The Science of Everything Podcast - Episode 159: Quantum Electrodynamics Part 2

Episode Date: April 1, 2026

Continuing from quantum electrodynamics part 1, here we explore the mathematical machinery used to compute interactions between particles, including propagators, Feynman diagrams, cross sections. We then walk through a simple example calculation to illustrate how these tools are applied. I conclude with an introduction to the problem of divergent loop integrals and how these can be resolved using renormalisation. Recommended pre-listening is Episode 158: Quantum Electrodynamics Part 1.  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

you're listening to The Science of Everything podcast, episode 159, Quantum ElectroDynamics Part 2. I'm your host, James Fodor. In this episode, we are continuing quantum electrodynamics, as the name indicates, and we're picking up straight from where we left off last time, so please do listen to that episode before we listen to this one, otherwise it's not going to make very much sense. I will very briefly recap what we discussed then.

I talked about how quantum electrodynamics is the first of the quantum field theories, and it's an attempt to generalize the Schroeniger equation to applying to cases where special relativity is important, so very high velocities. And to do that, we need to make a variety of, essentially, modifications to the Schroenger equation to produce first we talked about the Klein-Gordon equation, and then I talked about the Derrach equation. and the gamma matrices and the complicated algebra that that introduces as we need to deal with electrons spin. I then talked about the S matrix and perturbation theory,

which is essentially a way that we are able to describe the interactions between particles in quantum field theory, particularly like electrons and photons. The Dirac equation describes the free propagation of electrons and the Klein-Gordon equation or a modification, or a modification of the Klein-Gordon equation describes the free propagation of photons, but when we want to describe the interaction of both of those, we need to use the S-matrix and perturbation theory. And there's a long series of complicated approximations that needs to be made there. And we talked about Wix theorem and propagators.

We finished up just leading up to introducing the approximations that are needed to actually calculate any of this stuff. And so I'll begin then by introducing those and talking about fine. diagrams. You've very likely heard of these if you know anything about quantum field theory or particle physics because they're widely sort of displayed and used as a visualization of the interactions that occur. So a Feynman diagram is it's literally a picture, a diagrammatic representation of elementary processes that contribute to matrix elements in the S matrix expansion. So specifically each term in the WIC expansion, so each different contraction element that appears there,

corresponds to one Feynman diagram. So remember I said, Wicks theorem says that the time ordering of a bunch of fields is equal to the sum of the normal orderings of those fields with every possible combination of contractions. So if I have four fields, just to make this concrete, if I have four fields in my field operators, in my time ordering, inside my time ordering, I can contract the first and the second field, or the first and the third field, or the first and the third field, or the first and the fourth field. Or I can contract the second and the third field and the fourth field, or the third and the fourth field. So those are all possible two-field contraction. But then there's also double contractions, right? So I could contract the first and the second and the third and the third and the fourth,

or the first and the third and the fourth, and so forth. So that's what means by all possible contractions. Now, the benefit of Feynman diagrams is that each one of these terms in the WICS expansion corresponds to exactly one Feynman diagram. So we can draw out these diagrams and they help us to calculate the terms in the Wix expansion, which in turn help us to calculate all of the terms in the Dyson series expansion. Don't get confused. We've got two levels of expansion. There's the Dyson series series and then Wix theorem. Usually we'll calculate the Dyson series expansion up to a certain order, and then so within that we'll write out Wix theorem maybe once or twice for each order in the series, and then we've got a whole bunch of series within Wix theorem, and then we'll draw a fine,

diagram for each. Or at least we could draw a Feynman diagram for each, as it turns out, you don't always need to. But that's the idea. So what does a Feynman diagram actually look like? Well, basically they're a series of lines and squiggles, but it might be helpful to look some up if you've never seen one before. They can become quite complicated, but the simplest ones are quite straightforward. So what we're going to do is I'm going to talk about a specific example and work through how we can use that to compute a simple calculation for a matrix element. That's what we're trying to get. Remember, all we're trying to do is get the probability amplitude of transitioning from an initial state to a final state. That's all this is about, and we're churning through the maths to help us get there. The Feynman diagrams can help with this. So there's really four main elements to a Feynman diagram. The first are outgoing and incoming particles, so those are your initial and final states. The second element are your propagators, so they essentially transition or move your particles from an initial state to a final state.

We've already met those. Then there's the vertices that connect interacting particles together at a site where interaction occurs. And finally, there are internal loops. The internal loops are the trickiest ones. I'll come back to those in a moment because the first example that we're doing here doesn't have any internal loops, so we don't need to worry about that one. So let's consider a very simple process, which is called electron-mulon scattering. Our initial state here will consist of an electron and a positron, each of which will have a particular

on momentum. The final state will consist of a muon and an anti-muon. By the way, if you don't know what a muon is, it's basically like an electron but heavier. Other than that, they have the same properties. So electron muon scattering happens when an electron and a positron annihilate with each other, so they come close enough together to cancel each other out, producing a photon, which then propagates for a while and then is destroyed or annihilated to produce two muons, specifically a muon and an antimium. So we have an initial annihilation process, propagation, and then a creation process. So that's the process that we're trying to explain, and we want to know what the probability, we want to derive a formula for the probability of this transition happening. So obviously we'll

have an initial momentum for each of our initial states. The initial state consists of the positron plus the electron, so each of those has its own momentum. So our initial state is fairly simple. we'll just have our positron and its momentum, our electron and its momentum, and then each of those will also have a spin state associated with that. Remember our spinners? Well, those have come back there here as well. We need those to describe the spins of our initial electrons and then the positron. We then have our final state. So the final state is just the muon and its momentum and then the anti-mewon and its momentum, and likewise both of those have spin states as well. And so for each of those initial particles and also final particles, you essentially just draw a line.

with an arrow on it indicating the direction of the momentum. So we have two lines, the incoming electron and then the two outgoing lines, the outgoing muon and the antimium. So those are the incoming and outgoing states. We then need to add in a propagator. So the propagator is what connects the incoming to the outgoing states. We've got two lines that join each other. Those are the incoming electron and the positron, and then two lines that diverge from each other,

the outgoing muon and the anti-mulon, a line which normally we draw as a squiggly line, just to indicate that it's a photon. The squiggly line that connects them is our propagator or our photon propagator in this case. So this is the photon field that mediates the interaction between the electrons and the muons. In this case, it sort of transfers the energy from the annihilated electron and positron to then the newly created muon and antimian. because the propagator, the photon propagator, connects the incoming to the outgoing particles, there is two vertices in this diagram.

Each vertex has three lines that join together, so there's the two incoming lines when they connect to the photon propagator, and then there's two outgoing lines and where they diverge from the photon propagator. So that's all there is to this diagram. It's two incoming lines that join together at the photon propagator, then the photon propagated the squid wood line and then two a vertex that then the two outgoing nuanced diverge from.

And that's all there is to it, but this is a very useful way to describe the interaction we're interested in as well as how to compute the probability amplitude for this interaction. So the Feynman rules that go along with these diagrams tell us how to construct the

corresponding term in Wix theorem that will help us to calculate the transition amplitude. So each incoming fermion gets, as well as the antifermione, so here the electron and positrons, they get an appropriate spinner to describe its spin. Then at the vertex, where those lines meet and join the propagator, we get an interaction term, which specifies the strength of that interaction. So here that's actually just going to be the charge of the electron, so that's represented with an E. We also get a gamma matrix or a set of gamma matrices that basically ensures that spin is correctly incorporated into the rest of the calculation. And we have something called a delta

function. Don't worry about exactly what that is. All it does is that it helps to ensure, or it ensures that we have conservation of momentum. So the total momentum going into each vertex must be equal to the momentum coming out of that vertex. That's conservation of momentum. The delta functions help ensure that we have that, so you just throw one into each vertex. So the vertices, we have the interaction magnitude of the interaction, which is called the coupling strength, that's the charge of the electron E. You've got the gamma matrices that help preserve spin appropriately, and then you've got the delta function, which ensures that you have your conservation and momentum. We have likewise one of those for the outgoing vertex as well, for the muons, and their momentum

is conserved, and their spins are incorporated. And so then, Finally, we have a term for the propagator, and the propagator just has a particular mathematical expression. It's an integral with a fraction in it that is dependent on momentum. So we put all those elements together, and we have a term for, we have a mathematical description for a term in the WIC expansion of the matrix element. Now in general, WIC's theorem gives us lots and lots of these terms that we have to add together. But as I said, the benefit of the Feynman diagram process, or the formalism, is that each diagram corresponds to exactly one term in the WIC expansion.

And we can tell what terms they are based on which fields are contracted. So let's think about this diagram, this Feynman diagram that I've just been describing. We have two incoming electrons, electron and anti-electron, which then annihilate to produce a photon which propagates, which then annihilates and then produces two muons, milon and anti-mune. So in terms of the field content of what that's going to look like, we're going to have two, we're going to have two field operators corresponding to the incoming states, two corresponding to the outgoing states, and then we're going to have two electromagnetic field operators, one for each interaction vertex that the photon interacts between. So there's two interaction vertices that the photon mediates between, and so we need to have

two fields where the photon interacts between. And so the term in the WIC expansion that this will correspond to is simply one in which we have six field operators overall, two of them being the electromagnetic field, and then four of them are the fermion fields, and where the contraction occurs over the two electromagnetic fields, and so that forms the propagator. There are other diagrams in which instead of there being a photon propagator, we have a fermion propagator, like an electron filling that role, in which case we'd contract over two Fermion fields. So see, it all fits together. We have our WIC expansion, and each term in that

WIC expansion corresponds to one diagram, and the diagram will have exactly the right operations in it, or right components in it, so that we know what field components to write down to construct an expression for that corresponding element of the WIC expansion, which in turn will give us one of the components we need to compute the S matrix for the transition amplitude. So at this point, we're almost there. We're almost ready to actually compute this matrix element. There are a few extra mathematical tricks that we need to use. So at this point, what we have is in our matrix element, we have, from the Feynman rules, remember I talked about, we've got all these spinners and gamma matrices that are floating around, plus the photon propagator, which in this case is basically one over a

momentum term. You might wonder what we do with these spinners. I mean, how do you actually evaluate those in terms of, how does that come out to something that's actually measurable? Typically the way to do this is to use a trick called spin sums. Usually we don't actually measure the final and initial spin states. We normally measure an interaction amplitude, which is kind of a sum over all of the final spin states and an average over the initial spin states. So basically we can sum over these spinners. And it turns out that when you do that, what you get is a term that just depends on the momentum plus gamma matrices and the mass. So we can substitute these spin sums into our matrix element and do a bunch of algebra. We get

some traces that we evaluate if you know what those are. We use trace identities on the gamma matrices. Don't worry about that if it's a bit unfamiliar to you. But what it all comes out to in the wash, when we chug through this algebra in the matrix element for this Feynman diagram is this is all from my electron muon diagram that I was talking about before. What we get at the end of the day is a term that just has a bunch of momentum value, so the momentum of the initial electrons and the final muons, plus the electric charge of the electron and some mass terms for the masses of the particles. And that's pretty much it. So all of the other complicated field elements that we had before have all gone away. Now, some of them have been moved around into other mathematical,

other components of the maths, and some of them have cancelled each other out, and it's a bit hard to try to explain that here. But what it all comes down to is that this very complicated expression that we started with, remember, which was the exponential of our Hamiltonian, to describe the time evolution, while we expanded that out using the Dyson series expansion, and we took just the second order in that, which consists of two Hamiltonian interaction amortons, multiplied together. We then used Wix theorem to expand that out, and it turned out that the process that the process that we are interested in, this very specific process of two electrons going to two muons with a photon propagator, we only actually need one of the terms in the

in the WIC expansion to compute this. The other terms will describe different processes. So depending on what initial and final states we are interested in, we may need different terms in the WIC expansion. But here we only needed one of these terms, which corresponded to Feynman diagram that describes this process. Many more complicated interactions will need multiple Feynman diagrams, in fact it could require dozens, even hundreds in complicated cases if you're going to higher orders of the Dyson series and you're looking at more complex processes. But here we only needed one actually. And it all reduced down with some algebra to an expression which just has a bunch of momentum terms. So what do we do with this expression with momentum terms?

how do we combine that with something that's actually measurable? The specific quantity that we actually measure in quantum field theory and particle physics is called the cross-section. A cross-section is equal to the rate of interactions between particles or whatever it is that we're interested in, like a decay process. But usually it's going to be an interaction. So that's measured as a rate, so like number of interactions per second. The cross-section is the interaction rate divided by the incoming intensity of incident

particles. Essentially what you'll have is you'll have, imagine, for example, one setup would be a solid chunk of matter that you stick into particle accelerator and then you fire high energy particles at it. The high energy particles often go straight through the mass, but sometimes they'll interact with some of the particles that are within the mass, and then you'll get a scattering event occurring. Or other times you may actually just fire those two particles directly at each other. It doesn't really matter for our purposes here. What's important is that there'll be some sort of intensity of incident particles that you're firing at something else. And the cross-section is just how many interactions you have relative to the intensity of the

incident particles. So it's obvious why we might want to measure that because that's something that's easy to measure. You just count the number of interactions and then normalize that by how often you've been shooting particles to get the interactions, basically, and that allows you to compare across scanners that have different intensities of incident radiation and so forth. So the cross-section is a measurement of how readily the interaction occurs for a specific process. In another episode, I'll probably talk a bit more detail about how these experiments are set up and how we determine whether an interaction has taken place, for example. That's something I'm not explaining here.

I'm just assuming we have some way to know whether an interaction has occurred. What we're interested in is assessing how often interactions occur for a particular process and then how we can compare that to our theoretical calculation. In order to make a direct comparison between the cross-section, which is what we experimentally measure, and the matrix element, which is the thing that we've been computing, there's a little bit of extra algebra that needs to happen. So we need to ensure that we adjust for essentially the size of the area that we're looking at, like how big an area are we considering for where the interactions are allowed to occur in order to count.

And we also need to ensure that we have conservation of momentum, something that I mentioned before. So we have a term that incorporates that, our delta functions that I mentioned before that do that job for us. And we have to make sure that our field operators that go into our Hamiltonian that are then used to compute the matrix elements. We have to ensure that those are properly normalized so that the probabilities work out properly. But suffice it to say once we chug through that algebra, what we come out with is an expression called the differential cross-section. And the differential cross-section basically tells us for a given angle, of incidents, like when we look at different angles of interaction, because you can fire particles

at different angles relative to each other. So for a given angle, how many interactions do we expect to occur relative to the incident radiation? So that's the differential cross-section. And what we find is that in general, it depends on the relative momentum of the two particles. So suppose this is for a two-particle case. There are more complicated interactions, but we'll just look at the two-particle case. It depends on their relative momentum, and in general, the higher the momentum of the particles, the less likely they are to interact with each other, which kind of makes sense, the faster they're moving, essentially, or the more kinetic energy they have, then the more likely they're just to whiz past each other without

interacting. So that kind of makes sense. So we have these terms like that there, but we also have the matrix element, or the square of the matrix element technically. And that matrix element is the thing that we computed when we were going through the S matrix and the Wix theorem and phymer diagrams and all that. That's what we compute there. It's That's where the physics it really is. It's in the matrix element term. And remember, before we talked about at the end, that simplifies to a bunch of momentum terms, plus mass of the particles and then plus the electron charge, which determines the interaction strength.

So you take that matrix element and square it and then plug it into the expression for the differential cross section for the electron muon scattering process and do some simplifications. And you come to an expression which is widely quoted. So in the ultra-relativistic limit where the particles are moving very, very quickly such that their mass is not very significant compared to their kinetic energy. And you have that the differential cross-section is equal to pi over 16. That's basically just a geometric term, so don't worry too much about that. Multiplyed by alpha squared. Now, alpha is the fine structure constant. That is a measure of the strength of the interaction.

Basically, the E-term goes in there, the charge of the electron, it goes into the alpha. There's other constants in there as well. So alpha squared, multiplied by 1 over E squared, E being the energy of the incident particles, multiplied by 1 plus cos squared fate of theta. So that cos squared of theta is an angle term that specifies the angle of incidence of the particles as they interact with each other, or equivalently the angle at which the particles, the outgoing particles are produced. But basically what this term tells us in a specific quantitative way is that the high

the energy of the incident particles, the less likely they are to interact, which we're kind of knew that intuitively, so it's good that we get that answer. But it gives us a very precise numerical relationship, not just between the energy of the incident states and the cross-section, but also how the angle of those particles affects that. And this is just one example, a differential cross-section for a very simple process. Of course, there are many, many other processes that can be calculated. So what they do in these big particle accelerators, is that they smash different types of elements, they smash different types of particles together and see what, well, see what happens, but specifically they look for different types of processes

and compute how often these processes happen, like relative to the intensity of the incident particles. And then that allows us to measure a differential cross-section, and then we compare that to all of the differential cross-sections that we compute using quantum field theory that we've just been talking about. And we can compare how well the theory does at computing, predicting the experimental results. And for quantum electrodynamics, what we find is that the agreement is spectacular to many, many decimal places, like many significant figures. So quantum electrodynamics is the most stringently verified scientific theory that exists. It's been used to predict with incredible precision, many, many different processes, scattering processes,

as well as decay processes and other corrections as well to calculations that we can make in non-relativistic quantum mechanics, but turn out to have slight inaccuracies due to relativistic effects. We can make those corrections with quantum field theory and show extremely high precision accuracy of the theory. So it's quite remarkable how well the theory works, especially for how complicated it is. Now, before we close out, well, before I summarize and then close out, I did promise that I would come back to talk about these loops, this closed loops. Remember I said that there's sort of four main components to these Feynman diagrams. There's incoming and outgoing particles.

There's the propagators. There's the vertices that connect the incoming and outgoing particles to the propagators. And then there's these internal loops. Now, the electron-meon scattering process doesn't have any loops, so I didn't worry about them. But I said I'd come back to them. And here's now where we're coming back to them. So these loops happen. I mean, they're literally like loops or circles or boxes that appear on these diagrams and the

Feynman diagrams, and they correspond to various forms of sort of essentially self-interaction. So these are instances, for example, you might have a propagating photon, a photon that's propagating along, but instead of just drawing one squiggly line, you could actually draw one squiggly line which then turns into a fermion loop, which is like a circle with an electron and an anti-electron going in opposite directions, and then that turns back into the photon again, which then continues propagating. One way to describe this would be that the photon has spontaneously produced an electron positron pair, which then quickly annihilate each other, and then the photon keeps going.

This is a form of the photon field or the electromagnetic field interacting with the Fermion field. And in fact, there's an infinite number of these self-interactions that occur all of the time. The quantum fields do not just sort of exist either as having a particle, which is propagating or not having a particle. In fact, there's all of these different processes that are happening all of the time with sort of one way to describe it is virtual particles being created and destroyed, like the quantum C of fluctuations. These internal loops describe these processes. And the processes, you can't measure them directly, but they have measurable effect on, for example, the probability amplitudes of a given process occurring. And so we need to take these loops into consideration. The high the order of perturbation theory you go to with your S matrix expansion, then the more of these loops you tend to get.

And so one of the important things about these loops is that, so when there's a single propagator, like the single photon propagator, we had in an electron-mulan scattering example, remember I said that at the vertex that connects the incoming lines to the photon propagator, we have this special term called the delta function, which makes sure that the momentum is conserved going from the incoming electrons to the photon. And that means that we know what the momentum of the photon has to be. It's just the sum of the momentum of the two incident, the electron and the positron. So that's easy, right? We know what the photon's momentum must be. However, if there's a loop, there's actually two, well, two photons or two electrons. Could actually be more than that even, but we'll focus just on two.

There's actually two of them. Therefore, we don't actually know what the momentum is. It's not constrained. We know that overall momentum is, momentum is conserved, but because there's a single photon that splits into two, we don't know what the momentum of each of those is, only what the total momentum is. And so that means that we can't actually use these delta functions to specify exactly what the momentum is. That gets us part of the way there, but we also need to integrate over, basically sum over, all of the possible momentum values that these

intermediate virtual particles could have. Because effectively that they can have any of those values, right? And so we need to add all of those processes in. Loops with small momentum, medium, and very high momentum, they all need to contribute. They all do contribute, and therefore in their mathematics of it, we need to make sure that we reflect that. Well, you might wonder, okay, what's the problem? We do the integral, and we get the contribution of the loop, and then we, you know, we keep computing our matrix element. Well, it turns out that when quantum field theory was first developed and people started to try to compute these integrals for these internal loops, they ran into a problem.

And that is that the integrals did not converge. Or in other words, the value of the integrals was infinite. And that's very common in integrals, which range over an infinite domain. Basically, if you're summing over an infinite number of things, you can get infinity as a value, to put it crudely. Or in other words, the integral does not converge. So this is a problem because these integrals contribute to measurable quantities, and you can't measure an infinite magnitude. So something's clearly gone wrong here. And people were quite worried about

this for some time as to what to do with these infinities. Well, these days it's generally agreed, certainly for quantum electrodynamics, that we understand now why these infinities occurred and how to fix them. And this process is called renormalization. So renormalization is how we deal with these infinities and effectively make them go away. A physicist initially developed this as a sort of an ad hoc procedure where they figured out, well, we can kind of make the answers finite and agree with experiment if we do this, but it wasn't very well motivated. More recently, we've come up with better understandings of how and why this process works, and so it's generally accepted that it's not just a mathematical trick, that it's actually

sort of necessary that you would do this and makes good sense. So let me explain briefly what's going on here. To illustrate the process, let's consider a very simple potential type of interaction, which is just a single electron that's just sort of chilling by itself. if you write the potential, like the potential energy due to the charge of that electron, the denominator is going to be R squared, which is just the distance away that you are. If you're a charged particle, this is just a koolombic potential, right? The closer you are, the higher the potential is because of the negative charge at that point location

of the electron. Now, even in classical electromagnetism, people realize that this was an issue, because if you plug in R equals zero, you get infinity. and that doesn't really make sense, that the potential is infinite at the location of the electron itself. Does that mean that the electron feels its own negative repulsion and infinite amount? It kind of doesn't really make sense. Quantum field theory actually provides us with a way to resolve this, because we have many instances like this that occur in the integrals where we have these infinities occurring.

To illustrate how that works, let's consider modifying the classical Kulomik potential. So instead of 1 over R squared, we're now going to. to write that with an adjustment, which is effectively incorporating this loop integral from the Feynman diagram, this integral over these momentum states. So this integral is infinite, so now our potential is also infinite, but the point is we're going to try to use this to correct the theory. So what people realized is that, or what the physicists are still working on this realized, is that the electric charge, as well as the mass, but let's focus on the electric charge here. The electrocharge that we put into our Lagrangian, which determines the strength

of the interaction, remember, between electrons as well as coupling with the electromagnetic field, the value for that E that we write there, it's actually wrong. It's not the right value. Or, to put it another way, it's not actually the same as the physical constants that we measure in the laboratory when we do experiments and we experimentally measure the magnitude of this interaction. So these terms that appear in the Lagrangean are called bare quantities, because they are relevant only for free particles that are just propagating by themselves, not for interacting particles, and particularly not for particles that have these virtual loop effects, virtual particle loop effects that we're concerned about.

So we need to actually fix the formula so that it's written in terms of the correct measurable or renormalized quantities, not the bare quantities that are sort of a theoretical postulate that we don't actually measure in an interaction context. So the way we do this is that we measure the charge of the electron E at a specific momentum scale. So that basically refers to a specific distance away from the electron, because you may remember that distance and momentum are like a Fourier transform pair, so they're directly related to each other. That's why you have the uncertainty of relationship between position and momentum. So instead of talking about distance away from the electron, we can talk about the momentum scale. And basically, we're talking about, as we get

closer and closer, we're talking about smaller and smaller momentum scales. So we can measure the electric charge at some small, but specific momentum scale. Then what we can do is we can substitute this expression into the equation that we derived for the correction to our electric potential. We derived that based on assuming that there was a correction that was needed and that therefore we incorporated that loop integral from our quantum fifth, quantum, quantum. quantum field theory calculation. And we do a bit of rearranging. And what we come up with is a renormalization condition that on the left-hand side has the bare charge, just the E-value that appears in our Lagrangian. And then on the right-hand side, we have the experimentally measured value, again, at some

momentum scale. So that's the renormalized constant, multiplied by the integral from our quantum field theory calculation. So notice here that both sides of this equation, the left-hand side, which includes the bare electron charge, as well as the right-hand side, which includes our loop integral, they're both sort of theoretically infinite. Our integral is divergent, and the left-hand side is expressed in terms of that integral, so they're both divergent. However, this is not a problem by itself, because the bare electron charge, as well as the loop integral, neither of those are physical quantities. They're not measurable. And so it doesn't really matter if we have this sort of theoretical relationship between two things that are infinite. I mean, that's fine. The problem

is if we have a theory that's telling us that some physically measurable quantity is infinite, that's where the problem lies. And so this is where the physicists are sort of worked out how to make sense of this renormalization. What the infinities are telling us is that we're using the wrong constants in our theory. And so these renormalization conditions tell us how to fix that by substituting out the old infinite coupling constants, like the E-value, the charge of the electron, for finite ones, the renormalized ones. So what we do is we take this equation and substitute it into our, the equation that I mentioned, the equation that gives us the bare electron strength in terms of the renormalized electron charge.

We substitute that into our potential. And now, instead of the old film, which remember was just like the electron charge squared divided by R squared. We now have the renormalized electron charge over, or p squared, because we've moved to momentum space, plus, and then we've got the difference between two different loop integrals. At this point, what we have, we still have two infinities in the equation, but in some sense, two infinities is better than one, because one infinity, well, that's just a problem. But two infinities might cancel each other out. Now, it's not quite as simple as infinity minus infinity is zero. infinity minus infinity is not actually mathematically defined.

The way that we actually do it is we first regularize the theory, which means that we place an upper energy limit on it, or equivalently a lower momentum limit. And normally that would be sort of where we make our renormalization measurement that I talked about before. We physically measure at a particular momentum scale. So we sort of cap it at that scale, and then we renormalize it so that it's expressed

in terms of this experimentally measurable electron charge now. And now we have this difference of these two integrals, but they're finite because we've restricted it so that instead of trying to integrate over an infinite range of momentum, we only integrate over a finite range, and so now it's no longer infinite. And at this finite scale, now the two terms cancel each other out. And so it's all good. The two terms go away that we didn't want, and the original theory is restored. All that we have to do now is take the limit to the maximum, if we basically allow the maximum minimum, energy cut off to go to infinity, we take the limit back, as long as the difference between these two infinite integrals, as long as the difference between them trends to zero, then it's fine.

Mathematically, that's well defined. So to put it a bit more specifically, infinity minus infinity is not well defined, but the limit, as say momentum approaches zero, of one big number minus another big number, that can be equal to zero. And so to get around this problem of trying to subtract infinity is what we do, as I said, is we regularize, take a cutoff, then do the subtraction with finite values, and then take the limit as those values become bigger and bigger at approach infinity. And then the difference between them, if it tends to zero, then the theory is renormalizable and everything's worked. We've got a finite answer back, which has removed this nasty infinity. And the way we've done it is through reinterpreting

what the correct physical constants or that the interaction coupling strengths are that we to use. It's actually not these bare charges, or bare masses as well, that we have in our Lagrangian. Those are actually become infinite in an interaction context. So that's actually wrong. We made a mistake by doing it that way in the first place, but we just didn't realize it. The correct form is actually to substitute those out for a renormalized form and then adjust the integral to account for that, so that then when we take the limit, the infinities sort of cancel each other out and the theory is restored. So this is a clever but sort of elegant process for resolving the problem of these infinite loops, which appears in many higher higher order calculations. And there's actually a lot

more to say about renormalization through, which I won't get into here. We may talk a little bit more about that when we get into the standard model. So this has been quite a packed full episode and a difficult one to explain. So let me just summarize here. Remember, we began by trying to generalize the Schrodinger equation to make it relatively invariant. The Schrodinger equation can be derived by starting with the classical energy momentum relationship and then quantizing that and then you come up with an equation in terms of the partial derivatives of time of the wave function. We can generalize that by instead of using the classical energy and momentum relation, using

the relativistic one. And then when we apply the same process, we get a different equation called the Klein-Gordon equation. But it turns out that was the right equation for photons, but it's not the right equation for electrons because electrons have spin. And so there's a different equation that applies to electrons. It's called the Dirac equation. The derac equation, because it needs to describe spin, has these complicated mathematical elements to it, called these gamma matrices, which essentially encode how spin transforms in different reference frames. And we need that in order to make the theory relatively invariant.

But that allows us to have a relativistic form of, the Schroenger equation which describes electrons now, that's the Dirac equation. And we can expand that into its plane wave series solutions, which are our electron fields, essentially, field operators. And so they have the standard creation and annihilation operators, which create and annihilate particles with the particular momentum, as well as these extra spinner constructs, which describe the spin state as well. But the fields themselves aren't measurable. What we really wanted to do was describe transition probabilities from some initial state, some initial set of particles to some final state. And that's this S matrix, these matrix

elements where the S matrix describes that transition. And the energy state of a system, the Hamiltonian or Lagrangian, kind of the same thing in this context, the Hamiltonian of the system describes how it evolves in time. Specifically, the S matrix is equal to the exponential of the Hamiltonian. But we can't compute the exponential of a Hamiltonian. It's too complicated. We need to find a way to kind of get rid of the exponential. The exponential appears because it's a continuous process, but we need to kind of break it up into discrete bits, so to speak. And that's what the Dyson series expansion is for. It is an infinite series which isolates out different components of the S matrix so that we can actually compute it. This is an approximation called perturbation theory,

because we won't actually compute an infinite number of terms. We just take a finite number and then compute those, usually only a few. So this Dyson series expansion is written in terms of now products of the Hamiltonian. So one Hamiltonian, two Hamiltonians multiply together, three Hamiltonians, and so forth. And there's an additional complication because we actually can't just multiply the Hamiltonians together. We need to make sure that the time of all of the events that we're considering is correctly ordered so that we can meaningfully talk about going from the initial states to the final states. So we need to add this time ordering operator as well. And that makes it quite mathematically complicated. Because the Hamiltonian is comprised of a bunch of field operators

multiplied together. In quantum electrodynamics, it's two Fermion field operators, a set of gamma matrices, and then the electromagnetic field. So there's sort of four, three fields and the gamma matrices all multiplied together. And then in the Dyson series expansion, we'll have multiple of those multiplied together. So, for example, we'll have three fields multiplied together, and then six and then the next one will be nine and so on and so forth. So the time ordering operator then apply to all of these fields multiplied together. It gets quite complicated. Wicks theorem came to the rescue in allowing us to separate out different components to this and specifically allows us to identify, allows us to rewrite the time ordering of a bunch of fields as the normal ordering

of two fields contracted multiply by all of the other fields, but we have to then add all of the possible contractions. So remember, you contract like the first field with the second field, the first with the third, the first with the fourth. So those are the three contractions and then so on for all other possible combinations. And the great thing about WIC's theorem is that each term in the WIC expansion corresponds to a Feynman diagram. So in order to actually do to compute the transition matrix for the S-matrix for the process you're interested in, all you have to do is draw all of the Feynman diagrams that contribute to that process. So those will be all the diagrams that have the right ingoing and outgoing states and conserve momentum and other things like that. There's a few other

rules too. But you just draw all the Feynman diagrams and then you use the Feynman rules to work out the corresponding terms in the WIC series. And then you plug it into the S matrix expansion formula, which has some other terms as well, and do the algebra. And we talked about spin sums and traces and a few other things. And that actually allows you to get a term, which at the end of the day has just like momentum and mass and electron charge and a few other constants in there. all of the spinners and gamma matrices and propagates and complicated things, all in the end sort of cancel out or interact in various ways that allow you to simplify them. And that allows us to compute this experimentally measurable construct called the cross-section

and compare our predictions to the measurements, and as has been done for quantum field theory for decades now, we find incredible levels of accuracy to many, many decimal places. We finished off by talking about renormalization, which is a series of techniques, that have been developed to deal with these pesky infinite integrals that don't converge when we have loops in our Feynman diagrams corresponding to self-interactions. And what we've realized is that these occur because we were actually using the wrong interaction constants, that they're coupling constants, as well as masses. And we shouldn't be using the bare forms of these that come from the Lagrangean. We actually need to adjust it for renormalized versions. And when we do this, we sort of have these

two infinities that in the limit can't flage each other out. and return a finite result. So this is not an issue of mathematically sweeping problems under the rug. It's actually us realizing that we were doing it wrong in the first place and fixing that mistake by rewriting the integrals in a way where they actually converge. So I know that was quite technical. I hope you found it somewhat useful. If it was a bit much, again, well, I mean, you've already got to the end of it now, but the recommendation will be to listen to the prerequisites, which will help you a lot in making sense of this. This will, the theory that we talked about, in this episode will be very useful for the next one of the upcoming episodes where we'll talk

about the standard model. So we'll talk about the other forces and other fields that exist apart from just electromagnetism and the electrons. So that's all I have for today. Thanks very much for listening. If you would like to support the show, there's a few things you could do. You could rate the podcast favorably on Spotify or whatever aggregator you use. You can also go to our YouTube channel, Science of Everything podcast on YouTube and give a like to some of our videos there. I've been putting up more and more of those, and any support that you can provide there is very much appreciated to help push those in the algorithm and get our podcast out to a new audience. You can also become a Patreon supporter. So just Google Science of Everything podcast, Patreon, and you can become a regular donor.

I really appreciate all of the donations that have been made there to help me to keep up the website as well as help to bring the content to YouTube and devote time to it as well for heavily researched episodes like this one, which took a very long time. If you'd like to get in touch with me, you can send me an email. My address is Fods12 at gmail.com. F-O-D-S-1-2 at g-mall.com. Thanks very much for listening, and I'll talk to you next time.