Understanding Quantum Mechanics #7: Energy Levels

[This is a transcript of the video embedded below. Parts of the text will not make sense without the graphics in the video.]

Today I want to tell you what these plots show. Has anybody seen them before? Yes? Atomic energy levels, right! It’s one of the most important applications of quantum mechanics. And I mean important both historically and scientifically. Today’s topic also a good opportunity to answer a question one of you asked on a previous video “Why do some equations even actually need calculating, as the answer will always be the same?” That’s a really good question. I just love it, because it would never have occurred to me.

Okay, so we want to calculate what electrons do in an atom. Why is this interesting? Because what the electrons do determines the chemical properties of the elements. Basically, the behavior of the electrons explains the whole periodic table: Why do atoms come in particular groups, why do some make good magnets, why are some of them good conductors? The electrons tell you.

How do you find out what the electrons do? You use quantum mechanics. Quantum mechanics, as we discussed previously, works with wave-functions, usually denoted Psi. Here is Psi. And you calculate what the wave-function does with the Schrödinger equation. Here is the Schrödinger equation.

Now, the way I have written this equation here, it’s completely useless. We know what Psi is, that’s the thing we want to calculate, and we know how to take a time-derivative, but what is H? H is called the “Hamiltonian” and it contains the details about the system you want to describe. The Hamiltonian consists of two parts. The one part tells you what the particles do when you leave them alone and they don’t know anything of each other. So that would be in empty space, with no force acting on them, with no interaction. This is usually called the “kinetic” part of the Hamiltonian, or sometimes the “free” part. Then you have a second part that tells you how the particle, or particles if there are several, interact.
 
In the simplest case, this interaction term can be written as a potential, usually denoted V. And for an electron near an atomic nucleus, the potential is just the Coulomb potential. So that’s proportional to the charge of the nucleus, and falls with one over r, where r is the distance to the center of the nucleus. There is a constant in front of this term that I have called alpha, but just what it quantifies doesn’t matter for us today. And the kinetic term, for a slow-moving particle is just the square of the spatial derivatives, up to constants.

So, now we have a linear, partial differential equation that we need to solve. I don’t want to go through this calculation, because it’s not so relevant here just how to solve it, let me just say there is no magic involved. It’s pretty straight forward. But there some interesting things to learn from it.

The first interesting thing you find when you solve the Schrödinger equation for electrons in a Coulomb potential is that the solutions fall apart in two different classes. The one type of solution is a wave that can propagate through all of space. We call these the “unbound states”. And the other type of solution is a localized wave, stuck in the potential of the nucleus. It just sits there while oscillating. We call these the “bound states”. The bound states have a negative energy. That’s because you need to put energy in to rip these electrons off the atom.

The next interesting thing you find is that the bound states can be numbered, so you can count them. To count these states, one commonly uses, not one, but three numbers. These numbers are all integers and are usually called n, l, and m.

“n” starts at 1 and then increases, and is commonly called the “principal” quantum number. “l” labels the angular momentum. It starts at zero, but it has to be smaller than n.

So for n equal to one, you have only l equal to zero. For n equal to 2, l can be 0 or 1. For n equal to three, l can be zero, one or two, and so on.

The third number “m” tells you what the electron does in a magnetic field, which is why it’s called the magnetic quantum number. It takes on values from minus l to l. And these three numbers, n l m, together uniquely identify the state of the electron.

Let me then show you how the solutions to the Schrödinger equation look like in this case, because there are more interesting things to learn from it. The wave-functions give you a complex value for each location, and the absolute value tells you the probability of finding the electron. While the wave-function oscillates in time, the probability does not depend on time.

I have here plotted the probability as a function of the radius, so I have integrated over all angular directions. This is for different principal quantum numbers n, but with l and m equal to zero.

You can see that the wave-function has various maxima and minima, but with increasing n, the biggest maximum, so that’s the place you are most likely to find the electron, moves away from the center of the atom. That’s where the idea of electron “shells” comes from. It’s not wrong, but also somewhat misleading. As you can see here, the actual distribution is more complicated.

A super interesting property of these probability distributions is that they are perfectly well-behaved at r equals zero. That’s interesting because, if you remember, we used a Coulomb potential that goes as 1 over r. This potential actually diverges at r equal zero. Nevertheless, the wave-functions avoids this divergence. Some people have argued that actually something similar can avoid that a singularity forms in black holes. Please check the information below the video for a reference.

But these curves show only the radial direction, what about the angular direction? To show you how this looks like, I will plot the probability of finding the electron with a color code for slices through the sphere.

And I will start with showing you the slices for the cases of which you just saw the curves in the radial direction, that is, different n, but with the other numbers at zero.

The more red-white the color, the more likely you are to find the electron. I have kept the radius fix, so this is why the orbitals with small n only make a small blip when we scan through the middle. Here you see it again. Note how the location of the highest probability moves to a larger radius with increasing n.

Then let us look at a case where l is nonzero. This is for example for n=3, l=1 and m equals plus minus 1. As you can see, the distribution splits up in several areas of high probability and now has an orientation. Here is the same for n=4, l=2, m equals plus minus 2. It may appear as if this is no longer spherically symmetric. But actually if you combine all the quantum numbers, you get back spherical symmetry, as it has to be.

Another way to look at the electron probability distributions is to plot them in three dimensions. Personally I prefer the two-dimensional cuts because the color shading contains more information about the probability distribution. But since some people prefer the 3-dimensional plots, let me show you some examples. The surface you see here is the surface inside of which you will find the electron with a probability of 90%. Again you see that thinking of the electrons as sitting on “shells” doesn’t capture very well what is going on.

Now that you have an idea how we calculate atomic energy levels and what they look like, let me then get to the question: Why do we calculate the same things over and over again?

So, this particular calculation of the atomic energy levels was frontier research a century ago. Today students do it as an exercise. The calculations physicists now do in research in atomic physics are considerably more advanced than this example, because we have made a lot of simplifications here.

First, we have neglected that the electron has a spin, though this is fairly easy to integrate. More seriously, we have assumed that the nucleus is a point. It is not. The nucleus has a finite size and it is neither perfectly spherically symmetric, nor does it have a homogeneous charge distribution, which makes the potential much more complicated. Worse, nuclei themselves have energy levels and can wobble. Then the electrons on the outer levels actually interact with the electrons in the inner levels, which we have ignored. There are further corrections from quantum field theory, which we have also ignored. Yet another thing we have ignored is that electrons in the outer shells of large atoms get corrections from special relativity. Indeed, fun fact: without special relativity, gold would not look gold.

And then, for most applications it’s not energy levels of atoms that we want to know, but energy levels of molecules. This is a huge complication. The complication is not that we don’t know the equation. It’s still the Schrödinger equation. It’s also not that we don’t know how to solve it. The problem is, with the methods we currently use, doing these calculations for even moderately sized molecules, takes too long, even on supercomputers.

And that’s an important problem. Because the energy levels of molecules tell you whether a substance is solid or brittle, what its color is, how good it conducts electricity, how it reacts with other molecules, and so on. This is all information you want to have. Indeed, there’s a whole research area devoted to this question, which is called “quantum chemistry”. It also one of the calculations physicists hope to speed up with quantum computers.

So, why do we continue solving the same equation? Because we are improving how good the calculation is, we are developing new methods to solve it more accurately and faster, and we are applying it to new problems. Calculating the energy levels of electrons is not yesterday’s physics, it’s still cutting edge physics today.

If you really want to understand how quantum mechanics works, I recommend you check out Brilliant, who have been sponsoring this video. Brilliant is a website that offers a large variety of interactive courses in mathematics and science, including quantum mechanics, and it’s a great starting point to dig deeper into the topic. For more background on what I just talked about, have a look for example at their courses on quantum objects, differential equations, and linear algebra.  

To support this channel and learn more about Brilliant go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get twenty percent off the annual premium subscription.

Thanks for watching, see you next week.

---

You can join the chat about this video today (Saturday, Nov 7) at 6pm CET or tomorrow at the same time.

Guestpost: "Launch of the Initiative 'For a Smarter Science'" by Daniel Moreno

[This post was written by Daniel Moreno]

I remember when I first told a professor at my home University that I wanted to do a PhD and become a researcher. I was expecting him to react with enthusiasm and excitement, yet his response was a warning. That was the first clue I received about the academic world's inefficiencies, although I did not realize to what extent until many years later.

I was a postdoctoral researcher for six years after finishing my PhD in 2013, working on areas such as holographic QCD, numerical General Relativity and gravity dualities. It all came to an end last year, and I decided to rekindle an idea I developed with Sabine back in 2016, one that never came to fruition, but which has now turned into my current project, under the name 'For a Smarter Science'.

The precariousness of our scientific system is a topic of common discussion among informal circles in academic events and institutions. It should be familiar to readers of this blog, as well as it is to researchers from the many fields of Science. During lunch with colleagues, dinners with invited speakers, conference coffee breaks… conversation commonly drifts toward some version of it.

Increasing numbers of publications of decreasing relevance. Increasing numbers of temporary contracts of decreasing stability. Acceptance of bad scientific practice to the benefit of bare productivity. These are some of the criticisms and complaints, prompting warnings to young researchers.

Our initiative brings no original solution to any of these problems. It actually brings questions: What would happen if there existed a formal platform to discuss the current state of academic culture? What if there was a scientific method to approach issues about the way Science is done today?

If unsupervised, human structures tend to evolve in a predictable way. Issues such as the ones mentioned above are sociological issues, they form a sociological trend. This can be, and is, studied by experts who have been writing on the topic for decades. Educated analyses published in peer-reviewed journals.

These studies largely go unnoticed and/or dismissed by the people involved in the very scientific fields they talk about, for a variety of reasons. The assumption that social topics are naturally subjects of informal conversation only, the belief that intelligent people are not affected by cognitive biases, or the selection of like-minded people by the academic system itself, leading to communal reinforcement.

And so, the academic wheel continues running along the railway already set in front of it, with no one there to steer its course. Short-term thinking permeates research. Researchers go from one application deadline to the next. Academic metrics go unquestioned.

Our proposal is simple. One conference. One formal event bringing together experts from a specific subfield of Science (high energy theoretical physics) and from Sociology of Science, to elevate these questions to well-informed discussion. We trust that such an event has the potential to trigger a positive change in the historical development of all of Science.

Of course, not everyone necessarily agrees with the idea that scientific progress is being discouraged by the current peculiarities of the academic culture. This is why we find it fitting to fund the event by means of a crowd-funding call. Its success will be an effective measure of how worthy of consideration the scientific community considers these discussions. Are they just light topics for chit-chat during coffee breaks, or is it time to pause and actually examine the social evolution that scientific careers have been following?

This initiative is for all those who have been waiting for a chance to do something about the state of modern scientific research. Writing posts and discussing with our colleagues can only take us so far. If you believe in the place of Science as a shining light for human progress, make your voice be heard. #ForaSmarterScience

 

What is Energy? Is Energy Conserved?

Why save energy if physics says energy is conserved anyway? Did Einstein really say that energy is not conserved? And what does energy have to do with time? This is what we will talk about today.


I looked up “energy” in the Encyclopedia Britannica and it told me that energy is “the capacity for doing work”. Which brings up the question, what is work? The Encyclopedia says work is “a measure of energy transfer.” That seems a little circular. And as if that wasn’t enough, the Encyclopedia goes on to say, well, actually not all types of energy do work, and also energy is always associated with motion, which actually it is not because E equals m c squared. I hope you are sufficiently confused now to hear how to make sense of this.

A good illustration for energy conservation is a roller-coaster. At the starting point, it has only potential energy, that comes from gravity. As it rolls down, the gravitational potential energy is converted into kinetic energy, meaning that the roller-coaster speeds up. At the lowest point it moves the fastest. And as it climbs up again, it slows down because the kinetic energy is converted back into potential energy. If you neglect friction, energy conservation means the roller-coaster should have just exactly the right total energy to climb back up to the top where it started. In reality of course, friction cannot be neglected. This means the roller-coaster loses some energy into heating the rails or creating wind. But this energy is not destroyed. It is just no longer useful to move the roller coaster.

This simple example tells us two things right away. First, there are different types of energy, and they can be converted into each other. What is conserved is only the total of these energies. Second, some types of energy are more, others less useful to move things around.

But what really is this energy we are talking about? There was indeed a lot of confusion about this among physicists in the 19th century, but it was cleared up beautifully by Emmy Noether in 1915. Noether proved that if you have a system whose equations do no change in time then this system has a conserved quantity. Physicists would say, such a system has time-translation invariance. Energy is then by definition the quantity that is conserved in a system with time-translation invariance.

What does this mean? Time-translation invariance does not mean the system itself does not change in time. Even if the equations do not change in time, the solutions to these equations, which are what describe the system, usually will depend on time. Time-translation invariance just means that the change of the system depends only on the amount of time that passed since you started an experiment, but you could have started it at any moment and gotten the same result. Whether you fall off a roof at noon or a midnight, it will take the same time for you to hit the ground. That’s what “time-translation invariance” means.

So, energy is conserved by definition, and Noether’s theorem gives you a concrete mathematical procedure to derive what energy is. Okay, I admit it is a little more complicated, because if you have some quantity that is conserved, then any function of that quantity is also conserved. The missing ingredient is that energy times time has to have the dimension of Pla()nck’s constant. Basically, it has to have the right units.

I know this sounds rather abstract and mathematical, but the relevant point is just that physicists have a way to define what energy is, and it’s by definition conserved, which means it does not change in time. If you look at a simple system, for example that roller coaster, then the conserved energy is as usual the kinetic energy plus the potential energy. And if you add air molecules and the rails to the system, then their temperature would also add to the total, and so on.

But. If you look at a system with many small constituents, like air, then you will find that not all configurations of such a system are equally good at causing a macroscopic change, even if they have the same energy. A typical example would be setting fire to coal. The chemical bonds of the coal-molecules store a lot of energy. If you set fire to it, this causes a chain reaction between the coal and the oxygen in the air. In this reaction, energy from the chemical bonds is converted into kinetic energy of air molecules. This just means the air is warm, and since it’s warm, it will rise. You can use this rising air to drive a turb(ain), which you can then use to, say, move a vehicle or feed it into the grid to create electricity.

But suppose you don’t do anything with this energy, you just sit there and burn coal. This does not change anything about the total energy in the system, because that is conserved. The chemical energy of the coal is converted into kinetic energy of air molecules which distributes into the atmosphere. Same total energy. But now the energy is useless. You can no longer drive any turbine with it. What’s the difference?

The difference between the two cases is entropy. In the first case, you have the energy packed into the coal and entropy is small. In the latter case, you have the energy distributed in the motion of air molecules, and in this case the entropy is large.

A system that has energy in a state of low entropy is one whose energy you can use to create macroscopic changes, for example driving that turbine. Physicists call this useful energy “free energy” and say it “does work”. If the energy in a system is instead at high entropy, the energy is useless. Physicists then call it “heat” and heat cannot “do work”. The important point is that while energy is conserved, free energy is not conserved.

So, if someone says you should “save energy” by switching off the light, they really mean you should “save free energy”, because if you let the light on when you do not need it you convert useful free energy, from whatever is your source of electricity, into useless heat, that just warms the air in your room.

Okay, so we have seen that the total energy is by definition conserved, but that free energy is not conserved. Now what about the claim that Einstein actually told us energy is not conserved. That is correct. I know this sounds like a contradiction, but it’s not. Here is why.

Remember that energy is defined by Noether’s theorem, which says that energy is that quantity which is conserved if the system has a time-translation invariance, meaning, it does not really matter just at which moment you start an experiment.

But now remember, that Einstein’s theory of general relativity tells us that the universe expands. And if the universe expands, it does matter when you start an experiment. And expanding universe is not time-translation invariant. So, Noether’s theorem does not apply. Now, strictly speaking this does not mean that energy is not conserved in the expanding universe, it means that energy cannot be defined. However, you can take the thing you called energy when you thought the universe did not expand and ask what happens to it now that you know the universe does expand. And the answer is, well, it’s just not conserved.

A good example for this is cosmological redshift. If you have light of a particular wavelength early in the universe, then the wave-length of this light will increase when the universe expands, because it stretches. But the wave-length of light is inversely proportional to the energy of the light. So if the wave-length of light increases with the expansion of the universe, then the energy decreases. Where does the energy go? It goes nowhere, it just is not conserved. No, it really isn’t.

However, this non-conservation of energy in Einstein’s theory of general relativity is a really tiny effect that for all practical purposes plays absolutely no role here on Earth. It is really something that becomes noticeable only if you look at the universe as a whole. So, it is technically correct that energy is not conserved in Einstein’s theory of General Relativity. But this does not affect our earthly affairs.

In summary: The total energy of a system is conserved as long as you can neglect the expansion of the universe. However, the amount of useful energy, which is what physicists call “free energy,” is in general not conserved because of entropy increase.

Thanks for watching, see you next week. And remember to switch off the light.

---

We have two chats about this video’s topic, one today (Saturday, Oct 31) at noon Eastern Time (5pm CET). And one tomorrow (Sunday, Nov 1) also at noon Eastern Time (6pm CET).

A new model for the COVID pandemic

I spoke with the astrophysicist Niayesh Afshordi about his new pandemic model, what he has learned from it, and what the reaction has been to it.



You find more information about Niayesh's model on his website, and the paper is here.

You can join the chat with him tomorrow (Oct 29) at 5pm CET (noon Eastern Time) here.

Herd Immunity, Facts and Numbers

Today, I have a few words to say about herd immunity because there’s very little science in the discussion about it. I also want to briefly comment on the Great Barrington Declaration and on the conversation about it that we are not having.


First things first, herd immunity refers to that stage in the spread of a disease when a sufficient fraction of the population has become immune to the pathogen so that transmission will be suppressed. It does not mean that transmission stops, it means that on the average one infected person gives the disease to less than one new person, so outbreaks die out, instead of increasing.

It’s called “herd immunity” because it was first observed about a century ago in herds of sheep and, in some ways we’re not all that different from sheep.

Now, herd immunity is the only way a disease that is not contained will stop spreading. It can be achieved either by exposure to the live pathogen or by vaccination. However, in the current debate about the pursuit of herd immunity in response to the ongoing COVID outbreak, the term “herd immunity” has specifically been used to refer to herd immunity achieved by exposure to the virus, instead of waiting for a vaccine.

Second things second, when does a population reach herd immunity? The brief answer is, it’s complicated. This should not surprise you because whenever someone claims the answer to a scientific question is simple they either don’t know what they’re talking about, or they’re lying. There is a simple answer to the question when a population reaches herd immunity. But it does not tell the whole story.

This simple answer is that one can calculate the fraction of people who must be immune for herd immunity from the basic reproduction number R_0 as 1- 1/R_0.

Why is that? It’s because, R_0 tells you how many new people one infected person infects on the average. But the ones who will get ill are only those which are not immune. So if 1-1/R_0 is the fraction of people who are immune, then the fraction of people who are not immune is 1/R_0.

This then means that average number of susceptible people that one infected person reaches is R_0 * 1/R_0 which is 1. So, if the fraction of immune people has reached 1 – 1/R_0, then one infected person will on the average only pass on the disease to one other person, meaning at any level of immunity above 1 – 1/R_0, outbreaks will die out.

R_0 for COVID has been estimated with 2 to 3, meaning that the fraction of people who must have had the disease for herd immunity would be around 50 to 70 percent. For comparison, R_0 of the 1918 Spanish influenza has been estimated with 1.4 to 2.8, so that’s comparable to COVID, and R_0 of measles is roughly 12 to 18, with a herd immunity threshold of about 92-95%. Measles is pretty much the most contagious disease known to mankind.

That was the easy answer.

Here’s the more complicated but also more accurate answer. R_0 is not simply a property of the disease. It’s a number that quantifies successful transmission, and therefore depends on what measures people take to protect themselves from infection, such as social distancing, wearing masks, and washing hands. This is why epidemiologists use in their models instead an “effective R” coefficient that can change with time and with people’s habits. Roughly speaking this means that if we would all be very careful and very reasonable, then herd immunity would be easier to achieve.

But that R can change is not the biggest problem with estimating herd immunity. The biggest problem is that the simple estimate I just talked about assumes that everybody is equally likely to meet other people, which is just not the case in reality.

In realistic populations under normal circumstances, some people will have an above average number of contacts, and others below average. Now, people who have many contacts are likely to contribute a lot to the spread of the disease, but they are also likely to be among the first ones to contract the disease, and therefore become immune early on.

This means, if you use information about the mobility patterns, social networks, and population heterogeneity, the herd immunity threshold is lower because the biggest spreaders are the first to stop spreading. Taking this into account, some researchers have estimated the COVID herd immunity threshold to be more like 40% or in some optimistic cases even below 20%.

How reliable are these estimates? To me it looks like these estimates are based on more or less plausible models with little empirical data to back them up. And plausible models are the ones one should be especially careful with.

So what do the data say? Unfortunately, so far not much. The best data on herd immunity so far come from an antibody study in the Brazilian city of Manaus. That’s one of the largest cities in Brazil, with an estimated population of two point one million.

According to data from the state government, there have been about fifty five thousand COVID cases and two thousand seven hundred COVID fatalities in Manaus. These numbers likely underestimate the true number of infected and deceased people because the Brazilians have not been testing a lot. Then again, most countries did not have sufficient testing during the first wave.

If you go by the reported numbers, then about two point seven percent of the population in Manaus tested positive for COVID at some point during the outbreak. But the study which used blood donations collected during this time found that about forty-four percent of the population developed antibodies in the first three months of the outbreak.

After that, the infections tapered off without interventions. The researchers estimate the total number of people who eventually developed antibodies with sixty-six percent. The researchers claim that’s a sign for herd immunity. Please check the information below the video for references.

The number from this Brazilian study, about 44 to 66 percent seems consistent with the more pessimistic estimates for the COVID herd immunity threshold. But what it took to get there is not pretty.

2700 dead of about two million that’s more than one in a thousand. Hospitals run out of intensive care units, people were dying in the corridors, the city was scrambling to find ways to bury the dead quickly enough. And that’s even though the population of Manaus is pretty young; just six percent are older than sixty years. For comparison, in the United States, about 20% are above sixty years of age, and older people are more likely to die from the disease.

There are other reasons one cannot really compare Manaus with North America or Europe. Their health care system was working at almost full capacity even before the outbreak, and according to data from the world bank, in the Brazilian state which Manaus belongs to, the state of Amazonas, about 17% of people live below the poverty line. Also, most of the population in Manaus did not follow social distancing rules and few of them wore masks. These factors likely contributed to the rapid spread of the disease.

And I should add that the paper with the antibody study in Manaus has not yet been peer reviewed. There are various reasons why the people who donated blood may not be representative for the population. The authors write they corrected for this, but it remains to be seen what the reviewers think.

You probably want to know now how close we are to reaching herd immunity. The answer is, for all can tell, no one knows. That’s because, even leaving aside that we have no reliable estimates on the herd immunity threshold, we do not how many people have developed immunity to COVID.

In Manaus, the number of people who developed antibodies was more than twenty times higher than the number of those who tested positive. As of date in the United States about eight point five million people tested positive for COVID. The total population is about 330 Million.

This means about 2.5% of Americans have demonstrably contracted the disease, a rate that just by number is similar to the rate in Manaus, though Manaus got there faster with devastating consequences. However, the Americans are almost certainly better at testing and one cannot compare a sparsely populated country, like the United States, with one densely populated city in another country. So, again, it’s complicated.

For the Germans here, in Germany so far about 400,000 people have tested positive. That’s about 0.5 percent of the population.

And then, I should not forget to mention that antibodies are not the only way one can develop immunity. There is also T-cell immunity, that is basically a different defense mechanism of the body. The most relevant difference for the question of herd immunity is that it’s much more difficult to test for T-cell immunity. Which is why there are basically no data on it. But there are pretty reliable data by now showing that immunity to COVID is only temporary, antibody levels fall after a few months, and reinfections are possible, though it remains unclear how common they will be.

So, in summary: Estimates for the COVID herd immunity threshold range from roughly twenty percent to seventy percent, there are pretty much no data to make these estimates more accurate, we have no good data on how many people are presently immune, but we know reinfection is possible after a couple of months.

Let us then talk about the Great Barrington Declaration. The Great Barrington Declaration is not actually Great, it was merely written in place called Great Barrington. The declaration was formulated by three epidemiologists, and according to claims on the website, it has since been signed by more than eleven thousand medical and public health scientists.

The supporters of the declaration disapprove of lockdown measures and instead argue for an approach they call Focused Protection. In their own words:

“The most compassionate approach that balances the risks and benefits of reaching herd immunity, is to allow those who are at minimal risk of death to live their lives normally to build up immunity to the virus through natural infection, while better protecting those who are at highest risk. We call this Focused Protection.”

The reaction by other scientists and the media has been swift and negative. The Guardian called the Barrington Declaration “half baked” “bad science” and “a folly”. A group of scientists writing for The Lancet called it a “dangerous fallacy unsupported by scientific evidence”, the US American infectious disease expert Fauci called it “total nonsense,” and John Barry, writing for the New York Times, went so far to suggest it be called “mass murder” instead of herd immunity. Though they later changed the headline.

Some of the criticism focused on the people who wrote the declaration, or who they might have been supported by. These are ad hominem attacks that just distract from the science, so I don’t want to get into this.

The central element of the criticism is that the Barrington Declaration is vague on how the “Focused Protection” is supposed to work. This is a valid criticism. The declaration left it unclear just to how identify those at risk and how to keep them efficiently apart from the rest of the population, which is certainly difficult to achieve. But of course if no one is thinking about how to do it, there will be no plan for how to do it.

Why am I telling you this? Because I think all these commentators missed the point of the Barrington Declaration. Let us take this quote from an opinion piece in the Guardian in which three public health scientists commented on the idea of focused protection:

“It’s time to stop asking the question “is this sound science?” We know it is not.”

It’s right that arguing for focused protection is not sound science, but that is not because it’s not sound, it’s because it’s not science. It’s a value decision.

The authors of the Great Barrington Declaration point out, entirely correctly, that we are in a situation where we have only bad options. Lockdown measures are bad, pursuing natural herd immunity is also bad.

The question is, which is worse, and just what do you mean by “worse”. This is the decision that politicians are facing now and it is not obvious what is the best strategy. This decision must be supported by data for the consequences of each possible path of action. So we need to discuss not only how many people die from COVID and what the long-term health problems may be, but also how lockdowns, social distancing, and economic distress affect health and health care. We need proper risk estimates with uncertainties. We do not need scientists who proclaim that science tells us what’s the right thing to do.

I hope that this brief survey of the literature on herd immunity was helpful for you.

---

I have a video upcoming later today with astrophysicist (!) Niayesh Afshordi from Perimeter Institute about his new pandemic model (!!), so stay tuned. He will also join the Thursday chat at 5pm CET. Note that this is the awkward week of the year when the NYC-Berlin time shift is only 5 hours, so that's noon Eastern Time.

How can climate be predictable if weather is chaotic?

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

Today I want to take on a question that I have not been asked, but that I have seen people asking – and not getting a good answer. It’s how can scientists predict the climate in one hundred years if they cannot make weather forecasts beyond two weeks – because of chaos. The answer they usually get is “climate is not weather”, which is correct, but doesn’t really explain it. And I think it’s actually a good question. How is it possible that one can make reliable long-term predictions when short-term predictions are impossible. That’s what we will talk about today.


Now, weather forecast is hideously difficult, and I am not a meteorologist, so I will instead just use the best-known example of a chaotic system, that’s the one studied by Edward Lorenz in 1963.

Edward Lorenz was a meteorologist who discovered by accident that weather is chaotic. In the 1960s, he repeated a calculation to predict a weather trend, but rounded an initial value from six digits after the point to only three digits. Despite the tiny difference in the initial value, he got wildly different results. That’s chaos, and it gave rise to the idea of the “butterfly effect”, that the flap of a butterfly in China might cause a tornado in Texas two weeks later.

To understand better what was happening, Lorenz took his rather complicated set of equations and simplified it to a set of only three equations that nevertheless captures the strange behavior he had noticed. These three equations are now commonly known as the “Lorenz Model”. In the Lorenz model, we have three variables, X, Y, and Z and they are functions of time, that’s t. This model can be interpreted as a simplified description of convection in gases or fluids, but just what it describes does not really matter for our purposes.

The nice thing about the Lorenz model is that you can integrate the equations on a laptop. Let me show you one of the solutions. Each of the axes in this graph is one of the directions X, Y, Z, so the solution to the Lorenz model will be a curve in these three dimensions. As you can see, it circles around two different locations, back and forth.

It's not only this one solution which does that, actually all the solutions will end up doing circles close by these two places in the middle, which is called the “attractor”. The attractor has an interesting shape, and coincidentally happens to look somewhat like a butterfly with two parts you could call “wings”. But more relevant for us is that the model is chaotic. If we take two initial values that are very similar, but not exactly identical, as I have done here, then the curves at first look very similar, but then they run apart, and after some while they are entirely uncorrelated.

These three dimensional plots are pretty, but it’s somewhat hard to see just what is going on, so in the following I will merely look at one of these coordinates, that is the X-direction. From the three dimensional plot, you expect that the value in X-direction will go back and forth between two numbers, and indeed that’s what happens.

Here you see again the curves I previously showed for two initial values that differ by a tiny amount. At first the two curves look pretty much identical, but then they diverge and after some time they become entirely uncorrelated. As you see, the curves flip back and forth between positive and negative values, which correspond to the two wings of the attractor. In this early range, maybe up to t equals five, you would be able to make a decent weather forecast. But after that, the outcome depends very sensitively on exactly what initial value you used, and then measurement error makes a good prediction impossible. That’s chaos.

Now, I want to pretend that these curves say something about the weather, maybe they describes the weather on a strange planet where it either doesn’t rain at all or it pours and the weather just flips back and forth between these two extremes. Besides making the short-term weather forecast you could then also ask what’s the average rainfall in a certain period, say, a year.

To calculate this average, you would integrate the curve over some period of time, and then divide by the duration of that period. So let us plot these curves again, but for a longer period. Just by eyeballing these curves you’d expect the average to be approximately zero. Indeed, I calculated the average from t equals zero to t equals one hundred, and it comes out to be approximately zero. What this means is that the system spends about equal amounts of time on each wing of the attractor.

To stick with our story of rainfall on the weird planet, you can imagine that the curve shows deviations from a reference value that you set to zero. The average value depends on the initial value and will fluctuates around zero because I am only integrating over a finite period of time, so I arbitrarily cut off the curve somewhere. If you’d average over longer periods of time, the average would inch closer and closer to zero.

What I will do now is add a constant to the equations of the Lorenz model. I will call this constant “f” and mimics what climate scientists call “radiative forcing”. The radiative forcing is the excess power per area that Earth captures due to increasing carbon dioxide levels. Again that’s relative to a reference value.

I want to emphasize again that I am using this model only as an analogy. It does not actually describe the real climate. But it does make a good example for how to make predictions in chaotic systems.

Having said that, let us look again at how the curves look like with the added forcing. These are the curves for f equals one. Looks pretty much the same as previously if you ask me. f=2. I dunno. You wouldn’t believe how much time I have spent staring at these curves for this video. f=3. Looks like the system is spending a little more time in this upper range, doesn’t it? f=4. Yes, it clearly does. And just for fun, If you turn f up beyond seven or so, the system will get stuck on one side of the attractor immediately.

The relevant point is now that this happens for all initial values. Even though the system is chaotic, one clearly sees that the response of the system does have a predictable dependence on the input parameter.

To see this better, I have calculated the average of these curves as a function of the “radiative forcing”, for a sample of initial values. And this is what you get. You clearly see that the average value is strongly correlated with the radiative forcing. Again, the scatter you see here is because I am averaging over a rather arbitrarily chosen finite period.

What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.

This video was sponsored by Brilliant, which is a website that offers interactive courses on a large variety of topics in science and mathematics. In this video I showed you the results of some simple calculations, but if you really want to understand what is going on, then Brilliant is a great starting point. Their courses on Differential Equations I and II, probabilities and statistics cover much of the basics that I used here.

To support this channel and learn more about Brilliant, go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get 20 percent off the annual premium subscription.

---

You can join the chat about this week’s video, tomorrow (Sunday, Oct 25) at 5pm CET, here.

Particle Physicists Continue To Make Empty Promises

[This is a transcript of the video embedded below]

Hello and welcome back to my YouTube channel. Today I want to tell you how particle physicists are wasting your money. I know that’s not nice, but at the end of this video I think you will understand why I say what I say.


What ticked me off this time was a comment published in Nature Physics, by CERN Director-General Fabiola Gianotti and Gian Giudice, who is Head of CERN's Theory Department. It’s called a comment, but what it really is is an advertisement. It’s a sales pitch for their next larger collider for which they need, well, a few dozen billion Euro. We don’t know exactly because they are not telling us how expensive it would be to actually run the thing. When it comes to the question what the new mega collider could do for science, they explain:

“A good example of a guaranteed result is dark matter. A proton collider operating at energies around 100 TeV [that’s the energy of the planned larger collider] will conclusively probe the existence of weakly interacting dark-matter particles of thermal origin. This will lead either to a sensational discovery or to an experimental exclusion that will profoundly influence both particle physics and astrophysics.”

Let me unwrap this for you. The claim that dark matter is a guaranteed result, followed by weasel words about weakly interacting and thermal origin, is the physics equivalent of claiming “We will develop a new drug with the guaranteed result of curing cancer” followed by weasel words to explain, well, actually it will cure a type of cancer that exists only theoretically and has never been observed in reality. That’s how “guaranteed” this supposed dark matter result is. They guarantee to rule out some very specific hypotheses for dark matter that we have no reason to think are correct in the first place. What is going on here?

What’s going on is that particle physicists have a hard time understanding that when Popper went on about how important it is that a scientific hypothesis is falsifiable, he did not mean that a hypothesis is scientific just because it is falsifiable. There are lots of falsifiable hypotheses that are clearly unscientific.

For example, YouTube will have a global blackout tomorrow at noon central time. That’s totally falsifiable. If you give me 20 billion dollars, I can guarantee that I can test this hypothesis. Of course it’s not worth the money. Why? Because my hypothesis may be falsifiable, but it’s unscientific because it’s just guesswork. I have no reason whatsoever to think that my blackout prediction is correct.

The same is the case with particle physicists’ hypotheses for dark matter that you are “guaranteed” to rule out with that expensive big collider. Particle physicists literally have thousands of theories for dark matter, some thousandths of which have already been ruled out. Can they guarantee that a next larger collider can rule out some more? Yes. What is the guaranteed knowledge we will gain from this? Well, the same as the gain that we have gotten so far from ruling out their dark matter hypotheses, which is that we still have no idea what dark matter is. We don’t even know it is a particle to begin with.

Let us look again at that quote, they write:

“This will lead either to a sensational discovery or to an experimental exclusion that will profoundly influence both particle physics and astrophysics.”

No. The most likely outcome will be that particle physicists and astrophysicsts will swap their current “theories” for new “theories” according to which the supposed particles are heavier than expected. Then they will claim that we need yet another bigger collider to find them. What makes me think this will happen? Am I just bitter or cynical, as particle physicists accuse me? No, I am just looking at what they have done in the past.

For example, here’s an oldie but goldie, a quote from a piece written by string theorists David Gross and Edward Witten for the Wall street journal

“There is a high probability that supersymmetry, if it plays the role physicists suspect, will be confirmed in the next decade.”

They wrote this in 1996. Well, clearly that didn’t pan out.

And because it’s so much fun, I want to read you a few more quotes. But they are a little bit more technical, so I have to give you some background first.

When particle physicists say “electroweak scale” or “TeV scale” they mean energies that can be tested at the Large Hadron Collider. When they say “naturalness” they refer to a certain type of mathematical beauty that they think a theory should fulfil.

You see, particle physicists think it is a great problem that theories which have been experimentally confirmed are not as beautiful as particle physicists think nature should be. They have therefore invented a lot of particles that you can add to the supposedly ugly theories to remedy the lack of beauty. If this sounds like a completely non-scientific method, that’s because it is. There is no reason this method should work, and it does as a matter of fact not work. But they have done this for decades and still have not learned that it does not work.

Having said that, here is a quote from Giudice and Rattazzi in 1998. That’s the same Guidice who is one of the authors of the new nature commentary that I mentioned in the beginning. In 1998 he wrote:

“The naturalness (or hierarchy) problem, is considered to be the most serious theoretical argument against the validity of the Standard Model (SM) of elementary particle interactions beyond the TeV energy scale. In this respect, it can be viewed as the ultimate motivation for pushing the experimental research to higher energies.”

Higher energies, at that time, were the energies that have now been tested at the Large Hadron Collider. The supposed naturalness problem was the reason they thought the LHC should see new fundamental particles besides the Higgs. This has not happened. We now know that those arguments were wrong.

In 2004, Fabiola Gianotti, that’s the other author of the new Nature Physics comment, wrote:

“[Naturalness] arguments open the door to new and more fundamental physics. There are today several candidate scenarios for physics beyond the Standard Model, including Supersymmetry (SUSY), Technicolour and theories with Extra-dimensions. All of them predict new particles in the TeV region, as needed to stabilize the Higgs mass. We note that there is no other scale in particle physics today as compelling as the TeV scale, which strongly motivates a machine like the LHC able to explore directly and in detail this energy range.”

So, she claimed in 2004 that the LHC would see new particles besides the Higgs. Whatever happened to this prediction? Did they ever tell us what they learned from being wrong? Not to my knowledge.

These people were certainly not the only ones who repeated this story. Here is for example a quote from the particle physicist Michael Dine, who wrote in 2007:

“The Large Hadron Collider will either make a spectacular discovery or rule out supersymmetry entirely.”

Well, you know what, it hasn’t done either.

I could go on for quite some while quoting particle physicists who made wrong predictions and now pretend they didn’t, but it’s rather repetitive. I have collected the references here. Let us instead talk about what this means.

All these predictions from particle physicists were wrong. There is no shame in being wrong. Being wrong is essential for science. But what is shameful is that none of these people ever told us what they learned from being wrong. They did not revise their methods for making predictions for new particles. They still use the same methods that have not worked for decades. Neither did they do anything about the evident group think in their community. But they still want more money.

The tragedy is I actually like most of these particle physicists. They are smart and enthusiastic about science and for the most part they’re really nice people.

But look, they refuse to learn from evidence. And someone has to point it out: The evidence clearly says their methods are not working. Their methods have led to thousands of wrong predictions. Scientists should learn from failure. Particle physicists refuse to learn.

Particle physicists, of course, are entirely ignoring my criticism and instead call me “anti-science”. Let that sink in for a moment. They call me “anti-science” because I say we should think about where to best invest science funding, and if you do a risk-benefit assessment it is clear that building a bigger collider is not currently a good investment. It is both high risk and low benefit. We would be better off if we'd instead invest in the foundations of quantum mechanics and astroparticle physics. They call me “anti-science” because I ask scientists to think. You can’t make up this shit.

Frankly, the way that particle physicists behave makes me feel embarrassed I ever had anything to do with their field.

I Can’t Forget [Remix]

In the midst of the COVID lockdown I decided to remix some of my older songs. Just as I was sweating over the meters, I got an email out of the blue. Steven Nikolic from Canada wrote he’ be interested in remixing some of my old songs. A few months later, we have started a few projects together. Below you see the first result, a remake of my 2014 song “I Can’t Forget”.


If you want to see what difference 6 years can make, in hardware, software, and wrinkles, the original is here.

David Bohm’s Pilot Wave Interpretation of Quantum Mechanics

Today I want to take on a topic many of you requested, repeatedly. That is David Bohm’s approach to Quantum Mechanics, also known as the Pilot Wave Interpretation, or sometimes just Bohmian Mechanics. In this video, I want to tell you what Bohmian mechanics is, how it works, and what’s good and bad about it.

Ahead, I want to tell you a little about David Bohm himself, because I think the historical context is relevant to understand today’s situation with Bohmian Mechanics. David Bohm was born in 1917 in Pennsylvania, in the Eastern United States. His early work in physics was in the areas we would now call plasma physics and nuclear physics. In 1951, he published a textbook about quantum mechanics. In the course of writing it, he became dissatisfied with the then prevailing standard interpretation of quantum mechanics.

The standard interpretation at the time was that pioneered by the Copenhagen group – notably Bohr, Heisenberg, and Schrödinger – and is today usually referred to as the Copenhagen Interpretation. It works as follows. In quantum mechanics, everything is described by a wave-function, usually denoted Psi. Psi is a function of time. One can calculate how it changes in time with a differential equation known as the Schrödinger equation. When one makes a measurement, one calculates probabilities for the measurement outcomes from the wave-function. The equation by help of which one calculates these probabilities is known as Born’s Rule. I explained in an earlier video how this works.

The peculiar thing about the Copenhagen Interpretation is now that it does not tell you what happens before you make a measurement. If you have a particle described by a wave-function that says the particle is in two places at once, then the Copenhagen Interpretation merely says, at the moment you measure the particle it’s either here or there, with a certain probability that follows from the wave-function. But how the particle transitioned from being in two places at once to suddenly being in only one place, the Copenhagen Interpretation does not tell you. Those who advocate this interpretation would say that’s a question you are not supposed to ask because, by definition, what happens before the measurement is not measureable.

Bohm was not the only one dismayed that the Copenhagen people would answer a question by saying you’re not supposed to ask it. Albert Einstein didn’t like it either. If you remember, Einstein famously said “God does not throw dice”, by which he meant he does not believe that the probabilistic nature of quantum mechanics is fundamental. In contrast to what is often claimed, Einstein did not think quantum mechanics was wrong. He just thought it is probabilistic the same way classical physics is probabilistic, namely, that our inability to predict the outcome of a measurement in quantum mechanics comes from our lack of information. Einstein thought, in a nutshell, there must be some more information, some information that is missing in quantum mechanics, which is why it appears random.

This missing information in quantum mechanics is usually called “hidden variables”. If you knew the hidden variables, you could predict the outcome of a measurement. But the variables are “hidden”, so you can only calculate the probability of getting a particular outcome.

Back to Bohm. In 1952, he published two papers in which he laid out his idea for how to make sense of quantum mechanics. According to Bohm, the wave-function in quantum mechanics is not what we actually observe. Instead, what we observe are particles, which are guided by the wave-function. One can arrive at this interpretation in a few lines of calculation. I will not go through this in detail because it’s probably not so interesting for most of you. Let me just say you take the wave-function apart into an absolute value and a phase, insert it into the Schrödinger equation, and then separate the resulting equation into its real and imaginary part. That’s pretty much it.

The result is that in Bohmian mechanics the Schrödinger equation falls apart into two equations. One describes the conservation of probability and determines what the guiding field does. The other determines the position of the particle, and it depends on the guiding field. This second equation is usually called the “guiding equation.” So this is how Bohmian mechanics works. You have particles, and they are guided by a field which in return depends on the particle.

To use Bohm’s theory, you then need one further assumption, one that tells what the probability is for the particle to be at a certain place in the guiding field. This adds another equation, usually called the “quantum equilibrium hypothesis”. It is basically equivalent to Born’s rule and says that the probability for finding the particle in a particular place in the guiding field is given by the absolute square of the wave-function at that place. Taken together, these equations – the conservation of probability, the guiding equation, and the quantum equilibrium hypothesis – give the exact same predictions as quantum mechanics. The important difference is that in Bohmian mechanics, the particle is really always in only one place, which is not the case in quantum mechanics.

As they say, a picture speaks a thousand words, so let me just show you how this looks like for the double slit experiment. These thin black curves you see here are the possible ways that the particle could go from the double slit to the screen where it is measured by following the guiding field. Just which way the particle goes is determined by the place it started from. The randomness in the observed outcome is simply due to not knowing exactly where the particle came from.

What is it good for? The great thing about Bohmian mechanics is that it explains what happens in a quantum measurement. Bohmian mechanics says that the reason we can only make probabilistic predictions in quantum mechanics is just that we did not exactly know where the particle initially was. If we measure it, we find out where it is. Nothing mysterious about this. Bohm’s theory, therefore, says that probabilities in quantum mechanics are of the same type as in classical mechanics. The reason we can only predict probabilities for outcomes is because we are missing information. Bohmian mechanics is a hidden variables theory, and the hidden variables are the positions of those particles.

So, that’s the big benefit of Bohmian mechanics. I should add that while Bohm was working on his papers, it was brought to his attention that a very similar idea had previously been put forward in 1927 by De Broglie. This is why, in the literature, the theory is often more accurately referred to as “De Broglie Bohm”. But de Broglie’s proposal did, at the time, not attract much attention. So how did physicists react to Bohm’s proposal in fifty-two. Not very kindly. Niels Bohr called it “very foolish”. Leon Rosenfeld called it “very ingenious, but basically wrong”. Oppenheimer put it down as “juvenile deviationism”. And Einstein, too, was not convinced. He called it “a physical fairy-tale for children” and “not very hopeful.”

Why the criticism? One of the big disadvantages of Bohmian mechanics, that Einstein in particular disliked, is that it is even more non-local than quantum mechanics already is. That’s because the guiding field depends on all the particles you want to measure. This means, if you have a system of entangled particles, then the guiding equation says the velocity of one particle depends on the velocity of the other particles, regardless of how far away they are from each other.

That’s a problem because we know that quantum mechanics is strictly speaking only an approximation. The correct theory is really a more complicated version of quantum mechanics, known as quantum field theory. Quantum field theory is the type of theory that we use for the standard model of particle physics. It’s what people at CERN use to make predictions for their experiments. And in quantum field theory, locality and the speed of light limit, are super-important. They are built very deeply into the math.

The problem is now that since Bohmian mechanics is not local, it has turned out to be very difficult to make a quantum field theory out of it. Some have made attempts, but currently there is simply no Pilot Wave alternative for the Standard Model of Particle Physics. And for many physicists, me included, this is a game stopper. It means the Bohmian approach cannot reproduce the achievements of the Copenhagen Interpretation.

Bohmian mechanics has another odd feature that seems to have perplexed Albert Einstein and John Bell in particular. It’s that, depending on the exact initial position of the particle, the guiding field tells the particle to go either one way or another. But the guiding field has a lot of valleys where particles could be going. So what happens with the empty valleys if you make a measurement? In principle, these empty valleys continue to exist. David Deutsch has claimed this means “pilot-wave theories are parallel-universes theories in a state of chronic denial.”

Bohm himself, interestingly enough, seems to have changed his attitude towards his own theory. He originally thought it would in some cases give predictions different from quantum mechanics. I only learned this recently from a Biography of Bohm written by David Peat. Peat writes

“Bohm told Einstein… his only hope was that conventional quantum theory would not apply to very rapid processes. Experiments done in a rapid succession would, he hoped, show divergences from the conventional theory and give clues as to what lies at a deeper level.”

However, Bohm had pretty much the whole community against him. After a particularly hefty criticism by Heisenberg, Bohm changed course and claimed that his theory made the same predictions as quantum mechanics. But it did not help. After this, they just complained that the theory did not make new predictions. And in the end, they just ignored him.

So is Bohmian mechanics in the end just a way of making you feel better about the predictions of quantum mechanics? Depends on whether or not you think the “quantum equilibrium hypothesis” is always fulfilled. If it is always fulfilled, the two theories give the same predictions. But if the equilibrium is actually a state the system must first settle in, as the name certainly suggests, then there might be cases when this assumption is not fulfilled. And then, Bohmian mechanics is really a different theory. Physicists still debate today whether such deviations from quantum equilibrium can happen, and whether we can therefore find out that Bohm was right."" This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I always try to show you some of the key equations, but if you really want to understand how to use them, then Brilliant is a great starting point. For this video, for example, I would recommend their courses on differential equations, linear algebra, and quantum objects. To support this channel and learn more about Brilliant, go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get 20 percent off the annual premium subscription.

---

You can join the chats on this week’s topic using the Converseful app in the bottom right corner:

• Sunday, Oct 18 at 6pm CEST
• Tuesday, Oct 20 at 6pm CEST

You don’t have free will, but don’t worry.

Today I want to talk about an issue that must have occurred to everyone who spent some time thinking about physics. Which is that the idea of free will is both incompatible with the laws of nature and entirely meaningless. I know that a lot of people just do not want to believe this. But I think you are here to hear what the science says. So, I will tell you what the science says. In this video I first explain why free will does not exist, indeed makes no sense, and then tell you why there are better things to worry about.


I want to say ahead that there is much discussion about free will in neurology, where the question is whether we subconsciously make decisions before we become consciously aware of having made one. I am not a neurologist, so this is not what I am concerned with here. I will be talking about free will as the idea that in this present moment, several futures are possible, and your “free will” plays a role for selecting which one of those possible futures becomes reality. This, I think, is how most of us intuitively think of free will because it agrees with our experience of how the world seems to works. It is not how some philosophers have defined free will, and I will get to this later. But first, let me tell you what’s wrong with this intuitive idea that we can somehow select among possible futures.

Last week, I explained what differential equations are, and that all laws of nature which we currently know work with those differential equations. These laws have the common property that if you have an initial condition at one moment in time, for example the exact details of the particles in your brain and all your brain’s inputs, then you can calculate what happens at any other moment in time from those initial conditions. This means in a nutshell that the whole story of the universe in every single detail was determined already at the big bang. We are just watching it play out.

These deterministic laws of nature apply to you and your brain because you are made of particles, and what happens with you is a consequence of what happens with those particles. A lot of people seem to think this is a philosophical position. They call it “materialism” or “reductionism” and think that giving it a name that ends on –ism is an excuse to not believe it. Well, of course you can insist to just not believe reductionism is correct. But this is denying scientific evidence. We do not guess, we know that brains are made of particles. And we do not guess, we know, that we can derive from the laws for the constituents what the whole object does. If you make a claim to the contrary, you are contradicting well-established science. I can’t prevent you from denying scientific evidence, but I can tell you that this way you will never understand how the universe really works.

So, the trouble with free will is that according to the laws of nature that we know describe humans on the fundamental level, the future is determined by the present. That the system – in this case, your brain – might be partly chaotic does not make a difference for this conclusion, because chaos is still deterministic. Chaos makes predictions difficult, but the future still follows from the initial condition.

What about quantum mechanics? In quantum mechanics some events are truly random and cannot be predicted. Does this mean that quantum mechanics is where you can find free will? Sorry, but no, this makes no sense. These random events in quantum mechanics are not influenced by you, regardless of exactly what you mean by “you”, because they are not influenced by anything. That’s the whole point of saying they are fundamentally random. Nothing determines their outcome. There is no “will” in this. Not yours and not anybody else’s.

Taken together we therefore have determinism with the occasional, random quantum jump, and no combination of these two types of laws allows for anything resembling this intuitive idea that we can somehow choose which possible future becomes real. The reason this idea of free will turns out to be incompatible with the laws of nature is that it never made sense in the first place. You see, that thing you call “free will” should in some sense allow you to choose what you want. But then it’s either determined by what you want, in which case it’s not free, or it’s not determined, in which case it’s not a will.

Now, some have tried to define free will by the “ability to have done otherwise”. But that’s just empty words. If you did one thing, there is no evidence you could have done something else because, well, you didn’t. Really there is always only your fantasy of having done otherwise.

In summary, the idea that we have a free will which gives us the possibility to select among different futures is both incompatible with the laws of nature and logically incoherent. I should add here that it’s not like I am saying something new. Look at the writing of any philosopher who understand physics, and they will acknowledge this.

But some philosophers insist they want to have something they can call free will, and have therefore tried to redefine it. For example, you may speak of free will if no one was in practice able to predict what you would do. This is certainly presently the case, that most human behavior is unpredictable, though I can predict that some people who didn’t actually watch this video will leave a comment saying they had no other choice than leaving their comment and think they are terribly original.

So, yeah, if you want you can redefine “free will” to mean “no one was able to predict your decision.” But of course your decision was still determined or random regardless of whether someone predicted it. Others have tried to argue that free will means some of your decisions are dominated by processes internal to your brain and not by external influences. But of course your decision was still determined or random, regardless of whether it was dominated by internal or external influences. I find it silly to speak of “free will” in these cases.

I also find it unenlightening to have an argument about the use of words. If you want to define free will in such a way that it is still consistent with the laws of nature, that is fine by me, though I will continue to complain that’s just verbal acrobatics. In any case, regardless of how you want to define the word, we still cannot select among several possible futures. This idea makes absolutely no sense if you know anything about physics.

What is really going on if you are making a decision is that your brain is running a calculation, and while it is doing that, you do not know what the outcome of the calculation will be. Because if you did, you wouldn’t have to do the calculation. So, the impression of free will comes from our self-awareness, that we think about what to do, combined with our inability to predict the result of that thinking before we’re done.

I feel like I must add here a word about the claim that human behavior is unpredictable because if someone told you that they predicted you’d do one thing, you could decide to do something else. This is a rubbish argument because it has nothing to do with human behavior, it comes from interfering with the system you are making predictions for. It is easy to see that this argument is nonsense because you can make the same claim about very simple computer codes.

Suppose you have a computer that evaluates whether an equation has a real-valued root. The answer is yes or no. You can predict the answer. But now you can change the algorithm so that if you input the correct answer, the code will output the exact opposite answer, ie “yes” if you predicted “no” and “no” if you predicted “yes”. As a consequence, your prediction will never be correct. Clearly, this has nothing to do with free will but with the fact that the system you make a prediction for gets input which the prediction didn’t account for. There’s nothing interesting going on in this argument.

Another objection that I’ve heard is that I should not say free will does not exist because that would erode people’s moral behavior. The concern is, you see, that if people knew free will does not exist, then they would think it doesn’t matter what they do. This is of course nonsense. If you act in ways that harm other people, then these other people will take steps to prevent that from happening again. This has nothing to do with free will. We are all just running software that is trying to optimize our well-being. If you caused harm, you are responsible, not because you had “free will” but because you embody the problem and locking you up will solve it.

There have been a few research studies that supposedly showed a relation between priming participants to not believe in free will and them behaving immorally. The problem with these studies, if you look at how they were set up, is that people were not primed to not believe in free will. They were primed to think fatalistically. In some cases, for example, they were being suggested that their genes determine their future, which, needless to say, is only partly correct, regardless of whether you believe in free will. And some more nuanced recent studies have actually shown the opposite. A 2017 study on free will and moral behavior concluded “we observed that disbelief in free will had a positive impact on the morality of decisions toward others”. Please check the information below the video for a reference.

So I hope I have convinced you that free will is nonsense, and that the idea deserves going into the rubbish bin. The reason this has not happened yet, I think, is that people find it difficult to think of themselves in any other way than making decisions drawing on this non-existent “free will.” So what can you do? You don’t need to do anything. Just because free will is an illusion does not mean you are not allowed to use it as a thinking aid. If you lived a happy life so far using your imagined free will, by all means, please keep on doing so.

If it causes you cognitive dissonance to acknowledge you believe in something that doesn’t exist, I suggest that you think of your life as a story which has not yet been told. You are equipped with a thinking apparatus that you use to collect information and act on what you have learned from this. The result of that thinking is determined, but you still have to do the thinking. That’s your task. That’s why you are here. I am curious to see what will come out of your thinking, and you should be curious about it too.

Why am I telling you this? Because I think that people who do not understand that free will is an illusion underestimate how much their decisions are influenced by the information they are exposed to. After watching this video, I hope, some of you will realize that to make the best of your thinking apparatus, you need to understand how it works, and pay more attention to cognitive biases and logical fallacies.

---

You can join the chat about this week's post using these links:

Chat #1 - Sunday, October 11 @ 9 AM PST / 12PM EST / 6PM CEST
Chat #2 - Tuesday, October 13 @ 9 AM PST / 12PM EST / 6PM CEST

[Guest Post] New on BackRe(action): Real-Time Chat Rooms

[This post is written by Ben Alderoty.]

For those who’ve been keeping tabs, my team and I have been working with Sabine since earlier this year to give commenters on her site more ways to talk. Based on your feedback, we’re launching a new way to make that happen: real-time chat rooms. Here’s how they’ll work.

Chat rooms (chats) live in the bottom right corner of the blog. For the time being, they are only available on Desktop with support for mobile devices to come soon. Unlike traditional, always-available chat rooms, chats on BackRe(action) happen at scheduled times. This ensures people will be there at the same time as you and the conversation can happen in real-time. Chats start at their scheduled times and end when everyone has left.

You’ll see the first couple of chats have already been scheduled when you open the app. The topic for these chats is Sabine’s upcoming post on free will she is releasing on Saturday. If you’re interested in attending, you can set up a reminder by clicking ‘Remind me’ and selecting either Email or Calendar. You can also share links to the chat by clicking the icon next to the chat name. We’ll be trying out different topics and times for chats based on feedback we receive. 

 

The chats themselves happen right here on BackRe(action). You won’t need an account to participate, just a name (real, fake, pseudonym… anything works). Depending on how many people join, the group may be split into separate rooms to allow for better discussion. Chats will remain open for late joiners as long as there’s an active discussion taking place. Spectators are welcome too! All of the messages will disappear when the chat ends, so you’ll have to be there to see what’s said.

As a reminder, the first two chats are happening on:

Chat #1 - Sunday, October 11 @ 9 AM PST / 12PM EST / 6PM CEST
Chat #2 - Tuesday, October 13 @ 9 AM PST / 12PM EST / 6PM CEST

Come to one or come to both! New chats will be up mid-next week for the week after.

So, what do you think? Are you ready for chat rooms on BackRe(action)? What topics do you want to talk about? Let us know what you think in the comments section or in the app via the ‘Give Feedback’ button below the chats.

What are Differential Equations and how do they 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.]

Today I want to talk about that piece of mathematics which describes, for all we currently know, everything: Differential Equations. Pandemic models? Differential equations. Expansion of the universe? Differential equations. Climate models? Differential equations. Financial markets? Differential equations. Quantum mechanics? Guess what, differential equations.


I find it hard to think of anything that’s more relevant for understanding how the world works than differential equations. Differential equations are the key to making predictions and to finding out what is predictable, from the motion of galaxies to the weather, to human behavior. In this video I will tell you what differential equations are and how they work, give you some simple examples, tell you where they are used in science today, and discuss what they mean for the question whether our future is determined already.

To get an idea for how differential equations work, let us look at a simple example: The spread of a disease through the population. Suppose you have a number of people, let’s call it N, which are infected with a disease. You want to know how N will change in time, so N is a function of t, where t is time. Each of the N people has a certain probability to spread the disease to other people during some period of time. We will quantify this infectiousness by a constant, k. This means that the change in the number of people per time equals that constant k times the number of people who already are infected.

Now, the change of a function per time is the derivative of the function with respect to time. So, this gives you an equation which says that the derivative of the function is proportional to the function itself. And this is a differential equation. A differential equation is more generally an equation for an unknown function which contains derivatives of the function. So, a differential equation must be solved not for a parameter, say x, but for a whole function.

The solution to the differential equation for disease spread is an exponential function, where the probability of infecting someone appears in the exponent, and there is a free constant in front of the exponential, which I called N₀. This function will solve the equation for any value of this free constant. If you put in the time t equals zero, then you can see that this constant N₀ is simply the number of infected people at the initial time.

So, this is why infectious diseases begin by spreading exponentially, because the increase in the number of infected people is proportional to the number of people who are already infected. You are probably wondering now how these constants relate to the basic reproduction number of the disease, the R naught we have all become familiar with. When a disease begins to spread, this constant k in the exponent is (R₀-1)/ τ, where τ is the time an infected person remains infectious.

So, R naught can be interpreted as the average number of people someone infects. Of course in reality diseases do not continue spreading exponentially, because eventually everyone is either immune or dead and there’s no one left to infect. To get a more realistic model for disease spread, one would have to take into account that the number of susceptible people begins to decrease as the infection spreads. But this is not a video about pandemic models, so let us instead get back to differential equations. Another simple example for a differential equation is one you almost certainly know, Newton’s second law, F equals m times a. Let us just take the case where the force is a constant. This could describe, for example, the gravitational force near the surface of the earth, in a range so small you can neglect that the force is actually a function of the distance from the center of Earth. The equation is then just a equals F over m, which I will rename to small g, and this is a constant. a is the acceleration, so the second time-derivative of position. Physicists typically denote the position with x, and a derivative with respect to time with a dot, so that is double-dot x equals g. And that’s a differential equation for the function x of t.

For simplicity, let us take x to be just the vertical direction. The solution to this equation is then x(t)= gt²/2 + vt +x₀, where v and x₀ are constants. If you take the first derivative of this function, you get g times t plus v, and another derivative gives just g. And that’s regardless of what the two constants were.

These two new constants in this solution, v and x₀, can easily be interpreted, by looking at the time t=0. x₀ is the position of the particle at time t = 0, and, if we look at the derivative of the function, we see that v is the velocity of the particle at t=0. If you take an initial velocity that’s pointed up, the curve for the position as a function of time is a parabola, telling you the particle goes up and comes back down. You already knew that, of course. The relevant point for our purposes is that, again, you do not get one function as a solution to the equation, but a whole family of functions, one for each possible choice of the constants.

Physicists call these free constants which appear in the possible solutions to a differential equation “initial values”. You need such initial values to pick the solution of the differential equation which fits to the system you want to describe. The reason we have two initial values for Newton’s law is that the highest order of derivative in the differential equation is two. Roughly speaking, you need one initial value per order of derivative. In the first example of disease growth, if you remember, we had one derivative and correspondingly only one initial value.

Now, Newton’s second law is not exactly frontier research, but the thing is that all theories we use in the foundations of physics today are of this type. They are given by differential equations, which have a large number of possible solutions. Then we insert initial values to identify the solution that actually describes what we observe.

Physicists use differential equations for everything, for stars, for atoms, for gases and fluids, for electromagnetic radiation, for the size of the universe, and so on. And these differential equations always work the same. You solve the equation, insert your initial values, and then you know what happens at any other moment in time.

I should add here that the “initial values” do not necessarily have to be at an initial time from which you make predictions for later times. The terminology is somewhat confusing, but you can also choose initial values at a final time and make predictions for times before that. This is for example what we do in cosmology. We know how the universe looks today, that are our “initial” values, and then we run the equations backwards in time to find out what the universe must have looked like earlier.

These differential equations are what we call “deterministic”. If I tell you how many people are ill today, you can calculate how many will be ill next week. If I tell you where I throw a particle with what initial velocity, you can tell me where it comes down. If I tell you what the universe looks like today, and you have the right differential equation, you can calculate what happens at every other moment of time. This consequence is that, according to the natural laws that physicists have found so far, the future is entirely fixed already; indeed, it was fixed already when the universe began.

This was pointed out first by Pierre Simon Laplace in 1814 who wrote:

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”

This “intellect” Laplace is referring to is now sometimes called “Laplace’s demon”. But physics didn’t end with Laplace. After Laplace wrote those words, Poincare realized that even deterministic systems can become unpredictable for all practical purposes because they are “chaotic”. I talked about this in my earlier video about the Butterfly effect. And then, in the 20th century, along came quantum mechanics. Quantum mechanics is a peculiar theory because it does not only use an differential equations. Quantum mechanics uses another equation in addition to the differential equation. The additional equation describes what happens in a measurement. This is the so-called measurement update and it is not deterministic.

What does this mean for the question whether we have free will? That’s what we will talk about next week, so stay tuned.