PROFESSOR PENGUIN SINGH NGUYEN Good afternoon. Welcome to April Fool's Day. But we'll be serious about atomic physics. I hope you enjoyed your spring break. At least I did. It's just a little bit change of pace. So let me remind you what we have been discussing.

We have been talking about interactions between neutral atoms. And that can be trivial and very profound. Trivial because you've heard about van der Waals forces,

polarization forces between neutral atoms. And if you describe them phenomenologically, you just have 1 over r to the 6 or whatever power law of an interaction. On the other hand, if you take the position that everything which happens between neutral objects has to be mediated, has to be created by the photon field,

then it becomes a question of the vacuum, the vacuum which surrounds the two neutral atoms. And I've tried in this chapter to show you both sides, to show similar or the same physics from two different

perspectives. We talked mainly about the van der Waals interaction between two neutral atoms. I don't want to, we pretty much finished it, so I just give you a bird's view. I don't really repeat a lot of things. Because today we want to talk about the forces which happen between neutral atoms and metal

plates, and then two metal plates, which is the Casimir force. But just to remind you that we showed that the forces between neutral objects are actually created by quantum fluctuations.

Classical objects would not create fluctuating dipole moments, would therefore not give rise to any forces between neutral atoms. And I showed you that there are two aspects where quantum physics comes in, namely through the fluctuations

or the zero point motion of the atoms as oscillators. And we even had a circuit model for that. But then I also said that, well, assume you have two neutral atoms and you have the vacuum fluctuations of the field.

This vacuum fluctuation is driving coordinated synchronized dipole moment between the two atoms, and then the two atoms can attract each other. So we have seen the dual role which quantum physics plays here in terms of quantum fluctuations of the atomic oscillators and quantum fluctuation

of the electromagnetic field, which is another harmonic oscillator. And at least when I used very simple pictures, it seemed that the 1 over r to the sixth potential, the standard van der Waals potential at short range, comes from the fluctuations of the oscillator,

whereas we, I didn't do the derivation, but the simplest derivation for the long range, Casimir-Polder retarded potential, 1 over r to the seventh, focused on the fluctuations of the electromagnetic field. So these were some preliminaries.

Then I thought I want to show you the power of diagrams by showing to you that there are two different kinds of diagrams, dominant at short range and at long range. And with that, we realized diagrammatically that because of the nature of virtual photons which

are exchanged by the two neutral atoms, there is a change in power law. The power, the inverse power of the distance between the two neutral atoms is one higher at long range. And long range meant that also became clear diagrammatically. Long range means the distance is

longer than the wavelengths of the resonant radiation for the atom.

Any questions about that? So the goal for the first hour today is to go from two atoms to many, many atoms, which then form a metal plate and make the transition

from forces between neutral atoms to forces between two metal plates, which is a famous Casimir force. And I want you to sort of hold the thought about what is really the quantum aspect here. Is it the vacuum? Is it the zero point fluctuations of the electromagnetic field?

Or are the atoms the zero point fluctuations of the atomic oscillators responsible for those forces? So we'll come back to that after we have discussed the Casimir interaction between two neutral plates. So maybe one highlight for you today

is that maybe in an hour you should at least have an expert opinion whether zero point fluctuations

of the electromagnetic field are real or are just a convenient tool to describe some physics. But let's get there. So the way how I want to make the transition from two neutral atoms to two metal plates

is by reminding you that the potential was proportional at long range to 1 over r to the 6, at short range at 1 over r to the 6.

And if you had the model that the long range interaction comes because the vacuum fluctuations drive dipole moments in the two atoms, then we know that the potential is proportional to the polarizability of atom one times the polarizability of atom two.

So let's now extend that to the interaction between an atom and a wall. Well, since I don't know, at least not yet,

how to describe a wall, I take the position that, well, we have an atom close to a wall. But now I extend the wall to a sphere of radius z.

And it's sort of obvious that if an atom interacts with the wall, if the wall were flat, it is, of course, this part of the wall, which is most important. So we should actually get a quantitative or semi-quantitative result out of that. But now I can use a nice result

from electrostatics, which I also discuss in 8.21 in the AM01 course. Namely, when we have a conducting sphere of radius z,

do you know what the polarizability of a sphere is? You must have all solved this problem in classical electrodynamics. You apply an electric field. You calculate with the boundary condition of the sphere some spherical harmonic function. You calculate what the dipole moment is.

And you find the proportionality constant. Yes, the one thing I want is that it's proportional to z cubed. It's the volume of the sphere. And the reason why I discuss z in atomic physics is when we talk about the dipole moment and the polarizability of the hydrogen atom,

we find that the polarizability of the hydrogen atom is the Bohr radius cubed. So it seems that when it comes to electric fields and to polarizability, an atom, an hydrogen atom, behaves exactly like a conducting sphere. And the size of the sphere is now the size of the atom.

But that means now that the long range potential will now be proportional to the polarizability of the atom.

And we lose three powers in the power law because we have three powers of z in the polarizability of the sphere. OK, if that worked so well once, we

can now use that at least as an ridge to discuss what happens when we have two metal balls. Well, we can now say we have two spheres.

And the polarizability of each sphere is proportional to z cubed. Well, when we have two walls, we want to use the potential per area.

So what we had is we had 1 over r to the 7. We lose three powers of r or z because of one sphere, another three powers of the other sphere. So that's 1 over z. But when we normalize by area, it becomes 1 over z squared. And now we find that the potential is 1 over z cubed.

And at least with that, I have, I wouldn't say derived, but motivated for you, the famous Casimir potential, which is 1 over z cubed. And I want to give you an exact derivation of the Casimir potential in the next few minutes.

But before I do that, yes? The upside. I have the same size as the distance between them. Well, there's always a trick. You know what would happen is, let me just go back here.

If I would choose the sphere smaller, I would not have given the atom the whole exposure. I have to make it at least larger than z to make sure that the curvature of the sphere doesn't dominate. And what the atom sees is it sees that there is a wall.

But then you can say, when I make this sphere larger than z, then I delocalize the dipole moment. And the bigger sphere has a big polarizability. But the atom which is so close to the sphere doesn't care about it. So you would say the right choice is just

when you think either picture gets corrupted. But that's so often what it is. So if you think about it, it's the only choice. Both other choices would lead to an unphysical result. And you would immediately realize why. In one case, it's the curvature. In the other case, you're not really sampling the dipole of the sphere

because you want to do a dipole field. And if you have a dipole of size z, the field with the dipole starts to be an approximation if you go a distance z away. Other questions?

Yes? Last class, we had the result that the retarded van der Waals

potential is 1 over r to the 7. But with the physical picture in mind, that the van der Waals potential at long range, which is 1 over r to the 7 comes because the vacuum fluctuation drive two dipoles. And the two dipoles attract each other. Then each dipole is alpha 1 times the vacuum field.

The other one is alpha 2 times the vacuum field. But now for the sphere, alpha is z cubed. So therefore, three powers in the denominator cancel. And I get z to the 4. So you can say the alpha 2 I absorbed. I used the result that alpha 2 is z cubed.

And then the alpha 2 canceled with three powers of the distance in the denominator. Other questions? I have a question for you now. If you have two metal plates, what about short range and long range?

We have swept something under the rock. If we assume a metal plate, what is the resonant? You know, we had the distinction between long range and short range, which came from the resonant excitation of the atom. And when we are closer than the wavelengths,

we are in short range. If we are further than the wavelengths of the resonant transition, we are in long range. Between an atom and a plate, well, we still have the resonance of the atom. But what happens when we have two plates? There is clearly no resonance. So therefore, there is only one power law, 1 over z cubed.

But I want you to sort of think about it for a moment, whether everything is long range or everything is short range. In other words, it's the effective resonant radiation. It's the effective wavelengths of the plate.

Lambda equals 0 or lambda equals infinity. Any opinion?

Go ahead. If you have the DC forward ability, so it has to be the same volume of free space,

the energy difference, then the high frequency components cancel out. OK. We will actually do exactly what

you suggest to compare vacuum energies in the next few minutes. It's not quite obvious. But I want to tell you now, we have made, by assuming the metallic boundary condition, by assuming that we have idealized metal plates which fulfill the boundary condition that the electric field is

0 at the surface, the parallel component of the electric field 0 at the surface, we've made an assumption here. And the assumption is actually that the resonant wavelength goes to 0 and the resonance frequency goes to infinity.

And the power law we obtain is actually the long distance power law, so to speak. And for two metal plates, there is no short distance. I'm not sure if I can give you a simple explanation because the boundary condition is so idealized

that you don't recognize, you don't smell any atomic resonance anymore. But maybe the one thing I would say is that you set a boundary condition, metallic boundary condition, for infinitely high frequency. And sometimes that would also mean that the plasma frequency of the metal is pushed to infinity.

We'll come back to that in the next chapter.

Any questions? Excellent question. I will go for you now through its many classical E and M.

This is why I use pre-written slides. We sum up over all modes with a metallic boundary condition. If you have, in general, a dielectric or semiconductor which is not an ideal metal, has different boundary conditions, the Casimir force will change.

And this is actually a pretty outstanding, difficult problem where people have really chewed on hard. I think Lifshitz has some famous solution on it. What is the Casimir force for an arbitrary dielectric constant of the surface material? But absolutely, yes. If you don't have a metallic boundary condition,

your Casimir force changes. But let's just, I think this is sort of an interesting point and it may trigger immediately discussions. What does it really mean? But let me go now through the derivation where we have metallic boundary condition. And this actually leads to a quantitative derivation

of the Casimir force, quantitative in the way that we're not just getting the power law. We get the pre-factor. We get everything 100% out of this derivation. Colin. So what is the mechanism for the lower density of modes

inside and outside? Even with the arbitrary boundary condition, I have the same density of modes. What is the physical mechanism? To the best of my knowledge, it is the density of modes. But it's very subtle. I suggest, since not everybody is probably

familiar with the way how I will count all the modes in the metal plate, let me just do that. And then we should, and I hope then it's obvious or we should then discuss that boundary conditions been modified. Nancy, you had a question? Yes, but maybe we can discuss it afterwards. So the idea is the following.

If the Casimir force is only due to zero point fluctuations of the electromagnetic field, let me see up front. This is a valid assumption. I get a valid result. But I will tell you in half an hour that you can take another perspective and say,

this is just a holistic description. There are other derivations of the Casimir force which do not use the concept of zero point energy at all. So in other words, I'm presenting you one presentation of the problem, one solution of the problem, but there's a whole different formulation. But here I assume, I mean, I can easily

motivate myself, OK, well, here we have two metal plates. We want to understand if there is potential energy and attraction between the two metal plates. And let's just calculate what is the total energy of the system. And if I have two metal plates which are two mathematical boundary conditions, all I have to calculate is the ground state of the electromagnetic field in between.

And that means I have to find all the modes. And each mode contribute h bar omega over 2. So this is our agenda now. We say the metal plates are not real objects. They are mathematical boundary conditions. Looks like a good assumption. And now based on those boundary conditions,

we calculate the zero point energy of the world, at least of the electromagnetic world. And then we ask, if the plates are a little bit closer, has the total zero point energy changed? And when it has changed, that means there is a force between the atoms, because the potential changes as a function of distance.

So this is the agenda I want to execute now. Any questions about what we want to do? As you know, when you write down modes in E and M and you want to do it exactly, there are many, many indices. But this is a result which could actually come straight out of Jackson.

For your convenience, this is also a wonderful discussion by Serge Jarosz in some Lesouche summer school notes from the 70s or 80s. I've posted those on the web. So if there is a mathematical detail you want to look up, this is posted on the web.

So anyway, we have two metal plates with a distance L. And we have metallic boundary conditions. That means that when we calculate the modes, we get TE modes and TM modes. TE modes means there is no longitudinal electric field.

TM modes mean there is no longitudinal magnetic field. And the frequency of each mode is discrete in the z

direction. In the z direction, we have standing waves, which are labeled with an integer value of M. Whereas parallel to the plate, the modes can have an arbitrary transverse wave factor, k. And when we look at the dispersion relation,

the frequency of those modes is in quadrature. You can say the component k square c square of the transverse propagation plus the component of the standing wave. So for each integer value, we have a TE and a TM mode,

with the exception of M equals 0. For M equals 0, we don't have a TE mode. We only have a TM mode. So that's just fresh out of Jackson. What is very important now, and I want to spend a little bit more time on it also. It's just a purely classical result. What is the density of states?

Well, let's assume we have a given M. So we have a standing wave perpendicular to the capacitor plates. And therefore now, the density of modes only comes, the density of modes only comes from the transverse k.

Let me just get my toolbar back. So it is from this k square. And you know that the number of modes is proportional to the volume in k space.

And the important part is that I can write this as dk square. But k square is d omega square. And this gives me omega d omega. So for each M value, for each discrete mode perpendicular

to the plates, we have now a density of modes which goes as omega d omega. Before I show you the mathematical expression, let me just sort of plot that, what that means. The density of modes was, the number of modes

was omega d omega. That means the density of modes was proportional with omega. That means for M equals 0, we start at omega equals 0, and we get a straight line.

For M equals 1, or M equals 1, shown in green, and M equals 2, shown in purple, the frequency omega, sorry, the transfer, we have omega square is the quadrature sum of k square and M square.

So therefore, if you want to ask, what is the density of modes for M equals 1 or M equals 2, you have to start at the lowest frequency for M equals 1,

at the lowest frequency for M equals 2. But then, as we just derived, the density of modes is proportional to omega. So what we want to do in the end, we want to sum those modes all up. And what we will observe is that here, we have sort of a line with slope 1.

Here, we have to sum two lines with slope 1. Here, we have to sum three lines with slope 1. So progressively, the slope gets higher and higher. But a slope which increases linearly with the coordinate axis, if it were continuously, would just be a parabola.

So in other words, if I would coarse-grain the density of states which comes from the exact mode spectrum, if I would coarse-grain, it would be parabola. And a parabolic density of state, the number of modes is omega squared d omega. This is three-dimensional space.

So in other words, what we observe is what exactly the effect of the boundary condition is. Instead of getting this smooth parabola omega squared, we have sort of broken up the parabola into piecewise linear pieces. And the fact that we have a boundary condition,

that we have plates in free space, and not just free space, is actually the difference between the two. So it is the difference between the two which leads to the fact that the boundary condition modifies the vacuum energy.

Questions? OK. With that, the rest is mathematics. But I want to show you the steps, because it is ingenious mathematics. It makes a few approximations. It creates a few infinities and gets rid of it.

It's actually rather wild, but amazing that in the end we have a simple exact result. So the most important thing is we have a density of states. And the density of states for 1m is proportional to omega.

But now we have to multiply, when we want to know what is the density of state at a given omega, we have to multiply with the integer number of discrete modes, which are available at this omega. So this is because there is only one m equals 0 node,

we have a 1 here. And then for m equals 1 and higher, we have a twofold degeneracy between te and tm. So this is just counting the modes. And this is now a mathematical trick that instead of counting the modes, it's an integer,

we can sum up Heaviside step functions and sum over all m. This is just an exact rewrite, but this is something which helps us in the next step to do an integral. So now so far we haven't made any approximation. We have just reminded ourselves what

is the exact result of classical e and m. But now we want to calculate the zero point energy. So now it's getting interesting because we take the classical result and take it into quantum physics. The first step is very innocent.

We say each mode contributes h bar omega over 2. And then we integrate with the density of states. And the next line is also an exact result, just using the density of modes from the previous page. But of course, now we are asking for trouble. We are integrating with omega squared d omega,

and there is a UV catastrophe. Well, now I can say, I can put on my mathematical head and say, let's add a convergent term, something which exponentially decreases with omega, which is just cutting off the high frequency part.

And I hope that in the end, and this will come out too, that I get a result which is independent of the cutoff. So by choosing this cutoff parameter lambda to 0, if this is 0, I have no cutoff. So maybe I can introduce the cutoff, and then at the end,

let lambda go to 0. And if this gives me a physical result, I've sort of mathematically tricked around. But I also like the physical justification for that. I mean, you know that a metal plate has not metallic boundary conditions at infinitely high frequency. If you go deep to the x-rays and gamma rays,

it becomes transparent. So therefore, we would argue that once we go to very, very high frequencies, the modes do not feel that there is a boundary condition. And therefore, we may be allowed to do some physical cutoff.

OK, so we do this cutoff. So now we have to find the zero point energy. The zero point energy has a sum over m. And I call each integral I sub m. And this integral is the omega squared from the density of states, but now multiplied

with the convergence term, whether it's mathematical or physical. And by introducing this cutoff, I can now exactly solve the integral. The omega squared can be taken out of the integral

by taking the second derivative. And this is just an exponential function which can be integrated. And therefore, we have a mathematical exact result. So we are now able to do the integral because we have avoided the divergence. OK, now we come back and say we are really

interested if the cutoff is pushed to high and higher frequency. And this means lambda equals zero. So therefore, we want to take this expression and expand it around lambda equals zero. Well, you could say that's physically very well motivated.

But if I take this function which is lambda e to the lambda or x times e to the x and expand it, I get divergences one over x squared. So you would say, well, how can you expand it? Well, let's see what happens. Let's just do it.

Let's say we are interested around lambda or x equals zero. So we have some divergences now, but we have to find a way to deal with that. So we get rid of some of the divergences in the following way by saying, well, if you simply calculate the zero point

energy between those two capacitor plates, and then when, let's say, if the plates come closer and there is less zero point energy, we would say the plates attract each other. But what happens is the plates are not moving

and there is no world behind them. The world is sort of as it is. And that means we have to sort of now look more carefully at the zero point energy of our capacitor plates where L is small and the rest of the world. So we are now representing the world

as a big capacitor of size L0. And then our plate is just moving within that, because we are not assuming that when the two capacitor plates attract each other and move towards each other, that the size of the universe is changing. So we want to move a boundary condition within the universe.

So therefore, the correct expression for the zero point energy is not the zero point energy of a capacitor L. It's the zero point energy of a capacitor of size L plus of a capacitor L0 minus L. And so we take the result and simply add the expressions

for L and L0 minus L. And what is now nice is that the first highly divergent term with 1 over lambda to the 4, reminder, we want to let lambda go to 0, it becomes independent of L. So we would say, well, if there is an infinity,

it doesn't change with L. It doesn't provide a potential. It's just sort of one of those infinities which is constant, which is not affecting the physics. Also here, this is independent of L. And then the next term gives us the famous Casimir

potential 1 over L cubed, and it is independent of lambda. So we have a result which was independent of the cutoff we introduced earlier. So therefore, we have derived, with a little bit

of hint waving to navigate around infinities, that the potential between two metal plates is 1 over L cubed. A squared was the area, the size of the capacitor plates. And this is the exact brief factor. Or if you want to figure out what

is the pressure of the vacuum, pressure is, of course, force per unit area. So you take the derivative of the potential, that's a force. That's a force per unit area. And this is now the vacuum pressure. Sort of funny now you can put in a unit. It's 10 to the minus 5 millibar at a distance of 1

micrometer. So anyway, this is, I think, the classic derivation of the Casimir potential by, I would say, exactly summing up the zero point energy

within a certain boundary condition. Good. Maybe to address Colin's question,

what would happen if the metal plates were maybe semiconductors or had a different dielectric constant? The mode spectrum would change. Yeah, I don't know how to calculate. That's a hard problem. A lot of people have really worked very hard on it.

It seems for idealized boundary condition, you can do it easily. But it has been clearly a challenge to mathematical physics. And I'm not even sure if it has been solved in all generality or only in special examples to extend that to arbitrary surfaces. But we can, for instance, assume if you have a conductor which has not

infinite conductance, zero resistivity, then the electric field can penetrate a little bit into the conductor. And if the electric field penetrates into it, it changes the frequency of the mode. So it's clear that the boundary conditions have an effect

here. But let's now have some discussion.

We had focused before on the force between two atoms. And if we use a microscopic model for the metal plates, it must be possible to obtain the Casimir potential by just summing up all the forces between atoms.

Because ultimately, if you have no metal plates, you don't have any force. Whereas in the last few minutes, we focused on a derivation which was simply summing up the zero point energies. And there are two papers which I've posted on the wiki which

argue that both results are equivalent. So I would really recommend to you, he expresses it much better than I can, read the introduction, the conclusion, Bob Jaffe's paper.

He clearly says in the introduction that there are a lot of books and references by famous people who have said the Casimir force is the manifestation of the zero point energy. The fact that people have observed the Casimir force means that h bar omega over 2 is real.

Those authors take the opposite approach. They say this is just heuristic. It's convenient. You can get the correct result. But there is an equivalent derivation which gives exactly the same result, which focuses only on pairwise interactions between atoms.

And it does not make any reference to the zero point energy of the electromagnetic field. So therefore, I would agree with those authors that the conclusion is there is absolutely no observational evidence for the zero point energy,

that the zero point energy is sort of not just convenient counting, but that it is a real energy. And the Casimir force, of course, also it has often been quoted as a counter example that this is a direct observation, should not be regarded as such.

Now, people at MIT probably know Bob Jaffe. He's a high energy physicist. And why is he interested in atomic physics? Why is he interested in the Casimir force? Well, the question of zero point energies

is very, very important in these days for dark energy. I mean, you know that 80% of the energy of the universe is dark energy. Then there's dark matter, and then there is baryonic matter. But this dark energy is really mysterious. And one candidate for dark energy is that dark energy simply comes

from the zero point energy. So therefore, people are now very interested, is the dark energy real? And if it is real, does it have a gravitational effect? Does it change the metric of space? And can it accelerate the expansion of the universe?

So well, of course, you have a problem. If you calculate the zero point energy, you don't introduce a cut off. You have a spectacular UV divergence. You can maybe now truncate the divergence by saying, OK.

Eventually, if the wavelengths become shorter and shorter and shorter, there is a cut off. And if you have no idea at all, you would introduce a cut off at the Planck lengths. Because beyond the Planck lengths, all physical descriptions break down. But if you cut it off at the Planck lengths, you find that the zero point energy of all modes

is larger than the observed dark energy in the universe through acceleration of the cosmic expansion by 124 orders of magnitude. And if you think you want to introduce a cut off, which is maybe the classical electron radius,

the classical electron radius is the radius of a charged sphere for which the electrostatic energy would just be the rest energy of the electron, well, then you're off by 43 orders of magnitude. So some people have called those discrepancies, which are completely unresolved as of now,

the biggest failure ever of theoretical physics. So anyway, that's why there is renewed interest in the Casimir force. And at least in electromagnetics, we have a very, very simple model. And we have experiments which measure the Casimir force. But as I've said, it cannot be regarded as a direct

observation of the zero point energy. OK, let me now come to the question we started discussing earlier, namely, here we have atoms.

Here we have a metalized boundary condition. So maybe let's just talk for one minute about it. What kind of assumptions do we make about atoms when we replace them by a metallic boundary condition?

From the whole derivation of the Casimir force, it's not obvious that there are no atomic properties which have entered the calculation of the Casimir potential. What Bob Jaffe argues in this paper,

and again, it's a wonderful reading. I really recommend just if you want to enjoy some physics, open up the document and read the first and the last page. The Casimir force from an idealized boundary condition is completely independent of atomic properties. So therefore, it must be a limit where, well, we talked

about the resonant wavelengths, but we can also talk about the fine structure constant, because the fine structure constant in a hydrogenic model provides what the resonant transitions in hydrogen are. So you can argue if something is completely independent of atomic properties, somehow

by sweeping things under the rug, we must have set alpha to either 0 or infinite. Because if it is set to any value, it would correspond to some version of the hydrogen atom which has a resonant radiation. So the question is, is alpha set to 0? Is alpha set to infinity when you derive the Casimir force?

And I think this is now easy to discuss, because the fine structure constant alpha is e squared over h bar c. If you set alpha to 0, you do it by setting the charge of the atoms to 0, and then you have no interaction at all. So therefore, if you want to have the Casimir force

in a limit which is independent of atomic properties, the alpha equals 0 limit gives no force. So therefore, the equivalence between this picture summing up over real atoms and obtaining

the Casimir force for the metallic boundary condition, this equivalence happens when you do this derivation and set the fine structure constant to infinity. How big is the fine structure constant?

1 over 137. So it's actually, in some expansions of alpha and alpha squared, it's actually a small parameter. And now I'm telling you, when we have two metal plates and measure the Casimir force, this corresponds to the infinite alpha limit. Well, this is now getting complicated, and I can refer here only to Bob Jaffe's paper.

But he shows that when you go to distances of 0.5 micron, where people have measured the Casimir force, you are already in the alpha equals infinity limit if you are minus 5, if alpha is larger than 10

to the minus 5. So therefore, alpha equals 1 over 137 is, for that purpose, already very, very well in the large alpha limit.

But I just take this result here. This is rather complicated, and Bob Jaffe says that to get the Casimir potential by summing up over atoms is actually a quite challenging mathematical exercise.

Questions? Yes? If the calculations of summing over atoms has been worked through, then what is it? We can just talk about the plates being dielectrics from that atom, like summing over atoms, which is. There are solutions for dielectric plates.

And sometimes when people do experiments, the best surfaces you can get is a silicon surface. Because they wanted to do quantitative measurements, they're discussing corrections due to the dielectric constant. So I know the problem has been worked out

for dielectric surfaces. I assume not by doing the transition from summing up atom by atom, but by rather looking at the boundary condition for electromagnetic fields in the presence of a dielectric. But I don't know more details about that.

Yes, Timo? It may not make sense if you haven't thought. Again, when we considered two plates, if you remember from two atoms, we got the two, one over r to the sixth, one over r to the seventh cases, one of which is the retarded Casimir potential.

But for two plates, it was, again, we concluded that it was more like the short range. No, long range. We are always retarded. OK, but then I'm a bit confused, because then if you say alpha goes to infinity, don't you set the H bar or C to 0? Oh, wait, that makes sense. Alpha infinity is we know that this

is the rest mass of the electron. The Rydberg constant is alpha squared times smaller. And the Rydberg constant, well, a quarter of the Rydberg constant is the Lyman alpha radiation. So anyway, the Rydberg constant is the scale factor

in the transition frequencies of the hydrogen atom. And so therefore, it is consistent that if you make alpha large, we actually get, we push the frequency of the atomic transition to very, very high frequencies. And that would mean the resonant wavelength

goes to 0. And therefore, any finite distance between the capacitor plate is in the long range, is longer than the wavelengths of the transition. I think equivalently, you could also say if the speed of light is really slow, you always will have the Rydberg constant. Yeah, sure. You can do the transition in many ways,

by tuning the speed of light or tuning the charge. But it all has the same implication. Other questions? Nancy? Is there any other observable reason why we should be thinking about gas and wave forces and dark energy experimentally?

As far as I know, this question about dark energy and whether it is related or not to zero point energy of the field is completely open. I think if you had an idea how to do a decisive, if any of you had an idea how to make a decisive experiment measuring

whether zero point energies are real or not, that would have a big impact. As far as I know, it's completely open. Nobody has a really good idea. So some people feel that mathematically, the zero point energy has the right correct physical characteristics.

Zero point energy could explain the extra term in Einstein's field equation, which is often called the cosmological constant term. But the quantity, it's so many orders of magnitude off that people don't even know if that is correct or not.

One could also say, well, maybe zero point energies do not give rise at all to any gravitational potential. And therefore, maybe if energy doesn't give rise to gravitational potential, maybe the energy is not real. And what we find, what is responsible for the expansion of the universe is gravity is sort of,

so Einstein's, just the, what's the constant called again? I just mentioned it. The cosmological constant, but the cosmological constant reflects something else in the universe and is not related at all to zero point energy.

To the best of my knowledge, these are the possibilities. OK, so let's go back to atoms. Our next big chapter 12 is on resonance scattering.

Just that you see the structure of the course,

I wanted to introduce sort of some mysteries and subtleties of electromagnetic interactions to you by using diagrams, by using an exact formulation of QED. And then I gave you one example where we could use those diagrams. And these were van der Waals forces. So then we got a little bit sidetracked with a vacuum,

with van der Waals and Casimir forces. But now we want to go back to the diagrams. And there is one aspect of the diagrams which is sort of fascinating. And this is where we really need a treatment which goes beyond perturbation theory. And this is when we have resonant radiation, when

resonant light interacts with atoms. Because as I want to show you, we have then in any perturbative expression zero in the energy denominator. And we have a divergence. So I want to show you now, partially also for the beauty of the physical theory, how can we deal,

how can we fix those divergences, how can we fix those infinities we encounter in perturbation theory? The answer will be we get rid of these infinities by summing up an infinite number of diagrams. So I want to show you now, it actually sounds so much more harder than it

is, how with rather modest mathematical effort, we can perform an infinite sum over diagrams. And I want to show you what those diagrams are. And the result of that is that the divergences disappear. And we have a valid description what happens when atoms encounter resonant radiation.

So I'm following here almost religiously the atom-photon interaction book. So this introduction is a summary of those pages.

So when we discussed diagrams, we wanted to figure out what is the amplitude for a system to go from an initial to a final state.

And those were the matrix elements of the t matrix, the transition matrix. And if we do it in second order, well,

remember the structure is in second order. We sort of emit a photon, virtual or real. We have another photon. This gives us second order. And in between, the system propagates

with its eigen energy. So this was the structure of the diagrams we had.

And this term here, let's expand it in wave functions of the unperturbed Hamiltonian. And then we obtain this following structure.

So this is just rewriting in a very suggestive way what we had discussed before. And the case I focus on now is the interaction of atoms with resonant light, because this is when we have divergences. If we have state A, state B, we have a laser

and we scatter light. If the frequency is approximately resonant, then those perturbative results have a divergency.

So for resonant light scattering, the typical matrix element we are interested in, we start and end, of course, in the ground state. We scatter a photon. We go up to the excited state and go back to the ground state.

But we start initially with a photon of wave vector k and polarization epsilon. And this may be different for the scattered light. The interaction happens by the interaction operator.

Which, for the purpose of this discussion, can either be the A dot P term or the D dot E term. We've discussed about the special role of the A squared term. We've discussed it earlier. I don't want to include that in the discussion right now.

And then you would say, I just want to make sure that you're not confused. In second order perturbation theory, you always have an energy denominator. But as I also explained to you, the energy denominator

comes from the free propagator. You had a dot propagate and a dot. And what is the time evolution of this free propagation when you integrate it with time? This gave rise to the energy denominator. Just a side remark. So it should be very familiar that we have an energy denominator.

Now this energy denominator, the initial energy is the initial atomic state plus the photon. And then if you want to sum over all states, I can just put in here the operator h0. But you may immediately think about it. The only state which really matters is the state b.

And there will be a, and what matters is only when we go through intermediate state b, then h0 is eb, and then we have a divergence because the denominator is 0. So let's see how we can fix it.

But let me first give you the perturbative result, which you have seen many, many times. It is the product of two matrix elements. We have an intermediate state, the excited state without photon. We start it with a photon, and we end up with a photon.

And here we have our dipole or a dot p term. And our energy denominator diverges when the frequency of the photon

approaches the resonance frequency. Well, how can we fix it? You know that in many cases you are just adding an imaginary part to it, which

reflects that due to the coupling with the vacuum of the electromagnetic field, the excited state is broadened. And it gives its energy an imaginary part. This, of course, is very phenomenological. But at least technically, it avoids the divergence.

Now what I want to show you is in the next 15 minutes, and we continue on Wednesday, is how I can get this result of a very rigorous approach.

I want to use a diagrammatic approach. I want to sum an infinite number of diagrams. And I will tell you what are the approximations to get this. For instance, as you know, when you have an imaginary part, when you have just an imaginary part of the energy, the physics of it is exponential decay.

Well, I hope not today, but on Wednesday you will realize what assumptions lead to exponential decay, and that in principle, non-exponential decay is rather the standard than the exception. But the effects are small. But I sort of want to show you, so in some sense,

I want to give, I mean, we have to go through half an hour of work to derive this result. So we'd say, what's about it? You just get the phenomenological result. Well, we obtain, we have put it on a rigorous basis. And number two is we do understand what assumptions really go into this phenomenological result.

So the way how we will actually obtain this result is that in the second order diagram is that.

You start in state a, you go in state b, and you're back in state a, and you exchange photons. So the way how the physical picture we want to use now

is after the photon is absorbed right here, the system is not propagating with the Hamiltonian H naught. It will propagate with the Hamiltonian H.

If we do that, then we avoid the divergence. We get a result which can be applied to the formalism can be applied to many situations. And in one limiting case, we obtain that. So the physics behind is the following.

The resonance, sorry, the divergence which we encounter in resonant physics is only a divergence because we assume the atom in state b is naked. It's an excited state of an isolated atom. But in reality, the state b interacts by virtual photon

emission, interacts with all the modes of the vacuum. And once we throw that in by saying that between absorbing and emitting a photon, the state b propagates, not with the free Hamiltonian, but with the total Hamiltonian, we avoid the divergence. But this is no longer a perturbative result.

And let me sort of illustrate that to you. If you take the energy denominator, omega minus omega naught, and we add the imaginary part to it,

I can Taylor expand that, assuming the imaginary part is small.

Well, that's sort of a trivial expansion. But now you should know that gamma, the rate of spontaneous emission, the simplest expression is obtained in Fermi's golden rule and is second order

in the interaction Hamiltonian. Spontaneous emission is proportional to the atomic dipole matrix element squared. So it's second order in the Hamiltonian. But that means that this harmless addition of an imaginary part means, if you now look at the Taylor expansion, that we are using now

infinite orders, infinite orders n in this expansion.

So the phenomenological addition of width gamma to the excited level b is actually highly non-trivial, because it corresponds technically or physically to a non-perturbative result, which involves infinite order in the interaction matrix element.

Question about it? So this is more setting up the agenda. I wanted to remind you of the perturbative result. I want to sort of show you how we can fix it. I want to show you what it means.

But with that motivation, I want to go now back to our diagrammatic approach, our formal solution, our exact formal solution of the time evolution of a quantum system, and show you how I can implement this agenda, in particular,

by not allowing the atom to propagate with the naked Hamiltonian H0, but sort of staying dressed with photons and propagating with the Hamiltonian H. Questions about that?

OK, so the derivation now is focusing on two chapters in API.

And I need one result, which I hope you remember from about a week and a half or two weeks ago. We formulated the time evolution of the system

using the time propagation operator u. And we found an exact expression. I'm dropping all the arguments here, t and t prime. You can look them up either in the book or in the previous lecture notes. We found that the propagator in the interaction representation

is the unperturbed operator u0. And then, now I need actually the terms. It was a time integral over some time t1, which involved u0, v, and u.

And I introduced that to you as you could plug that into whatever is the equivalent of Schrodinger's equation for the time evolution operator. And you really find this is a solution for u. But well, it's not really a solution

because it's a solution of u expressed in terms of u. But then I showed you that it allows an iterative approach. And this is why we did it. If you have a first order result, the zeros order result is u0. When you plug the zeros order result in here, you get the first order correction. When you put the first order result here,

you get the second order correction. So you get a recursive formula which solves the problem. And this is now our starting point. This is an exact formulation of QED. And this is now our starting point to address the problem of divergences. We want to describe what many of you do in the lab,

namely have resonant laser light interact with atoms. And I want to show you how can you describe it with QED with the formalism we have introduced. OK.

Now, for this chapter, we want to have one big simplification. Things are only getting complicated. They're getting simpler. If you have a time integral, which is a convolution,

and you can often simplify it by Fourier transforming it. Because the Fourier transform of a convolution is simply the product of the two Fourier transforms. There is one glitch, so one subtlety I want to feed in a moment. But let's just assume for a moment we go from time

we Fourier transform to frequency or energy. So g of e is the Fourier transform of, I have to fill in a few corrections, of u.

That would mean, if that were true, that the equation above would now be an equation where the integral be sort of constant instead of a convolution of u0 and u,

we get simply the product of the two Fourier transforms. And that's much, much simpler. It's an algebraic equation instead of an integral equation. And remember, when we came to n-th order, we had n integrals with some time ordering. We were able to do it, but it's mathematically more complicated.

The problem is that this is not a convolution. It would be a convolution if the time limits were infinite. So what we can do instead is we

can define g of e, not the Fourier transform of u. If we say it is the Fourier transform of k, which uses a step function, then if you insert now k in the formula above, you see that because of the step

function, we can now take the integral limits from minus infinity to plus infinity. So therefore, when I said g, g is called the resolvent.

It's really the mathematical quantity we have to work on now. And I wanted to tell you very simply, hey, it's just the Fourier transform of the time evolution operator. Not quite. It's the Fourier transform of the time evolution operator with a theta function just to make sure

that formally we can extend the upper and lower integration limits to infinity. Or this is also a name what I'm really kind of discussing here are Laplace transformations.

But I want to convey to you the physics. It would take many weeks to do the mathematic accurately. But you can get the idea and a really profound feeling for how QED works and what we are doing with not so much mathematical rigor. So with that grain of salt, we

have Fourier transformed the equation. We have the Fourier transform of the time evolution operator. But if you look at it mathematically more exact, it's a Laplace transform. And we are a little bit navigating in complex place. The Fourier transform does not exist for real values. We always have to add a small imaginary part to make things converge. But you've seen that in many other places, I assume.

OK. So the quantity which allows us to describe resonant interaction in a very straightforward way

is called the resolvent. I can simply define the resolvent. All I gave you was some motivation. But if I had wanted, I could just say the resolvent is defined by this operator equation.

Z is the complex number. H is the Hamiltonian including interaction between atoms in electromagnetic field. And actually, if you would take that,

and if you take this and plug it into the equation above.

So if you plug this into the equation, you find this expression solves the equation. So I just want to say, you can forget about all the subtleties with Laplace transform and such. You can just say, let me define some interesting object. But it is connected to what I said you earlier, because this solves this equation. And also, I like to sort of show you

the mathematical structure if you write the time evolution operator as e to the minus the Hamiltonian. And now you Fourier transform it.

If you Fourier transform it, you multiply it with e to the i omega t, and you integrate. Let me integrate from 0 to infinity. There are some issues with 0 or infinities. But if you simply solve this integral with time,

you have sort of an exponent e to the i omega minus h. So if you Fourier transform it, what you obtain is nothing else than the energy minus h.

And this looks very different. This looks very, very similar to the definition of the resolvent. The only mathematical thing I'm not fully discussing here is, is z real, or has it a small imaginary part, the small imaginary part? Actually, you have to add a small imaginary part here

to make sure the Fourier transform exists. But you should clearly see how things are connected. I'm focusing now on the resolvent. It is equivalent to the time evolution operator. It is a sort of Fourier transform of it. And we want to formally write down

a diagrammatic solution for the resolvent. And with that, I think I should stop now. We start on Wednesday. Any questions to that point? OK, then see you on Wednesday.