6 · 9 = 42 My Search for the Theory of Everything Prof. Dr. rer. nat. Peter Gerwinski 10 May 2021
Good evening. Welcome to my speech about: 6 · 9 = 42. My Search for the Theory of Everything.
That's a weird title. I'll start by explaining where it comes from.
That's a quote from the novel The Hitchhiker's Guide to the Galaxy by Douglas Adams, a science fiction parody.
There, philophers ask a computer: “What's the answer to the ultimate question of life, the universe, and everything.”
The computer calculates for 7½ million years and then answers: “42.”
The philosophers aren't really enthusiastic about this. Then the computer explains:
“I think the problem is that you've never actually known what the question is.”
Then the philosophers build an even bigger computer, and …
Okay, now I'm spoilering the contents of the novel and of the movies.
At the end of the second book the question appears, and it reads: “What is 6 times 9?”
After that the leading character says: “I always thought something was fundamentally wrong with the universe.”
Thus ends volume 2 of the 5-volume trilogy.
This is about physics. My question is somewhat more humble: “What are the laws of nature?” That's what physics deals with.
The current state of research reads: One theory explains everything which is big.
That's general relativity. It explains the universe as a whole, black holes and things of comparable size.
And we have quantum field theory, which explains the smallest parts of the universe.
Quarks, leptons, and the Higgs boson fall within this scope.
Both theories describe medium-sized objects.
A microbe or a planet – seen from both theories they are about medium-sized.
You can choose one of both theories to describe medium-sized things.
Both contain so-called classical physics.
This raises the question why we cannot use both theories together.
As everyone knows, noone has yet succeeded on this.
You cannot combine these theories.
When you ask an expert why not you get the answer:
“Quantisation of gravity leads to non-renormaliseable divergences.”
Uh-huh. What is this supposed to mean?
Where precisely lies the insurmountable contradiction?
I studied physics, and I asked this question to several people, but I didn't get a satisfying answer.
In the year when I turned 42 I decided that I want to find out by myself.
I want to actually know what the problem is. What is the question?
My approach: I try to unify both theories.
Then I'll see where I run into a dead end.
And what I found out thereby is what I want to speak about today.
This implies that I must summarise all of physics in this speech.
It is scheduled for about 1 hour.
I'll explain all of physics today, classical physics, special and general relativity, and quantum physics, up to quantum field theory.
After that we can address the question: What is quantum gravity? Where's the problem?
And then I can tell what I found out and how my research might continue after this.
Okay. We start in ancient Greece.
Back then, Aristotle was one of the first who asked how things move.
Why do things fall down, but fire goes up, air goes up, water and earth go down.
His explanation: These are the four elements everything consists of.
Birds can fly because their feathers contain a high amount of the element air.
This theory explains, more or less, how things move on Earth.
But when you look at the sky you see things orbiting around the Earth: the stars, the Sun, and the Moon.
Aristotle's explanation: This is something entirely different.
Sky objects don't consist of fire, water, earth, and air.
They consist of a fifth element, quintessence, or aether. Things consisting of aether move in circles.
It took over 2000 years until somone succeeded to resolve this contradiction that sky objects move entirely differently than objects on Earth.
In 1686, Newton described in his Mathematical Principles of Natural Philosophy that both is in fact the same.
The fall of a thing on the Earth and the orbit of the Moon around the Earth are in fact the same thing, caused by gravity.
Here we see Newton's law of gravity. Don't worry, you don't need to understand and to memorise all these formulas, …
… except if you are studying at our university.
Gravity depends on the distance between the two things which attract each other, …
… and from the two masses. This G is the gravitational constant. The important thing is that it depends on the distance.
On the left we see a force. Using this, Newton calculated how things move.
For this purpose he developed a new calculation method, the so-called differential equation.
“Differential equation” means: How it is now tells us how it changes.
Here we see an important special case of Newton's equation of motion: force = mass · acceleration.
Acceleration tells us how the velocity changes. Velocity tells us how the position changes.
How it is now, the force, tells us, how it changes.
Newton had great difficulties to solve this differential equation using pen and paper. Of course. He had just invented this.
Today this is much easier. I can simply ask my computer to solve this differential equation for me.
That's what I'm doing now.
Here we see the simulation of a ball lying on top of a tower.
When I poke it, it flies to the side in a characteristic curve, a so-called parabola, the ballistic trajectory.
The computer has calculated this using the differential equation we just have seen.
Now I can make the tower higher, so the Earth's curvature comes into play.
Again the ball falls down, essentially describing a parabola – not entirely, but very similar.
I can make the tower even higher.
When I poke the ball with sufficient thrust, I can reach regions behind the horizon. In some sense, I can throw the ball around the earth.
When I go even further, I can literally throw the ball around the Earth. Now the ball describes an elliptic orbit around the Earth.
It precisely returns to the point where I threw it, and it will orbit around the Earth forever.
When I do this in the right distance and with the right velocity I get a circular orbit of an object, for instance the Moon, around the Earth.
This way Newton has shown that both is indeed the same: motion on Earth and motion in the sky.
What else is part of classical physics? A very important thing is the principle of relativity, established by Galilei in 1632.
Imagine you are in a train. Someone is walking inside the train – in hurry, at 10 km/h, in the direction of travel.
Sitting in the train I can say: He's walking at 10 km/h, rather fast.
Now the train is moving as well, at 100 km/h.
Then an observer at the station can say: He's passing the station at 110 km/h.
And the walking person can say: When I look outside I can see the ground departing backwards at 110 km/h.
And all three of them are right. That's the principle of relativity: Everything is relative.
That's quite simple, but there is a problem. Light doesn't act that way.
In 1810 Arago was the first to measure that …
When I depart from a star, its light should reach me more slowly.
Or when a star approaches me, or the Earth is approaching the star along its way around the Sun, the light should be faster.
He measured, however, that the light has always the same speed, whatever I do.
In 1887, Michelson and Morley ascertained this extremely precisely in their famous experiment.
It is really like this: The light has always the same speed, whatever I do, however I try to accelerate it.
The light in the vacuum. It's slower in glass, but I cannot make it faster, whatever I do.
This contradicts Galilei's theory of relativity.
An explanation was found by Einstein in 1905. He said: Our measurements weren't wrong.
Instead our perceptions of space and time are wrong.
Space and time depend on the observer. In particular: When you move, your time goes slower.
This sounds incredible at first, but it fits the measurements. And later it has been checked directly.
In 1971 there was the Hafele-Keating experiment.
They put an atomic clock into an aircraft and let it fly around the whole earth, both clockwise and counterclockwise.
After that they compared the time shown by this atomic clock to the time shown by another atomic clock which had remained on the ground.
The atomic clocks showed that the time passed in the aircraft was one tenth of a millionth part of a second shorter than the time passed on ground.
This sounds incredible. That's a very short time. Probably their measurement was errorneous … nope. Atomic clocks are this precise.
In fact, seen from today, this is a quite normal time interval.
Today's computers are calculating in the range of some gigahertz.
A gigahertz corresponds to 1 billionth of a second. This means that within a tenth of a millionth of a second, a computer can do 100 things.
This is clearly measurable.
In the International Space Station which circles around the Earth every 90 minutes, this effect is much bigger.
There you can measure it using an ordinary stop watch. Over there, 1/100th second per year passes less than on Earth.
So it seems to be correct what Einstein came up with to explain the weird behaviour of the light.
Now you can carry on calculating. When you move, your time passes slower. And when you move with the speed of light, your time comes to a hold.
And when you move even faster than light, your time passes backwards.
This can be a problem. From “Back to the Future” we know what can happen when you travel to the past and disrupt the romance of your own parents.
This can reduce the probability of your own birth. A time travel to the past can lead to paradoxa.
Travelling faster than light is the same as travelling to the past.
When you travel faster than light you end up in the past, which can lead to problems.
The same holds when it is not a human travelling faster than light, but just information.
For instance if I send a message to my father telling “Keep the hands off that woman!” this might lead to problems.
Information and objects may travel at most with the speed of light, otherwise we end up in the past and run into problems.
But what happens if I'm in a train which already moves with 75% of the speed of light, and I run forward with another 75% of the speed of light?
Then I should have, seen from the station, 150% of the speed of light.
I don't. We can measure it. We can calculate it and measure it.
Seen from the station you “only” run with 96% of the speed of light. In no event you can surpass the speed of light.
This can be measured.
That's how nature is like, whether we understand it or not. Of course our brains are not made for understanding such things, but we can calculate it.
But now we have a problem with Newton. The gravitational force, for instance that of the Sun, only depends on the distance.
If the Sun jumps by 1m, its gravity which affects me must change without any delay.
But that mustn't be. Then it would move faster than light.
So we need a new theory of gravity. That's general relativity, developed by Einstein as well.
Special relativity plus gravity yields general relativity. How to approach this?
Einstein used the equivalence principle, which says that inertial and gravitational mass are the same.
What does that mean?
When an aircraft takes off you feel like pressed into your seat due to the acceleration.
“Inertial and gravitational mass are the same.” means: “Acceleration feels like gravity.”
This is taken advantage of in, for instance, flight simulators.
When I tilt the simulator, raising its nose, I feel like being pressed into my seat, so I feel that the aircraft is accelerating.
Now Einstein says: This doesn't just feel alike. This is indeed the same.
This is the so-called equivalence principle, and it has enourmous consequences.
When I move without forces, for instance when I go by car straight ahead, then I feel that I'm not following a bend. I would feel the bend.
Moving straight ahead means to move without forces, and moving without forces means to move straight ahead.
But we have orbits. There is zero gravity. That's without forces. This should imply that you move straight ahead.
Yes, that's what you do. An orbit is straight ahead. Not in normal space, but in a four-dimensional, curved space-time.
The curvature of four-dimensional space-time is a gemetrical effect, but we perceive it as a force, gravity.
How can we imagine this?
Imagine you are on the surface of a big sphere, and you cannot depart from that surface just like that.
Now we start two aircrafts from the equator precisely heading south.
Both take care that they don't fly any bends. Then they will, of course, meet at the south pole.
Then the two pilots can communicate: “Did you fly a bend?” “No. Did you?” “No. Then maybe there is an attracting force between our aircrafts.”
But we know better. This is not an attractive force. It's a geometric effect. This is geometry on the surface of a sphere.
That's how gravity works. It's just not the two-dimensional curved surface of a sphere, but four-dimensional, curved space-time.
This can easily overcharge our brains, but we can calculate everything, and the results are amazingly correct.
Here we see again Newton's formulas. What did Einstein improve?
The problem is this “normal” equation. The differential equation is okay, for it contains the time.
How it is tells us how it changes. A change implies time.
This equation says: How it is tells us how it is. So we must replace this equation by a differential equation.
That's what Einstein did. The equations he found we are now calling Einstein's field equations.
This is “force = mass · acceleration”. This didn't change much. However this second equation looks totally different.
For the experts: This is a differential equation because these terms contain the derivatives of g.
You don't need to memorise now how these equations look like or what they mean. Just remember: I like differential equations.
In particular you can use this equation to calculate gravity without getting into trouble with the speed of light.
So this works. We can describe it mathematically using an equation. Now, what can we do with it? Can we use it to calculate something?
Yes, we can. One year later, Schwarzschild was the first to solve Einstein's field equations. He found out how gravity around a star works.
Good news: When the star is not too strong, for instance our Sun is too small for that, everything is essentially the same as with Newton.
There are small deviations. We can measure them, and they are correct.
That's good news. But what happens when the star is much heavier than our Sun? Then very strange things happen.
When you approach the star to its so-called Schwarzschild radius, then space and time interchange their meanings.
You fall on the star, and when you have passed the Schwarzschild radius, then the direction of the fall is the new direction of time.
The fall into the black hole, that's how this kind of stars is called, becomes the new time. What used to be time, becomes normal space.
Now you could travel to the future and to the past just like that, but you don't get anything out of it since you are inevitably falling into the black hole.
You cannot escape because for that you would have to travel backwards in time.
So you reach the centre of the black hole, and there is a so-called singularity in the equations.
The formulas yield infinity. You cannot reasonably calculate with this.
This is where the validity of general relativity ends.
If we want to calculate further, we need a new theory.
People looked at these results and concluded: We must have miscalculated. This cannot be.
So we are searching for errors in our calculations. We are searching for errors in general relativity.
Instead we found – black holes.
In 1973 this object got discovered. We call it Cygnus X-1.
It is the invisible companion of a blue giant star, and it absorbs that star.
It's extremely massive. It weights 21 times as much as our Sun.
But it's scarily small. We can calculate its Schwarzschild radius as 50km. Within this radius of 50km there is 21 times the mass of our Sun.
There is no other choice. This cannot be anything else but a black hole.
Meanwhile we know of more black holes. A very big one is in the centre of the Milky Way. We call it Sagittarius A-star.
In 2002 one has cleared the last doubts that this is a black hole.
One has measured the orbit of a star named S2 and found that it made a U-turn within just a few months.
S2 is much heavier than our Sun, but it got tossed around like a ping pong ball by something in the centre, which is invisible.
Since it is that mighty, it can be nothing else than a black hole.
It has 4.3 million times the mass of our Sun.
It's also bigger than 50km. It has a Schwarzschild radius of 22 millions of km.
For comparison: The orbit of the Earth around the Sun has a distance of about 150 millions of km between the Earth and the Sun.
But there are even much bigger ones, for instance this one.
This black hole got photographed in 2019. We call it M87-star.
This black hole got photographed in 2019. Why this one? Why not Sgr A*? That would be just next door. Or Cygnus X-1, which is even much closer.
This thing is 1000 times as far as Sgr A*, but it is also 1000 times as big.
Its mass is 6.5 billion times the mass of our Sun.
Its Scharzschild radius is 120 astronomical units, 120 times the diameter of the orbit of the Earth around the Sun, bigger than the orbit of Pluto.
One has interconnected the whole Earth to form one big radio telescope in order to take this photo.
It is so big that it appears from Earth at the same size as Sgr A* in our own galaxy, although it is located in the far galaxy M87. Thus we call it M87*.
What else can we calculate using Einstein's field equations?
We can calculate backwards in time. How did everything around us come into existence? This leads us to the so-called Big Bang.
When you calculate back far enough into the past, you reach, again, a so-called singularity. Then the equations yield infinity.
This means that there was a second 0. From that point you cannot calculate backwards any further. At that point, time begins.
At least we cannot use general relativity to calculate further into the past. That's when our universe emerged.
The state just 1 thousandth of a second after the Big Bang can be calculated very well using general relativity.
We can use general relativity to calculate how our universe came into existence.
However there are also things we cannot yet calculate using general relativity.
One of our unsolved mysteries is the so-called dark matter. I'm now trying to show you a film. Let's hope that you can see it in a moment …
Okay, now you should at least see the two galaxise on my desktop, as still images.
Now you can see two galaxies. They look alike because they don't move.
The left image has been calculated using general relativity. The core is rotating fast. I'm trying to visualise this using my cursor.
The outer stars are moving slowly. They are going round gently.
This is also visualised as a curve. In the middle the stars are fast, in the outer regions they are slow.
This was calculated using general relativity. However what we actually measure is an entirely different behaviour.
The core is rotating fast, and the outer stars are moving as fast as the core.
The velocity of the outer stars is the same as that of the inner ones. This contradicts general relativity.
Here we have a mystery of our time, which we cannot yet explain. There must be something.
We don't know what it is, but we call it dark matter.
It might be actual matter. According to some other theories there are corrections to general relativity. In any case it is an unsolved mystery of our time.
Thus we have reached the end of the first part of my speech, about general relativity.
It says: Gravity is curvature of the four-dimensional space-time, and thus geometry.
We have seen the limits of the theory. One is the emergence of the universe, the Big Bang.
Another one is the centre of a black hole, where space and time simply end. Here the theory yields infinity, and we cannot calculate any further.
And one of the unsolved mysteries is dark matter.
Are there any comprehension questions so far, before we proceed to dealing with the smallest parts of the universe?
You may ask questions now.
This doesn't sound like any need of questions.
So I suppose that you have grasped general relativity.
We now leave the biggest things in the universe and proceed to the smallest ones.
Again we start in ancient Greece, with Democritus.
In about 420 BC, Democritus observed that when things decay, always something remains.
For instance when an animal dies, it decays to earth, out of which plants can grow.
From this Democritus concluded that there is something which cannot be devided any further, of which everything consists of.
He named it “átomos” in his language, which means “indivisible”. That's why we still refer to the smallest parts of matter as “atoms”.
Democritus: “By convention sweet is sweet, bitter is bitter, hot is hot, cold is cold, color is color; but in truth there are only atoms and the void.”
It took over 2000 years until evidence was found that these atoms actually exist.
In 1803 Dalton investigated the ratios of quantities in chemical reactions.
For instance, when I let oxygen and hydrogen react, how much water do I get?
When I take too much oxygen, that oxygen will be left over.
I can implement chemical reactions only with the same ratios of quantities.
Dalton took this as an evidence that these “atoms” actually exist.
He found out that a carbon atom can bond with either one or two oxygen atoms.
He found out there there are carbon monoxide and carbon dioxide.
So there are specific rules how these atoms relate to each other, how they can react with each other, and what are the reaction products.
During the next decades these rules have been investigated further.
In the 1860s, various scientists found several relations and rules for the atoms.
This culminated in the periodic table of elements, which is valid until today. It is depicted here in its modern form.
Today we know 118 atoms and the rules how they relate to each other and which ones bond with which ones.
On the right there are the inert gases. They are inert and essentially don't do anything, chemically.
On their left there are the halogens. They are rather aggressive and can be used to make, for instance, bleach.
There is a system behind this, from which scientists concluded even back then, that our “atoms” in fact consist of even smaller parts.
The first pointers to these constituents of atoms have also been found in 19th century.
In 1897, Thomson found out that the so-called cathod rays consist of particles.
Those of us who, like me, had contact to CRT televisions know that it contains a glass tube, essentially with vacuum inside.
One can place this metal cross inside and connect the tube to high voltage. Then there are these characteristic rays.
Then this metal cross casts a shadow. Thomson concluded that something inside is moving on straight lines.
Thomson found out: When we place a magnet close to the tube, the shadow moves.
Then the particles move differently, on a curved path.
When I remove the magnet, the shadow goes up again. When I put it back, the shadow goes down.
Whatever it is, it flies straight ahead, it reacts to magnetic fields, and it casts a shadow.
From this, Thomson concluded that these are particles, which we now call electrons.
But where do the electrons come from? We just did connect it to a voltage.
They must have been already inside the wires.
Thus Thomson has identified the electrons as one ingredient of the atoms.
From this Thomson made the first atomic model, also called the plum pudding model.
An atom is a big cloud of electrons. For instance you need about 1800 electrons to balance a hydrogen atom.
They are in kind of a bag, a massless background charge.
The electrons are negatively charged, the background charge is positive. This holds everything together.
Thomson was aware that this is just a rudimentary model, but we can already use it to explain several thigns.
For instance, atoms can gather energy. The electrons in the “plum pudding” can vibrate. When they don't there is no energy.
This got investigated further. In 1909, Rutherford, to put it simply, collided atoms.
He found out that in most cases they simply passed each other.
Only if the hit was precisely central something got reflected, but then in a very hard way.
So an atom mostly consists of empty space, with a small core in the centre, the nucleus, which essentially contains all of the atom's mass.
From this Rutherford created a new atomic mode. The nucleus is positively charged, the electrons negatively, thus they attract each other.
Then the electrons can orbit the nucleus in a similar way as the planets are orbiting the Sun.
This is a very nice atomic model. It has one further advantage: It also explains how atoms can gather energy.
There are orbits with different heights. On this higher orbit the electron has more energy than on this lower one.
When an electron jumps from a higher orbit to a lower one, the atom emits energy in the form of light.
The height of the fall determins the colour.
When the electron falls down a small height, the light is red. When it falls further down, it is violet.
When it falls down even further, we get UV or X-rays. When it falls down by a very small height, we get IR or radio waves.
That's a nice explanation, but there is one thing we cannot explain this way.
Why does the electron stop falling? Why doesn't it fall down to the nucleus?
Why aren't all fall heights allowed? If they were, every object could emit light of every colour.
Then everything would contain every colour. Everything would be white. But that's not the case.
Thus something is missing, which we need to explain everything.
In 1913 Bohr found sort of an explanation.
He just postulated that only specific orbits are allowed. “That's how it is like.”
He set up mathematical formulas which orbits are allowed and which are not.
This made it possible to explain the actual colours of the atoms.
By the way, this is one of many meanings of the term “quantisation”. The orbits of the electrons around the nucleus are “quantised”.
This means: Only specific orbits are allowed.
What he couldn't explain: Why? Why are only specific orbits allowed? What's up there?
An explanation for this was found later with the emerge of quantum theory.
In 1926, Schrödinger set up an equation named after him.
This equation describes the “orbits” of the electrons around the nucleus in a better way than ellipses do: as waves engulfing the nucleus.
On the right we see an electron resting comfortably around a nucleus as a spherical cloud.
On the left we see a wave consisting of two anti-nodes and two nodes.
Down we see a wave consisting of four anti-nodes and four nodes, which, in some sense, orbits the nucleus.
In fact there can be much more complicated structures.
Now this thing has little energy and this other thing has more energy. When it changes from this state to the other one, light emerges.
Now the question why the orbits are quantised does not even arise.
Of course there is nothing between these two states, except if we can calculate it using the equation.
The Schrödinger equation made it possible to comprehend the properties of the atoms very precisely.
One could explain things already measured, and one could predict new things, measure them, and they turned out correct. That's a big success.
Here we see the mathematical description of the waves. It is again a differential equation. I like differential equations.
The left-hand side says how it changes. That depends on how it currently is.
The Schrödinger equation is a differential equation, too, and it describes these waves.
We can solve this equation. Then we get results which coincide with experiments.
But the Schrödinger equation has one problem. It allows a wave, for instance an electron, to move with infinite speed.
Now we have a problem with special relativity, which says that the speed of light is an upper bound.
So the Schrödinger equation, and thus quantum mechanics, contradicts special relativity. How to deal with this contradiction?
Two years later, Dirac solved this contradiction and set up the Dirac equation, named after him.
It is a differential equation, too: How it is tells us how it changes.
This equation harmonises with special relativity. It has the speed of light as an upper bound.
Now we can precalculate how atoms behave even more precisely, and measure that.
One thing which now could be explained that way is the so-called hyperfine structure.
When you measure spectral lines very precisely, each of them splits into two lines.
He predicted even more. The Dirac equation has four solutions, two of which have negative kinetic energy.
What does that mean?
To imagine kinetic energy, I sit down, in my thoughts, in my electric car.
I operate … well, not the gas pedal, but the electricity button.
I accelerate as fast as I can, and after that the battery is empty, because I had to extract the kinetic energy from the battery.
If kinetic energy is negative, that implies that I accelerate my electric car, and the battery gets fuller and fuller.
That would be nice, but this doesn't coincide with reality. So something is wrong with this equation.
Now Dirac could say: Okay, I miscalculated.
Instead he said: This is something new. This is so-called antimatter.
These solutions with negative kinetic energy must be interpreted as “holes” of particles with positive kinetic energy.
They whoosh around as holes, and they behave in every respect like normal matter, but somehow reversed.
For instance an electron has negative charge, so an anti-electron should have positive charge.
Well, four years later this got actually discovered.
Anderson discovered a particle in the cosmic rays. This is the first-ever photo of a so-called positron.
It behaves in every respect like an electron, but it doesn't carry a negative charge, but a positive one.
This was a big triumph for physics. Anderson discovered it. Dirac had predicted it. That's perfect. What else cold we ask for?
Now we can explain the whole world. We consist of atoms, which consist of a nucleus and electrons orbiting it, somehow, as waves.
I didn't address that the nucleus consists of protons and neutrons. This was also discovered at the beginning of the 20th century.
And there is the positron, discovered in 1932 by Anderson. But that's enough. Now we can explain the whole world.
Well, we still need some “glue” for the nucleus to make the protons and the neutrons stick together.
We call this the pion. Now we search for the pion.
Instead we find – the muon, discovered in 1936, again by Anderson and again in the cosmic rays.
The muon is like an electron, but it is much heavier, and it decays after 2.2 microseconds.
The muon is not stable like the electron, but it decays, just by itself.
This was unexpected. This doesn't fit into the picture.
But actually we were looking for the pion. And we find it.
But we also find the neutrino, and the kaon, and the sigma hyperon, and the lambda hyperon, and several mesons, and so on.
Until the 1960s over 100 of these so-called elementary particles had been found, the so-called particle zoo.
That's more than we wanted. Actually we just needed atoms.
Instead we found a whole particle zoo. What do we do with it?
First of all, let's examine how they relate to each other. Maybe they react with each other and form new particles, or similar.
Yes, they are some rules how they relate. This already suggests that they might consist of something even smaller.
This was actually found in the 1960s. One of the protagonists was Gell-Mann.
In 1964 he set up the so-called quark hypothesis.
Here we see, in some sense, the successor of the periodic table of elements.
These are the smallest parts of the universe, quarks, leptons, and interaction particles, including the Higgs boson.
We have found all of them, and there are no signs that there could be something even smaller.
We are looking for smaller things, which is good, but up to now all indications are that these are indeed the smallest parts of the universe.
This has been the case for over 40 years. Back then when the atoms got discovered it was clear very soon that they must have some ingredients.
In this case it is clear, at least as of today, that these are indeed the smallest things.
Now we can build the other elementary particles out of them.
For instance we can build protons and neutrons out of up quarks and down quarks.
When we take 1 quark of one type and 2 of the other type, we get either one or the other.
The electron is one of the leptons.
This means that we consist only of these three particles.
Everyhing else we can only find in cosmic rays, in particle colliders, and in the centre of the Sun, but not in our everyday life.
Just this corner dominates our lives. Everything else is, well, very interesting, but it doesn't affect our lives.
Now we have the Standard Model of elementary particles, the Standard Model of quantum field theory, in front of us.
As a theoretical physicist I want to know the details. How to do calculations with it?
Can we use this to calculate what combines with what to form other things, how that behaves, and so on?
Now we wish for a differential equation. We'll get one in some sense, but it's going to be very long.
When we describe all of this using a differential equation, it gets very, very long. That's why we use an abbreviation.
The Lagrangian density is, to put it simply, an extreme abbreviation for a differential equation.
And when I abbreviate all of this massively, I get this nice, little formula.
This formula describes the smallest parts of the universe, quarks, leptons, …, everything is in there, for instance the Higgs boson down here.
It's good that the Higgs boson is there because if it weren't there, this formula would continue, than it would be about twice as long.
This was a very great discovery. In 2012 at LHC the Higgs boson was discovered. That's good. The formula is long enough as it stands.
Now what can we do with this? How can we apply this to calculate how the smallest particles behave?
It is, of course, possible to convert this to a differential equation, for which it is an abbreviation.
As an example I brought in yet another Lagrangian density, the one of general relativity.
Then I can calculate Einstein's field equations from the Lagrangian density.
That's what Lagrangian densities are for. We can convert them to differential equations.
Of course for this Lagrangian density, it will become very long and confusing. And in fact that's not done usually.
What to do instead? Instead we use time-ordered perturbation theory.
This approximative method allows us to calculate something reasonable from this monstrous formula.
“Approximative” sounds as if it was imprecise. No, just the opposite. This is in fact the most precise theory we ever had.
We can use this to calculate how electrons behave in magnetic fields up to 11 decimals.
When we measure it, the result coincides precisely with these 11 decimals.
If we predicted a time 3000 years in advance, this accuracy would be below 1 second. That's extremely accurate.
To calculate that precisely I need so-called Feynman diagrams. I decide in advance how precisely I want to know it.
For a rough result I can do with just two of these so-called Feynman diagrams.
This looks like colliding particls. Actually it is a mathematical abbreviation for integrals.
Using two of these diagrams I can rudimentarily describe what happens when two electrons collide, or one electron and a positron.
The positron is an anti-particle, thus it travels backwards with respect to this arrow.
The electron moves this way. They collide. They annihilate each other. A gamma quantum emerges, gamma rays.
Out of this gamma quantum emerges, for example, a pair of a muon and an anti-muon, which again moves backwards with respect to the arrow.
This is just an illustrative interpretation of the Feynman diagram. Actually, it stands for an integral.
If you don't apply 2, but 15000 Feynman diagrams you can calculate up to 11 decimals how electrons – or muons – behave in a magnetic field.
Recently there were interesting experiments with muons which are currently being examined.
This is the most successful physical theory we ever had.
We can use this to describe the universe in microcosm perfectly.
Up to now there is not a single experiment, maybe except the ongoing muon experiment, which would contradict all of this.
That's a giant triumph for theoretical physics.
Now we have reached the point where we can try to reconcile all this with gravity.
For one thing we have the Standard Model of elementary particles, of quantum field theory.
That's a very long Lagrangian density, which could fill all of this slide just by itself.
On the other side we have general relativity with Einstein's field equations, a differential equation. How can we get this together?
In fact that's very easy. I now describe general relativity using a Lagrangian density.
Lagrangian densities have a pleasant property: I may just add them.
Here it is, the Lagrangian density of quantum gravity. It's rather long. In particular these three dots are very long.
However, general relativity + quantum field theory = quantum gravity.
This has been well-known since the 1960s. So where is the problem of quantum gravity?
The problem: When I try to apply time-ordered perturbation theory, Feynman diagrams, to this combined formula, …
… then this works excellently for this part, but it doesn't work at all for that part. There it yields infinity.
Well, this part also yields infinity, but there is a mathematical trick how we can, in some sense, reel in that infinity.
That's the so-called renormalisation. However it doesn't work for this part. Here we still get infinity.
These are the non-renormaliseable divergences you always hear about when you ask:
Where is the problem of combining quantum field theory and general relativity.
It is not possible to describe gravity using Feynman diagrams, using so-called gravitons, particles, which convey gravity.
These are the non-renormaliseable divergences.
As a theoretical physicist I always want to know the details.
So I ask: It doesn't work. Is that just a technical problem, or is there a fundamental contradiction, causing that it doesn't work?
I want to fathom that fundamental contradiction.
What if we don't use Feynman diagrams, but instead we convert everything to a differential equation?
Of course it will become extremely long. However, 15000 Feynman diagrams aren't exactly short, either.
Let's just try. This leads us to the next problem.
Describing pair annihilation and creation only works, so far, using Feynman diagrams or other methods using time-ordered perturbation theory.
You need time-ordered perturbation theory, and as soon as you apply that to gravity, it yields infinity.
What's this supposed to mean – “only works so far”? I want to have a very close look at this.
What exactly goes wrong? Is this just a technical problem, or is there a fundamental contradiction?
Now I can phrase my research question, my question about the question.
Why isn't it possible to describe pair annihilation and creation using a differential equation?
Now we have arrived at the results of my own research.
To pursue this question I just try to describe pair annihilation and creation using a differential equation.
Then I'll see what goes wrong. I write that down. This will answer my question about the question.
I want to try that out. The first thing I encounter are mathematical obstacles. So I must do a lot of mathematical exercises.
That's what I did, and I claim that I have understood the mathematical structures, which show up when I set up that differential equation.
In fact it wouldn't be necessary to understand these structures for applying Feynman diagrams, for you can use them just as a recipe.
But I want to do something different than Feynman diagrams, so I had a very, very close look on those mathematical structures.
Then I tried to set up the differential equation. Now the big question reads: What precisely didn't work out?
On 5 May 2018 at 22:39h I saw the differential equation in front of me.
My attempt to find out why it is not possible to set up that differential equation has failed. I failed to fail.
I succeeded to set up the differential equation. That was unexpected.
Okay, this part seems to work. Then it must be something else which goes wrong.
So probably it's not possible to solve the differential equation. It might yield zero or infinity or something else, which is useless.
So let's see why the computer fails to solve it.
I'm afraid it will actually fail right now. I'll now try again to show you a film of what my computer made out of this differential equation.
Too bad. Perfect. I'll now switch to my slides and try to convey the film by some other means. We'll find a way.
So it won't be a film, but a sequence of still pictures.
Here we see the still picture again.
I'll now go a little to the future.
Now I let these particles approach each other.
On the left we see, to put it simply, an electron. Actually it is a many-particle wave function. And on the right we see, so to say, a positron.
Now I let them collide. Now they overlap. At first we don't see very much.
This movie isn't linked yet, I'll do that soon, so you can download it. We can do that.
The electron and the positron have touched. Now they permeate.
We can still identify them, but we also see some blue glow.
This blue glow are gamma quanta, which got produced because the two things have permeated each other. But there is more to happen.
There is not much left of the electron and the positron. This cyan and rose are gamma quanta.
From the electron and the positron there are only small remainders left, over here, and over there.
Most of them has annihilated. This is the so-called pair annihilation. I just removed the gamma quanta from the image.
This was a simulation of pair annihilation, calculated by solving a differential equation of quantum field theory.
This was pair annihilation. What about pair creation?
Here we see two gamma quanta which we let loose on each other. Two light rays.
Or rather light waves. I can imagine this as two stones dropped into water.
Waves emerge and spread. After some time they will overlap.
Now the waves overlap, and we see a small glow in the middle.
This glow are newly generated electrons and positrons.
Here we see a simulation of pair creation, simulated by solving a differential equation of quantum field theory.
So I tried to fail setting up and solving a differential equation, but I failed.
It doesn't fail. It works. We can describe pair annihilation and creation by a differential equation.
This is, at least from my point of view a huge sensation, which I announce today in public for the first time.
What does this imply? What is it good for?
This is a new method to calculate many-particle quantum wave functions. This is one of the many meanings of the word “quantisation”.
So I accidentally found a new method of quantisation, an alternative to Feynman diagrams.
Well, it isn't really an alternative to Feynman diagrams. Not yet.
Yes, I'm calculating without Feynman diagrams, and it works.
However the precision is far apart from what is possible using Feynman diagrams.
11 decimals … I would be glad if I had just one.
So far I have qualitative statements, that particles are annihilated and created.
But my statements how much is happening there are yet very rudimentary.
Anyway it is possible to describe many-particle quantum effects using a differential equation.
But what is it good for? We already have the Feynman diagrams, and they work perfectly. Yes, but not for gravity.
My method will still work when we include gravity.
How do we describe gravity? Via Einstein's field equations. That's a differential equation.
That we can describe general relativity via a differential equation is entirely clear.
That this also works for pair annihilation and creation was not clear.
I successfully implemented that and I accidentally developed a new quantisation method.
And it is compatible with general relativity.
And even more: It is also compatible with … Ah – sorry. One less.
I found a method how to investigate quantum gravity.
But I'm working without Feynman diagrams, so I won't be able to calculate gravitons.
“Graviton” means, in the language of time-ordered perturbation theory, that I describe gravity by mediating particles, which I call gravitons.
I can describe gravitons by Feynman diagrams, but that's precisely what I'm not doing. My simulation won't be able to show gravitons.
Maybe I get something similar which even behaves like gravitons and even can be, one day, described by Feynman diagrams.
However for now, my method is without gravitons.
Instead I can add general relativity. I don't think that anything can go wrong here.
But I can do even more. I can, for example, combine this with noncommutative geometry. Again, what is that?
Well that is, in a way, the theory of everything.
Noncommutative geometry was largely developed by Alain Connes, a French mathematician.
It is a logical advancement of general relativity.
General relativity says: Gravity is geometry. Noncommutative geometry says: All forces are geometry.
Not only gravity, but also electromagnetic, weak and strong interaction.
It's all geometry, but not on a curved space-time, but we have additional dimensions, so-called discrete extra-dimensions.
How to imagine this? Our well-known dimensions length, width, height, and time are continuous.
There I can measure not only 1, 2, 3, but also 1.5, 1.7, 1.72, …
That's continuous. A discrete dimension has only a few points.
For instance a discrete dimension of four points means that a particle can only be located at one of four positions.
When we consider this togethher with a “normal” universe it looks about like this.
Here we have a two-dimensional curved universe with a discrete extra dimension, which consists of four points.
Well, what is this good for?
In 1996 Chamseddine and Connes set up a formula which looks like this.
It describes a universe with a four-dimensional curved space-time and 11 discrete extra-dimensions.
For the experts: Here we see the 3x3 matrices over the complex numbers.
However they don't provide 9 extra-dimensions, but only 8.
These are Hamilton's quaternions, essentially vectors of two complex numbers, and these are the complex numbers themselves.
So here we see a normal curved space-time with 11 extra-dimensions.
Okay, so what? From this we can calculate a Lagrangian density. This one.
But we recognise it.
This is the Lagrangian of general relativity, and that, including the dots, is precisely the Lagrangian of the Standard Model of elementary particles.
This tiny formula implies both, general relativity and the Standard Model of quantum field theory.
They are together in one formula. So this formula describes quantum gravity.
But there is something leftover, this weird stuff. What's that? We don't know what that is.
I thought about this. This might be an explanation for dark matter.
There are advanced theories, advanced versions of MOND theory – modified Newtonian dynamics.
There are general-relativistic advancements, which can even explain the Bullet cluster.
I think it's worth a try to look at these terms, whether they can explain dark matter. That would be the next sensation.
So this formula is extremely powerful, and it has been known for 25 years.
Why didn't Connes and his co-workers get the Nobel price? They wrote down quantum gravity.
Yes, but this cannot be quantised. What does that mean?
This Lagrangian density has been known since the 1960s, but we cannot do calculations with it.
This can be calculated only with Feynman diagrams, and that cannot be calculated with Feynman diagrams.
It is not possible to calculate both parts together using the same method.
There is no calculation method for many-particle quantum wave functions which could be applied to investigate this Lagrangian density.
Now it exists. Using my method I see no obstacles to “quantise” this Lagrangian density with these extra terms.
We can calculate it using a many-particle quantum wave function.
I have reached the end of my speech. What did we see?
For the first time we can describe pair annihilation and creation using a differential equation.
This is a new method to handle many-particle quantum wave functions. I developed a new quantisation method.
It works without using Feynman diagrams. I'm not using time-ordered perturbation theory. Instead I directly solve the differential equation.
This implies that I won't see any gravitons as mediating particles.
But maybe we see something similar. We can investigate this now.
My new quantisation method is compatible with general relativity and with noncommutative geometry.
Thus this formula is a hot candidate for the theory of everything.
While trying to find out why there is no theory of everything I accidentally found it, and it is 25 years old.
I quantised it. That way I proved that it really is the theory of everything.
Well, I can tell you a lot at this place. Just a moment, I'll upload it … We can spare that second …
When you reload this URL now, you can look up all this in two scientific publications.
One has 6 pages, the other one 110. Both in English, of course.
When you have studied physics and you understood nonrelativistic quantum theory and special relativity, …
… you are well-placed to understand at least the longer one of both papers.
Okay, what comes next? How can we proceed from here?
First of all, using what I have developed you can do much more.
My objective was the combination with general relativity.
In fact this is a method to calculate electrons and positrons rather precisely.
This can be valuable for solid-state physics for examining new materials numerically.
For instance: How must I change this material to obtain specific electrical properties? Then we can, for example, build faster computers.
Then I have, at least as far as I can see, developed a new method to solve a partial differential equation.
This has applications inside mathematics.
Using this new method to solve differential equations we can do many things I cannot even guess right now.
This might become valuable in constructing machines and buildings.
For me as a theoretical physicists other things are more interesting.
I said several times that we can combine this with general relativity. Yes, that's possible, but one has to do it, and that's a lot of work.
I don't doubt that it will work out, because I developed this method in such a way that this will work out.
I take something of which I know that it works for general relativity and apply it to quantum theory, and I look how far I get.
Result: I get through. It should be feasible to combine this with general relativity, but it's a lot of work.
I'm somewhat more scared of the extension to the full Standard Model.
For one thing because it is huge. I must convert the Lagrangian density into a differential equation. It will be very long.
For another thing I only calculated electrons and positrons so far.
I'd like to calculate muons, and quarks, and Higgs bosons, and so on. There are many things which can go wrong.
Here again I only expect just a lot of work.
However it might be that it will become so much work for the computer that it cannot be calculated using today's computers.
That's the worst thing which can happen. But I don't see anything which could go wrong concerning the concepts.
I'm worrying mostly about the Higgs boson. Maybe this will require new algorithms, or we overwhelm everything by calculation power.
Than it can happen that the calculation power does not suffice.
What else can we do? We can apply everything to noncommutative geometry and look what kind of quantum gravity emerges.
Maybe this approach provides an explanation for one of our last big mysteries, dark matter.
This was my speech. I apologise for the technical problems.
I hope you could hear me and see my slides. The movies didn't work out. Next time.
Thank you for participating in this load test for our online learning platform.
I thank you for your interest, for listening, for coming, and I'm now available for questions. Many thanks.