The Science of Everything Podcast - Episode 158: Quantum Electodynamics Part 1

Episode Date: March 1, 2026

A detailed look into the physics and mathematics of quantum electrodynamics, the theory of how light and matter interact. We discuss the generalisation of the Schrodinger equation to the Klein-Gordon and Dirac equations, and howw these describe the propogation of light and fermions respectively. We then consider the process of computing transition probabilities between quantum states, including the S matrix and perturbation theory, and Wick's theorem. Recommended pre-listening is Episode 85: Introduction to Quantum Field Theory. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything

listening to The Science of Everything podcast episode 158, quantum electrodynamics. I'm your host, James Fodor. In today's episode, we are going to discuss the quantum field theory of quantum electrodynamics, or QED, as it's often called. So this theory describes the relativistic interactions between light and ordinary matter, so that's like electrons and similar types of particles. This episode follows on from one I did quite a few years ago now on an introduction to quantum field theory. So that's the recommended pre-listening episode 85, introduction to quantum field theory.

So that was an introductory conceptual overview of quantum field theory. This episode is going to be building on that and we'll review some of the concepts, but you should really listen to that first or have some background in quantum field theory to appreciate what we're going to be discussing today. So here we're going to go through the mathematics and the formal in much more detail, and I'm going to discuss particularly how we use the formulism, including Feynman diagrams, to calculate experimentally measurable quantities in QED. This episode will also form the basis for a subsequent episode, which hopefully will come out

within the next few months, where we'll talk about the standard model, so we'll look at other interactions beyond. So as I said, Recommended Prelisting is episode 85 on introduction to quantum field theory. The two episodes immediately prior to that 83 and 84, advanced quantum mechanics would also be useful. I mean, they're in turn prerequisites for 85. So hopefully you'll listen to that first. So just be warned, this is going to be a more technical episode. And prerequisites will be important here.

So, for example, the first thing I'm going to do is start by talking about how we can make the Schrodinger equation apply at relativistic energy scale. So if you don't understand what that means, then the episode's going to be. bit hard to get into. So do check out those earlier episodes if you are not really sure what any of that is about. So that being said, let's dive in and start with the good old Schrodinger equation from non-relativistic quantum mechanics. We know that the Schrodinger equation describes how a quantum system, or like a particle, for example, evolves over time. And specifically, what the Schroenger equation actually is, is it says that the evolution of the wave function, so that the partial derivative with respect to time of the wave function,

is equal to the energy operator applied to the wave function. So the energy of the particle determines how it evolves over time effectively, or energy of the particle system, because it can be more than one particle. You can actually derive the Stroudanager equation by starting with the classical total energy momentum relationship, which says that the energy of a system is equal to the mass energy, so MC squared, plus the kinetic energy. So that's one-half mv squared. That's the non-relativistic formula for the kinetic energy. If you then effectively transcribe those energy relations into the quantum mechanics length,

So you take the energy to refer to the partial derivative with the system with respect to time and the momentum referring to the partial derivative with respect to space, multiplied by Planck's constant, of course. If you make that transcription of the energy terms, substitute them in and rearrange, you end up with Schrodinger's equation. So that's kind of how he came up with the equation. But we should be emphasized that that's sort of a heuristic derivation because there's no, formal set of principles that says you should sort of start with that. Ultimately, Schroenger's equation is justified experimentally in terms of how useful it is. The point is, though, that this blueprint provides us with a way to understand how we can make the Schroeniger equation apply in relativistic cases, because as it is, the Schrodinger equation does not incorporate relativistic effects.

So when the velocity of the particle becomes very large, and we start looking at velocities close to the speed of light, relativistic effects become important, and and the Schrodinger equation is not going to appropriately describe those. So one attempt that we can make to develop a relativistically invariant form of the Schroenger equation is to, instead of starting with the classical energy momentum relationship, which remember says that energy is equal to like mass energy plus kinetic energy, E equals MC squared plus half mv squared, right? Instead of starting with that and then translating that into quantum realm and getting the Schrodinger equation,

we can start with the relativistic energy equation and then translate that into quantum realm and see what we get. So the relativistic energy equation, which I've discussed in episodes we talked about relativity, e squared equals m squared c to the power of four plus c squared p squared. So note that this is actually slightly different because this is the equation that gives energy squared. So that's sort of that will become important later. So the mc squared part is the same. what's different is that instead of having the kinetic energy, we now have a term for the total momentum. So C squared, p squared is the momentum, the magnitude of the momentum, and we multiply that by C squared.

The difference between the two, the half mv squared transforming into C squared p squared, the difference there is entirely due to relativistic effects. And you can actually prove that if you restrict two speeds much, much less than the speed of light, that the relativistic form of the energy relationship at the limit with low velocities turns back into the classical energy relationship. So if it's sort of not obvious to you how you go from C squared p squared down to half mv squared, it's just due to relativistic effects. And you can you can prove that using some fairly simple, if a bit tedious algebra. So the point of all this is that the relationship between momentum and energy changes in classical compared to relativistic

context. And so now when we do our conversion into quantum mechanics realm, we replace energy with the partial derivative with respect to time times planks constant, and momentum with the partial derivative with respect to space times blanks constant. We substitute that into the relativistic energy equation and rearrange. What we come up with is a new equation, which is now relativistically invariant, which means it's applicable to relativistic context, and it's called the Klein Gordon equation. The Klein-Gordon equation, if I just sort of read it out mathematically, it says that some of the partial derivatives of the wave function is equal to the mass squared times the wave function with some extra planks constants and c squared's thrown in. Note that this is a relativistically invariant formulation,

which means that it doesn't single out and differentiate derivatives with respect to time with derivatives with respect to space. And this is important because, as you may recall, relativity, we treat space and time as connected in four-dimensional space time. And when you change frames of reference, components that in one frame of reference contributed to spatial terms can, in another frame of reference, contribute to time-like terms and vice versa. They transform into each other. There's no strict separation between space and time. It depends on your, it depends on your velocity and your frame of reference. So that's as expected when we move from the Schroen equation, which separates out space in time, you know, it's got a partial derivative with respect

to time and a second partial derivative of respect to space on the other side of the equation, which gives rise to the kinetic energy. That won't do as a relativistic description. We're going to need a formulation which merges them together, and that's what we get with the Klein-Gordon equation. We actually have an operation which kind of sums together all of the partial derivatives and treats them on equal footing. So that's good. Now we, so the Klein-Gordon equation is what we want. it's relativistically invariant, so we can transform between different frames of reference appropriately, and it will obey the Lorentzian variance that we need from relativity theory. It turns out that the Klein-Gordon equation actually does correctly describe the dynamics of a

spinless particle. So remember, spin is this intrinsic property that most particles have that describes some kind of intrinsic angular momentum. It doesn't really have a classic, analog, so it's sort of hard to give an intuition for what it is, but it affects how it interacts with other types of angular momentum as well as a range of other types of interactions, which we'll discuss further in this episode. But the point is the Klein-Gordon equation describes a spinless particle. So you may recall in previous episodes I've talked about spin. The electrons have a spin of one-half. So the Klein-Gordon equation does not describe the dynamics of electrons. However, photons have no spin, so photons are spinless. So the Klein-Gordon equation does

describe the evolution of the electromagnetic field. We have to just make a couple of minor modifications because the electromagnetic field is a vector field. It has four components. So typically the way the Klein-Gordon equation is written is just for a scalar field that has only one component, but that's a minor modification. The point is the Klein-Gordon equation was sort of developed as a way to generalize the Schroenger equation to become relativistically invariant to incorporate time and space together and use the proper energy relations that we have in relativity. And it turns out that when we do that, we actually get an equation that describes with some modifications the evolution of, or the electromagnetic

field in free space, so how it evolves over time. The problem is the Klein-Gordon equation is not suitable for describing particles with spin one-half, which includes electrons. Or any other fermions as well, including quarks. But, mostly I'll focus on electrons just because they're a bit simpler to understand. We'll talk about quarks more in a future episode. So we need a different equation if we're going to describe in a relativistically invariant way the motion of electrons. So our quest to find a relativistic replacement to the Schrodinger equation that's applicable to electrons is still ongoing. It turns out that the equation we're looking for is called the Dirac equation. And this was developed

shockingly by Dirac. Now, the way he developed it's sort of interesting. What he did is that he was sort of thinking that the Klein-Gordon equation is on the right track, but the problem is that it is an equation in the second partial derivatives of the wave function. I mentioned that before, and the reason that that comes about is because we started with the squared energy equation, the relativistic equation, which is energy squared equals mass squared times C to the four, like that's MC squared, all squared, plus C-squared, right? So everything is squared. And that is how energy is described relativistically.

But Darac was thinking that, you know, that's not what the Schrodinger equation looks like. The Schrodinger equation has first order derivatives with time particularly. And he was thinking that maybe the way to describe thermion, like electrons, is going to need to look a bit more like that. So kind of what he did is found a way to sort of take the square root of the Klein-Gordon equation. I've got to put that in quotes because you can't exactly take the square root of the whole equation. But what he did is he found a way to sort of break it up into two and then cancel one part out so that he was left with sort of the square root. But in order to do that, though, he needed to introduce this new mathematical device, which is a matrix. So it's a four by four matrix.

It's called the gamma matrices. There's actually several. There's actually four of them. And that was necessary in order to make the algebra work in order to sort of do this square root operation. you can't do that with ordinary numbers. He needed to introduce this matrix. So he turned the Klein-Gordon equation into a matrix equation. And so what you end up with is this equation that instead of, instead of in being second-order derivatives, it's now first-order derivatives. So the Dirac equation essentially says that the sum of the first-order derivatives of the wave function is

equal to the mass times the wave function. But there's this extra component in the gamma matrices. So it's not just the derivatives, like the partial derivatives with respect to time and all of the spatial dimensions, it's these partial derivatives multiplied by the gamma matrices. Now, why do we need to introduce this sort of weird four-by-four matrices into the picture? Well, essentially, it turns out that these matrices are necessary for describing spin and how the spin of an electron changes in different frames of reference and also as the velocity of the system changes. So this isn't necessary in the Klein-Gordon equation.

We don't have the gamma matrices there because that describes the evolution of spinless particles, so they don't have spin. It's also not necessary in the Schrodinger equation because the Schrodinger equation ignores spin. There are ways of incorporating spin into the Schrodinger equation, but it's sort of done separately. And critically, when you're not dealing with relativistic effects, spin is easier because, again, you can separate out time and space, and spin doesn't have as complicated interrelated, actions as it does when we go into relativity land, right? So these gamma matrices are four by four matrices. You can write them out in explicitly. It gets technical because you can't write the matrices

out in an explicit form without choosing a basis for it. And I just described it in the Dirac basis, which is one particular way you can write them out. There's other ways to write them. But what it all comes down to is that it doesn't actually matter which basis you choose. The physics all comes out the same. In the end, it's just sort of different ways of doing the calculations. And in fact, often you don't even need to write out the explicit form at all. It's just a way to understand what it is that you're dealing with in terms of the nature of these mathematical beasts. You can write them out explicitly if that helps. The point is, though, that these gamma matrices are going to be very important going forward

because the gamma matrices describe effectively how the spin of an electron transforms in different frames of reference and with changing our velocity. I mean, that comes down to the same thing because different frames of reference differ in terms of the velocity that they ascribe different particles because velocity is all relative, right? You know, speed relative to what different reference frames will ascribe different velocities to things, but the physics all is the same in the end. That's what relativity tells us, right? So we need these gamma matrices in order to make sure that the spin of an electron is correctly described when we make these changes between reference frames. And that's their job. And so we will see them crop up throughout the different calculations that we do. I mean, I'm not always going to describe exactly the mathematics.

But the point is that you need to know about these gamma matrices because they play an important role in the formalism. So I mentioned before that our Dirac equation describes the evolution of, say, an electron over time, and it does so by introducing these gamma matrices, which helps explain how the spin transforms in different reference reps and keeps track of that. It turns out that in order for this to work, the equation is now going to describe the evolution of not just a single field value like we had in the Tronager equation. The wave function, at least in the simple case for a single particle system for the Troniger equation, it just gives the probability of finding the electron in a single particular location, right? You square the wave function, you get a probability of finding the electron in a certain location. So the wave function is a scalar. It just has one value.

It's a complex number. You square it, you get a probability. That's no longer true in the Dirac equation. The wave function is not a scalar anymore. It's actually a special mathematical object called a spinner. S-P-I-N-O-R, spinner. The spinner has four elements. So you can write it as a vector that has like four elements,

but technically it's a spinner, not a vector, because it transforms a bit differently. I won't try to describe the exact sort of mathematical formalism behind that, but it's a known mathematical construct. And the reason it takes this form is effectively because there are two spin polarizations of an electron. There's spin up and they're spin down. Also, and it turns out that this is essential to the formalism, which is interesting, there are two different types of electrons, or two different versions, if you like.

There's electrons and positrons, or the negatively charged electrons and polytron. positively charged electrons, matter and anti-matter versions. You've probably heard of that before, that there are positrons that are positively charged versions of electrons, but have all the same, all the other properties the same. Now, what's interesting is that at the time Dirac formulated his equation, I can't remember if they had just, if positrons had just been discovered or if it was just after that, but it was around the same time. And Dirac's equation essentially predicts that positrons must exist because the spinors have four elements to them, and two of describe the spin of the electron, like essentially whether it's a spin up or a spin down.

One element will be you spin up and the other is spin down. And then the other two elements describe the spin status of your anti-electron, your positron, the antimatter version. And you can't actually get the direct equation to work if you ignore the antimatter. Effectively, this is because matter and antimatter are affected by relativistic transformations. And so we need to fit them all into the same formulism in order to make the equation relativistically invariant. So loosely, you can think that this spinner that we have here, this mathematical object that describes the spin state of an electron, has four elements. And those four elements essentially say how much of the spin state of that electron is in the spin upstate, how much is it in the spin down state, and then how much is in the antimatter spin upstate and the antimatter spin down state? That's not quite right, but it's sort of close enough for our purposes here.

The spin of an electron doesn't change as you change different frames of reference, but the way you describe it does. And so the spinners keep track of that, how the different components change in different frames of reference, because different frames of reference describe the same underlying physics in different ways. So in our formalism, we need to keep track of that, and that's what the spinners do. Now, you may recall from the previous episode, the introduction to quantum field theory, that's the prerequisite. I talked about how in quantum field theory we change the interpretation of the, effectively what is the wave function in non-relativistic quantum mechanics. In that context, we think of the wave function as describing a particular particle or group

of particles, and we square the wave function, we get a probability that the particles are in a particular configuration. In quantum field theory, we don't really work with a wave function anymore. We work with these field states, and a field is actually an operator. So we say that the fields are now operator valued. And what the field does is it acts on some sort of quantum. quantum state, which is more the analog of the wave function, that the quantum state is an abstract description of the system, basically based on how many particles it has. So the vacuum quantum state, which is often just written as like a zero in a bra rocket notation. I talked about that in the previous episode as well, the advanced quantum mechanics ones. In quantum field theory, an operator value

field will act on that vacuum state and either create or destroy a particle. I mean, in the case of the vacuum state, there is no particles to destroy, so the destruction operator won't do anything. The creation operator, however, will create one particle of a specific momentum in that state. And so a field in quantum field theory is actually constructed of a bunch of these operators, each of which will create a particle of a particular momentum when acting on a given quantum state. And you can think of this in terms of the idea that there's like these springs that are oscillating at each point in space, and the operator acts on a particular spring. exciting it to a particular value and number of degrees of excitation that it has

corresponds to the number of particles in that energy state. Again, I talked all about this idea in the episode 85, so go back to that one if you're sort of not really familiar with this, but this idea of a quantum field as an operator that acts on a given state to create a certain number of particles of a given momentum, we're going to be appealing to that again. We're going to still use that idea. The only difference is last time I didn't talk about spin. Here we're going to add spin into the mix because as we've seen that that's essential in order to describe relativistically the behavior of an electron. So we need to add in spin, and that comes in the form of our spinners. So we still have these creation and destruction operators. They're separate, right? They're not the

spinners. They're a different thing. So now we have a field operator, which is the solution to the Dirac equation. The Durac equation explains how the field operator evolves over time and over space. The field operator includes the creation and destruction operators, each, you know, one for each momentum state. And it also includes the spinners, which specify for each of the momentum states, what spin it has. And so overall, what the field operator does is it allows us to describe different quantum states in terms of a momentum basis and also incorporate spin into the description in an appropriate way. Now these quantum fields are again different from the wave function of the Schroenger equation because the way function is not directly observable,

but pretty close to it because the square of the wave function gives a probability of observing the particle in a particular configuration. However, in quantum field theory, the field operator that we now have doesn't actually describe a quantum state at all. It's an operator, which really just allows us to express a quantum state in a momentum basis. What is actually observable is not the field state, or even the square of the field state. Rather, what we can usually observe in it's relevant to quantum field theory are probabilities of particles interacting in a particular context, like scattering off each other in a particle accelerator, or the decay amplitudes, like the rate of decay of a massive particle. And so the way the formalism works is that we can describe the initial

and the final states of whatever it is, whatever process we're interested in. Let's suppose it's a scattering state. So we collide some electrons together and then we sort of see what happens. There'll be some initial particles that collide or not even necessarily collide, but they could just interact with each other. They could come near each other and then repel each other if they're two electrons, for example. Or there could be some interaction which produces, which leads to one particle decaying into other particles. Whatever the process is, we'll have some initial state, which describes the particles that existed initially, and there's some final state which describes the particles as they exist after the interaction or the decay process has occurred.

There is a special operator in quantum field theory called the S matrix. The S matrix is an evolution operator. It describes the probability amplitude of transitioning from this initial state to the final state. There's some extra caveats there that I'll get into in a second, But the point is that this S matrix will, at the end of the day, be made up of field operators, as well as some various physical constants and other numbers, but field operators form the basis of it. So if we consider a very simple example, suppose we just have a single electron with a particular momentum, and it's just cruising along, it's just propagating through space. That's our initial state.

So we can describe that KET formalism, but we talked about in advanced. quantum mechanics, that's just the vertical line and then you describe the state and then you have the angular bracket. So that's just a way of describing some sort of quantum state. So here you could just describe that by an electron with a certain momentum. And it's just propagating through space, right? It's not interacting with anything. So in order to describe that mathematically, what we can do is the, we don't actually need the full S matrix here. We just use a single field state. So the field operator applied to this initial state and then on the, the, the, the, the final state here will just be the vacuum state, effectively because what we're describing

as an initial state of a propagating electron and the final state is the vacuum. Like, it's not interacting with anything, it's just propagating through the vacuum, and you can imagine that it's eventually annihilated when it's measured. At some point, it will be annihilated. So, what we're doing is we're sort of sandwiching the field operator in between this initial singular momentum state and this final vacuum state, which doesn't have any electron in it. And it turns out what that actually is when you work out the mathematics is you apply the field operator and you sandwich it between these two state descriptions. That actually gives you a plain wave. That's a, that's a solution to the Dirac equation which describes an electron propagating through space without interacting

with anything. So a plain wave is just a simple way to describe a wave that propagates through space and gradually spreads out over time. So the Dirac equation describes the dynamics the dynamic of a free non-interacting fermions, like an electron, as it propagates through space. And we can also describe that using the matrix element formalism that I mentioned before. We have the vacuum state and we have the initial state of the propagating electron, and we sandwich either side of the field operator. And so the field operator then gives you the plane wave solution. The field operator is really just a way of, again, describing mathematically using a momentum basis.

a given quantum state. And so these field operators are solutions to the Dirac equation. So that's all good. So at this point, what we have is a method for describing the evolution of, say, electrons as they propagate through space, just like the Schroedinger equation did, except now we've turned it into the Dirac equation. We've made it relatively invariate. We've incorporated spin in doing so. So that's all good. The main thing that we still need to do here is we need to incorporate interaction effects, because so far we've just been talking about a free electron that's not interacting with anything. It's just propagating through space. It's described by a plane wave solution. That's not really very interesting. We need to be able to describe how it

interacts with other particles, either electrons or other photons as well. So quantum electrodynamics, which is the theory that we're looking at here, describes how fermions, like electrons, interact with photons, so light and matter. So how do we incorporate interaction effects? Well, non-relativistic quantum mechanics, we modify the Schrodinger equation effectively. I mean, it's not really modifying the equation, but what we do is we add a potential energy term, which describes, well, it can describe lots of things, but generally an interaction effect between one particle and another, for example, an electron, which is affected by the potential energy of another electron, or interacting with a photon field or something like that.

As we did that in the Schrodinger equation, we have a variety of ways of then solving the resulting equation so that we can have a way function that describes an interacting particle. In the episode I did on quantum, on computational chemistry a while back, I talked about how complicated this is to solve the Schrodenger equation when there are even more than one electron surrounding a central like atomic nucleus, like in an atom. So we can do that in the Schroenger equation, but it become quite complicated very quickly. What about in the Dirac equation? Well, the Durac equation is more complicated again than the Schrodenger equation, and the

trouble is that we can't modify it in quite the same way just to include a potential. Essentially, there's no solution to that. But there are other ways so that we can try to describe interactions. So the formalism is rather different here. This is where we diverge even further from non-relativistic quantum mechanics. So as I said before, usually in quantum field theory, we can't actually describe, or at least in general, we don't actually have the ability to describe bound states, like an electron orbiting an atom, orbiting an atom, orbiting as part of an atom. Instead, normally what we describe is transition processes, so like scattering processes or decay. where we have an initial state, there's a transition, and it converts into a final state.

We can describe the probability amplitude, you square the probability amplitude to get the probability. So we can describe the probabilities of transitioning from an initial state to a final state through various processes. And that's what the matrix elements that I mentioned before, that's what they do. But we have to be able to describe how you get from the initial state to the final state. And the Dirac equation is not sufficient for that because that just describes freely propagating electrical. We want to describe interacting electrons. Now there is a formulism for doing this, so let me explain a bit how it works. This is called the S matrix.

The S matrix allows us to describe the probability of transitioning from that initial state to that final state in the presence of interactions. And we do that by using a function called the Hamiltonian. I've talked about the Hamiltonian before. The Hamiltonian is the total energy function for a particular system or a particular particle. In this case it will actually be multiple particles because we're talking about particles that interact with each other. I mentioned at the very start of this episode that energy is directly related to change with respect to time and momentum changes with respect to space. And so we're applying that principle here as well. What we want to do is calculate how a system

changes over time, how it evolves from the initial state to the final state. So this S matrix that I just mentioned, this is a time evolution operator. It tells us how a system changes from an initial state to a final state, or it computes that probability. by taking the exponential of the Hamiltonian. So literally the exponential function of that Hamiltonian function, which describes the total energy of the system. By the way, if you're wondering where the exponential function comes in, that effectively arises because of time evolution.

The exponential function often occurs when we have continuous time evolution of something, or continuous evolution of one variable in terms of another. It's the solution to a simple differential equation, right? So that's where the exponential comes from. It's basically saying that the function that describes the energy of a system also describes how that system evolves over time, but continuously. So hence the exponential. The trouble, of course, with the exponential is that it makes it very hard to solve for. So that's why we need to use the series approximation, the Dyson series expansion.

Now, this is a very complicated, and this is why we can't solve these equations exactly because the exponential of a very complicated function is a very, very complicated function. And so we're going to need to use approximation methods. And the particular method that's typically used in quantum field theory is called perturbation theory. Again, I did discuss this in a previous episode as well. I think advanced quantum mechanics we discussed this as well as the quantum field theory episode. perturbation theory is when you expand your complicated function as a series of polynomials. So like increasing powers of, or powers of X in the simple case. But here it's going to be essentially powers of our Hamiltonian function.

It's technically an infinite series, so you have infinite many powers, but you truncated at some point, and so you just calculate the first few and use that as an approximation. How good the approximation is, well, that will depend on the theory in question. For quantum electrodynamics, the interaction strength is relatively low, like the strength of the interaction between electrons and photons. And so the perturbation theory gives a very accurate approximation to the full theory, which is, again, you can't compute. But in the case of perturbation theory, you only need the first few orders. a few terms of the infinite expansion to actually give quite accurate results. So this expansion of the S matrix in terms of increasing powers of the Hamiltonian is called

the Dyson series expansion. And so when I say powers of the Hamiltonian, effectively what I mean is we take one Hamiltonian function. That's like the first order, and then the second order will be two multiplied together, and then the third order would be like three multiplied together and so forth. And so this is a way of approximating the exponential function. Okay, that's fine, But where does this Hamiltonian function come from? Well, just to confuse you a little bit more, there's a bit of terminology here. There are two functions which are very similar to each other, the Lagrangian and the Hamiltonian. So so far I've been talking about the Hamiltonian, the Hamiltonian is the total energy function,

which generates evolution in time. The Lagrangian is very similar. As it turns out, in quantum electrodynamics, they're pretty much identical to each other. The Hamiltonia is actually just the negative of the Lagrangian, or the interaction components, I should say, one is the negative of the other. So look, we don't really need to worry about this negative sign very much, and I'm just kind of going to talk about them interchangeably. They're both described the energy of the system, and that's what's important for us here. So the Hamiltonian is technically what is used in the S-matrix and generates the motion over time.

The Lagrangian is something we'll talk in more detail about when we talk about quantum quantum field theories applied to the standard model, because the Lagrangian is what we use to describe the different interactions in the standard model. But for here, it's just a bit of terminology because interactions are usually described using the Lagrangian, not the Hamiltonian, but you can convert between the pretty easily. So we need to worry too much about this. At any rate, we need to know the form of this Hamiltonian, or Lagrangian, in order to be able to compute the terms of the Dyson series expansion in order to compute the transition probabilities, right? That's the whole point of this exercise that we're trying to do here. We want to be able to describe the probability of one state transitioning into another state.

for that we need the interaction Lagrangian. How do we get the form of that? Well, that is something that we'll need to postpone until I talk about the standard model in general. Although, to some extent, finding the right form of the Lagrangian is an exercise in like guessing and then, you know, testing experimentally whether that generates the correct predictions. But there are many principles that can be appealed to to narrow the search base. For our purposes here, I'm just going to tell you that the interaction Lagrangian, that the part of the energy that is related to the interaction between the electrons and photons is given by two Fermion fields multiplied by a gamma matrix, or the gamma matrices, I should say, all four of them, and a photon field. So it's four

things multiplied together. So we've got two Fermion fields and a photon field plus the gamma matrices there. Remember the gamma matrices? There are four by four matrices, which help to describe the transformations of the spinner terms in different frames of reference and ensure that the spin is properly incorporated into our description. So that's why that is here, because of the spin of the electrons. As to why there are two fermion fields and one photon field, the fundamental reason for that is because fermions and photons are quite different types of things. That photons are a type of particle, which is called a gauge boson, and they mediate the forces between particles. a type of energy that mediates forces, whereas the fermion fields, the electrons, they are matter,

which experience the forces that are mediated by the photons. So they're different types of things, and therefore they appear in our equations differently. That's the best I can do for the moment. I'll get into more detail about that in the standard model episode. For now, we'll just take it as a fact that the type of the interaction that appears in our Lagrangian is two Fermion, fields multiplied by two electron fields, multiplied by a photon field, and then the gamma matrices is thrown in as well. So that's the form of the energy function, which is going to appear in our Hamiltonian, which then in turn will appear in the S matrix and the Dyson series expansion, which will then use to calculate these transition probabilities. So let me summarize where we're

at here. I've just described to you the mathematical form that the Lagrangian, and hence the Hamiltonian takes for an interaction between electrons and photons. This whole Lagrangian, the two electron fields, the gamma matrices and the electromagnetic field, this is really one giant operator, because remember all of the fields in quantum field theory are operators. They don't describe a quantum system. They're mathematical functions that act on some description of a system. So remember you've got your bras and your kets, the angular brackets that I mentioned in previous episodes, and I mentioned before just now as well, what the Lagrangian-Hamiltonian functions will do is they'll act on these quantum states and return effectively a description of the energy

of that state. What we want to do here is use this Lagrangian function, or the Hamiltonian, in order to construct the S-matrix, which is really another operator, that acts on an initial state and generates a probability amplitude of transitioning from that initial state to the final state. So the initial state could be two electrons with certain momentum and the final state could be two electrons with different momentum states, for example. Although there could be more particles created, particles destroyed. There's all sorts of things that it could potentially be. We'll go through a specific example later about like the initial and final states. But that's what the S matrix is doing. It's telling us the probability amplitude of going from the initial

to the final, and it does that using our fields. I mentioned before that that's the ultimate purpose of fields that allows us to describe how our quantum states change over time, because the field doesn't describe the state. The field describes how a state changes, really, and how it can be represented in terms of a momentum basis. So, at this point, what we have is our S-matrix equation, which is basically a bunch of different fields multiplied together. Remember I said that a single Hamiltonian or the Grangian consists of four of these two electron fields, one electromagnetic field and the gamma matrices. But then in order to use the dice and series expansion, you have multiple terms in the series. So the first one has one Hamiltonian, then the next has two multiplied together,

and then the next has three, and so forth. Now, I should mention that here we're actually dealing with something called a Hamiltonian density, which means we need to integrate it over the three dimensions of space and one dimension of time. That's fine. We just have extra integrals that then appear in our S-matrix Dyson series expansion. That's okay. But it does introduce a complication because as we're integrating over the different dimensions, particularly the time dimension, we need to ensure that the Hamiltonian states evaluated at different points in time are placed in the correct order, specifically when we have more than one Hamiltonian term in our orders of our series expansion, like if we have two multiplied together, for example, at second order, or three in third order,

we need to make sure that we place the fields evaluated at different times in the correct temporal ordering. Because the whole point of this expression is that we're trying to describe the probability amplitude of an initial state transitioning into a final state. And that takes place through a sort of a series of intermediate, interactions, and we need to make sure those interactions occur in the right time ordering. And so we introduced something called a time ordering operator, which basically just rearranges the Hamiltonian terms in the integral, so that they always appear in the correct time order. So earlier is to the right, and then progressively later as you go towards the left in the order

that is written. And just a note on how that interacts with relativity, because you may recall in relativity, the order of events can actually change in different frames of reference. However, that's only true for space-like separated events. And basically that means it's only true for events that are so far apart from each other in space that light could not, even light, or anything traveling less than the speed of light, wouldn't be able to get from one event to the other. They're very far away effectively. Any events that are closer together than that always preserve the temporal ordering regardless of what frame of reference you're in.

And so in order to account for this, we actually make an additional postulate called microcausality in quantum field theory, which basically says that the Hamiltonian functions evaluated at events that have a space-like separation, always commute with each other, meaning that it doesn't matter what order they're in. So for the times when it does matter what order they're in, namely when the events are not space-like separated, they're like or time-like separated, we can always unambiguously specify what order. order there in because events that are not very far apart from each other in space relative to time always have the same ordering in every reference frame. It's only for those events that don't have an unambiguous ordering, that the ordering changes in different reference frames. In those cases, it doesn't matter what order the Hamiltonian is placed in. So it all works out. Either way, if the order matters, you can specify in an invariant way what order the events occur in, or if you can't specify in an unambiguous way, what order they're occurring, then the order

it doesn't matter. So this extra assumption allows us to always apply this time ordering operator when we're integrating over the different times that the events can occur. Now, the trouble with this time ordering operator, though, is that it does make the mathematics a bit more complicated. So what we have then is the time ordering of these multiple Hamiltonians multiplied together, two Hamiltonians for the second order level expansion three for the third, and then so forth in our Dyson series. And remember, each of these Hamiltonians consist of several different field operators multiplied together. So particularly two Fermion fields, electromagnetic field, plus the gamma matrices. So what we end up with is this time-ordering operator applied to this big long list

or big-long multiplication of all of these field operators. And that's very mathematically difficult to manage. So how do we then make progress here in calculating the transition amplitudes, the probability the amplitudes from the initial to the final state. Well, thankfully, there's a mathematical theorem that helps us out, and it's called Wix theorem. I mentioned this in the previous episode on quantum field theory, but I didn't really explain what it is. Wix theorem states that the time ordering of any number of field operators multiplied

together is equal to the normal ordering of all possible contractions of two of those fields together. So we need to break that down as to what? that means. First of all, the normal ordering is a way of ordering the operators, the creation and annihilation operators that we have. Remember, those are part of our fields. Each of our field operators, in turn, is built out of annihilation and creation operators, which create and annihilate particles at particular memento. In order for that to make sense, we need to make sure that we put those operators in the right order. So specifically, we have all the creation operators

put to the left and all the annihilation operators on the right. Basically, that means that then we do all the annihilations first, and then we do all the creations to make sure that we end up with essentially going from the initial state to the final state, which is what we want. Specifically, the way that we think about it is that you start with whatever you have in your initial state, you annihilate all those particles, and then you propagate them, and then you create them all back again at the very end. So that's why we want to.

That's actually how this description works. You sort of imagine starting with some initial state, you destroy everything there and then propagate the fields and then create everything back at the end. That's a bit of an oversimplification, but that's kind of how the formalism actually works. And so the normal ordering ensures that the order of the creation and annihilation operators is consistent with that description. Now, in terms of a contraction, a contraction is a special operation which just takes two fields, so there could be many, many fields multiplied together, but it just takes two of these

field operators and computes the time-ordered vacuum expectation value of those two fields. The vacuum expectation value was just when we sandwich the two fields in between a vacuum state. Remember, you've got your bras and your ket states, so the vacuum state is just the state with no particles in it. And so we put one of those on either side, a zero in the angular brackets on the right and one on the left as well. and then in between we put the time-ordered product of two different field states. This special construction here, a two-field vacuum expectation value, is what we get when we do a contraction,

and it has a special name. They're called propagators. So propagators play a very important role in quantum field theory. We've kind of been talking about a generalized form of propagators all along, as we've been talking about the computing the matrix elements because they're very similar ideas.

So what a propagator does is it gives the probability amplitude for a particle to be created at one location, propagate to another location and then be destroyed at that location. And so that's why there are two field operators in the propagator. One of them essentially does the creating at one location and the other

does the destroying at the other location. So that's what a contraction is. A contraction turns two fields. into a propagator into this vacuum expectation value that gives a probability amplitude. So coming back to Wix theorem, what Wix theorem does is it takes this time ordering of many, many fields all multiplied together, it turns it into the sum of all of the possible ways of creating a single propagator from all of these fields, plus all of the ways of creating two propagators from these fields, and all the ways of creating three propagators from all these

fields, and so on until we've exhausted all of the fields. Now, why would we want to do so? I mean, how is it advantageous to turn one product into a whole bunch of products? Well, the answer is because time ordering is very complicated to do when we have lots of fields, but it's much easier to do if you only have two fields. And particularly, we already have a formalism for this. It's the propagators, right? We can describe quite clearly what the propagator is. So it's very handy to turn the time ordering of this whole bunch of fields into the normal

ordering of two fields and then multiply by the other fields. And it turns out that we can actually readily interpret all of the elements in our WIC expansion in a very simple way. And so this will turn out to be a very handy theorem indeed. Don't worry if it's still a little unclear. All the pieces are coming together. So at this point, what we have is a way of describing the transition from an initial state to the final state.

That's our S matrix. We can compute the S matrix by using the Hamiltonian. so that's the energy of our system and taking the exponential of that Hamiltonian. But an exponential of a Hamiltonian is mathematically intractable, so we use the Dyson series expansion, which expands it out into one Hamiltonian, and then two Hamiltonians multiply together, three Hamiltonians, and then so forth. So there's an infinite series expansion there.

Usually we just focus on the first few terms in the series. But we do need to add a time ordering when we start multiplying these multiple Hamiltonians together because we need to make sure that the different interactions happen in the correct. ordering. The time evolution operator causes a bit of mathematical trouble, and so we use Wix theorem to simplify it, Wix theorem turning a time ordering of a bunch of field operators multiplied together into a series of contractions, where we have, well, no contractions, and then two fields contracted together, all the possible ways of doing that with two fields, and then two different contractions, so all possible ways of doing two contractions with two fields,

and then three contractions and then so forth into we've exhausted all of the fields that are appearing inside the time order. If it combined together the Dyson series expansion plus the WIC expansion, we actually turn a very, very complicated expression into relatively simple expressions. And we can put all these terms together and actually finally do the calculation to get our matrix element, the S matrix that gives the probability amplitude from transitioning from the initial to the final states. So the next task is to actually calculate the terms in the WIC expansion and use that to determine our transition probability amplitude. But we're going to put a pause in that and resume that discussion in the next part of the episode.

Originally this was going to be a single episode, but we ended up covering quite a lot of territory, and so I'll split it here. So in the next episode, we're going to be talking about Feynman diagrams and how we use those to calculate the transition matrix probabilities, and how we use those to predict the cross sections for the scattering amplitudes, which are the things that we can experimentally measure, and we'll also talk about renormalization. So stay tuned for that when we finish out this interesting discussion in quantum field theory. If you enjoy this episode, you can support the podcast by going to your aggregator of choice, such as Spotify, leaving a favorable review.

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