What is quantum cryptography and how does it work?

[This is a transcript of the video embedded below. Some parts of the text may not make sense without the graphics in the video.]

If you punch your credit card number into a website and hit “submit”, I bet you don’t want to have twenty fraudulent charges on your bank account a week later. This is why all serious online retailers use encryption protocols. In this video, I want to tell you how quantum mechanics can help us keep secrets safe.


Before I get to quantum cryptography, I briefly have to tell you how the normal, non-quantum cryptography works, the one that most of the internet uses today. If you know this already, you can use the YouTube tool bar to jump to the next chapter.

The cryptographic codes that are presently being used online are for the most part public key systems. The word “key” refers to the method that you use to encrypt a message. It’s basically an algorithm that converts readable text or data into a mess, but it creates this mess in a predictable way, so that the messing up can be undone. If the key is public, this means everybody knows how to encrypt a message, but only the recipient knows how to decrypt it.

This may sound somewhat perplexing, because if the key is public and everybody knows how to scramble up a message, then it seems everybody also knows how to unscramble it. It does not sound very secure. But the clever part of public key cryptography is that to encode the message you use a method that is easy to do, but hard to undo.

You can think of this as if the website you are buying from gives you, not a key, but an empty treasure chest that locks when you close it. You take the chest. Put in your credit card number, close it. And now the only person who can open it, is the one who knows how to unlock it. So your message is safe to send. In practice that treasure chest is locked by a mathematical problem that is easy to pose but really hard to solve.

There are various mathematical problems that can, and that are being used, in cryptographic protocols for locking the treasure chest. The best known one is the factorization of a large number into primes. This method is used by the algorithm known as RSA, after its inventors Rivest (i as in kit), Shamir, and Adleman. The idea behind RSA is that if you have two large prime numbers, it is easy to multiply them. But if you only have the product of the two primes, then it is very difficult to find out what its prime-factors are.

For RSA, the public key, the one that locks the treasure chest, is a number that is derived from the product of the primes, but does not contain the prime factors themselves. You can therefore use the public key to encode a message, but to decode it, you need the prime factors, which only the recipient of your message has, for example the retailer to whom you are sending your credit card information.

Now, this public key can be broken, in principle, because we do know algorithms to decompose numbers into their prime factors. But for large numbers, these algorithms take very, very long, to give you a result, even on the world’s presently most powerful computers. So, maybe that key you are using can be broken, given a hundred thousand years of computation time. But really who cares. For all practical purposes, these keys are safe.

But here’s the thing. Whether or not someone can break one of these public keys depends on how quickly they can solve the mathematical problem behind it. And quantum computers can vastly speed up computation. You can see the problem: Quantum computers can break cryptographic protocols, such as RSA, in a short time. And that is a big security risk.

I explained in a previous video what quantum computers are and what to expect from them, so check this out if you want to know more. But just how quantum computers work doesn’t matter so much here. It only matters that you know, if you had a powerful quantum computer, it could break some public key cryptosystems that are currently widely being used, and it could do that quickly.

This is a problem which does not only affect your credit card number but really everything from trade to national security. Now, we are nowhere near having a quantum computer that could actually do such a computation. But the risk that one could be built in the next decades is high enough so that computer scientists and physicists have thought of ways to make public key cryptography more secure.

They have come up with various cryptographic protocols that cannot be broken by quantum computers. This is possible by using protocols which rely on mathematical problems for which a quantum computer does not bring an advantage. This cryptography, which is safe from quantum computers is called “post-quantum cryptography” or, sometimes, “quantum resistant cryptography”.

Post-quantum cryptographic protocols do not themselves use quantum effects. They have the word “quantum” in their name merely to say that they cannot be broken even with quantum computers. At least according to present knowledge. This situation can change because it’s possible that in the future someone will find a way to use a quantum computer to break a code currently considered unbreakable. However, at least at the moment, some cryptographic protocols exist for which no one knows how a quantum computer could break them.

So, computer scientists have ways to keep the internet safe, even if someone, somewhere develops a powerful quantum computer. Indeed, most nations already have plans to switch to post-quantum cryptography in the coming decade, if not sooner.

Let us then come to quantum cryptography, and its application for “quantum key distribution”. Quantum key distribution is a method for two parties to securely share a key that they can then use to encode messages. And quantum physics is what helps keep the key safe. To explain how this works, I will again just use the simplest example, that’s a protocol known as BB Eighty-four, after the authors Bennett and Brassard and the year of publication.

When physicists talk about information transfer, they like to give names to senders and receivers. Usually they are called Alice and Bob, so that’s what I will call them to. Alice wants to send a secret key to Bob so they can then have a little chat, but she does not want Bob’s wife, Eve, to know what they’re talking about. In the literature, this third party is normally called “Eve” because she is “eavesdropping”, hahaha, physics humor.

So, Alice creates a random sequence of particles that can have spin either up or down. She measures the spin of each particle and then sends it to Bob who also measures the spin. Each time they measure spin up, they note down a zero, and each time they measure spin down, they note down a one. This way, they get a randomly created, shared sequence of bits, which they can use to encode messages.

But this is no good. The problem is, this key can easily be intercepted by Eve. She could catch the particle meant for Bob in midflight, measure it, note down the number, and then pass it on to Bob. That’s a recipe for disaster.

So, Alice picks up her physics textbooks and makes the sequence of particles that she sends to Bob more complicated.

That the spin is up or down means Alice has to choose a direction along which to create the spin. Bob has to know this direction to make his measurement, because different directions of spins obey an uncertainty relation. It is here where quantum mechanics becomes important. If you measure the direction of a spin into one direction, then the measurement into a perpendicular direction is maximally uncertain. For a binary variable like the spin, this just means the measurements in two orthogonal directions are uncorrelated. If Alice sends a particle that has spin up or down, but Bob mistakenly measures the spin in the horizontal direction, he just gets left or right with fifty percent probability.

Now, what Alice does is to randomly choose whether the particles’ spin goes in the up-down or left-right direction. As before, she sends the particles to Bob, but – and here is the important bit – does not tell him whether the particle was created in the up-down or left-right direction. Since Bob does not know the direction, he randomly picks one for his measurement. If he happens to pick the same direction that Alice used to create the particle, then he gets, as previously, a perfectly correlated result. But if he picks the wrong one, he gets a completely uncorrelated result.

After they have done that, Alice sends Bob information about which directions she used. For that, she can use an unencrypted channel. Once Bob knows that, he discards the measurements where he picked the wrong setting. The remaining measurements are then correlated, and that’s the secret key.

What happens now if Eve tries to intersect the key that Alice sends? Here’s the thing: She cannot do that without Bob and Alice noticing. That’s because she does not know either which direction Alice used to create the particles. If Eve measures in the wrong direction – say, left-right instead of up-down – she changes the spin of the particle, but she has no way of knowing whether that happened or not.

If she then passes on her measurement result to Bob, and it’s a case where Bob did pick the correct setting, then his measurement result will no longer be correlated with Alice’s, when it should be. So, what Alice and Bob do is that they compare some part of the sequence they have shared, again they can do that using an unencrypted channel, and they can check whether their measurements were indeed correlated when they should have been. If that’s not the case, they know someone tried to intercept the message. This is what makes the key safe.

The deeper reason this works is that in quantum mechanics it is impossible to copy an arbitrary state without destroying it. This is known as the no-cloning theorem, and this is ultimately why Eve cannot listen in without Bob and Alice finding out.

So, quantum key distribution is a secure way to exchange a secret key, which can be done either through optical fiber or just free space. Quantum key distribution actually already exists and is being used commercially, though it is not in widespread use. However, in this case the encoded message itself is still sent through a classical channel without quantum effects.

Quantum key distribution is an example for quantum cryptography, but quantum cryptography also more generally refers to using quantum effects to encode messages, not just to exchange keys. But this more general quantum cryptography so far exists only theoretically.

So, to summarize: “Post quantum cryptography” refers to non-quantum cryptography that cannot be broken with a quantum computer. It exists and is in the process of becoming widely adopted. “Quantum key distribution” exploits quantum effects to share a key that is secure from eavesdropping. It does already exist though it is not widely used. “Quantum cryptography” beyond quantum key distribution would use quantum effects to actually share messages. The theory exists but it has not been realized technologically.

I want to thank Scott Aaronson for fact-checking parts of this transcript, Tim Palmer for trying to fix my broken English even though it’s futile, and all of you for watching. See you next week.

Path Dependence and Tipping Points

[This is a transcript for the video embedded below. Part of the text may not make sense without the graphics in the video.]



Most of the physics we learn about in school is, let’s be honest, a little dull. It’s balls rolling down slopes, resistance proportional to voltage, pendula going back and forth and back and forth... wait don’t fall asleep, that’s what I will not talk about today. Today I will talk about weird things that can happen in physics: path dependence and tipping points.

I want to start with chocolate. What’s chocolate got to do with physics? Chocolate is a crystal. No, really. A complicated crystal, alright, but a crystal, and a truly fascinating one. If you buy chocolate in a store you get it in this neat smooth and shiny form. It melts at a temperature between thirty-three and thirty-four degrees Celsius, or about ninety-two degrees Fahrenheit. That’s just below body temperature, so the chocolate will melt if you stuff it into your mouth but not too much earlier. Exactly what you want.

But suppose your chocolate melts for some other reason, maybe you left it sitting in the sun, or you totally accidentally held a hair drier above it. Now you have a mush. The physicist would say the crystal has undergone a phase transition from solid to liquid. But no problem, you think, you will just put it into the fridge. And sure enough, as you lower the temperature, the chocolate undergoes another phase transition and turns back into a solid.

Here’s the interesting thing. The chocolate now looks different. It’s not only that it has lost some of its original shape, it actually has a different structure now. It’s not as smooth and shiny as it previously wass. Even weirder, it now melts more easily! The melting point has dropped from about thirty-four to something like twenty-eight degrees Celsius. What the heck is going on?

What happens is that if the chocolate melts and becomes solid again, it does not form the same crystal structure that it had before. Instead, it ends up in a mixture of other crystal structures. If you want to get the crystal structure that chocolate is normally sold in, you have to cool it down very carefully and add seeds for the structure you want to get. This process is called “tempering”. The crystal structure which you get with tempering, the one that you normally buy, is actually unstable. Even if you do not let it melt, it will decay after some time. This is why chocolate gets “old” and then has this white stuff on the surface. Depending on what chocolate you have, the white stuff is sugar or fat or both, and it tells you that the crystal structure is decaying.

For our purposes the relevant point is that the chocolate can be in different states at the same temperature, depending on how you got there. In physics, we call this a “path dependence” of the state of the system. It normally means that the system has several different states of equilibrium. An equilibrium state is simply one that does not change in time. Though, as in the case of chocolate these states may merely be long-lived and not actually be eternally stable.

Chocolate is not exactly the example physicists normally use for path dependence. The go-to example for physicists is the magnetization of a ferromagnet. A ferromagnet is a metal that can be permanently magnetized. It’s what normal people call a “magnet”, period. The reason ferromagnets can be magnetized is that the electron shell structure means the atoms in the metal are tiny little magnets themselves. And these tiny magnets like to align their orientation with that of their neighbors.

Now, if you find a ferromagnetic metal somewhere out in the field, then its atomic magnets are almost certainly disordered and look somewhat like this. To make the illustration simpler, I will pretend that the atomic magnets can point in only one of two directions. If the little magnets are randomly pointing into one of these directions, then the metal has no overall magnetization.

If you apply a magnetic field to this metal, then the atoms will begin to align with the field because that’s energetically the most favorable state. At some point they’re just all aligned in the same direction, and the magnetization of the metal saturates. If you now turn off the magnetic field, some of those atoms will switch back again just because there’s some thermal motion and so on. However, at room temperature, the metal will keep most of the magnetization. That’s what makes ferromagnets special.

If you turn on the external magnetic field again but increase its strength into the other direction, then the atomic magnets will begin to line up pointing into that other direction until saturated. If you turn down the field back to zero, again most of them will continue to point there. Turn the external field back to the other side and you go back to saturating the magnetization in the first direction.

We can plot this behavior of the magnet in a graph that shows the external magnetic field and the resulting magnetization of the magnet. We started from zero, zero, saturated the magnetization pointing right, turned the external field to zero, but kept most of the magnetization. Saturated the magnetization pointing left, turned the field back to zero but kept most of the magnetization. And saturated the magnetization again to the right.

This is what is called the “hysteresis loop”. Hysteresis means the same as “path dependence”. Whether the magnetization of the metal points into one direction or the other does not merely depend on the external field. It also depends on how you got to that value of the field. In particular, if the external field is zero, the magnet has two different, stable, equilibrium states.

This path-dependence is also why magnets can be used to store information. Path-dependence basically means that the system has a memory.

Path-dependence sounds like a really peculiar physics-y thing but really it’s everywhere. Just to illustrate this I have squeezed myself into this T-shirt from my daughter. See, it has two stable equilibrium states. And they keep a memory of how you got there. That’s a path-dependence too.

Another common example of a path dependence are air conditioning units. To avoid a lot of switching on and off, they’re usually configured so that if you input a certain target temperature, they will begin to cool if the temperature rises more than a degree above the target temperature, but will stop cooling if the temperature has dropped to a degree below the target temperature. So whether or not the air condition is running at the target temperature depends on how you got to that temperature. That’s a path-dependence.

A common property of path-dependent systems is that they have multiple stable equilibrium states. As a reminder, equilibrium merely means it does not change in time. In some cases, a system can very suddenly switch between different equilibrium states. Like this parasol. It has a heavy weight at the bottom, so if the wind sways it a little, it will stay upright. That’s an equilibrium state. But if the wind blows too hard, it will suddenly top over. Also an equilibrium state. But a much more stable one. Even if the wind now blows into the other direction, the system is not going back to the first state.

Such a sudden transition between two equilibrium states is called a “tipping point”. You have probably heard the word “tipping point” in the context of climate models, where they are a particular pain. I say “pain” because by their very nature they are really hard to predict with mathematical modeling, exactly because there are so many path-dependencies in the system. A glacier that melts off at a certain level of carbondioxide will not climb back onto the mountain if carbondioxide levels fall. And that’s one of the better understood path-dependencies.

A much discussed tipping point in climate models is the Atlantic meridional overturning circulation. That’s a water cycle in the atlantic ocean. Warm surface water from the equator flows north. Along the way it cools and partly evaporates, which increases the density of salt in the water and makes the water heavy. The cool, salty water sinks down to the bottom of the ocean, comes back up where it came from, warms, and the cycle repeats. Why does it come back up in the same place? Well, if some water sinks down somewhere, then some water has to come up elsewhere. And a cycle is a stable configuration, so once the system settles in the cycle, it just continues cycling.

But. This particular cycle is not the only equilibrium configuration and the system does not have to stay there. In fact, there’s a high risk this water cycle is going to be interrupted if global temperatures continue to rise.

That’s because ice in the arctic is mostly fresh water. If it melts in large amounts, as it presently does, this reduces the salt content of the water. This can prevent the water in the atlantic overtuning circulation from sinking down and thereby shut off the cycle.

Now, this circulation is responsible for much of the warm wind that Europe gets. Did you ever look at a world map and noticed that the UK and much of middle Europe is North of Montreal? Why is the climate in these two places so dramatically different? Well, that atlantic overturning circulation is one of the major reasons. If it shuts off, we’re going to see a lot of climate changes very suddenly. Aaaand it’s a path-dependent system. Reducing carbondioxide after we’ve crossed that tipping point will not just turn the circulation back on. And some evidence suggests that this cycle is weakening already.

There are many other tipping points in climate models, that, once crossed can bring sudden changes that will stay with us for thousands of years, even if we bring carbondioxide levels back down. Like the collapse of the Greenland and West Antarctic Ice Sheet. If warming continues, the question is not whether it will happen but just when. I don’t want to go through this whole list, I just want to make clear that tipping points are not fear mongering. They are a very real risk that should not be dismissed easily.

I felt it was necessary to spell this out because I recently read an article by Michael Shellenberger who wrote: “Speculations about tipping points are unscientific because levels of uncertainty and complexity are too high, which is exactly why IPCC does not take such scenarios seriously.”

This is complete rubbish. First, tipping points are covered in the IPCC report, it’s just that they are not collected in a chapter called “tipping points,” they are called large scale singular events. I found this out by googling “tipping points IPCC”, so not like it would have taken Shellenberger much of an effort to get this right. Here is a figure from the summary for policy makers about the weakening of the atlantic overturning circulation, that’s the tipping point that we just talked about. And here they are going on about the collapse of ice sheets, another tipping point.

Having said that, tipping points are not emphasized much by the IPCC, but that’s not because they do not take them seriously, but because the existing climate models simply are not good enough to make reliable predictions for exactly when and how tipping points will be crossed. That does not mean tipping points are unscientific. Just because no one can presently put a number to the risk posed by tipping points does not mean the risk does not exist. It does mean, however, that we need better climate models.

Path-dependence and tipping points are cases where naïve extrapolations can badly fail and they are common occurrences in non-linear systems, like the global climate. Just because we’ve been coping okay with climate change so far does not mean it will remain that way.

I want to thank Michael Mann for checking parts of this transcript.

What is a singular limit?

Imagine you bite into an apple and find a beheaded worm. Eeeh. But it could have been worse. If you had found only half a worm in the apple, you’d now have the other half in your mouth. And a quarter of worm in the apple would be even worse. Or a hundredth. Or a thousandth. If we extrapolate this, we find that the worst apple ever is one without worm.

Eh, no, this can’t be right, can it? What went wrong?

I borrowed the story of the wormy apple from Michael Berry, who has used it to illustrate a “singular limit”. In this video, I will explain what a singular limit is and what we can learn from it.


A singular limit is also sometimes called a “discontinuous limit” and it means that if some variable gets closer to a certain point, you do not get a good approximation for the value of a function at this point. In the case of the apple, the variable is the length of the worm that remains in the apple, and the point you are approaching is a worm-length of zero. The function is what you could call the yuckiness of the apple. The yuckiness increases the less worm is left in the apple, but then it suddenly jumps to totally okay. This is a discontinuity, or a singular limit.

You can simulate such a function on your smartphone easily if you punch in a positive number smaller than one and square it repeatedly. This will always give zero, eventually, regardless of how close your original number was to 1. But if you start from 1 exactly, you will stay at 1. So, if you define a function from the limit of squaring a number infinitely often, that would be f(x) is the limit n to infinity of x²ⁿ, where n is a natural number, then this function makes a sudden jump at x equals to 1.

This is a fairly obvious example, but singular limits are not always easy to spot. Here is an example from John Baez that will blow your mind, trust me, even if you are used to weird math. Look at this integral. Looks like a pretty innocent integral over the positive, real numbers. You are integrating the function sin(t) over t, and the result turns out to be π/2. Nothing funny going on.

You can make this integral a little more complicated by multiplying the function you are integrating with another function. This other function is just the same function as previously, except that it divides the integration variable by 101. If you integrate the product of these two functions, it comes out to be π/2 again. You can multiply these two functions by a third function in which you divide the integration variable by 201. The result is π/2 again. And so on.

We can write these integrals in a nicely closed form because zero times 100 plus 1 is just one. So, for an arbitrary number of factors, that we can call N, you get an integral over this product. And you can keep on evaluating these integrals, which will give you π/2, π/2, π/2 until you give up at N equals 2000 or what have you. It certainly looks like this series just gives π/2 regardless of N. But it doesn’t. When N takes on this value:

15,341,178,777,673,149,429,167,740,440,969,249,338,310,889

The result of the integral is, for the first time, not π/2, and it never becomes π/2 for any N larger than that. You can find a proof for this here. The details of the proof don’t matter here, I am just telling you about this to show that mathematics can be far weirder than it appears at first sight.

And this matters because a lot of physicists act like the only numbers in mathematics are 2, π, and Euler’s number. If they encounter anything else, then that’s supposedly “unnatural”. Like, for example, the strength of the electromagnetic force relative to the gravitational force between, say, an electron and a proton. That ratio turns out to be about ten to the thirty-nine. So what, you may say. Well, physicists believe that a number like this just cannot come out of the math all by itself. They called it the “Hierarchy Problem” and it supposedly requires new physics to “explain” where this large number comes from.

But pure mathematics can easily spit out numbers that large. There isn’t a priori anything wrong with the physics if a theory contains a large number. We just saw one such oddly specific large number coming out of a rather innocent looking integral series. This number is of the order of magnitude 10⁴³. Another example of a large number coming out of pure math is the dimension of the monster group that is about 10⁵³. So the integral series is not an isolated case. It’s just how mathematics is.

Let me be clear that I am not saying these particular numbers are somehow relevant for physics. I am just saying if we find experimentally that a constant without units is very large, then this does not mean math alone cannot explain it and it must therefore be a signal for new physics. That’s just wrong.

But let me come back to the singular limits because there’s more to learn from them. You may put the previous examples down as mathematical curiosities, but they are just very vivid demonstrations for how badly naïve extrapolations can fail. And this is something we do not merely encounter in mathematics, but also in a lot of physical systems.

I am here not thinking of the man who falls off the roof and, as he passes the 2nd floor, thinks “so far, so good”. In this case we full well know that his good luck will soon come to an end because the surface of earth is in the way of his well-being. We have merely ignored this information because otherwise it would not be funny. So, this is not what I am talking about. I am talking about situations where we observe sudden changes in a system that are not due to just willfully ignoring information.

An example you are probably familiar with are phase transitions. If you cool down water, it is liquid, liquid, liquid, until suddenly it isn’t. You cannot extrapolate from the water being liquid to it being a solid. It’s a pattern that does not continue. There are many such phase transitions in physical systems where the behavior of a system suddenly changes, and they usually come along with observable properties that make sudden jumps, like entropy or viscosity. These are singular limits.

Singular limits are all over the place in condensed matter physics, but in other areas, physicists seem to have a hard time acknowledging their existence. An example that you find frequently in the popular science press are calculations in a universe with a negative cosmological constant, that’s the so-called Anti-de Sitter space, which falsely raise the impression that these calculations tell us something about the real world, which has a positive cosmological constant.

A lot of physicists believe the one case tells us something about the other because, well, you could take the limit from a very small but negative cosmological constant to a very small but positive cosmological constant, and then, so they argue, the physics should be kind of the same. But. We know that the limit from a small negative cosmological constant to zero and then on to positive values is a singular limit. Space-time has a conformal boundary for all values strictly smaller than zero, but no longer for exactly zero. We have therefore no reason to think these calculations that have been done for a negative cosmological constant tell us anything about our universe, which has a positive cosmological constant.

Here are a few examples of such misleading headlines. They usually tell stories about black holes or wormholes because that’s catchy. Please do not fall for this. These calculations tell us nothing, absolutely nothing, about the real world.

Do we really travel through time with the speed of light?

[Note: This transcript will not make much sense without the equations that I show in the video.]

Today I want to answer a question that was sent to me by Ed Catmull who writes:

“Twice, I have read books on relativity by PhDs who said that we travel through time at the speed of light, but I can’t find those books, and I haven’t seen it written anywhere else. Can you let me know if this is right or if this is utter nonsense.”

I really like this question because it’s one of those things that blow your mind when you learn about them first, but by the time you have your PhD you’ve all but forgotten about them. So, the brief answer is: It’s right, we do travel through time at the speed of light. But, as always, there is some fine-print to what exactly this means.

At first, it does not seem to make much sense to even talk about a speed in time. A speed is distance per time. So, if you travel in time, a speed would be time per time, and you would end up with the arguably correct but rather lame insight that we travel through time at one second per second.

This, however, is not where the statement that we travel through time at the speed of light comes from. It comes from good, old Albert Einstein. Yes, that guy again. Einstein based his theory of special relativity on an idea from Hermann Minkowski, which is that space and time belong together to a common entity called space-time. In space-time, you do not only have the usual three directions of space, you have a fourth direction, which is time. In the following, I want to show you a few equations, and for that I will, as usual, call the three directions of space, x, y, and z, and t stands for time.

Now, here’s the problem. You can add directions like North and West to get something like North-West. But you cannot add space and time because that’s like adding apples and oranges. Space and time have different units, so if you want to add them, you have to put a constant in front of one of them. It does not matter where you put that constant, but by convention we put it in front of the time-coordinate. The constant you have to put here so that you can add these directions must have units of space over time, so that’s a speed. Let’s call it “c”.

You all know that c is the speed of light, but, and this is really important, you do not need to know this if you formulate special relativity. You can put a dummy parameter there that could be any speed, and you will later find that it is the speed of massless particles. And since we experimentally know that the particles of light are to very good precision massless, that constant is then also the speed of light.

Now, of course there is a difference between time and space, so that can’t be all there is to space-time. You can move around in space either which way, but you cannot move around in time as you please. So what makes time different from space in Einstein’s space-time? What makes time different from space is the way you add them.

If you want to calculate a distance in space, you use Euclid’s formula. A distance, in three dimension, is the square-root of the of the sum of the squared distances in each direction of space. Here the Δx is a difference between two points in direction x, and Δy and Δz are likewise differences between two points in directions y and z.

But in space-time this works differently. A distance between two points in in space-time is usually called Δs, so that’s what we will call it too. A distance in space-time is now the square-root of minus the squares of the distances in each of the dimensions of space, plus c square times the squared distance in time.

Maybe let me mention that some old books on Special Relativity use a different notation, in which, instead of just putting a minus in the space-time distance, one uses a prefactor for the time-coordinate that is i times c. This has the exact same effect because the i square will give you a minus. The I turns out to be useless otherwise though, so this notation is not used today any more.

But why would you define a space-time distance like this, why not just all plusses? Well, for one, if you do it differently it doesn’t work. It would not correctly describe observation. That’s an answer, but not a very insightful one, so here is a better answer.

Einstein based special relativity on the idea that the speed of light is the same for all observers. You cannot do this in a Euclidean space where all the signs are plusses. But you can do it if one of the signs is different relative to the others. 

That’s because a space-time distance that is zero for one observer is zero for all observers. This is also the case in Euclidean space, but in Euclidean space, this just means zero in each of the directions of space. But what does a zero distance mean in space-time? Well, let’s find out. For simplicity, let us look at only one dimension of space. So if the distance in space-time is zero, this means that the distance in space divided by the distance in time equals plus or minus c. And that’s the same for all observers. So this speed, c, is an invariant speed.

But, well, we are not light, so we do not travel with the speed of light through space, and we do actually cover a distance in space-time. So let us look at this equation for the space-time distance again. Now let us divide this by the time difference. Now what you have on the left side is the space-time distance per time. And under the square root you have roughly something like the squares of the velocities in each of the directions of space. Plus c².

And there you have it. Relative to yourself, you do not move through space, so these velocities are zero. You then only move into the time-like direction, and in this direction, you move with the speed of light. So, we indeed all travel through time with the speed of light.

I always try to show you equations because physics is all about equations. But to really understand what these equations mean, you have to use them yourself. A great place to do this is Brilliant, who have been sponsoring this video. Brilliant offers a large variety of interactive courses on topics in science and mathematics. They do for example have a course on Special Relativity, that will teach you all you need to know about space-time diagrams, Lorentz-transformations, and 4-vectors.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Your sudden enthusiasm for virtual meetings is beginning to worry me

Screenshot from Zoom meting. Image Source: Reshape.

I live about 100 kilometers away from my workplace. A round trip takes at least 2 hours, up to 4 by public transport. That’s why, for the past 5 years, I’ve had a home-office contract which allows me to do part of my job remotely.

My husband works for a company that has sections in several other countries, including India, the USA, and Great Britain. He, too, is used to teleconferences with participants from several time-zones.

This makes me think my family was probably better prepared for the covid lockdown than many others. For the same reason though, we also had more time to contemplate the pros and cons of remote collaboration.

The pros are clear: Less time wasted in transit. Less carbon dioxide emitted. Less germs circulated.

And with more people in the same situation, the pros have proliferated. I have, for example, been thrilled to see the spike in online seminars. Suddenly, even I am able to find seminars that are actually interesting for my research! Better still, if it turns out they’re not as interesting as anticipated, no one notices if I virtually sneak out. Also, asking for a virtual meeting has become routine. Everyone is now familiar with screen sharing and prepared to tolerate the hassle of lagging vids or chopped audios.

These have been positive developments, and many of them deserve to be carried forward. Traveling for seminars or colloquia has long been absurdly out-of-date. We all know that a lot of speakers will give the same seminar dozens of times to largely disinterested audiences, when those who actually wanted to hear it could as well have called into the same online meeting, or watched a recording.

Or consider this. I have frequently gotten invitations from overseas institutions that were prepared to fly me in and out for giving a one-hour talk. This isn’t only ecologically insane, it’s also a bad use of researchers’ time. A lot of my colleagues work while on planes and in airports, and of course I do, too, but let’s be honest: It’s not quality time. Traveling is disruptive, both mentally and metabolically. And that’s leaving aside that it screws up the work-life balance.

So, yes, scientists could certainly slim down those seminar series and cut back traveling quite a bit. But as researchers are becoming more familiar with virtual meetings and teleconferences, I fear some of them are getting carried away.

I’ve seen scientists on social media seriously discussing that seminar series should remain online-only even post-pandemic. Virtual conferences are supposedly better than the real thing. And if you listen to them, there’s nothing, it seems, you can’t get done on Zoom.

Let us therefore talk about the cons.

Virtual collaborations work well as long as you know the people in real life already. Even with both audio and video, a lot of information that humans draw on to efficiently communicate is missing. Through a screen, you neither get body language nor the context from chatter in the hallway or just from physically being in the same room. These cues are important for deliberation and argumentation to work properly.

I know this sounds somewhat Neanderthal, but fact is that evolution didn’t prepare us to communicate  through webcams.

This has long been known to sociologists who therefore recommend that teams which collaborate remotely meet in person at least a few times a year, a recommendation that my husband’s employer strictly follows. The occasional in-person meeting, so the idea, provides team members with the required information to understand where the others are coming from. It is especially important to introduce new members to a group.

A good starting point to get a sense of the troubles that remote collaboration can bring is the 2005 report by the (US-American) National Defense Research Institute on “Challenges in Virtual Collaboration”. Summarizing the published literature, they find that during video- and audio-conferences “local coalitions can form in which participants tend to agree more with those in the same room than with those on the other end of the line” and that computer-mediated communication has “shown to increase polarization, deindividuation, and disinhibition. That is, individuals may become more extreme in their thinking, less sensitive to interpersonal aspects of their messages, and more honest and candid.”

Online-only scientific collaboration and conferences would therefore most likely work well for some time, but eventually communication would suffer. Especially those who currently praise the zoomiverse for its supposed inclusivity, as this recent piece in SciAm, simply have not thought it through.

You see, regardless of how much effort we put into online conferencing and meeting, there will still be people who know each other in real life. These will be those who just happen to work or live near each other, or who have the funds to travel. Unless you actually want to forbid everyone to meet in real life, this will create a two-class community. Those who can meet. And those who can’t.

At present, most funding agencies acknowledge the need to occasionally see each other in person to collaborate effectively. If that would no longer be the case, then it would be especially the already disadvantaged people who would suffer because they would become remote-only participants. The Ivy League, I am sure, would find a way to continue having drinks together one way or the other.

None of this is to say that I am against virtual conferences or remote collaboration. But international collaboration has been a boon to science. And abstract ideas, like the ones we deal with in the foundations of physics, are hard to get across; having to pipe them through glass fiber cables doesn’t help. As we discuss how to reduce traveling, let us not forget that communication is absolutely essential to science. 

Flat Earth “Science”: Wrong, but not Stupid

I, as many people in science communication, am fascinated with flat earthers. Here you have a group of people steadfastly rejecting evidence that’s right in their face. Today, I want to tell you why I nevertheless think flat earthers are neither stupid nor anti-scientific. Most of them, anyway. More importantly, I also want to explain why you should not be embarrassed if you can’t remember how we know that the earth is round.

But first I have to tell you what flat earthers actually believe and how they got there. The most popular flat earth model is that of a disk where the North pole is in the middle and the south pole is an ice wall on the edge of the disk. But not all flat earthers sign up to this. An alternative is the so-called bipolar model where both poles are on the disk, surrounded by water that’s held by a rim of something, maybe ice or rocks. And a minority of flat earthers believe that earth is really an infinite plane.

They mostly agree though that gravity does not exist, and that the observations we normally attribute to gravity come instead from the upward acceleration of the flat earth. As a consequence, the apparent gravitational acceleration is the same everywhere on earth. I explained last week that this is in conflict with evidence – we know that the gravitational acceleration is most definitely not the same everywhere on earth.

The idea that gravity is due to upward acceleration also causes other problems. For example, you have to assume that the moon and the sun accelerate along with the flat earth so we don’t just run into them. That’s an ad-hoc assumption which disfavors the flat earth hypothesis against models where the orbits of the moon and the sun can be calculated from the gravitational law.

But that’s not the only problem. You also have to get the moon and the sun to somehow circle around over the disk to explain day and night and the phases of the moon. To get the day-night cycle to be noticeable, you have to shrink the sun and move it closer to the earth. 

You also have to somehow get the radiation of the sun to be directional. That’s many more ad hoc assumption. But even with those assumptions, the size of the sun will change during the day more than we observe. And no one has ever successfully predicted solar eclipses on a flat earth, or calculated the observed motions of the planets.

The bottom line is: it’s not easy to improve on today’s scientific standard. It was for good reasons that the hypothesis of a flat earth was abandoned more than two thousand years ago.

Some people suggested to me that flat earthers do not actually believe the earth is flat, they are just mocking people who take scientific evidence on trust. And that, let us to be honest, is something we all do to some extent every now and then. And it is probably the case that some flat earthers are indeed just pretending. But I find it exceedingly implausible they are all just faking it. 

To begin with, they would all have to be excellent actors. Just look at some of the videos on YouTube. Also, they’re putting quite some time and, in some cases, money behind their conviction. And that’s while most of them full well know coming out as flat earther will make others doubt their sanity. All that makes it unlikely they are just in for the fun.

Now, you may want to discard flat earthers as conspiracy theorists, which some fraction of them arguably are. But I think that would be somewhat unfair to most of them. To understand why, it helps to have a look at the history of the flat earth society.

The flat earth society goes back to an Englishman by name Samuel Rowbotham, who lived in the 19th century. He was a medical doctor who believed he had proved that the earth is flat and then complained for the rest of his life that the supposed scientific authorities ignored him. He referred to his methodology as “Zeteticism” after the Greek word zeteo, “to seek”. 

By “Zeteticism” he meant an extreme version of the philosophy of empiricism. Rowbotham’s philosophy, which is still the philosophy of flat earthers today, is that if you want to understand nature, you should only rely on information from your own senses. You can for example read on the website of the flat earth society:

“The world looks flat, the bottoms of clouds are flat, the movement of the Sun; these are all examples of your senses telling you that we do not live on a spherical heliocentric world. This is using what’s called an empirical approach, or an approach that relies on information from your senses. “

That flat earthers insist on evidence from your own senses only really is key to understanding their problem; I will come back to this. But first, let me tell you the rest of their history.

After Rowbotham’s death in 1884, the flat earth idea was carried forward by another British guy, Samuel Shenton, who once explained to a journalist:

“No man knows the ultimate shape of the earth, but that portion we life on is definitely flat. No one will ever know what the whole complexity is like, I suppose, because it goes beyond his sphere of observation, investigation and comprehension.”

Again, note the emphasis on personally collected evidence. In 1954, Shenton created the International Flat Earth Society. Few people cared. He died in 1971.

After his death, the Flat Earth Society was taken over by the US-American Charles Johnson. But even after the advent of the internet, flat earthers did not attract much attention. Johnson died in 2001, at which point the flat Earth society had 3500 or so members. The job then fell to another American, Daniel Shenton, who is not related to the earlier Shenton but whose logic falls right in line. He said in an interview with the Guardian in 2010:

“I don't think there is solid proof. I'm not intentionally being stubborn about it, but I feel our senses tell us these things, and it would take an extraordinarily level of evidence to counteract those. How many people have actually investigated it? Have you?”

Shenton had the idea to set up a wiki page for the flat earth community. Still no one cared. But in 2016, everything changed.

What happened in 2016 is that a few devout flat earthers put up videos, here, on YouTube. And that really got things going, by way of recruiting new believers. These videos have meanwhile been watched by millions of people. And that had consequences: In a 2018 poll in the United States, two percent of the respondents said they believe the earth is flat, while another 5 percent are not quite sure. 

Reliable numbers are hard to come by, but we are meanwhile probably talking about more than ten-thousand people in the developed world who reject science that was settled by the middle ages. Let that sink in for a moment.

How does someone end up rejecting something as scientifically well-established as the fact that the earth is round? 

There is not only one reason, of course. Some flat earthers find the idea is appealing for religious reasons, others are of the crowd who think NASA is evil, space a fake, and the moon landing didn’t happen. But mostly it’s because they think they are merely being rational skeptics. They have not themselves been able to prove the earth is round, so they believe they are only reasonable when they request evidence. CNN for example reports from a flat earth conference:

“Like most of the speakers at the event CNN spoke to, he was convinced after he decided he couldn’t prove the Earth’s roundness.”

I want to leave aside here that, of course, you cannot strictly speaking prove any empirical fact; you can only prove mathematical identities, so more precisely we should speak of seeking evidence that disfavors the hypothesis that the earth is flat. Of which there is plenty, starting with the historical evidence about how stellar constellations shift if you travel, how the length of shadows changes, to Newton’s 1/R² force law that is the law for a sphere, not a disk, not to mention Einstein and gravitational redshift and the perihelion precession of mercury, and so on, and so forth.

The problem that flat earthers have is that they cannot do most of these observations themselves. So if you buy the idea that it’s only your personally collected evidence that you should accept, then it seems you cannot refute the idea that the earth is round, and so flat earthers philosophy forbids them to accept scientific fact. 

Needless to say, if you want to hold on to your convictions it helps if you refuse to do observations that could speak against them. There are actually many ways to convince yourself that the earth is round which are not that technically difficult. Buy a telescope and try to explain the motions of the moons of Jupiter, for example.

So what’s wrong with flat earther’s attitude? Isn’t it asking for evidence exactly what rational thinkers should do? Sure, evidence is key to scientific progress, but flat earthers’ philosophical approach by which they reject certain types of evidence is inconsistent and, ultimately, logically wrong. 

See, the only evidence anyone ever has of anything is evidence you collect with your own senses. Except, as Descartes pointed out, evidence of your own ability to think, but this is not relevant here. Relevant is that the distinction which flat earthers are trying to draw between different types of evidence does not exist.

All evidence you have is sensory input. If you hear an explanation of someone else’s experiment, if you read a paper laying out someone else’s argument, that’s your own sensory input. A distinction which does exist, however, is that some of our sensory input requires very little decoding, while some requires a lot. Flat earthers’ problem is that they refuse to decode difficult sensory input.

A good example for the need to decode sensory input by conscious thought are optical illusions. Your brain tries to interpret visual input in ways that sometimes gives a misleading result as in this example. You almost certainly think square A is darker than square B. It is not. 

To understand your sensory input correctly you need to draw on other information, in this case your knowledge about optical illusions. Your brain interprets this image as if it was a natural, 3-dimensional scene, and therefore calculates back to the original color of the squares taking into account what appears to be a shadow. This is the wrong interpretation if you want to know the actual color of pixels on the screen. The lesson is, if you do not think about your sensory input, if you do not properly decode it, you arrive at a wrong conclusion.

Flat earthers similarly arrive at the wrong conclusion by failing to decode evidence, indeed by simply ignoring a lot of evidence that their own senses deliver. This is evidence about how society and science works.

Whether we are scientists or not, we all constantly use this evidence to navigate life. And I am sure flat earthers are no exception. Just consider going to the supermarket and buying canned soup. Do you have evidence that what’s in the can is edible? Probably not. For one, the can’s closed. And if you are anything like me, you probably have no idea how or where or by whom it was produced. Why then are you not afraid of eating canned soup? Isn’t this entirely irrational?

No, because you do have evidence that canned soup is edible. You know how the legal system in your country works, you know that there are regulations on what can be sold as food in a supermarket, you know that if what’s in that can was harmful to you, then a lot of people along the food chain would be punished for their mistake, and they don’t want that. Your trust in canned food is an entirely reasonable inference from evidence, evidence that you collected with your own sense, because what else could you possibly have collected it with?

Now let’s come back to flat earthers. Most of you don’t have a physics degree and chances are that after learning in school how we know the earth is round you didn’t think much about it ever again. By and large you are probably confident it’s correct because what you learned in school was plausible, and you know it is widely taught to children, and you know that your government strives to give children in your country a scientifically accurate education. So you have good reason to think the knowledge you were taught is backed by solid scientific evidence.

There is no appealing to authority here. You have totally yourself collected all this evidence about how society works. You have also yourself collected lots of evidence that science works. Any airplane, any laptop, any pair of glasses is evidence that science works. It’s evidence that the system works. It’s evidence for how the whole world works.

So, if you cannot recall just what experiments demonstrate that the earth is not flat, or if you cannot immediately figure out what’s wrong with flat earther’s arguments, there’s no shame in rejecting their claims, because your rejection is based on evidence, evidence that science works.

What’s wrong is that flat earthers’ claim they are leading a scientific argument. But there is no scientific argument about whether the earth is flat. This argument was settled long ago. Instead, flat earthers’ argument is about whether you should trust evidence that other people have collected before you. And it’s an important argument because this trust is essential for society and science to progress. The only alternative we have is that each and every one of us has to start over from scratch with birth. You see, flat earthers would eventually figure out the earth is round. But it might take them a thousand years until they’ve reinvented modern science.

This is why I think scientists should take flat earthers’ philosophical problem seriously. It’s a problem that any scientifically advanced society must address. It is not possible for each and every one of us to redo all experiments in the history of science. It therefore becomes increasingly important that scientists provide evidence for how science works, so that people who cannot follow the research itself can instead rely on evidence that the system produces correct and useful descriptions of nature.

To me, therefore, flat earthers, are a warning sign that scientists should take seriously. The more difficult scientific experiments and arguments are to follow for non-experts, the more care we must take to explain how we lead those arguments.

Theories of Everything [I've been singing again]

Understanding Quantum Mechanics #5: Decoherence

[Note: This transcript will not make much sense without the graphics in the video.]


I know I promised I would tell you what it takes to solve the measurement problem in quantum mechanics. But then I remembered that almost one of two physicists believes that the problem does not exist to begin with. So, I figured I should first make sure everyone – even the physicists – understand why the measurement problem has remained unsolved, despite a century of effort. This also means that if you watch this video to the end, you will understand what half of physicists do not understand.

That about half of physicists do not understand the measurement problem is not just anecdotal evidence, that’s poll results from 2016.This questionnaire was sent to a little more than one thousand two hundred physicists, from which about twelve percent responded. That’s a decent response rate for a survey, but note that the sample may not be representative for the global community. While the questionnaire was sent to physicists of all research areas, forty-four percent of them were Danish.

With those warnings ahead, a stunning seventeen percent of the survey-respondents said the measurement problem is a pseudoproblem. Even worse: twenty-nine percent erroneously think it has been solved by decoherence. So, this is what I want to explain today: What is decoherence and what does it have to do with quantum measurements? For this video, I will assume that you know the bra-ket notation for wave-functions. If you do not know it, please watch my earlier video.

In quantum mechanics, we describe a system by a wave-function that is a vector and can be expanded in a basis, which is a set of vectors of length one. The wave-function is usually denoted with the greek letter Psi. I will just label these basis vectors with numbers. A key feature of quantum mechanics is that the coefficients in the expansion of the wave-function, for which I used the letter a, can be complex numbers. Technically, there can be infinitely many basis-vectors, but that’s a complication we will not have to deal with here. We will just look at the simplest possible case, that of two basis vectors.

It is common to use basis vectors which describe possible measurement outcomes, and we will do the same. So, |1> and |2>, stand for two values of an observable that you could measure. The example that physicists typically have in mind for this are two different spin values of a particle, say +1 and -1. But the basis vectors could also describe something else that you measure, for example two different energy levels of an atom or two different sides of a detector, or what have you.

Once you have expanded the wave-function in a basis belonging to the measurement outcomes, then the square of the coefficient for a basis vector gives you the probability of getting the measurement outcome. This is Born’s rule. So if a coefficient was one over square root two, then the square is one half which means a fifty percent probability of finding this measurement outcome. Since the probabilities have to add up to 100%, this means the absolute squares of the coefficients have to add up to 1.

With these two basis vectors you can describe a superposition, which is a sum with factors in front of them. For more about superpositions, please watch my earlier video. The weird thing about quantum mechanics now is that if you have a state that is in a superposition of possible measurement outcomes, say, spin plus one and spin minus one, you never measure that superposition. You only measure either one or the other.

As example, let us use a superposition that is with equal probability in one of the possible measurement outcomes. Then the factor for each basis vector has to be the square root of one half. But this is quantum mechanics, so let us not forget that the coefficients are complex numbers. To take this into account, we will put in another factor here, which is a complex number with absolute value equal to one. We can write any such complex number as e to the I times theta, where theta is a real number.

The reason for doing this is that such a complex number does not change anything about the probabilities. See, if we ask what is the probability of finding this superposition in state |1>, then this would be (one over square root of two) times (e to the I theta) times the complex conjugate, which is (one over square root of two) times (e to the minus I theta). And that comes out to be one half, regardless of what theta is.

This theta also called the “phase” of the wave-function because you can decompose the complex number into a sine and cosine, and then it appears in the argument where a phase normally appears for an oscillation. There isn’t anything oscillating here, though, because there is no time-dependence. You could put another such complex number in front of the other coefficient, but this doesn’t change anything about the following.

Ok, so now we have this superposition that we never measure. The idea of decoherence is now to take into account that the superposition is not the only thing in our system. We prepare a state at some initial time, and then it travels to the detector. A detector is basically a device that amplifies a signal. A little quantum particle comes in one end and a number comes out on the other end. This necessarily means that the superposition which we want to measure interacts with many other particles, both along the way to the detector, and in the detector. This is what you want to describe with decoherence.

The easiest way to describe these constant bumps that the superposition has to endure is that each bump changes the phase of the state, so the theta, by a tiny little bit. To see what effect this has if you do a great many of these little bumps, we first have to calculate the density-matrix of the wave-function. It will become clear later, why.

As I explained in my previous video, the density matrix, usually denoted with the greek letter rho, is the ket-bra product of the wave-function with itself. For the simple case of our superposition, the density matrix looks like this. It has a one over two in each entry because of all the square roots of two, and the off-diagonal elements also have this complex factor with the phase. The idea of decoherence is then to say that each time the particle bumps into some other particle, this phase randomly changes and what you actually measure, is the average over all those random changes.

So, understanding decoherence comes down to averaging this complex number. To see what goes on, it helps to draw the complex plane. Here is the complex plane. Now, every number with an absolute value of 1 lies on a circle of radius one around zero. On this circle, you therefore find all the numbers of the form e to the I times theta, with theta a real number. If you turn theta from 0 to 2 \Pi, you go once around the circle. That’s Euler’s formula, basically.

The whole magic of decoherence is in the following insight. If you randomly select points on this circle and average over them, then the average will not lie on the circle. Instead, it will converge to the middle of the circle, which is at zero. So, if you average over all the random kicks, you get zero. The easiest way to see this is to think of the random points as little masses and the average as the center of mass.

Now let us look at the density matrix again. We just learned that if we average over the random kicks, then these off-diagonal entries go to zero. Nothing happens with the diagonal entries. That’s decoherence.

The reason this is called “decoherence” is that the random changes to the phase destroy the ability of the state to make an interference pattern with itself. If you randomly shift around the phase of a wave, you don’t get any pattern. A state that has a well-defined phase and can interfere with itself, is called “coherent”. But the terminology isn’t the interesting bit. The interesting bit is what has happened with the density matrix.

This looks utterly unremarkable. It’s just a matrix with one over two’s on the diagonal. But what’s interesting about it is that there is no wave-function that will give you this density matrix. To see this, look again at the density matrix for an arbitrary wave-function in two dimensions. Now take for example this off-diagonal entry. If this entry is zero, then one of these coefficients has to be zero, but then one of the diagonal elements is also zero, which is not what the decohered density matrix looks like. So, the matrix that we got after decoherence no longer corresponds to a wave-function.

That’s why we use density matrices in the first place. Every wave-function gives you a density matrix. But not every density matrix gives you a wave-function. If you want to describe how a system loses coherence, you therefore need to use density matrices.
br> What does this density matrix after decoherence describe? It describes classical probabilities. The diagonal entries tell you the probability for each of the possible measurement outcomes, like in quantum mechanics. But all the quantum-ness of the system, that was in the ability of the wave-function to interfere with itself, have gone away with the off-diagonal entries.

So, decoherence converts quantum probabilities to classical probabilities. It therefore explains why we never observe any strange quantum behavior in every-day life. It’s because this quantum behavior goes away very quickly with all the many interactions that every particle constantly has, whether or not you measure them. Decoherence gives you the right classical probabilities.

But it does not tell you what happens with the system itself. To see this, keep in mind that the density matrix in general does not describe a collection of particles or a sequence of measurements. It might well just describe one single particle. And after you have measured the particle, it is with probability 1 either in one state, or in the other. But this would correspond to a density matrix which has one diagonal entry that is 1 and all other entries zero. The state after measurement is not in a fifty-fifty probability-state, that just isn’t a thing. So, decoherence does not actually tell you what happens with the system itself when you measure it. It merely gives you probabilities for what you observe.

This is why decoherence only partially solves the measurement problem. It tells you why we do not normally observe quantum effects for large objects. It does not tell you, however, how it happens that a particle ends up in one, and only one, possible measurement outcome.

The best way to understand a new subject is to actively engage with it, and as much as I love doing these videos, this is something you have to do yourself. A great place to start engaging with quantum mechanics on your own is Brilliant, who have been sponsoring this video. Brilliant offers interactive courses on a large variety of topics in science and mathematics. To make sense of what I just told you about density matrices, for example, have a look at their courses on linear algebra, probabilities, and on quantum objects.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Really Big Experiments That Physicists Dream Of

This week, I have something for your intellectual entertainment; I want to tell you about some really big experiments that physicists dream of.


Before I get to the futuristic ideas that physicists have, let me for reference first tell you about the currently biggest experiment in operation, that is the Large Hadron Collider, or LHC for short. Well, actually the LHC is currently on pause for an upgrade, but it is scheduled to be running again in May 2021. The LHC accelerates protons in a circular tunnel that is 27 kilometer long. Accelerating the protons requires powerful magnets that, to function properly, have to be cooled to only a few degrees above absolute zero. With this, the LHC reaches collision energies of about 14 Tera Electron Volt, or TeV.

Unless you are a particle physicist, this unit of energy probably does not tell you much. It helps to know that the collision energy is roughly speaking inversely proportional to the distances you can test. So, with higher collision energies, you can test smaller structures. That’s why particle physicists build bigger colliders. The fourteen TeV that the LHC produces correspond to about ten to the minus nineteen meters. For comparison, the typical size of an atom is ten to the minus ten meters, and a proton roughly has a size of ten to the minus fifteen meters. So, the LHC tests structures a thousand times smaller than the diameter of a proton.

As you may have read in the news recently, CERN announced that particle physicists want a bigger collider. The new machine, called the “Future Circular Collider” is supposed to have a tunnel that’s one-hundred kilometers long and it should ultimately reach one-hundred TeV collision energy, so that’s about six times as much as what the LHC can do. What do they want to do with the bigger collider? That’s a very good question, thanks for asking. They want to measure more precisely some properties of some particles. What is the use given that these particles live some microseconds on the outside? Nothing, really, but it keeps particle physicists employed.

The former Chief Scientific Advisor of the British government, Prof Sir David King, commented on the new collider plans in a BBC interview: “We have to draw a line somewhere, otherwise we end up with a collider that is so large that it goes around the equator. And if it doesn't end there perhaps there will be a request for one that goes to the Moon and back.”

Particle physicists don’t currently have plans for an accelerator around the equator, but some of them proposed we could place a collider with one-thousand-nine-hundred kilometer circumference in the gulf of Mexico. What for? Well, you could reach higher collision energies.

However, even particle physicists agree that a collider the size of the Milky Ways is not realistic. That’s because, as the particle physicist James Beachman explained in an interview with Gizmodo, unfortunately even interstellar space, with a temperature of about 3 degrees above absolute zero, is still too warm for the magnets. This means you’d need a huge amount of Helium to cool the magnets. And where would you get this?

But even a collider around the equator would be a technological challenge. Not only because of the differences in altitude, also because the diameter of Earth pulses with a period of about 21 minutes. That’s one of the fundamental vibrational modes of the Earth and, by the way, more evidence that the earth is not flat. The fundamental vibrational modes get constantly excited through earthquakes. But, as a lot of physicists have noticed, this is a problem which you would not have --- on the moon. The moon has very little seismic activity, and there’s also no life crawling around on it, so, except for the occasional asteroid impact, it’s very quiet there.

Better still, the moon has no atmosphere which can cloud up the view of the night sky. Which is why physicists have long dreamed of putting a radio telescope on the far side of the moon. Such a telescope would be exciting because it could measure signals from the “dark ages” of the early universe. This period has so-far been studied very little due to lack of data.

The dark ages begin after the emission of the cosmic microwave background but before the formation of the first stars, and they could tell us much about both, the behavior of normal matter and that of dark matter.

The dark ages, luckily, were not entirely dark, just very, very dim. That’s because back then the universe was filled mostly by lots of hydrogen atoms. If these bump into each other, they can emit light at a very specific wavelength, 21 cm. This wavelength then stretches with the expansion of the universe and should be measureable to day with radio telescopes. Physicists call this “21 centimeter astronomy” and a few telescopes are already looking out for this signal from the dark ages. But the signal is very weak and hard to measure. Putting a telescope on the moon would certainly help.

This is not the only experiment that physicist would like to put on the moon, if we’d just let them. Just in February this year, for example, a proposal appeared to put a neutrino source on the moon and send a beam of neutrinos from there to earth. This would allow physicists to better study what happens to neutrinos as they travel. This information could be interesting because we know that neutrinos can “oscillate” between different types as they travel – for example an electron-neutrino can oscillate into a muon-neutrino – but there are some oddities in the existing measurements that could mean we are missing something.

And only a few weeks ago, some physicists proposed to put a gravitational wave interferometer on the moon, though this idea was originally proposed already in the 1990s. Again the reason is that the moon is far less noisy than our densely populated and seismically active planet. The downside is, well, there are no people on the moon to actually build the machine.

That’s why I am more excited about another proposal that was put forward some years ago by two physicists from Harvard University. These guys suggested that to better measure gravitational waves, we could leave a trail of atomic clocks behind us on our annual path around the sun. When a gravitational wave passes through the solar system, the time that it takes signals to travel between the atomic clocks and earth slightly changes. The cool thing about it is that this would allow physicists to detect gravitational waves with much longer wavelengths than what is possible with interferometers on earth or on the moon. Gravitational waves with such long wavelengths should be created in the collisions of supermassive black holes and therefore could tell us something about what goes on in galactic cores.

These experiments have in common that they would be great to have, if you are a physicist. They also have in common that they are big. And since they are big, they are expensive, which means chances are slim any of those will ever become reality. Unfortunately, ignoring economic reality is common for physicists. Instead of thinking about ways to make experiments smaller, easier to handle, and cheaper to produce, their vision is to do the same thing again, just bigger. But, well, bigger isn’t always better.

What is the equivalence principle?

Folks, I recently read the website of the Flat Earth Society. I’m serious! It’s a most remarkable collection of… nonsense. Maybe most remarkable is how it throws together physical facts that are correct – but then gets their consequences completely wrong! This is most evident when it comes to flat earthers’ elaborations on Einstein’s equivalence principle.


The equivalence principle is experimentally extremely well-confirmed, yes. But flat earthers misconstrue evidence for the equivalence principle as “evidence for universal acceleration” or what they call the “universal accelerator”. By this they mean that the gravitational acceleration is the same everywhere on earth. It is not. But, you see, they believe that on their flat earth, there is no gravity. Instead, the flat earth is accelerating upwards. So, if you drop an apple, it’s not that gravity is pulling it down, it’s that the earth comes up and hits the apple.

The interesting thing is now that flat earthers’ claim Einstein said you cannot distinguish upward acceleration from downward gravity. That’s the equivalence principle, supposedly. So, you see, Einstein said it and therefore the earth is flat.

You can read on their website:

“Why does the physics of gravity behave exactly as if the earth were accelerating upwards? The Universal Accelerator answers this long-standing mystery, which has baffled generations of scientists, by positing that the earth is accelerating upwards.”

Ingenious! Why didn’t Einstein think of this? Well, because it’s wrong. And in this video, I will explain why it’s wrong. So, what is the equivalence principle? The equivalence principle says that:

“Acceleration in a flat space-time is locally indistinguishable from gravity.”

Okay, that sounds somewhat technical, so let us go through this step by step. I assume you know what acceleration is because otherwise you would not be watching a physics channel. Flat space-time means you are dealing with special relativity. So, you have combined space and time, as Einstein told us to do, but they are not curved; they’re flat, like a sheet of paper. “Locally” means in a small region. So, the equivalence principle says: If you can only make measurements in a small region around you, then you cannot tell acceleration apart from gravity. You can only tell them apart if you can make measurements over a large enough distances.

This is what Einstein’s thought experiment with the elevator was all about. I talked about this in an earlier video. If you’re in the elevator, you don’t know whether the elevator is sitting on the surface of a planet and gravity is pulling down, or if the elevator is accelerating upward.

The historical relevance of the equivalence principle is that it allowed Einstein to make the step from special relativity to general relativity. This worked because he already knew how to describe acceleration in flat space – you can do that with special relativity. In general relativity then, space-time is curved, but locally it is flat. So you can use special relativity locally and get general relativity. The equivalence principle connects both – that was Einstein’s great insight.

So, the equivalence principle says that you cannot tell gravity from acceleration in a small region. That sounds indeed very much like what flat earthers say. But here’s the important point: How large the region needs to be to tell apart gravity from acceleration depends on how precisely you can measure and how far you are willing to walk. If you cannot measure very precisely, you may have to climb on a mountain top. You then find that the acceleration up there is smaller than at sea level. Why? Because the gravitational force decreases with the distance to the center of the earth. That’s Newton’s 1/R² force. Indeed, since the earth is not exactly a sphere, the acceleration also differs somewhat between the equator and the poles. This can and has been measured to great precision.

Yeah, we’ve know all this for some while. If the acceleration we normally assign to gravity was the same everywhere on earth, that would contradict a huge number of measurements. Evidence strongly speaks against it. If you measure very precisely, you can even find evidence for the non-universality of the gravitational pull in the laboratory. Mountains themselves, for example, have a non-negligible gravitational pull. This can, and has been measured, already in the 18th century. The gravitational acceleration caused by the ground underneath your feet has also local variations at constant altitude just because in some places the density of the ground is higher than in others.

So, explaining gravity as a universal acceleration is in conflict with a lot of evidence. But can you instead just give the flat earth a gravitational pull? No, that does not fit with evidence either. Because for a disk the gravitational acceleration does not drop with 1/R². It falls more slowly with the distance from the disk. Exactly how depends on how far you are from the edge of the disk. In any case, it’s clearly wrong.

The equivalence principle is sometimes stated differently than I put it, namely as the equality of inertial and gravitational mass. Physicists don’t particularly like this way of formulating the equivalence principle because it’s not only mass that gravitates. All kinds of energy densities and momentum flow and pressure and so on also gravitate. So, strictly speaking it’s not correct to merely say inertial mass equals gravitational mass.

But in the special case when you are looking at a slowly moving point particle with a mass that is very small compared to earth, then the equality of inertial and gravitational mass is a good way to think of the equivalence principle. If you use the approximation of Newtonian gravity, then you would describe this by saying that F equals m_i times a, with m_i the inertial mass and a the acceleration, and that must be balanced with the gravitational force that is m_g, the gravitational mass of the particle, times the mass of earth divided by R^2, where R is the distance from the center of earth which is, excuse me, a sphere. So, if the inertial mass is equal to the gravitational mass of the particle, then these masses cancel out. If you calculate the path on which the particle moves, it will therefore not depend on the mass.

In general relativity, the equivalence of inertial and gravitational mass for a point particle has a very simple interpretation. Remember that, in general relativity, gravity is not a force. Gravity is really caused by the curvature of space-time. In this curved space-time a point particle just takes the path of the longest possible proper time between two places. This is an entirely geometrical requirement and does not depend on the mass of the particle.

Let me add that physicists use a few subtle distinctions of equivalence principles, in particular for quantum objects. If you want to know the technical details, please check the information below the video for a reference.

In summary, if you encounter a flat earther who wants to impress you with going on about the equivalence principle, all you need to know is that the equivalence principle is not evidence for universal acceleration. This is most definitely not what Einstein said.

If this video left you wishing you understood Einstein’s work better, I suggest you have a look at Brilliant dot org, who have been sponsoring this video. Brilliant offers online courses on a large variety of topics in mathematics and science, including physics. They have interactive courses on special relativity, general relativity, and even gravitational physics, where they explore the equivalence principle specifically. Brilliant is a great starting point to really understand how Einstein’s theories work and also test your understanding along the way.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

Thanks for watching, see you next week.

Einstein’s Greatest Legacy: Thought Experiments

Einstein’s greatest legacy is not General Relativity, it’s not the photoelectric effect, and it’s not slices of his brain. It’s a word: Gedankenexperiment – that’s German for “thought experiment”.

Today, thought experiments are common in theoretical physics. We use them to examine the consequences of a theory beyond what is measureable with existing technology, but still measureable in principle. Thought experiments are useful to push a theory to its limits, and doing so can reveal inconsistencies in the theory or new effects. There are only two rules for thought experiments: (A) relevant is only what is measureable and (B) do not fool yourself. This is not as easy as it sounds.

The maybe first thought experiment came from James Maxwell and is known today as Maxwell’s demon. Maxwell used his thought experiment to find out whether one can beat the second law of thermodynamics and build a perpetual motion machine, from which an infinite amount of energy could be extracted.

Yes, we know that this is not possible, but Maxwell said, suppose you have two boxes of gas, one of high temperature and one of low temperature. If you bring them into contact with each other, the temperatures will reach equilibrium at a common temperature somewhere in the middle. In that process of reaching the equilibrium temperature, the system becomes more mixed up and entropy increases. And while that happens – while the gas mixes up – you can extract energy from the system. It “does work” as physicists say. But once the temperatures have equalized and are the same throughout the gas, you can no longer extract energy from the system. Entropy has become maximal and that’s the end of the story.

Maxwell’s demon now is a little omniscient being that sits at the connection between the two boxes where there is a little door. Each time a fast atom comes from the left, the demon lets it through. But if there’s a fast atom coming from the right, the demon closes the door. This way the number of fast atoms on the one side will increase, which means that the temperature on that side goes up again and the entropy of the whole system goes down.

It seems like thermodynamics is broken, because we all know that entropy cannot decrease, right? So what gives? Well, the demon needs to have information about the motion of the atoms, otherwise it does not know when to open the door. This means, essentially, the demon is itself a reservoir of low entropy. If you combine demon and gas the second law holds and all is well. The interesting thing about Maxwell’s demon is that it tells us entropy is somehow the opposite of information, you can use information to decrease entropy. Indeed, a miniature version of Maxwell’s demon has meanwhile been experimentally realized.

But let us come back to Einstein. Einstein’s best known thought experiment is that he imagined what would happen in an elevator that’s being pulled up. Einstein argued that there is no measurement that you can do inside the elevator to find out whether the elevator is in rest in a gravitational field or is being pulled up with constant acceleration. This became Einstein’s “equivalence principle”, according to which the effects of gravitation in a small region of space-time are the same as the effects of acceleration in the absence of gravity. If you converted this principle into mathematical equations, it becomes the basis of General Relativity.

Einstein also liked to imagine how it would be to chase after photons, which was super-important for him to develop special relativity, and he spent a lot of time thinking about what it really means to measure time and distances.

But the maybe most influential of his thought experiments was one that he came up with to illustrate that quantum mechanics must be wrong. In this thought experiment, he explored one of the most peculiar effects of quantum mechanics: entanglement. He did this together with Boris Podolsky and Nathan Rosen, so today this is known as the Einstein-Podolsky-Rosen or just EPR experiment.

How does it work? Entangled particles have some measureable property, for example spin, that is correlated between particles even though the value for each single particle is not determined as long as the particles were not measured. If you have a pair of particles, you can know for example that if one particle has spin up, then the other one has spin down, or the other way round, but you may still not know which is which. The consequence is that if one of these particles is measured, the state of the other one seems to change – instantaneously.

Einstein, Podolsky and Rosen suggested this experiment because Einstein believed this instantaneous ‘spooky’ action at a distance is nonsense. You see, Einstein had a problem with it because it seems to conflict with the speed of light limit in Special Relativity. We know today that this is not the case, quantum mechanics does not conflict with Special Relativity because no useful information can be sent between entangled particles. But Einstein didn’t know that. Today, the EPR experiment is no longer a thought experiment. It can, and has been done, and we now know beyond doubt that quantum entanglement is real.

A thought experiment that still gives headaches to theoretical physicists today is the black hole information loss paradox. General relativity and quantum field theory are both extremely well established theories, but if you combine them, you find that black holes will evaporate. We cannot measure this for real, because the temperature of the radiation is too low, but it is measureable in principle.

However, if you do the calculation, which was first done by Stephen Hawking, it seems that black hole evaporation is not reversible; it destroys information for good. This however cannot happen in quantum field theory and so we face a logical inconsistency when combining quantum theory with general relativity. This cannot be how nature works, so we must be making a mistake. But which?

There are many proposed solutions to the black hole information loss problem. Most of my colleagues believe that the inconsistency comes from using general relativity in a regime where it should no longer be used and that we need a quantum theory of gravity to resolve the problem. So far, however, physicists have not found a solution, or at least not one they can all agree on.

So, yes, thought experiments are a technique of investigation that physicists have used in the past and continue to use today. But we should not forget that eventually we need real experiments to test our theories.

Understanding Quantum Mechanics #4: It’s not as difficult as you think! (The Bra-Ket)

If you do an image search for “quantum mechanics” you will find a lot of equations that contain things which look like this |Φ> or this |0> or maybe also that <χ|. These things are what it called the “bra-ket” notation. What does this mean? How do you calculate with it? And is quantum mechanics really as difficult as they say? This is what we will talk about today.


I know that quantum mechanics is supposedly impossible to understand, but trust me when I say the difficulty is not in the mathematics. The mathematics of quantum mechanics looks more intimidating than it really is.

To see how it works, let us have a look at how physicists write wave-functions. The wave-function, to remind you, is what we use in quantum mechanics to describe everything. There’s a wave-function for electrons, a wave-function for atoms, a wave-function for Schrödinger’s cat, and so on.

The wave-function is a vector, just like the ones we learned about in school. In a three-dimensional space, you can think of a vector as an arrow pointing from the origin of the coordinate system to any point. You can choose a particularly convenient basis in that space, typically these are three orthogonal vectors, each with a length of one. These basis vectors can be written as columns of numbers which each have one entry that equals one and all other entries equal to zero. You can then write an arbitrary vector as a sum of those basis vectors with coefficients in front of them, say (2,3,0). These coefficients are just numbers and you can collect them in one column. So far, so good.

Now, the wave-function in quantum mechanics is a vector just like that, except it’s not a vector in the space we see around us, but a vector in an abstract mathematical thing called a Hilbert-space. One of the most important differences between the wave-function and vectors that describe directions in space is that the coefficients in quantum mechanics are not real numbers but complex numbers, so they in general have a non-zero imaginary part. These complex numbers can be “conjugated” which is usually denoted with a star superscript and just means you change the sign of the imaginary part.

So the complex numbers make quantum mechanics different from your school math. But the biggest difference is really just the notation. In quantum mechanics, we do not write vectors with arrows. Instead we write them with these funny brackets.

Why? Well, for one because it’s convention. But it’s also a convenient way to keep track of whether a vector is a row or a column vector. The ones we talked about so far are column-vectors. If you have a row-vector instead, you draw the bracket on the other side. You have to watch out here because in quantum mechanics, if you convert a row vector to a column vector, you also have to take the complex conjugate of the coefficients.

This notation was the idea of Paul Dirac and is called the bra-ket notation. The left side, the row vector, is the “bra” and the right side, the column vector, is the “ket”.

You can use this notation for example to write a scalar product conveniently as a “bra-ket”. The scalar product between two vectors is the sum over the products of the coefficients. Again, don’t forget that the bra-vector has complex conjugates on the coefficients.

Now, in quantum mechanics, all the vectors describe probabilities. And usually you chose the basis in your space so that the basis vectors correspond to possible measurement outcomes. The probability of a particular measurement outcome is then the absolute square of the scalar product with the basis-vector that corresponds to the outcome. Since the basis vectors are those which have only zero entries except for one entry which is equal to one, the scalar product of a wave-function with a basis vector is just the coefficient that corresponds to the one non-zero entry.

And the probability is then the absolute square of that coefficient. This prescription for obtaining probabilities from the wave-function is known as “Born’s rule”, named after Max Born. And we know that the probability to get any measurement outcome is equal to one, which means that that the sum over the squared scalar products with all basis vectors has to be one. But this is just the length of the vector. So all wave-functions have length one.

 The scalar product of the wave-function with a basis-vector is also sometimes called a “projection” on that basis-vector. It is called a projection, because it’s the length you get if you project the full wave-function on the direction that corresponds to the basis-vector. Think of it as the vector casting a shadow. You could say in quantum mechanics we only ever observe shadows of the wave-function.

The whole issue with the measurement in quantum mechanics is now that once you do a measurement, and you have projected the wave-function onto one of the basis vectors, then its length will no longer be equal to 1 because the probability of getting this particular measurement outcome may have been smaller than 1. But! once you have measured the state, it is with probability one in one of the basis states. So then you have to choose the measurement outcome that you actually found and stretch the length of the vector back to 1. This is what is called the “measurement update”.

Another thing you can do with these vectors is to multiply one with itself the other way round, so that would be a ket-bra. What you get then is not a single number, as you would get with the scalar product, but a matrix, each element of which is a product of coefficients of the vectors. In quantum mechanics, this thing is called the “density matrix”, and you need it to understand decoherence. We will talk about this some other time, so keep the density matrix in mind.

Having said that, as much as I love doing these videos, if you really want to understand quantum mechanics, you have to do some mathematical exercises on your own. A great place to do this is Brilliant who have been sponsoring this video. Brilliant offers courses with exercise sets on a large variety of topics in science and mathematics. It’s exactly what you need to move from passively watching videos to actively dealing with the subject. The courses on Brilliant that will give you the required background for this video are those on linear algebra and its applications: What is a vector, what is a matrix, what is an eigenvalue, what is a linear transformation? That’s the key to understanding quantum mechanics.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

You may think I made it look too easy, but it’s true: Quantum mechanics is pretty much just linear algebra. What makes it difficult is not the mathematics. What makes it difficult is how to interpret the mathematics. The trouble is, you cannot directly observe the wave-function. But you cannot just get rid of it either; you need it to calculate probabilities. But the measurement update has to be done instantaneously and therefore it does not seem to be a physical process. So is the wave-function real? Or is it not? Physicists have debated this back and forth for more than 100 years.