After Finding the Higgs Particle
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YearHypothetisches TeilchenOrder and disorder (physics)Elementary particleMassHiggs bosonLecture/Conference
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MassParticleHypothetisches TeilchenModel buildingStandard cellProzessleittechnikHiggs bosonHypothetisches TeilchenFinger protocolHull (watercraft)Club (weapon)Brake shoeHiggs bosonMassComputer animation
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QuantumGlassNear field communicationMusical ensembleLecture/ConferenceComputer animation
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QuantumElectronGeokoronaLungenautomatMechanismus <Maschinendynamik>StrahldivergenzMusical ensembleMusical developmentYearMaxwellsche TheorieBasis (linear algebra)Brake shoeProgressive lensLecture/ConferenceMeeting/InterviewComputer animation
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ElectronGeokoronaMechanismus <Maschinendynamik>LungenautomatStrahldivergenzProgressive lensBasis (linear algebra)Color chargeBrickyardElectronGeokoronaBook coverMeeting/InterviewComputer animationLecture/Conference
03:34
GeokoronaElectronMechanismus <Maschinendynamik>LungenautomatStrahldivergenzOrder and disorder (physics)DayColor chargeMechanicLungenautomatApparent magnitudeElectronGeokoronaAtmospheric pressureComputer animation
04:53
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QuantumElectronPair productionParticleStandard cellGangStrahldivergenzVertical integrationFeynman diagramLungenautomatMechanismus <Maschinendynamik>GeokoronaChandrasekhar limitEffects unitGauge blockEnergiesparmodusOrbital periodKette <Zugmittel>Antenna diversityLungenautomatElectronMechanicQuantumGeokoronaLinear motorHypothetisches TeilchenPhotonVertical integrationProgressive lensCell (biology)Corporal (liturgy)BecherwerkLecture/ConferenceComputer animation
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Mechanismus <Maschinendynamik>LungenautomatGeokoronaEffects unitGauge blockQuantumEnergiesparmodusElectricityChandrasekhar limitHalo (optical phenomenon)Local Interconnect NetworkColor chargeField-effect transistorFaraday cageLecture/Conference
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LungenautomatMechanismus <Maschinendynamik>GeokoronaEffects unitGauge blockQuantumEnergiesparmodusChandrasekhar limitLocal Interconnect NetworkOrbitRedshiftMechanicElectronAnomaly (physics)MagnetismElectricityMassAbsorption (electromagnetic radiation)Bill of materialsHydrogen atomMusical developmentLamb shiftMagnetic momentMechanicLungenautomatRedshiftOrbitQuantumElectronQuantum Hall effectMeasurementArc lampDrehmasseFACTS (newspaper)Bird vocalizationLecture/Conference
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ElectronElectricityMassBill of materialsTigerLinear motorBurr (edge)ElectronLadungstrennungMassCell (biology)WeightSpare partFerryMeasurementMeeting/Interview
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Elle (magazine)Lint (software)ElectronElectricityMassAbsorption (electromagnetic radiation)Cell (biology)MassQuantumMagnetic momentLamb shiftMagnetic coreElectronRedshiftAnomaly (physics)Lecture/ConferenceMeeting/Interview
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ElectronElectricityMassAbsorption (electromagnetic radiation)Feynman diagramRulerRulerCERNMassFeynman diagramCell (biology)Beta particleString theoryMaxwellsche TheorieRail profileMeasurementField-effect transistorPair productionSeries and parallel circuitsGentlemanComputer animation
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Feynman diagramRulerHypothetisches TeilchenParticleAmplitudeScoutingAvro Canada CF-105 ArrowElectronPositronQuantumRankingProzessleittechnikGloss (material appearance)AntiparticleHot workingProzessleittechnikDyeingOceanic climatePower (physics)Pair productionLecture/ConferenceComputer animation
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ParticleAmplitudeRulerFeynman diagramProzessleittechnikScoutingElectronPositronAvro Canada CF-105 ArrowQuantumAbsorption (electromagnetic radiation)Hose couplingFeynman diagramVertical integrationRulerMassMeasurementYearHammockElectronFACTS (newspaper)QuantumModel buildingRailroad carOrder and disorder (physics)Coach (bus)SensorEnergy levelVideoPositronPhotonCell (biology)Moving walkwayAvro Canada CF-105 ArrowScatteringMeeting/InterviewComputer animationLecture/Conference
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NeutronNeutrinoMuonCrystal structureSpin (physics)ParticleSchubvektorsteuerungMeasurementFeynman diagramRulerGreyRailroad carWeak interactionTypesettingRulerWeekElementary particleYearDrehmasseProzessleittechnikComputer animation
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NeutronMuonCrystal structureCougarSpin (physics)ParticleSchubvektorsteuerungRulerFeynman diagramGreyNeutrinoMeasurementRadioactive decayKaonVisible spectrumCombined cycleGreyHypothetisches TeilchenElectronCrystal structureSpin (physics)NeutrinoEffects unitElementary particleProzessleittechnikElectronic mediaDayCableA Large Ion Collider ExperimentCartridge (firearms)Computer animation
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RulerMeasurementFeynman diagramNeutronMuonNeutrinoCrystal structureSpin (physics)SchubvektorsteuerungParticleGreyHose couplingStud weldingProzessleittechnikScatteringSpin (physics)Hose couplingElementary particleSchubvektorsteuerungCell (biology)GameRadioactive decayScatteringStream bedEveningRelative datingProzessleittechnikSeries and parallel circuitsAbsorption (electromagnetic radiation)StagecoachElectronic mediaHeat exchangerComputer animation
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Hose couplingSonic boomProzessleittechnikStud weldingScatteringTurningHypothetisches TeilchenHiggs bosonMassHot workingField strengthQuantumMonthCartridge (firearms)Relative datingBrickyardSeries and parallel circuitsAerodynamicsLastHochbahnYearRefractive indexSpare partLecture/ConferenceComputer animation
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Hypothetisches TeilchenHiggs bosonProzessleittechnikMassLocal Interconnect NetworkRRS DiscoveryHose couplingHypothetisches TeilchenHiggs bosonSchubvektorsteuerungCERNTrainMassNoise figureEffects unitYearDaySpare partVolumetric flow rateCartridge (firearms)Lecture/ConferenceComputer animation
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RRS DiscoveryHiggs bosonHypothetisches TeilchenLocal Interconnect NetworkComputer animation
Transcript: English(auto-generated)
00:15
You know, getting a birthday gets boring after you get 80.
00:21
But there is nothing you can do to avoid it. Ah, here I am. I want to tell you about an adventure, something, and that is something that is here. You all have heard about the Higgs particle. What does it do?
00:41
It is supposed to give mass to other particles, and you wonder what the hell is that important for? Everyone gives mass to everybody, so what the hell? The Higgs, by itself, predicts nothing except its own existence, so it's not that.
01:01
It does not even predict its own mass. And I will try to tell you what the importance, the real importance, of the Higgs particle. In the end, let's get something clear too, when you do theoretical physics. In the end, you want to be able to do calculations, the outcome of which you can use this experiment.
01:24
That's the only and one criterion that you apply to physics theory. No, revolutionary ideas have to have been in this world since a long time. For example, in the old times, there were people who invented glasses, some monks near
01:42
Venice, and you can imagine how life was when there were no glasses. But even more stunning is the invention of music, the way you do music, which happened by some monk, somewhere in the 1600s or something, and very few things have been so important
02:02
as this one here. It's due to this, that we could have people like Mozart and Beethoven who would write music in their own house, and if you don't know how to write music, you have a problem, and therefore that happened in Europe, and outside Europe, nothing much
02:21
in the way of development of music happened. Well, field theory was something that was created around 1930, but it remained stuck till 1948, so 80 years long, nothing much happened, and then the very simple idea of
02:42
which you probably have heard very little happened, and that idea was due to a person named Kramus, so the idea of renormalization, and I will explain to you how renormalization has been on the basis of any progress that was made since then.
03:01
Now in 1904, talking about the energy of the electron, the self-energy of the problem of the electron, there was a theory about that of Abraham and Lorenz, and the idea that they had, these people, the question is, what do you think of an electron? An electron has the charge concentrated in a point, how must I see this?
03:24
How can you pile all the charge together in one point? Now one thing to do is to assume that the charge is in a little sphere, so the idea of Abraham and Lorenz is the electron is a little sphere, and there it is, that's
03:41
the electron, and then you see that the energy, the little sphere, you think of the charge coming from infinity and then being pressed together, the question, how much energy does it cost you to press all that charge together in a little sphere?
04:01
And the answer is, the energy that you need for that is inversely proportional to the magnitude to the radius of the sphere, so as you want smaller and smaller, the energy you need to apply is bigger and bigger. So you cannot really go to a point because one over the other, that means that energy
04:23
becomes infinite, so this is the problem, it starts surfacing in those old days. Now the smearing out, I use this model, as old as it is, in order to introduce some terminology, the smearing out of the charge of the electron over a little sphere,
04:43
that's called a regulator mechanism. You regulate something that otherwise would be infinite, so it's a regulated infinity, and so you have, here you have something of a little, with a radius r, and the next
05:00
thing, the radius r is the regulator parameter. It allows you to tell you precisely how you regulate, how you keep things finite. And if that regulator parameter goes to zero, meaning the sphere shrinks, then the energy going with it will go to infinity.
05:21
And you speak of, it goes proportional to that, so as the radius becomes twice as small, the energy becomes twice as big, and you describe that as a linear divergence. So the classical electromagnetism has a linear divergence, and this, I would say, rather
05:41
naive description of the theory, that remained part of the theory for 35 years. 35 years, no one knew very well what to do about the solvency of the electron. And then in the meantime, many things happened, quantum mechanics was born, and so people started to worry about the solvency of the electron within the context of quantum
06:05
mechanics, and that changes things. Here you see Abraham and Lorenz doing their first steps in the unknown of the electron. And then we got quantum field theory.
06:21
Quantum field theory was born at the end of the 1920s, made by Heisenberg and Pauli. And they came as the quantum version of the theory of the electron and the force, the electric force and everything. And the calculation of the electron solvency was done by Victor Weisskopf.
06:43
Victor Weisskopf was director of CERN when I got there, so that's the link that I feel is that time period. And Weisskopf discovered that at least within the quantum theory, the diversions of the electron, that thing inversely
07:02
proportion to the radius, that disappeared, and the only diversions was logarithmic. I can amplify that a little bit. You have something called the Feynman diagram, and in the Feynman diagram, you picture the electron as a particle that can emit a photon and reabsorb it.
07:24
And when you try to calculate it, out comes infinity. So that's what you have. And the corresponding expression, which you can write down in field theory, you don't have to understand it very well. That corresponding expression is like that, and you see it's odd in the momentum p,
07:43
meaning that the linear diversions by symmetric integration disappears. And that you can see in the last slide. So that does not much of progress, but in some progress, the diversions of the self-mass of the electron in that scheme was logarithmic.
08:04
And now, you see the following. If you want to make, if you want to deal, which you have to, if you want to deal with infinities, you have to invent a way to make that infinity finite. You have to make, you charge in the sphere, or what have you.
08:23
Yeah, and you have to introduce a regulator parameter in such a way that if that regulator parameter goes this way or that way, the situation, you approach the situation that you think happens in nature. And then a new complication arises. The regulator mechanism may violate important properties.
08:43
You come for some real difficulties. For example, in the theory of Abram and Lawrence, the sphere that an electron is doesn't remain a sphere if the electron starts moving, according to the theory of relativity.
09:00
It becomes an elliptic thing. So, then you would say the regulator method violates Lawrence invariance. And so, you get this problem, which is really your problem. And as you really have, you have other things too, an important property in quantum electrodynamics is gauge invariance.
09:25
You want that gauge invariance because it guarantees you that the theory will produce something that if its charge is conserved, and we all believe very much in the conservation of charge. Experimentally, it's something that works very well.
09:40
So, you have to have gauge invariance. Now Pauli and Villar, this is sort of an abstract thing. Yeah, so people thinking about the problem didn't realize the complexities of it. Pauli and Villar understood it, and they developed a regulator mechanism that respected gauge invariance
10:01
and that respected Lawrence invariance. So, they did something in that respect. And then finally, after that, there was the great revolution of 1948. In 1948, there was a conference in Shelter Island in America.
10:21
And at that conference, two people presented results which were absolutely surprising and new. And these two things were the Lamb shift. That's a small shift in the energy spectrum of the hydrogen atom. There's the Lamb shift discovered by Mr. Lamb using equipment that was developed in World War II.
10:44
And then there was the magnetic moment of the electron. That also was measured for that time unprecedented precision. And they found a little bit of difference from the magnetic moment that the Dirac theory predicts. That's called the anomalous magnetic moment.
11:03
And so these two things were there, and people knew, well, where do they come from? Probably it's a quantum effect at all. Quantum theory is not developed. And not only that, whenever one tried to do a calculation, you got infinity. So this was a real difficulty.
11:22
And then Mr. Kramers from Leiden, he was a chairman at some point of obsession, and he got up and he said, what you really should do, and he came with the idea of renormalization. And that I'm going to explain to you. And this is the very simple idea that I want you to understand as something
11:42
that has been crucial in the development of this theory. And the idea of renormalization is as simple as it can be. And you wonder why they didn't think of it before. There's always these big ideas. Once you know them, you say, yeah, of course.
12:01
But before you know them, you don't seem to be able to think of it. So Kramers says, let's look to the mass of the electron. Here's an electron that has a mass. What we see is the net result of a lot of things, or at least some things. Surely the mass of the electron has as an ingredient
12:22
the mass that the electron has before you give it an electric charge. So that's what you call the bare mass. And then the electron gets its mass, then you turn on the electric field, and you get the cell of energy of the electron is sitting in its own field,
12:40
that little sphere, remember? Bit more complicated with Pauli Villard, but that's it. So the mass that you see experimentally is the sum of the two. Well, says Kramers, okay, now it's clear that we don't have too much knowledge what goes on in the very inner part of the electron.
13:03
But for sure, the mass that we see is the sum of two masses. We don't really know what the mass of the electron is before we started this theory. So the bare mass of the electron is a fictitious parameter. It is not something that you can measure.
13:21
But then comes added to this bare mass of the electron comes the mass due to the cell of energy of the field, that field mass, so you get the addition. The only thing that you can observe experimentally is the sum of the two. Well, says Kramers, it's very simple. If you get out an infinite energy
13:42
for the cell of energy of the electron, just make that cell of mass minus infinity so that together they come out at the desired value. So you say, I know the result. He says, why are you arguing? I don't have to know the interior of the electron. The only thing I have to know is the sum total, that mass.
14:03
So this was the idea of renormalization, and it electrified the people, all the participants on this conference. And calculations started to go on, and people started to produce calculations of the Lamb shift and the anomalous magnetic moment. And quantum field theory, as we know it today,
14:21
was essentially born. But no one has gotten this idea before Kramers. It's sort of an amazing thing. So Kramers says, okay, a few corrections to the cell of mass of the electron. It's evident, well, just make the mass initially
14:41
minus infinity. Let's not worry where it comes from. But that's the idea. Now, the idea of Kramers initiated calculations of many people, Feynman, Bethe, Schwinger, Tomonaga. But Feynman, I would say, did something very special.
15:01
And again, it's something special. You have to appreciate it. We appreciate it only much later. But at the time, Feynman came up with the idea of Feynman diagrams. And Feynman diagrams are this. You make a little drawing, and then corresponding to the drawing, you write down an equation, a formula.
15:22
You say, hm, hm, hm, hm. That's the Feynman diagram. And that picture that you make can be a very easy way of reflecting on what that value should be. So what Feynman's method does is make it very easy for you
15:41
to write down what actually is going on, the actual value of that thing, whatever, the theory. So in other words, then for Feynman's diagrams, you have Feynman rules. And Feynman rules tells you how you should make those diagrams.
16:01
So knowing that Feynman rules, you have the theory. And it is easy. It's wonderfully easy. And Feynman himself, when he came from Stockholm, he only got the Nobel Prize. He came to CERN. And at the end, he stood at CERN.
16:21
He stood, he said, but what did I do? All I did is inventing a method of doing bookkeeping. The point is, that's true. But sometimes that's more important than anything else because it makes what you do transparent. So that's what Feynman did. And in a complicated series, such as field theory,
16:44
having a good method and having a simplification rules and nice way of putting it is essential. That's what Feynman did. So from here on, I will sort of work with
17:00
Feynman diagrams. And just remember, when I write down a Feynman diagram, they are intuitive things. And what they put down there gives you an idea what's going on in terms of a process. But the essential idea is that you write down that thing and it's easy to make it up. And then you can write an equation that goes with it.
17:22
And that works fine. Diagrams are made up of lines and vertices. And you see in this diagram here, an example of a Feynman diagram with two vertices. This is an experiment I do.
17:42
And some guys say, this is huge, my God. You see how wonderful this, who invented this is a genius. There were the two vertices. Here's vertex number one. Hey, look at this.
18:01
And that's vertex number two. And these are the lines. So that's a very simple diagram. And it gives you an idea. These are an electron and a positron scattering. And the arrow tells you which is the electron, which is the positron.
18:23
Given all this diagram, and given the Feynman rules, specifying the momenta of the ingoing particles, you can actually compute an equation that gives you precisely by squaring it on something that gives you the probability of this happening.
18:42
So it's a very nice way of doing a calculation. And here you see the Feynman rules. And I won't go any further than it. Just believe me when you say that once you know the rules, you can do the calculation. And there you saw something. This is the cell of energy of the electron.
19:01
And you might present it as something, a photon goes away from the electron, travels a little bit, and comes back. And it does it so fast that it escapes your detection. You cannot measure it. But it is there. It happens. Quantum mechanics allows that.
19:22
And a diagram like this is called a one-loop diagram. And there you do the calculation. And for God's sake, out comes infinity. So there is the infinity of quantum field theory. And there you get the remark of Mr. Kramers of renormalization.
19:40
He comes with that infinity, this one here, which is created by the electron sitting in its own field. And then you have here the bare mass, the mass that the electron would have if there was no electric field. And all you want is the sum to be finite. And that precisely is the idea of Kramers.
20:04
And in quantum electrodynamics, all infinities of the theory, and that is the wonderful thing, can be taken care of this way. And once you have done that, absorbed the infinities in the free parameter of the theory, you get the finite theory, you can compute everything.
20:22
But this is really very wonderful. And so then we go on to the next. Here's Mr. Feynman. He was very proud of his invention because we see here, he painted his car with it. And the Feynman rules are quite simple.
20:40
But then we go on to another type of interaction which is weak interactions. A weak interaction has had a lifetime, in some sense coinciding with my time that I was in the field. When I entered the field, we knew next to nothing of it. The only thing you knew,
21:00
and I'm not speaking about 1960 roughly, the only thing you knew were certain processes, decay of the neutron, the decay of the K-meson. You saw them decaying. You could measure the energy and the momentum of the particles coming out, but you had no idea what was going on.
21:22
And so you looked at these things, and you didn't know what the hell is this. And the only thing we knew at that time was that these processes were not going very strongly. They were called what we called weak interactions. Later on, we understood they are far from weak. But in those days, for the things that we knew,
21:42
they were weak. Then there was the thing which was very peculiar about the interactions. We discovered, the physicists of that time discovered that the interactions looked like there was an intermediary particle.
22:02
So you see where you go, where you look for this neutral decay here, then you think yourself, make a particle that is in between there, and this imposes a certain structure on the combination of electron and neutrino. So the kinematics, it has spin one.
22:23
And that was seen. So we saw special effects in the spectrum of those decays indicating that there was a spin one particle. And I remember very well that was the situation. And you looked at it, and you thought, what the hell is going on here? What is this spin one particle?
22:41
How heavy is it? You didn't know. Same thing happened here with the k-meson. Had the W, as we called it by that time, that was going that way. So this was the only thing known to those interactions. And so at this point, what do you do?
23:01
You do theory. You try to go further, but where the hell do you go? What do you do? Well, I tell you what you do. What you do is you start assuming things about these intermediary particles.
23:20
So you have something that comes from experiment, and then you start playing with it. And what kind of game are we playing? First of all, we make this thing interacting with the cell of SOTY and to use another, we have the W plus and the W minus that you could see here from the neutral decay
23:42
had the W minus, decay was the W plus, so you know these two are there. We didn't know anymore. But we invented another one. Not we, I mean not me. People invented another one, the W zero, and introduced an interaction between the three. Completely hypothetical.
24:02
And so what the hell do we do with this interaction? Well, start doing calculations. For a theoretical physicist, doing calculation is like inventing the paradise. So we started doing calculations like that diagram over there
24:20
that you could have a vector boson, scattering of two of these Ws. They exchange the W, they go on, and they reabsorb, and out they come. So this is a process. You can compute it. What do we find? We find infinity. So that's the first thing. You did this calculation.
24:42
You found infinity. But then you discovered, or you discovered, you realized there could be other possibilities. For example, you could exchange the two vertices below, and you would get another diagram, and that also is a possibility. So you get many more diagrams.
25:01
So you keep on doing the calculation, adding all these diagrams, and you hope you pray every evening. You go to bed and you pray that the result is finite, but it ain't. So what happened then is there are infinities, and you couldn't absorb them
25:21
in the known constants in the strengths, for example, of the coupling of the Ws to themselves. And so you had bad infinities that you could not absorb in the three parameters of the theory. And if you had such bad infinities, we call the theory not renormalizable.
25:42
And so we got a new goal in life. Can we make that theory renormalizable? Change it a little bit so that the infinities cancel and can be absorbed in the three parameters. And so that starts the hunt for what to do here,
26:01
where you want to make the theory finite. I mean, this is actually how it happened. I mean, I did this, and I didn't know what to do. And so the next step, you introduce a new interaction, and that new interaction is two Ws interacting in a point.
26:22
That can happen in Feynman diagrams, and then you get new diagrams. So what happens is you introduce a new interaction and try to make it in such a way that it makes the theory more decent, has less infinities.
26:42
Then you choose the strengths of that interaction. You choose very carefully, in such a way there is cancellation of infinities. That was a new thing that was not happening in quantum electrodynamics, but so that's the first idea that you put in. Maybe I can, by having different kind of interactions,
27:01
make them work together and make something that's finite. So that's what you can try. And by God, that works. And it turns out that if you have this thing here, you can get rid of almost all infinities. And the theory that you get them,
27:23
that four-point interaction, two-point interaction, was known. Not to me at the time. I didn't know what I was doing. I mean, most of the time, you don't know what you're doing. But there, the theory had this four-point interaction, a three-point interaction, and that was already studied before
27:42
in something called the Yang-Mills theory, who had done it as a sort of an exercise in elegance. It did something that had nothing to do with infinities. It was symmetry. No one knew that it had anything to do with infinities. And it was done long ago,
28:00
long before we started this work here. But in any case, it was what we call a Yang-Mills theory. That was pointed out to me by somebody. And I thought, well, we are lucky. We are teaming up with the known stuff. So you get a Yang-Mills theory,
28:20
and you get that purely out of the requirement that things sum up so that you don't get too many infinities, that you get a renormalizable theory. Did we get a renormalizable theory? Answer, almost. What happened is there was just one little infinity
28:42
somewhere left, and you couldn't get rid of that. You tried that, that didn't work. You would have to visit for a month this way, that way, and so on. Couldn't do it. Then it turns out another idea was needed. The other idea that that was needed was that you had to introduce another particle,
29:04
which I've indicated with red lines there. And that particle couples to the W and adds, by this factor that is there, it adds to the complications of the situation. And then it turns out that that particle
29:21
was known also, this particle, and the couplings that you could introduce was the Higgs particle introduced long ago, long before that. That is to say, five or six years. So out of the requirement, and now this is Higgs particle,
29:41
you made it in such a way that the theory, the infinities, would cancel as much as possible. And that in fact, the resulting theory would be what we call a renormalizable theory, an idea, that idea of Mr. Kramers. And so you see there is now,
30:01
not only has there been new interactions being introduced between the vector models, in addition, we first restarted by introducing the W zero, that was not seen, then you started making the interaction with four Ws, still got, you got almost there,
30:22
then the idea came to introduce the Higgs, and the Higgs resolved the whole problem. And there was only one parameter that occurred, only one new particle, which was the mass of the Higgs particle. That one, you couldn't figure out from the situation. Renormalizability requires you to read that to be a Higgs,
30:41
it doesn't tell you how heavy. So that was the situation, and so you can see this long train of speculation. It could have broken down at any point, but it is just continuing an idea, the idea of Kramers, up, up, up, up,
31:02
to the end, as far as you can. You do that, you get something. And whether that something is meaningful or not has to become clear, and the only way that it can become clear here is finding the Higgs.
31:20
And by God, that happened at CERN. They found Higgs, not Mr. Higgs, that's Mr. Higgs there. But they found it in the experiment done at CERN, and you see here, Mr. Higgs looking proudly to us, more or less at the H when he discovers,
31:40
or when he postulated his particle, and that was done. Mr. Kramers, so you stand there as if I have to stop. Well, okay, I stop.