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.

Do we need a Theory of Everything?

I get constantly asked if I could please comment on other people’s theories of everything. That could be Garrett Lisi’s E8 theory or Eric Weinstein’s geometric unity or Stephen Wolfram’s idea that the universe is but a big graph, and so on. Good, then. Let me tell you what I think about this. But I’m afraid it may not be what you wanted to hear.


Before we start, let me remind you what physicists mean by a “Theory of Everything”. For all we currently know, the universe and everything in it is held together by four fundamental interactions. That’s the electromagnetic force, the strong and the weak nuclear force, and gravity. All other forces that you are familiar with, say, the van der Waals force, or muscle force, or the force that’s pulling you down an infinite sequence of links on Wikipedia, these are all non-fundamental forces that derive from the four fundamental interactions. At least in principle.

Now, three of the fundamental interactions, the electromagnetic and the strong and weak nuclear force, are of the same type. They are collected in what is known as the standard model of particle physics. The three forces in the standard model are described by quantum field theories which means, in a nutshell, that all particles obey the principles of quantum mechanics, like the uncertainty principle, and they can be entangled and so on. Gravity, however, is described by Einstein’s theory of General Relativity and does not know anything about quantum mechanics, so it stands apart from the other three forces. That’s a problem because we know that all the quantum particles in the standard model have a gravitational pull. But we do not know how this works. We just do not have a theory to describe how elementary particles gravitate. For this, we would need a theory for the quantum behavior of gravity, a theory of “quantum gravity,” as it’s called.

We need a theory of quantum gravity because general relativity and the standard model are mathematically incompatible. So far, this is a purely theoretical problem because with the experiments that we can currently do, we do not need to use quantum gravity. In all presently possible experiments, we either measure quantum effects, but then the particle masses are so small that we cannot measure their gravitational pull. Or we can observe the gravitational pull of some objects, but then they do not have quantum behavior. So, at the moment we do not need quantum gravity to actually describe any observation. However, this will hopefully change in the coming decades. I talked about this in an earlier video.

Besides the missing theory of quantum gravity, there are various other issues that physicists have with the standard model. Most notably it’s that, while the three forces in the standard model are all of the same type, they are also all different in that each of them belongs to a different type of symmetry. Physicists would much rather have all these forces unified to one, which means that they would all come from the same mathematical structure.

In many cases that structure is one big symmetry group. Since we do not observe this, the idea is that the big symmetry would manifest itself only at energies so high that we have not yet been able to test them. At the energies that we have tested it so far, the symmetry would have to be broken, which gives rise to the standard model. This unification of the forces of the standard model is called a “grand unification” or a “grand unified theory”, GUT for short.

What physicists mean by a theory of everything is then a theory from which all the four fundamental interactions derive. This means it is both a grand unified theory and a theory of quantum gravity.

This sounds like a nice idea, yes. But. There is no reason that nature should actually be described by a theory of everything. While we *do need a theory of quantum gravity to avoid logical inconsistency in the laws of nature, the forces in the standard model do not have to be unified, and they do not have to be unified with gravity. It would be pretty, yes, but it’s unnecessary. The standard model works just fine without unification.

So this whole idea of a theory of everything is based on an unscientific premise. Some people would like the laws of nature to be pretty in a very specific way. They want it to be simple, they want it to be symmetric, they want it to be natural, and here I have to warn you that “natural” is a technical term. So they have an idea of what they want to be true. Then they stumble over some piece of mathematics that strikes them as particularly pretty and they become convinced that certainly it must play a role for the laws of nature. In brief, they invent a theory for what they think the universe *should be like.

This is simply not a good strategy to develop scientific theories, and no, it is most certainly not standard methodology. Indeed, the opposite is the case. Relying on beauty in theory development has historically worked badly. In physics, breakthroughs in theory-development have come instead from the resolution of mathematical inconsistencies. I have literally written a book about how problematic it is that researchers in the foundations of physics insist on using methods of theory development that we have no reason to think should work, and that as a matter of fact do not work.

The search for a theory of everything and for grand unification began in the 1980s. To the extent that the theories which physicists have come up with were falsifiable they have been falsified. Nature clearly doesn’t give a damn what physicists think is pretty math.

Having said that, what do you think I think about Lisi’s and Weinstein’s and Wolfram’s attempts at a theory of everything? Well, scientific history teaches us that their method of guessing some pretty piece of math and hoping it’s useful for something is extremely unpromising. It is not impossible it works, but it is almost certainly a waste of time. And I have looked closely enough at Lisi’s and Weinstein’s and Wolfram’s and many other people’s theories of everything to be able to tell you that they have not convincingly solved any actual problem in the existing fundamental theories. And I’m not interested to look any closer, because I don’t also want to waste my time.

But I don’t like commenting on individual people’s theories of everything. I don’t like it because it strikes me as deeply unfair. These are mostly researchers working alone or in small groups. They are very dedicated to their pursuit and they work incredibly hard on it. They’re mostly not paid by tax money so it’s really their private thing and who am I to judge them? Also, many of you evidently find it entertaining to have geniuses with their theories of everything around. That’s all fine with me.

I get a problem if theories that despite having turned out to be useless grow to large, tax-paid research programs that employ thousands of people, as it has happened with string theory and supersymmetry and grand unification. That creates a problem because it eats up resources and can entirely stall progress, which is what has happened in the foundations of physics.

People like Lisi and Weinstein and Wolfram at least remind us that the big programs are not the only thing you can do with math. So, odd as it sounds, while I don’t think their specific research avenue is any more promising than string theory, I’m glad they do it anyway. Indeed, physics can need more people like them who have the courage to go their own way, no matter how difficult.

The brief summary is that if you hear something about a newly proposed theory of everything, do not ask whether the math is right. Because many of the people who work on this are really smart and they know their math and it’s probably right. The question you, and all science journalists who report on such things, should ask is what reason do we have to think that this particular piece of math has anything to do with reality. “Because it’s pretty” is not a scientific answer. And I have never seen a theory of everything that gave a satisfactory scientific answer to this question.

[Guest Post] Update of Converseful comment feature now allows for more conversations

[This post is written by Ben Alderoty from Conversful.]

Conversful launched on BackRe(action) at the end of April. For those that aren’t familiar with our name, Conversful is the app in the bottom corner that allows you to have conversations with other readers. It is still only supported on computers, so if you’re on a phone or tablet right now, come back another time to see what I’m talking about. Based on the feedback we’ve received from many of you, we have been working on some big changes to Conversful and are excited to announce these changes are now live for everyone on BackRe(action).

To participate in Conversful you will now need to create an account. You’ll be able to see a preview of the conversations happening without an account, but to join one or start your own you’ll need to create one. Accounts make it easier to find the right people you want to talk to and maintain those conversations over time.

Conversations on Conversful are still 1-on-1 and start with questions (formerly topics). Questions can pertain to a specific article you might be reading or a more general physics question you are pondering. Ask specific questions as those will elicit the best replies. When you post a question, it will remain open for a week by default and can receive multiple responses. These responses will come in the form of multiple threads within your Conversations tab.

Conversful now works both in real-time and asynchronously. If you see another reader with a green circle next to their name that means they are currently online. We’ll notify you via email if you receive a new message on Conversful when you’re offline. This way you’ll know when to get back on BackRe(action) to continue the conversation. You can control this setting in your profile tab by clicking the avatar in the top left corner of the app.

And that’s it! The app is still pretty simple as we wanted to make it as easy as possible to start conversations and enjoy the one’s you’re already in. We hope that the 1-on-1, private nature of Conversful makes it easy to say something if you’ve ever wanted to, but didn’t want to do so publicly. For all of the active commenters out there, keep commenting! We hope you can use us to continue your conversations and stray into off topic discussions that don’t pertain to a specific post.

As always, please send feedback and suggestions to ben@conversful.com. For those that have given us feedback thus far, thank you so much!

What is Quantum Metrology?

Metrology is one of the most undervalued areas of science. And no, I am not just mispronouncing meteorology, I actually mean metrology. Think “meter” not “meteor”. Metrology is the science of measurement. Meteorology is about clouds and things like that. And the study of meteors, in case you wonder, is called meteoritics.



Metrology matters because you can’t do science without measuring things. In metrology, scientists deal with problems how to define conventions for units, how to do this most accurately, how to most reliably reproduce measurements, and so on. Metrology sounds boring, but it is super-important to move from basic research to commercial application.

Just consider you are trying to build a house. If you cannot measure distances and angles, it does not matter how good your mathematics is, that house is not going to come out right. And the smaller the object that you want to build, the more precisely you must be able to measure. It’s as simple as that. You can’t reliably produce something if you don’t know what you are doing.

But if you start dealing with very small things, then quantum mechanics will become important. Yes, quantum mechanics is in principle a theory that applies to objects of all sizes. But in practice its effects are negligibly tiny for large things. However, from the size of molecules downwards, quantum effects are essential to understand what is going on. So what then is quantum metrology? Quantum metrology uses quantum effects to make more precise measurements.

It may sound somewhat weird that quantum mechanics can help you to measure things more precisely. Because we all know that quantum mechanics is… uncertain, right? So how do these two things fit together, quantum uncertainty and more precise measurements? Well, quantum uncertainty is not something that applies to any measurement. It only sets a limit to the entirety of information you can obtain about a system.

For example, there is nothing in quantum mechanics that prevents you from measuring the momentum of an electron precisely. But if you do that, you cannot also measure its position precisely. That’s what the uncertainty principle tells you. So, you have to decide what you want to measure, but the uncertainty principle is not an obstacle to measuring precisely per se.

Now, the magic that allows you to measure things more precisely with quantum effects is the same that gives quantum computers an edge over ordinary computers. It’s that quantum particles can be correlated in ways that non-quantum particles can’t. This quantum-typical type of correlation is called entanglement. There are many different ways to entangle particles, so entanglement lets you encode a lot of information with few particles. In a quantum computer, you want to use this to perform a lot of operations quickly. For quantum metrology, more information in a small space means a higher sensitivity of your measurement.

Quantum computers exist already, but the ones which exist are far from being useful. That’s because you need a large number of entangled particles, as much as a million, to not only make calculations, but to make calculations that are actually faster than you could do with a conventional computer. I explained the issue with quantum computers in an earlier video.

But in contrast to quantum computers, quantum metrology does not require large numbers of entangled particles.

A simple example for how quantum behavior can aid measurement comes from medicine. Positron emission tomography, or PET for short, is an imaging method that relies on, yes, entangled particles. For PET, one uses a short-lived radioactive substance, called a “tracer”, that is injected into whatever body part you want to investigate. A typical substance that’s being used for this is carbon-11 which has a half-life of about 20 minutes.

The radioactive substance makes a beta-decay and emits a positron. The positron annihilates with one of the electrons in the neighborhood of the decay site which creates, here it comes, an entangled pair of photons. They fly off not in one particular direction, but in two opposite directions. So, if you measure two photons that fit together, you can calculate where they were emitted. And from this you can reconstruct the distribution of the radioactive substance which “traces” the tissue of interest.

Positron emission tomography has been used since the 1950s and it’s a simple example for how quantum effects can aid measurements. But the general theoretical basis of quantum metrology was only laid in the 1980s. And then for a long time not much happened because it’s really hard to control quantum effects without getting screwed up by noise. In that, quantum metrology faced the same problem as quantum computing.

But in the past two decades, physicists have made rapid progress in designing and controlling quantum states, and, with that, quantum metrology has become one of the most promising avenues to new technology.

In 2009, for example, entangled photons were used to improve the resolution of an imaging method called optical coherence tomography. The way this works is that you create a pair of entangled photons and let them travel in two different directions. One of the photons enters a sample that you want to study, the other does not. Then you recombine the photons, which tells you where the one photon scattered in the sample, which you can then use to reconstruct how the sample is made up.

You can do that with normal light, but the quantum correlations let you measure more precisely. And it’s not only about the precision. These quantum measurements require only tiny numbers of particles, so they are minimally disruptive and therefore particularly well suited to the study of biological systems, for example, the eye, for which you don’t exactly want to use a laser beam.

Another example for quantum metrology is the precise measurement of magnetic fields. You can measure a magnetic field by taking a cloud of atoms, splitting it in two, letting one part go through the magnetic field, and then recombining the atoms. The magnetic field will shift the phases of the atoms that passed through it – because particles are also waves – and you can measure how much the phases were shifted, which tells you what the magnetic field was. Aaaand, if you entangled those atoms you can improve the sensitivity to the magnetic field. This is called quantum enhanced magnetometry.

Quantum metrology has also be used to improve the sensitivity of the LIGO gravitational wave interferometer. LIGO uses laser beams to measure periodic distortions of space and time. Laser light itself is already remarkable, but one can improve on it by bringing the laser light into a particular quantum state, called a “squeezed state,” that is less sensitive to noise and therefore allows more precise measurements.

Now, clearly these are not technologies you will have a switch for on your phone any time soon. But they are technologies with practical uses and they are technologies that we already know do really work. I don’t usually give investment advice, but if I was rich, I would put my money into quantum metrology, not into quantum computing.