waves. But standing wave patterns can only support themselves at quantized values.

That is, for a given wavelength, I need to fit in a whole number, an integer multiple of wavelengths around my string, say, if you imagine a string, sort of vibrations around a string, a circular piece of string, and the vibrations are going around the string. In order to constructively interfere, I need to fit a whole number, integer multiple of vibrations around that string. Otherwise, if I have, say, a third of a vibration left, then the interference will not be regular, there will be destructive interference, and there'll be essentially chaotic motion. It won't be nice standing wave. So standing wave implies quantization. That's how you get standing waves. There are only certain wavelengths that can be supported. So the existence of standing waves or the need to

support standing waves directly leads to quantization. But Fourier transform pairs are effectively directly related to the concept of standing waves. Because remember, you've got the waves, you've got either your sign curve that goes out to infinity and defines exactly the momentum without defining the position or vice versa. You're precisely defined position with no idea what the momentum is. These can be considered to be standing wave patterns. Staining wave patterns imply quantization. So that's the loose way that I think about how commuting variables on, or more than the point, non-commuting variables, lead to quantization.

There's quite a lot of conceptual steps there. The basic idea is non-commuting, if two variables are non-commuting, position and momentum in this case, it means that if I specify one, then I don't have complete freedom about the value the other one takes. The order matters. It's not just like I can specify one, and then it completely, independently specify the other. The order that I specify the matters. Another way of saying that is that the two variables form a Fourier transform pair. Again, the value that one takes depends on the value that the other takes, and you can't arbitrarily define each to perfect precision at the same time.

But Fourier transform pairs are related to standing waves because of the wave pattern relationship of Fourier transformed variables. And standing waves apply or necessitate or directly lead to quantization because of the need to fit in a whole integer number number of vibrations around whatever space your system is constrained into. It doesn't have, you can think of a string wrapped around on itself and needing to fit in a whole number of vibrations around the string, but it doesn't have to be in one dimension like that. That's just a simple way of thinking about it. So, bottom line is to get the quantum bit into the quantum field theory, we need to have quantization. For that, we need to have these Fourier transforming pair

of variables, position and momentum, and in particular, we need them to be non-commuting. so that the order in which you specify them, or the order that you multiply them together, matters. The reason I spent so long on that is because this key idea of the commutation relationship, the commutator, the difference between doing it one way and doing it the other way, between position and momentum, is absolutely crucial to quantum field theory. Like, that's one of the most important ideas behind it. And whenever you try and define a new field theory or extend one to a new circumstance, you need to think about what is the commutation relationship that I'm going to use here.

and we have things called the canonical commutation relations that applies to position of momentum and so on. So that's why I spent so long on that, to understand, even if you didn't get all the details there, the key point to take away is that you need to get quantization, you need non-commuting position and momentum fields. And that really gets your quantum into the quantum field theory. Classical field theory does not have this. Classical field theory has commuting variables, so the commutator is zero. The commutator in quantum field theory, by the way, in the canonical computation, relations is directly related to Planck's constant, which is donated H-bar, if you're interested in looking that up.

Okay, so we've now got our two fields, position and momentum. Remember, the fields is a function of space time, and we define a position in space in a particular time, and then the field is a function that gives us back some number that tells us the value of the field at that point. We've now said, jumping from, or transitioning from classical to quantum field theory, that we're going to have a non-zero commutator between our position and momentum field, That's going to quantize the energy levels and other phenomena that we get at the end of after we've finished our calculations. But we're still not quite there yet because we need to build the structure of what the field actually looks like.

How do we define the quantum field? Okay, so at this point, we've got our fields, position and momentum. Well, we don't literally have them yet, but we've got the idea that we're going to have those fields, and we know that they're not going to commute with each other. they're going to have this non-zero commutation relation. That's going to give us the quantization that we need. But we still need to figure out what values are our fields going to take. Remember, a field is a function of a space time,

so we need to specify what that function is, how to specify the value that the field takes at each particular point in space. And generally, we're going to want to do this in terms of the energies involved with whatever the system is. Now, it turns out that a very general and useful way to do this is to define the fields in terms of harmonic oscillators. Now, what is a harmonic oscillator? Essentially, a harmonic oscillator is a weight or a mass attached to a spring.

The harmonic oscillator is the mathematical description of the dynamics of that system. So if you imagine a spring and a weight attached to one end, if you pull that away from its equilibrium position, there's going to be a restoring force that tries to pull it back. and in fact the restoring force in a classical harmonic oscillator is linear in the distance that you pull it away. So the further you pull it away, the greater the force is that pulls it back. That setup, that dynamic setup defines the energetics and therefore the mechanics, the dynamics of a harmonic oscillator. So what we imagine doing in a quantum field theory is taking a bunch of these classical harmonic oscillators,

quantizing them, so applying those commutation relations that I mentioned, so they're now quantum harmonic oscillators, and putting an infinite number of them at each point in space, actually each point in space time, because we're doing this in relativistically covariant way, so that means that we treat time analogous to space, but don't worry too much about that, because I haven't talked about relativity yet. I'll get there in a future episode. But basically, think about it, putting an infinite number of these quantum harmonic oscillators at each point in space. Now, why an infinite number? Essentially because we need one for each momentum that the particle could be in. We haven't specified what particle yet. Just think of it as an electron.

So each point in space has an infinite number of these quantum harmonic oscillators in it. And this extends in three dimensions. So think of a three-dimensional lattice of these little spring or weights on springs that are vibrating. And each possible momentum that the particle could have has its own corresponding oscillator. And essentially, the way you can think of it is that the stiffness of the spring corresponds to the value of the momentum of the particle in question. So really stiff springs vibrated at a very high frequency, so they have a high momentum. Very non-stiff springs vibrated in much lower frequency, and so have a lower momentum. So you've got at each point in space, an infinite number of these harmonic oscillators all sitting on top of each other, vibrating at different characteristic frequencies.

Each oscillator has exactly one specifically defined frequency, but there's an infinite number sitting in each point in space, all three dimensions of space. Now, the other thing you have to imagine is that these oscillators are all connected up to each other, so that they don't just vibrate independently of each other, but there's interactions between them. And the nature of these interactions turns out to be

the real guts of quantum field theory, the real tricky bit. But for the moment, just imagine they're sort of all tied together. So if you... So imagine these oscillators all start off in a state of rest, in practice they never do, but just imagine that they were in a state of rest. Then you picked up exactly one of them. So at exactly one point in space, you pick up exactly one of these specific momental, specific oscillator, so it has a characteristic stiffness, and then let it go.

So essentially give it some energy. Then that energy is going to propagate out through all of the connected oscillators, both those at the same point with different stiffnesses or different frequencies, and also those at different points in space. and you'll get a traveling wave which expands outwards and a complicated pattern of activity, depending on exactly how hard you gave the initial tug. This pattern of disturbances is essentially a wave,

it's a traveling wave, and we refer to this as a particle. So disturbances like this that travel through the array of oscillators are particles. You can have more than one particle in your array at the same time. In fact, you can have more than one particle in the same place. You just excite the oscillators appropriately, because a particle just is sort of a bundle of excitations, a localized bundle of excitations of your oscillators. It doesn't have to be at exactly one precisely defined position.

It can be spread over some spatial extent of oscillators around some area. And so you can put multiple particles on top of each other. And indeed, particles can be created and destroyed in this formalism, just by essentially changes, in the oscillation patterns of our oscillators. Now remember, when I say that there's this array of oscillators in three dimensions and that there's an infinite number at each point, you can imagine that there are tiny balls on springs at each point.

In fact, that's helpful visualization, but we don't imagine that literally there is any physical thing there. There's no literal oscillator. The idea is that the fields behave as if they were all of these oscillators an infinite number at each location spread throughout space in a three-dimensional lattice, that the field behaves as if it were comprised of these oscillators. We don't think there's an actual literal physical oscillator there, well, other than the field itself, but there's nothing material there outside the field. The field behaves as if it were the oscillators.

So you can think about it in terms of those oscillators, but don't take it too literally. It's just a visual model to help us see what we're doing. Another caveat is that I mentioned previously that imagining that we picked up exactly one of these oscillators at exactly one point in space and imparted it with some energy. Now, in practice you can't do that because that would be specifying exactly the position and exactly the momentum of a particle. Remember exactly one point in space and exactly one oscillator, which corresponds to exactly one stiffness, one momentum? You can't do that by the uncertainty principle. But that was just a thought experiment to help you understand what's going on here. In practice, if you had a singularly localized particle,

you'd have to have a superposition of all of the different possible momentum, which essentially would mean that all of the springs at that particular spatial location were all vibrating at their particular frequencies. So you'd have some energy spread throughout all of the different possible momentum states. Or conversely, if we specified exactly the momentum of the particle, then we would only have one oscillator with one characteristic momentum vibrating, except now we wouldn't just be vibrating at one point, but the oscillator of that same frequency would be vibrating all throughout space.

So we'd have the momentum exactly specified, but now the position would be completely unknown, so we'd have oscillators vibrating all throughout space. And in fact it turns out that I've said that we have an infinite number of oscillators at each point, each oscillator at each point corresponding to one particular momentum. But you can look at it, you can think of it as being the other way around as well. You can think about oscillators in momentum space and each one corresponds to, each point in that space corresponds to one particular value of momentum, and there's an infinite number of different position oscillators on that particular point. That's

sort of less visually helpful, I think, because that's not the way we perceive the world. But the point is, mathematically, it doesn't actually matter whether you think of it in terms of momentum oscillators in position space or position oscillators in momentum space because they're Fourier transform. They're a Fourier transform pair, so it's equivalent to look at it in other way. But I'll just talk about it in terms of momentum oscillators, that is, oscillators with a characteristic stiffness located throughout physical coordinate space, because that's easy to visualize. But bear in mind that mathematically, that they're actually equivalent. So, this formalism is really powerful because it allows us to explain particles as

vibrations or as disturbances in this array of oscillators. The number of particles is determined by essentially the excitation level, or the harmonic, if you want to think of it that way, of the oscillator. So again, think about my oscillator at a specific point in space, just for simplicity. Think about one momentum, one stiffness value in oscillator at one point in space. There's not just one frequency that that can vibrate at. There's one lowest fundamental frequency it can vibrate at. Think back to our episode on vibrations and waves if you're a bit fuzzy about fundamental frequencies and harmonics. But there's one lowest, one longest wavelength or one lowest frequency that this particular spring can vibrate at. But it can also vibrate at essentially

half that wavelength or half that again. It relates back to the standing wave. You have to have the integer multiples of wavelengths fitting into whatever space you've got available. But so long as you've got that, there are many different frequencies that the system can vibrate at. So we can't vibrate at any frequency that it likes, but there is more than one. So the point here is that if that particular spring, that particular oscillator, is vibrating at its lowest possible frequency, that is its fundamental frequency, then we say there's one particle with that particular momentum located at that particular point in space. If, on the other hand, that particular spring is vibrating at its first excitation or its first harmonic essentially, its first excited frequency,

then we say there are two particles with that momentum located at that specific point in space, and so on for three and four, and however many you like. So the number of particles is specified in terms of the excitation of a specific oscillator at a specific point in space. In practice, of course, you're not going to be able to localize a particle to exactly one point in space and exactly one value of momentum. It's going to be spread over many points in space and many different momentum because very a transform pair. But again, I was just specifying the example of one position and one momentum just so you can visualize what's happening. Because there's a lot of things happening here. There's an infinite number of these oscillators at every point in space. So there's different points in

space. Then there's the different oscillators at each point in space, each one corresponding to a different spring stiffness, or equivalently a different value of the momentum of the particle. And then there is also what frequency is each of those oscillators vibrating at? or more to the point what excitation level with regard to its fundamental frequency is each of those springs vibrating at. The high the excitation level of any particular frequency oscillator, the more particles of that momentum are located at that particular point in space. So this formalism gets us essentially all the things we want. It allows us to talk about particles, many different particles, that can be created or destroyed so we can vary the number of them, at different points in space with different values of momentum. while still retaining the quantization that we needed.

And remember, we get that by saying that the two fields, position momentum, don't commute with each other, and the resulting quantum harmonic oscillator. Now, it turns out that there's a bit more structure that we need to add to this. So far, we've been able to define a quantum field, but we haven't been able to incorporate the effects of spin, intrinsic angular momentum. That goes beyond what I can discuss in this episode, so I'm really not going to go through it any further. If you're interested, you can look up to the Dirac and

Klein-Gordon equations. Those are two separate equations, the Dirac equation and the Klein-Gordon equation, which are essentially relativistic versions of the Schroeniger equation, or partially relativistic versions of the Schroencher equation, and they relate to the spin of the particles equation, the intrinsic angular momentum. So it turns out that we have to add more complexity into the fields to incorporate that as well, but I'm not going to go into the details of that. So, so far, essentially, what we've got is a quantum field theory. We've got fields, and we've explained how they're quantized and we explained how, at least in the free case, the non-interacting case, the fields evolve with time. They evolve, their behavior is defined in terms of these

harmonic oscillators. So this whole process of essentially bullying out the, getting out the harmonic oscillators and producing quantum fields is sometimes called second quantization. And physicists don't always like that name because it's not very, it's not very descriptive, but you'll hear it. And so it's useful to have some sense of what's going on there. First quantization was what happened in non-relativistic quantum mechanics, the type of quantum mechanics I talked about in the previous two episodes. First quantization deals with the operators of quantum mechanics like position and momentum, and particularly it concerns single particles. It gives them commutation relations. The sort of characteristic phrase there is that when you do first quantization, you're saying

that particles behave like fields, or particles behave like waves, if you want to think of it that way, because of fields and waves are similar sort of idea. Second quantization deals with fields of quantum field theory and saying they don't commute. So remember, in first quantization, it was operators of position momentum don't commute. When in second quantization, it's actually fields of position momentum that don't commute. So instead of operators, we're now dealing with fields. That's a big difference. And second quantization essentially says that fields behave like particles,

or waves behave like particles, if you want to think of it that way. Again, because we're quantizing these fields and saying they can only take sort of discrete values and these discrete values correspond to the particles that are propagating through the system. That discretization, the commutation relations that we establish between the position of momentum fields mean that you can't just have, you can't have like one and a half particles in the system because the number of particles is defined by the excitation level of a particular of the particular momentum harmonic oscillator. So it can vibrate at its ground level or at its first excited level, but it can't vibrate at a frequency in between those. So you can have one

particle or two particles in that particular momentum, but not one and a half. That's where the quantization comes from. So this is saying that fields behave like particles. A field theory gives rise to the discretization in terms of the excitation of the energy levels that is, that corresponds to a particle. That's the idea of particles, right, that you can't have one and a half particles. They're discrete. They're spatially localized, so they don't exist everywhere, but also they're discrete in number. So that's the two quantizations, essentially. Very loosely speaking, particles behave like waves, and waves behave like particles, or fields behave like particles. Now, so far, all we've actually managed to describe is free field theories. Essentially,

that is when you've got a bunch of particles moving about in our field, in our lattice of oscillators, but not interacting with each other. So this would be an example of photons just moving about in, I don't know, a gas. Photons don't interact with each other. So they just pass through each other and move about as in accordance with their equations of motion. That's all we've been able to describe so far. We can describe particles moving about in our field, but not the interaction of particles in that field. And that's where we get to interacting field theories. And this is, as I said, where the guts of field theory are, because this is where the hard stuff begins, if you thought that all of the stuff beforehand was easy. And by comparison, it is.

But we won't talk too much about interacting theories, because that's where the really heavy mass comes in. The basic idea, though, of what we do in an interacting theory is that you need to have a way of describing the form of the interaction between different particles. Say an electron and a photon, for example. How do those interact with each other? What is the function that describes their interaction strength and the the form of this. How do we incorporate that into the theory? Well, if you recall before, I started the episode by talking about Hamiltonians and Lagrangians, particularly Hamiltonians. Now, there's a subtle distinction between Lagrangian and Hamiltonian. I'm just going

to talk about them as if there's the same thing for this little section because it's going to get too confusing otherwise. So there's some caveats here that are not strictly true, but it's good enough for our purposes. So we have a function called the Lagrangian, which is essentially our energy function. It essentially tells us the energy of the system. Technically the Hamiltonian is the energy and the Lagrangian is a bit different, but good enough for us. The Lagrangian is a function that tells you the energy of this system. The form of the Lagrangian depends on what system you're considering, obviously. If we just consider a free field theory that is no interactions,

that's the field we've been describing up till now, the Lagrangian just takes the form of a harmonic oscillator. Essentially there's your kinetic energy term and your basic potential energy term, which look exactly the same as they do for a harmonic oscillator. That's why we picked harmonic oscillators to construct our theory with, because a free field theory with no interactions, for pretty much anything, really, just looks like a harmonic oscillator. There'll be a few different constants here and there,

but essentially any particle that's not interacting will behave like a harmonic oscillator, and that's why we use this as the free field theory. So that's our sort of boring free-field Lagrangian. It tells us the energy of a particle that's not interacting with anything, other than essentially its own field. but what about the interactions? That's where the interesting stuff happens, like the scattering events. How do we define those?

Well, we need to add the term to our Lagrangian, or maybe multiple terms, to account for the interactions. So there'll be some terms in our Lagrangian, which is just the basic kinetic energy and the basic potential energy, which just goes along with the particle doing its thing, the free-filled components of the Lagrangian. But then we need to add in the interacting part or parts to the Lagrangian, which define the interactions between different types of particles. And that's the tricky bit. And there's no real rules for how to do this. Basically, you just have to come up with terms for your Lagrangian and do all your calculations and then test them against experiment to see whether your theory works.

Because there are many different potential forms of Lagrangian can take, and there are various theoretical arguments about, say, Lorenzov variance and other things that we would want our Lagrangians to satisfy. But at the end of the day, you don't really know unless you test. against the experiment. I emphasize this because the behavior of a quantum field theory really depends on the Lagrangian that you pick for the theory, essentially the energy function, how you say the particles behave when they interact with each other. And you can't derive that from anywhere. You just sort of guess it and then see how it goes against experiment.

But we're not going to worry too much about that, and I'm not going to talk through the form of different Lagrangians either. Of course, there are different ones to appear on what type of interactions you're trying to model. Is it an electron with a photon? Is it two electrons with each other? Is it an electron with a muon? All sorts of things you can have interactions for. And they're going to have different Lagrangians and different terms than Lagrangians. But for our purposes, you get your Lagrangian from wherever. The Lagrangian is an energy function. It tells you how your system behaves, including the free field theory bit and the interaction bits. Now, we then need a way of figuring out the equations of motion for this system. How do the particles behave

over time because that's ultimately what we want. Now that's not trivial to do. That's very difficult to do. It's easy enough to do in the free field theory, very difficult to do in the interacting theory because of these extra interaction terms in the Lagrangian. Harmonic oscillators are relatively simple to describe mathematically, which is why it's so neat that we're able to describe free fields in terms of harmonic oscillators. But the interaction terms complicate matters and we can't deal with the maths nearly so easily. So we use a bunch of math tricks in order to get essentially the interacting field expressions in terms of free fields only. Now, it's sort of a miracle that that's actually possible, but we actually can do that,

at least for many field theories, I don't know if you can for all of them, but one like quantum electrodynamics, for example, you can get the interacting field equations and you can rewrite them only in terms of free fields. And there's a bunch of math tricks that are required to do that, including the LSZ reduction formula, perturbation theory, which I talked about in the previous episode, so that's a very important tool. WIC contraction and other things as well, enable us to pull this feet off. I'm not going to talk about what any of those things are, apart from perturbation theory, I did talk about in the last episode, but if you're interested in the details, you can look up LSZ reduction formula and WIC contraction, that's WICK, if you're interested in some of

these details. But it's quite mathematically complicated. The long and short of it, though, is that we are able to go from complicated interacting field expressions to, okay, well, still complicated, but nevertheless, expressions of the interactions in terms of the free fields only, and these we can calculate. To help us calculate them, we use pictorial diagrammatic representations called Feynman diagrams. Now, it's likely you may have seen these if you're a bit of a physics enthusiast. I mean, you don't even really have to be much of a physics enthusiast to have encountered a Feynman diagram at some point. they're basically just straight or squiggly lines connected to each other. They don't look like a whole lot or post them up on the podcast Facebook page to have a look at. Really what they are is calculational devices that help us keep track of all of the different types of interactions that can happen

and so that we can do our maths correctly to calculate the terms that we need. Loosely, you can also think of them as if they sort of visualize the trajectories of the particles as they're moving together and interacting with each other and then moving apart. but you shouldn't take that too literally because particles don't have precisely defined, generally don't have precisely defined locations, and there are other details there that complicate the matter. But the main point is Feynman diagrams are a calculational device that help us to make calculations in quantum field theories.

So they're really useful. And what is it that we're actually trying to calculate at the end of the day? I've talked about doing these calculations and reducing the interacting fields to free fields, but, I mean, what are we trying to do? Well, what we want to do is be able to compare our results to experiment, to test our theories, particularly to test our Lagrangian, to see if the energy function that we're defined for our system is correct. And so usually what we're interested in calculating, or often what we're interested in calculating, is something called a cross-section.

Now, essentially, this represents loosely the probability of an interaction occurring in a particular region of space. And by an interaction, we mean some sort of, well, interaction event between particles, like, for example, two electrons repelling each other, or one particle decaying and giving off two other particles, something like that. That's an interaction event. Just a single particle moving along and doing its own thing, that's free-field theory. There's no interactions there. Or likewise, two particles that don't interact with each other,

which is generally the most common thing. They just sail past each other and nothing happens, like two photons, for example. That's also not an interaction. An interaction is when there's some change in the mementa of the particles and or the type of particles and or the number of particles. That's an interaction. That's the interesting stuff. A cross-section tells us essentially the probability of a particular type of interaction happening in a particular region of space.

That you can experimentally measure in experimental results from particle accelerators or other physical setups like that. So that's why physicists build these really expensive machines to accelerate particles at really high speeds because, well, if one thing to get the interactions to happen, you generally need very high energy. and that means very high velocities of the particles. Otherwise you'll see like one interaction every million years and you'll never get to publish your paper. So you need the high energies to get the interactions to happen so that you can measure enough events, interactions that is, to adequately test your theory. Your theory tells you how many of these events there should be and what probability they should occur. So you need to be able to compare that to an experiment. So you have to have enough of the experimental events to compare it to. And essentially at the end of the day, if you've done everything correct,

you should be able to compare your results of experiment and see how many of these interaction events did my theory predict and how many actually happened. Now, sort of the magic of it all is, and obviously it's not magic, but it seems magical, is that quantum field theories are unbelievably accurate. I believe that quantum field theories provide the most precise predictions that have been experimentally verified of any scientific theory. apart from possibly general relativity, the most precise test of any quantum field theory that I'm aware of relates to a parameter called alpha, or actually one over alpha, which essentially is related to the anomalous magnetic dipole moment of the electron. Now, we're not really worried about what this is.

It's just a property of the electron that relates to its spin, and it can be both measured through a number of different experimental techniques, but also it can be predicted by the results of quantum field theory, specifically quantum electrodynamics. All I want to do is give a sense of the amazing predictive ability of quantum electrodynamics. All of that complicated theoretical apparatus of the oscillators and the free fields, free fields and then interacting fields and all that stuff that I went through. So to simplify things and just get to the heart of the matter, the best theoretical calculation of the value of alpha, 1 over alpha,

is 137.035-99070. So that's to, what, nine decimal places. So that's a position better than one part and a billion. That's a pretty precise answer. Imagine if you came up with that answer as the result from some long calculation, you'd wonder how close it was. Well, this is the empirically measured value that it can be compared to. And it is, 37.0359987. So the difference there is in the sixth decimal place, and there are three significant figures before the decimal place. So essentially that's one part in, about one part in 100 million, which is an incredible degree of accuracy. To put that in perspective, that's roughly analogous to calculating your height to the nearest molecule width and getting it correct.

your height, that is the height of a person to the nearest width of a molecule and being correct. That's really some amazing precision. So it's, for this reason that physicists generally think that quantum electrodynamics is onto something. Nature really does behave something like in this way of these superposed harmonic oscillators. The idea is that the entire universe, essentially, or observable universe at least, is suffused with these fields, essentially one field for each different, type of particles. So one for your photons, one for electrons and so forth. There'd be another field for the Higgs boson, which you might have heard about, might talk about that in a future

episode. But these fields then are vibrating in accordance with all of these principles of harmonic oscillators and so on that we've talked about. And the vibrations are interacting with each other in accordance with the forms that we figure out in our Lagrangians that we put into our theory. And this is what gives rise to, well, essentially all of the particles that we know of and the interactions thereof, and all of the forces of nature as well are all built into the behavior of these interacting fields. Another payoff to quantum field theory as a conceptual laboratists is it enables us to explain why different particles

or why each instance of a given particle is identical, as far as we can tell to all of the others, say, why are all electrons identical? Obviously they can vary in momentum and position and so on, but apart from that, the actual intrinsic properties of the elements of the elements, electron are the same and exactly identical, completely indistinguishable between different electrons. And I talked about that in the previous episode when I talked about particle statistics. But why should that be the case?

Well, quantum field theory explains that in terms of they are all, that is, every electron in the universe is a vibration or disturbance in the same underlying electron field. There's only one electron field in the whole universe. And just different vibrations or disturbances in this field accounting for different electrons, but in a sense they're all the same because they're all part of the same field. They're all manifestations of disturbances in the same underlying field. You recall before, by the way, that I talked about conjugate fields, position and momentum as two different fields, and now here I'm talking about one electron field.

Well, remember, we can treat them as two mathematically distinct fields and have our commutation relation and so on, but they form a Fourier transform pair, the position of momentum field. So in some sense they're linked, and so we also talk about them as being one field, the electron field, but there's sort of two bits to it, if you like. Anyway, that's the technicality. Quantum field theory is also very useful because it unites quantum mechanics and special relativity. So it produces relativistically invariant results that it hold regardless of your frame of reference. That's not the case for Newtonian mechanics.

I haven't covered that a lot in this episode, but quantum field theory doesn't just incorporate the quantum aspects, it also incorporates the relativistic aspects as well. with one exception. There's no yet widely accepted quantum field theory of gravity. So we've got quantum field theories for electromagnetism, and we've got quantum field theories for the weak nuclear force, that's electro-weak theory, and we've got one for the strong force, that's quantum chromodynamics. So those are three of the fundamental forces of nature. But the fourth fundamental force, gravity, does not have an accepted quantum field theory for it yet. If we had one, we would be able to unify quantum mechanics and relativity, which we can't, because we have

haven't been able to come up with this field theory yet. One final conceptual take-home from quantum field theory, which sort of came up in regular old quantum mechanics, but really reaches its sort of full manifestations in quantum field theory, is the idea that empty space really isn't empty, because empty space still contains all of these harmonic oscillators, these fields of the different particles. And, although I hadn't mentioned this before,

the ground-level excitation of a quantum field, that that is the lowest excitation it can have, even when there's no particle there, there's still a residual amount of energy there. There's still some residual vibrations. So there's never the case that all of the oscillators are just sitting there still doing nothing, even if we're light years away from any atoms or stars or photons or anything. Even if there's no particles around, there's still some residual vibrations in the field. That's why I mentioned earlier in the episode that you can't actually have a perfectly still set of oscillators in your field.

You can imagine it, but it can't actually exist. Because there's this residual ground state energy that's always there. This is sometimes called the vacuum energy. And it can be thought of in terms of virtual particles that are constantly coming into existence and then being destroyed in very short fractions of time in terms of the vibrations in the field that permeates space. And the reason this can happen is because matter is composed of disturbances

in these underlying fields. The fields do have to obey conservation of energy, but over very short periods of time, essentially you can sort of borrow energy and repay it back into the fields, as long as the total amount of energy and time borrowed for is relatively small. Don't worry too much about that.

But the basic idea is the matter itself is comprised of disturbances in these fields. So you can have energy, and you can have even forces, that is feel a physical force exerted, even in the apparent absence of any actual material or substance. Have a look at the Casimir effect, if you're interested in the feeling of the forces,

example. That's the force exerted by a vacuum, and that's measurable, a measurable effect. So I think there's very profound implications that quantum field theory has for our picture of reality, that sort of the idea that we have about matter as sort of hard, substantial something versus the vacuum as empty nothingness is really not quite true. Essentially, matter is just a state of the fields that has more energetic vibrations and disturbances in the field than the vacuum, which has the minimum amount of disturbances in the field. So quite a profound way of thinking. To close out this episode, I just now want to recap, essentially, briefly, the different levels of theory that we've talked about in this episode, because I haven't sort of outlined them

carefully in the way that might be helpful. So we begin with Newtonian mechanics. This is, you know, Newton's Laws of Motion. This is the stuff you learn about physics at high school. Newton developed these in the 17th century. Then extending Newton's mechanics, not replacing them, but extending it and building upon it, you can sort of go in three different directions. One is classical field theory, which doesn't necessarily give you any new physics, or it does a little bit in electromagnetism, because Newton didn't come up with that. Electromagnetism is an example of classical field theory. La Grangian and Hamiltonian are other examples of a classical field theory. But classical field theory, essentially,

enriches the formalism of Newton's physics, and as I said, provides also a way of describing electromagnetism that Newton didn't describe. So that's one direction that you can go in, classical field theory. Another direction you can go in is very high energy, very high velocity, or close to the speed of light, which is relativity, or particularly special relativity. This was devised by Einstein in the early 20th century. Classical field theories belong to the 18th and 19th centuries, particularly Maxwell formulating classical electromagnetism in the 19th century. LaGrange came up with his Lagrangian mechanics at the end of the 18th century. So 17th century for Newton, classical field theory belongs to the 80th and 19th century.

Special relativity, early 20th century, and the third extension to Newtonian mechanics, non-relativistic quantum mechanics or quantum theory, which describes the very small, the very low energy scales, single particles of matter or small systems of matter, that also dates to Einstein in the early 20th century. century. Einstein actually came up essentially with the origins of both quantum mechanics and special relativity. And these levels of classical field theory, non-relativistic quantum mechanics and special

relativity, you'll typically study these as sort of an early university level if you study physics in a university, whereas you begin Newtonian mechanics at high school. The next stage that we can build on, the next level of complexity is to try and combine these three different extensions. If you imagine Newtonian mechanics sitting at the bottom of the trunk and three different branches sticking out, classical field theory, non-relativistic quantum mechanics, and special relativity, all coming out in their different directions. They all describe different things. Enriched formalism in classical field theory, including infinite degrees of freedom and ability

to create and destroy particles. Quantum mechanics gives you the ability to describe discrete and very small systems. Special relativity gives you ability to describe relativistically invariant, high-energy systems. ultimately we want to be able to describe all of those things. And so we want to be able to combine these together. So if you take two, if you combine special relativity and field theories, that gives you general relativity. A general relativity is an example of combining those things together. That also dates the early 20th century, but a bit after special relativity.

So general relativity is a field theory, a fully-fledged field theory, which incorporates special relativity, but not quantum effects. As I said before, we don't have a quantum theory of gravity yet. Another avenue you can take is to combine quantum mechanics with field theory, and there you get non-relativistic quantum field theory. That's not studied so much because it's less useful than the fully-fledged relativeistic field theory, which we'll get to. But you can have that. Interestingly, you might imagine just combining quantum mechanics and special relativity without bothering with a whole fully-fledged field theory. Now, it turns out this isn't possible to do consistently, which is why when you study old-fashioned,

non-relativistic quantum theory, you only really look at single particles or states where you have a defined, conserved number of particles. If you try to incorporate varying number of particles, like decay events, for example, or interactions that change the number of particles, into your quantum mechanical description, you end up in trouble. Essentially because when you get to very high velocities, the energies get very high. And when the energy is high enough, you can create new particles. For example, very high-energy photons can, well, essentially split into a positron and an electron. So two different particles. That won't happen at low energy, so in regular quantum theory, you're okay, but if you try to combine quantum mechanics with special relativity,

with the really high energies, this can start to happen, so the number of particles will not be conserved, and you'll have to deal with these extra degrees of freedom. So relativistic quantum mechanics without field theory is not actually possible. It's inconsistent. They're sort of tried and failed to do this around the sort of 30s. You can only combine quantum mechanics and special relativity in the context of a fully-fledged field theory. So you've got quantum mechanics plus special relativity plus field theory all coming together, and that's quantum field theory, which was developed from the 1930s to around the 1950s. So that's one way of thinking about what's so great about quantum field theory.

And really it's only taught at high level undergraduate or graduate level at universities because it's quite advanced, but it brings together all of the other aspects that you've learned about in physics. it brings together the non-conservation of particles and the many degrees of freedoms and the complexity that you can get with classical field theories added to the effects of discretization and very small energy scales and all the other weaknesses associated with quantum theory plus it's relativistically invariance so you get the effects of special relativity as well the only thing missing from quantum field theories or from the most developed quantum field theories

is general relativity or the effects of gravity because we don't yet have a quantum theory of gravity or to put it a different way, we have quantum field theories for all of the other fundamental forces of nature, but we don't have a quantum field theory for gravity. So that is still a missing piece to the puzzle. And physicists have been looking for a quantum theory of gravity, well, pretty much ever since Einstein, he didn't succeed. String theory is, I think, probably the most popular candidate. I know very little about string theory, but perhaps one day I'll do an episode on it. But it's not, it has not been experimentally verified, and so it's not considered to be established that that's sort of the right answer or not even the right approach to this. But quantum field theories,

in particular quantum electrodynamics, have been experimentally verified to very high degrees of precision, and so they are accepted to be, you know, as good as we can get in terms of accurately describing the way reality works. So that's been a pretty full-on episode. Hopefully you got a reasonable amount out of that, even if you didn't understand everything. If you enjoy the podcast, I'd invite you to leave a favorable review on iTunes or the aggregator of your choice. There are quite a few around now. You can also like the podcast Facebook page. Just go to Facebook and type in the Science of Everything podcast. Liking the page helps to get more publicity for the show, which is always good, and you'll also get some notifications

and extra visuals and so on, which might be of interest. Finally, you can also send me an email. My address is Fods12 at gmail.com. That's F-O-D-S-1-2 at gmail.com. Feel free to send me questions, suggestions, feedback, criticisms, whatever you like. I just, or even just say hi, I like the show. I love to hear from my listeners. Thanks for listening, and I'll talk to you next time.