Saturday, March 27, 2021

Is the universe REALLY a hologram?

[This is a transcript of the video embedded below.]

Do we live in a hologram? String theorists think we do. But what does that mean? How do holograms work, and how are they related to string theory? That’s what we will talk about today.

In science fiction movies, holograms are 3-dimensional, moving images. But in reality, the technology for motion holograms hasn’t caught up with imagination. At least so far, holograms are still mostly stills.

The holograms you are most likely to have seen are not like those in the movies. They are not a projection of an object into thin air – however that’s supposed to work. Instead, you normally see a three-dimensional object above or behind a flat film. Small holograms are today frequently used as a security measure on credit cards, ID cards, or even banknotes, because they are easy to see, but difficult to copy.

If you hold such a hologram into light, you will see that it seems to have depth, even though it is printed on a flat surface. That’s because in photographs, we are limited to the one perspective from which the picture was taken, and that’s why they look flat. But you can tilt holograms and observe them from different angles, as if you were examining a three-dimensional object.

Now, these holograms on your credit cards, or the ones that you find on postcards or book covers, are not “real” holograms. They are actually composed of several 2-dimensional images and depending on the angle, a different image is reflected back at you, which creates the illusion of a 3-dimensional image.

In a real hologram the image is indeed 3-dimensional. But the market for real holograms is small, so they are hard to come by, even though the technology to produce them is straightforward. A real hologram looks like this.

Real holograms actually encode a three-dimensional object on a flat surface. How is this possible? The answer is interference.

Light is electromagnetic waves, so it has crests and troughs. And a key property of waves is that they can be overlaid and then amplify or wash out each other. If two waves are overlaid so that two crests meet at the same point, that will amplify the wave. This is called constructive interference. But if a crest meets a trough, the waves will cancel. This is called destructive interference.

Now, we don’t normally see light cancelling out other light. That’s because to see interference one needs very regular light, where the crests and troughs are neatly aligned. Sunlight or LED light doesn’t have that property. But laser light has it, and so laser light can be interfered.

And this interference can be used to create holograms. For this, one first splits a laser beam in two with a semi-transparent glass or crystal, called a beam-splitter, and makes each beam broader with a diverging lens. Then, one aims one half of the beam at the object that one wants to take an image of. The light will not just bounce off the object in one single direction, but it will scatter in many different directions. And the scattered light contains information about the surface of the object. Then, one recombines the two beams and captures the intensity of the light with a light-sensitive screen.

Now, remember that laser light can interfere. This means, how large the intensity on the screen is, depends on whether the interference was destructive or constructive, which again depends on just where the object was located and how it was shaped. So, the screen has captured the full three-dimensional information. To view the hologram, one develops the film and shines light onto it at the same wavelength as the image was taken, which reproduces the 3-dimensional image.

To understand this in a little more detail, let us look at the image on the screen if one uses a very small point-like object. It looks like this. It’s called a zone plate. The intensity and width of the rings depends on the distance between the point-like object and the screen, and the wavelength of the light. But any object is basically a large number of point-like objects, so the interference image on the screen is generally an overlap of many different zone plates with these concentric rings.

The amazing thing about holograms is now this. Every part of the screen receives information from every part of the object. As a consequence, if you develop the image to get the hologram, you can take it apart into pieces, and each piece will still recreate the whole 3-dimensional object. To understand better how this works, look again at the zone plate, the one of a single point-like object. If you have only a small piece that contains part of the rings, you can infer the rest of the pattern, though it gets a little more difficult. If you have a general plate that overlaps many zone plates, this is still possible. So, at least mathematically, you can reconstruct the entire object from any part of the holographic plate. In reality, the quality of the image will go down.

So, now that you know how real holograms work, let us talk about the idea that the universe is a hologram.

When string theorists claim that our universe is a hologram, they mean the following. Our universe has a positive cosmological constant. But mathematically, universes with a negative cosmological constant are much easier to work with. So, this is what string theorists usually look at. These universes with a negative cosmological constant are called Anti-de Sitter spaces and into these Anti-de Sitter things they put supersymmetric matter. To best current knowledge, our universe is not Anti De Sitter and matter is not supersymmetric, but mathematically, you can certain do that.

For some specific examples, it has then been shown that the gravitational theory in such an Anti de Sitter universe is mathematically equivalent to a different theory on the conformal boundary of that universe. What the heck is the conformal boundary of the universe? Well, our actual universe doesn’t have one. But these Anti-De Sitter spaces do. Just exactly how they are defined isn’t all that important. You only need to know that this conformal boundary has one dimension of space less than the space it is a boundary of.

So, you have an equivalence between two theories in a different number of dimensions of space. A gravitational theory in this anti-De Sitter space with the weird matter. And a different theory on the boundary of that space, which also has weird matter. And just so you have heard the name: The theory on the boundary is what’s called a conformal field theory, and the whole thing is known as the Anti-de Sitter – Conformal Field Theory duality, or AdS/CFT for short.

This duality has been mathematically confirmed for some specific cases, but pretty much all string theorists seem to believe it is much more generally valid. In fact, a lot of them seem believe it is valid even in our universe, even though there is no evidence for that, neither observational nor mathematical. In this most general form, the duality is simply called the “holographic principle”.

If the holographic principle was correct, it would mean that the information about any volume in our universe is encoded on the boundary of that volume. That’s remarkable because naively, you’d think the amount of information you can store in a volume of space grows much faster than the information you can store on the surface. But according to the holographic principle, the information you can put into the volume somehow isn’t what we think it is. It must have more correlations than we realize. So it the holographic principle was true, that would be very interesting. I talked about this in more detail in an earlier video.

The holographic principle indeed sounds a little like optical holography. In both cases one encodes information about a volume on a surface with one dimension less. But if you look a little more closely, there are two important differences between the holographic principle and real holography:

First, an optical hologram is not actually captured in two dimensions; the holographic film has a thickness, and you need that thickness to store the information. The holographic principle, on the other hand, is a mathematical abstraction, and the encoding really occurs in one dimension less.

Second, as we saw earlier, in a real hologram, each part contains information about the whole object. But in the mathematics of the holographic universe, this is not the case. If you take only a piece of the boundary, that will not allow you to reproduce what goes on in the entire universe.

This is why I don’t think referring to this idea from string theory as holography is a good analogy. But now you know just exactly what the two types of holography do, and do not have in common.

Saturday, March 20, 2021

Whatever happened to Life on Venus?

[This is a transcript of the video embedded below.]

A few months ago, the headlines screamed that scientists had found signs of life on Venus. But it didn’t take long for other scientists to raise objections. So, just exactly what did they find on Venus? Did they actually find it? And what does it all mean? That’s what we will talk about today.

The discovery that made headlines a few months ago was that an international group of researchers said they’d found traces of a molecule called phosphine in the atmosphere of Venus.

Phosphine is a molecule made of one phosphorus and three hydrogen atoms. On planets like Jupiter and Saturn, pressure and temperature are so high that phosphine can form by coincidental chemical reactions, and indeed phosphine has been observed in the atmosphere of these two planets. On planets like Venus, however, the pressure isn’t remotely large enough to produce phosphine this way.

And the only other known processes to create phosphine are biological. On Earth, for example, which in size and distance to the Sun isn’t all that different to Venus, the only natural production processes for phosphine are certain types of microbes. Lest you think this means that phosphine is somehow “good for life”, I should add that the microbes in question live without oxygen. Indeed, phosphine is toxic for forms of life that use oxygen, which is most of life on earth. In fact, phosphine is used in the agricultural industry to kill rodents and insects.

So, the production of phosphine on Venus at fairly low atmospheric pressure seems to require life in some sense, which is why the claim that there’s phosphine on Venus is BIG. It could mean there’s microbial life on Venus. And just in case microbial life doesn’t excite you all that much, this would be super-interesting because it would give us a clue to what the chances are that life evolves on other planets in general.

So, just exactly what did they find?

The suspicion that phosphine might be present on Venus isn’t entirely new. The researchers first saw something that could be phosphine in two-thousand and seventeen in data from the James Clerk Maxwell Telescope, which is a radio telescope in Hawaii. However, this signal was not particularly good, so they didn’t publish it. Instead they waited for more data from the ALMA telescope in Chile. Then they published a combined analysis of the data from both telescopes in Nature Astronomy.

Here’s what they did. One can look for evidence of molecules by exploiting that each molecule reacts to light at different wave-lengths. To some wave-lengths, a molecule may not react at all, but others it may absorb because they cause the molecule to vibrate or rotate around itself. It’s like each molecule has very specific resonance frequencies, like if you’re in an airplane and the engine’s being turned up and then, at a certain pitch the whole plane shakes? That’s a resonance. For the plane it happens at certain wavelengths of sound. For molecules it happens at certain wave-lengths of light.

So, if light passes through a gas, like the atmosphere of Venus, then just how much light at each wave-length passes through depends on what molecules are in the gas. Each molecule has a very specific signature, and that makes the identification possible.

At least in principle. In practice… it’s difficult. That’s because different molecules can have very similar absorption lines.

For example, the phosphine absorption line which all the debate is about has a frequency of two-hundred sixty-six point nine four four Gigahertz. But sulfur dioxide has an absorption line at two-hundred sixty-six point nine four three GigaHertz, and sulfur dioxide is really common in the atmosphere of Venus. That makes it quite a challenge to find traces of phosphine.

But challenges are there to be met. The astrophysicists estimated the contribution from Sulphur dioxide from other lines which this molecule should also produce.

They found that these other lines were almost invisible. So they concluded that the absorption in the frequency range of interest had to be mostly due to phosphine and they estimated the amount with about seven to twenty parts per billion, so that’s seven to twenty molecules of phosphine per billion molecules of anything.

It’s this discovery which made the big headlines. The results they got for the phosphine amount from the two different telescopes are a little different, and such an inconsistency is somewhat of a red flag. But then, these measurements were made some years apart and the atmosphere of Venus could have undergone changes in that period, so it’s not necessarily a problem.

Unfortunately, after publishing their analysis, the team learned that the data from ALMA had not been processed correctly. It was not their fault, but it meant they had to redo their analysis. With the corrected data, the amount of phosphine they claimed to see fell to something between 1 and 4 parts per billion. Less, but still there.

Of course such an important finding attracted a lot of attention, and it didn’t take long for other researchers to have a close look at the analysis. It was not only that finding phosphine was surprising, not finding sulphur dioxide was not normal either; it had been detected many times in the atmosphere of Venus in amounts about 10 times higher than what the phosphine-discovery study claimed it was.

Already in October last year, a paper came out that argued there’s no signal at all in the data, and that said the original study used an overly complicated twelve parameter fit that fooled them into seeing something where there was nothing. This criticism has since been published in a peer reviewed journal. And by the end of January another team put out two papers in which they pointed out several other problems with the original analysis.

First they used a model of the atmosphere of Venus and calculated that the alleged phosphine absorption comes from altitudes higher than eighty kilometers. Problem is, at such a high altitude, phosphine is incredibly unstable because ultraviolet light from the sun breaks it apart quickly. They estimated it would have a lifetime of under one second! This means for phosphine to be present on Venus in the observed amounts, it would’ve to be produced at a rate higher than the production of oxygen by photosynthesis on Earth. You’d need a lot of bacteria to get that done.

Second, they claim that the ALMA telescope should not have been able to see the signal at all, or at least a much smaller signal, because of an effect called line dilution. Line dilution can occur if one has a telescope with many separate dishes like ALMA. A signal that’s smeared out over many of the dishes, like the signal from the atmosphere of Venus, can then be affected by interference effects.

According to estimates in the new paper, line dilution should suppress the signal in the ALMA telescope by about a factor 10-20, in which case it would not be visible at all. And indeed they claim that no signal is entirely consistent with the data from the second telescope. This criticism, too, has now passed peer review.

What does it mean?

Well, the authors of the original study might reply to this criticism, and so it will probably take some time until the dust settles. But even if the criticism is correct, this would not mean there’s no phosphine on Venus. As they say, absence of evidence is not evidence of absence. If the criticism is correct, then the observations, exactly because they probe only high altitudes where phosphine is unstable, can neither exclude, nor confirm, the presence of phosphine on Venus. And so, the summary is, as so often in science: More work is needed.

Wednesday, March 17, 2021

Live Seminar about Dark Matter on Friday

I will give an online simiar about dark matter and modified gravity on Friday at 4pm CET, if you want to attend, the link is here:

I'm speaking in English (as you can see, half in American, half in British English, as usual), but the seminar will be live translated to Spanish, for which there's a zoom link somewhere.

Saturday, March 13, 2021

Can we stop hurricanes?

[This is a transcript of the video embedded below.]

Hurricanes are among the most devastating natural disasters. That’s because hurricanes are enormous! A medium-sized hurricane extends over an area about the size of Texas. On a globe they’ll cover 6 to 12 degrees latitude. And as they blow over land, they leave behind wide trails of destruction, caused by strong winds and rain. Damages from hurricanes regularly exceed billions of US dollars. Can’t we do something about that? Can’t we blast hurricanes apart? Redirect them? Or stop them from forming in the first place? What does science say about that? That’s what we’ll talk about today.

Donald Trump, the former president of the United States, has reportedly asked repeatedly whether it’s possible to get rid of hurricanes by dropping nuclear bombs on them. His proposal was swiftly dismissed by scientists and the media likewise. Their argument can be summed up with “you can’t” and even if you could “it’d be a bad idea.” Trump then denied he ever said anything, the world forgot about it, and here we are, still wondering if not there’s something we can do to stop hurricanes.

Trumps idea might sound crazy, but he was not the first to think of nuking a hurricane, and he probably won’t be the last. And I think trying to prevent hurricanes isn’t as crazy as it sounds.

The idea to nuke a hurricane came up already right after nuclear weapons were deployed for the first time, in Japan in August 1945. August is in the middle of the hurricane season in Florida. The mayor of Miami Beach, Herbert Frink, made the connection. He asked President Harry Truman about the possibility to use the new weapon to fight against hurricanes. And, sure enough, the Americans looked into it.

But they quickly realized that while the energy released by a nuclear bomb was gigantic compared to all other kinds of weapons, it was still nothing compared to the energies that build up in hurricanes. For comparison: The atomic bombs dropped on Japan released an energy of about 20 kilotons each. A typical hurricane releases about 10,000 times as much energy – per hour. The total power of a hurricane is comparable to the entire global power consumption. That’s because hurricanes are enormous!

By the way, hurricanes and typhoons are the same thing. The generic term used by meterologists is “tropical cyclone”. It refers to “a rotating, organized system of clouds and thunderstorms that originates over tropical or subtropical waters.” If they get large enough, they’re then either called hurricanes or typhoons, or they just remain tropical cyclones. But it’s like the difference between an astronaut and a cosmonaut. The same thing!

But back to the nukes. In 1956 an Air Force meteorologist by name Jack W Reed proposed to launch a megaton nuclear bomb – that is about 50 times the power of the ones in Japan – into a hurricane. Just to see what happened. He argued: “Since a complete theory for the dynamics of hurricanes will probably not be derived by meteorologists for several years, argument pros and con without conclusive foundation will be made over the effects to be expected… Only a full-scale test could prove the results.” In other words, if we don’t do it, we’ll never know just how bad the idea is. For what the radiation hazard was concerned, Reed claimed it would be negligible: “An airburst would cause no intense fallout,” never mind that a complete theory for the dynamics of hurricanes wasn’t available then and still isn’t.

Reed’s proposal was dismissed by both the military and the scientific community. The test never took place, but the proposal is interesting nevertheless, because Reed went to some length to explain how to go about nuking a hurricane smartly.

To understand what he was trying to get at, let’s briefly talk about how hurricanes form. Hurricanes can form over the ocean when the water temperature is high enough. Trouble begins at around 26 degrees Celsius or 80 degrees Fahrenheit. The warm water evaporates and rises. As it rises it cools and creates clouds. This tower of water-heavy clouds begins to spin because the Coriolis force, which comes from the rotation of planet Earth, acts on the air that’s drawn in, and the more the clouds spin, the better they get at drawing in more air. As the spinning accelerates, the center of the hurricane clears out and leaves behind a mostly calm region that’s usually a few dozen miles in diameter and has very low barometric pressure. This calm center is called the “eye” of the hurricane.

Reed now argued that if one detonates a megaton nuclear weapon directly in the eye of a hurricane, this would blast away the warm air that feeds the cycle, increase the barometric pressure, and prevent the storm from gathering more strength.

Now, the obvious problem with this idea is that even if you succeeded, you’d deposit radioactive debris in clouds that you just blasted all over the globe, congratulations. But even leaving aside the little issue with the radioactivity, it almost certainly wouldn’t work because - hurricanes are enormous.

It’s not only that you’re still up against a power that exceeds that of your nuclear bomb by three orders of magnitude, it’s also that an explosion doesn’t actually move a lot of air from one place to another, which is what Reed envisioned. The blast creates a shock wave – that’s bad news for everything in the way of that shock – but it does little to change the barometric pressure after the shock wave has passed through.

So if nuclear bombs are not the way to deal with hurricanes, can we maybe make them rain off before they make landfall? This technique is called “cloud seeding” and we talked about this in a previous video. If you remember, there are two types of cloud seeding, one that creates snow or ice, and one that creates rain.

The first one, called glaciogenic seeding was indeed tried on hurricanes by Homer Simpson. No, not this Homer, but a man by name Robert Homer Simpson, who in 1962 was the first director of the American Project Stormfury, which had the goal of weakening hurricanes.

The Americans actually *did spray a hurricane with silver iodide and observed afterwards that the hurricane indeed weakened. Hooray! But wait. Further research showed that hurricane clouds contain very little supercooled water droplets, so the method couldn’t work even in theory. Instead, it turned out that hurricanes frequently undergo similar changes without intervention, so the observation was most likely coincidence. Project Stormfury was canceled in 1983.

What about hygroscopic cloud seeding, which works by spraying clouds with particles that absorb water, to make the clouds rain off? The effects of this have been studied to some extent by observing natural phenomena. For example, dust that’s blown up over the Sahara Desert can be transported by winds over long distances. Though much remains to be understood, some observations seem to indicate that interactions with this dust makes it easier for the clouds to rain off, which naturally weaken hurricanes.

So why don’t we try something similar? Again, the problem is that hurricanes are enormous! You’d need a whole army of airplanes to spray the clouds, and even then that would almost certainly not make the hurricanes disappear, but merely weaken them.

There’s a long list of other things people have considered to get rid of hurricanes. For example, spraying the upper layers of a hurricane with particles that absorb sunlight to warm up the air, and thereby reduce the updraft. But again, the problem is that hurricanes are enormous! Keep in mind, you’d have to spray an area about the size of Texas.

A similar idea is to prevent the air above the ocean from evaporating and feeding the growth of the hurricane, for example by covering the ocean surface with oil films. The obvious problem with this idea is that, well, now you have all that oil on the ocean. But also, some small-scale experiments have shown that the oil-cover tends to break up, and where it doesn’t break up, it can actually aid the warming of the water, which is exactly what you don’t want.

How about we cool the ocean surface instead? This idea has been pursued for example by Bill Gates, who, in 2009, together with a group of scientists and entrepreneurs patented a pump system that would float in the ocean and pump cool water from deep down to the surface. In 2017 the Norwegian company SINTEF put forward a similar proposal. The problem with this idea is, guess what, hurricanes are enormous! You’d have to get a huge number of these pumps in the right place at the right time.

Another seemingly popular idea is to drag icebergs from the poles to the tropics to cool the water. I leave it to you to figure out the logistics for making this happen.

Yet again other people have argued that one doesn’t actually have to blow apart a hurricane to get rid of it, one merely has to detonate a nuclear bomb strategically so that the hurricane changes direction. The problem with this idea is that no one wants multiple nations to play nuclear billiard on the oceans.

As you have seen, there are lots of ideas, but the key problem is that hurricanes are enormous!

And that means the most promising way to prevent them is to intervene before they get too large. Hurricanes don’t suddenly pop out of nowhere, they take several days to form and usually arise from storms in the tropics which also don’t pop out of nowhere.

What the problem then comes down to is that meteorologists can’t presently predict well enough and not long enough in advance just which regions will go on to form hurricanes. But, as you have seen, researchers have tried quite a few methods to interfere with the feedback cycle that grows hurricanes, and some of them actually work. So, if we could tell just when and where to interfere, that might actually make a difference.

My conclusion therefore is: If you want to prevent hurricanes, you don’t need larger bombs, you need to invest into better weather forecasts.

Saturday, March 06, 2021

Do Complex Numbers Exist?

[This is a transcript of the video embedded below.]

When the world seems particularly crazy, I like looking into niche-controversies. A case where the nerds argue passionately over something that no one knew was controversial in the first place. In this video, I want to pick up one of these super-niche nerd fights: Are complex numbers necessary to describe the world as we observe it? Do they exist? Or are they just a mathematical convenience? That’s what we’ll talk about today.

So the recent controversy broke out when a paper appeared on the preprint server with the title “Quantum physics needs complex numbers”. The paper contains a proof for the claim in the title, in response to an earlier claim that one can do without the complex numbers.

What happened next is that the computer scientist Scott Aaronson wrote a blogpost in which he called the paper “striking”. But the responses were, well, not very enthusiastic. They ranged from “why fuss about it” to “bullshit” to “it’s missing the point.”

We’ll look at the paper in a moment, but first I will briefly summarize what we’re even talking about, so that no one’s left behind.

The Math of Complex Numbers

You probably remember from school that complex numbers are what you need to solve equations like x squared equals minus 1. You can’t solve that equation with the real numbers that we are used to. Real numbers are numbers that can have infinitely many digits after the decimal point, like square root of 2 and π, but they also include integers and fractions and so on. You can’t solve this equation with real numbers because they’ll always square to a positive number. If you want to solve equations like this, you therefore introduce a new number, usually denoted “i” with the property that it squares to -1.

Interestingly enough, just giving a name to the solution of this one equation and adding it to the set of real numbers turns out to be sufficient to make all algebraic equations solvable. Doesn’t matter how long or how complicated the equation, you can always write all their solutions as a+ib, where a and b are real numbers. 

Fun fact: This doesn’t work for numbers that have infinitely many digits before the point. Yes, that’s a thing, they’re called p-adic numbers. Maybe we’ll talk about this some other time.

Complex numbers are now all numbers of the type a plus I time b, where a and b are real numbers. “a” is called the “real” part, and “b” the “imaginary” part of the complex number. Complex numbers are frequently drawn in a plane, called the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. i itself is by convention in the upper half of the complex plane. But this looks the same as if you draw a map on a grid and name each point with two real numbers. Doesn’t this mean that the complex numbers are just a two-dimensional real vector space?

No, they’re not. And that’s because complex numbers multiply by a particular rule that you can work out by taking into account that the square of i is minus 1. Two complex numbers can be added like they were vectors, but the multiplication law makes them different. Complex number are, to use the mathematical term, a “field”, like the real numbers. They have a rule both for addition AND for multiplication. They are not just like that two-dimensional grid.

The Physics of Complex Numbers

We use complex numbers in physics all the time because they’re extremely useful. There useful for many reasons, but the major reason is this. If you take any real number, let’s call it α, multiply it with I, and put it into an exponential function, you get exp(Iα). In the complex plane, this number, exp(Iα), always lies on a circle of radius one around zero. And if you increase α, you’ll go around that circle. Now, if you look only at the real or only at the imaginary part of that circular motion, you’ll get an oscillation. And indeed, this exponential function is a sum of a cosine and I times a sine function.

Here’s the thing. If you multiply two of these complex exponentials say, one with α and one with β, you can just add the exponents. But if you multiply two cosines or a sine with a cosine… that’s a mess. You don’t want to do that. That’s why, in physics, we do the calculation with the complex numbers, and then, at the very end, we take either the real or the imaginary part. Especially when we describe electromagnetic radiation, we have to deal with a lot of oscillations, and complex numbers come in very handy.

But we don’t have to use them. In most cases we could do the calculation with only real numbers. It’s just cumbersome. With the exception of quantum mechanics, to which we’ll get in a moment, the complex numbers are not necessary.

And, as I have explained in an earlier video, it’s only if a mathematical structure is actually necessary to describe observations that we can say they “exist” in a scientifically meaningful way. For the complex numbers in non-quantum physics that’s not the case. They’re not necessary.

So, as long as you ignore quantum mechanics, you can think of complex numbers as a mathematical tool, and you have no reason to think they physically exist. Let’s then talk about quantum mechanics.

Complex Numbers in Quantum Mechanics

In quantum mechanics, we work with wave-function, usually denoted Ψ, which are complex valued, and the equation that tells us what the wave-function does is the Schrödinger equation. It looks like this. You’ll see immediately, there’s an “i” in this equation, which is why the wave-function has to be complex valued.

However, you can of course take the wave-function and this equation apart into a real and an imaginary part. Indeed, one often does that, if one solves the equation numerically. And I remind you, that both the real and the imaginary part of a complex number are real numbers. Now, if we calculate a prediction for a measurement outcome in quantum mechanics, then that measurement outcome will also always be a real number. So, it looks like you can get rid of the complex numbers in quantum mechanics, by splitting the equation into a real and imaginary part, and that’ll never make a difference for the result of the calculation.

This finally brings us to the paper I mentioned in the beginning. What I just said about decomposing the Schrödinger equation is of course correct, but that’s not what they looked at in the paper, that would be rather lame.

Instead they ask what happens with the wave-function if you have a system that is composed of several parts, in the simplest case that would be several particles. In normal quantum mechanics, each of these particles has a wave-function that’s complex-valued, and from these we construct a wave-function for all the particles together, which is also complex-valued. Just what this wave-function looks like depends on which particle is entangled with which. If two particles are entangled, this means their properties are correlated, and we know experimentally that this entanglement-correlation is stronger than what you can do without quantum theory.

The question which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.

I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!

Well, kind of. Here is the fineprint: This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.

Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.

Or, of course, it might be that that the experiment will not agree with the predictions of quantum mechanics, which would be the most exciting of all possible outcomes. Either way, I am sure that this is a topic we will hear about again.

Tuesday, March 02, 2021

[Guest Post] Problems with Eric Weinstein's “Geometric Unity”

[This post is written by Timothy Nguyen, a mathematician and an author of the recently released paper “A Response to Geometric Unity”.]

On April 2, 2020, Eric Weinstein released a video of his 2013 Oxford lecture in which he presents his theory of everything “Geometric Unity” (GU). Since then, Weinstein has appeared in interviews alongside Sabine Hossenfelder, Brian Keating, Lee Smolin, Max Tegmark, and Stephen Wolfram to discuss his theory. 

In these interviews, Weinstein laments that the scientific community is dismissive of GU because he has not released a technical paper, but insists that scientists should be able to understand the substantive content of GU from the lecture alone (see here and here). In fact, Weinstein regards the conventional requirement of writing a paper to be flawed, since he questions the legitimacy of peer review, credit assignment, and institutional recognition (see here, here, here, and here).

Theo, my anonymous physicist coauthor, and I became aware of Weinstein and Geometric Unity through his podcast The Portal. We independently communicated with Weinstein on Discord and we both came to the conclusion that Weinstein was unable to provide an adequate explanation of GU or why it was a compelling theory. 

I also became increasingly skeptical of Weinstein’s claims when I pressed him about his alleged discovery of the Seiberg-Witten equations before Seiberg and Witten (see here, here, here, and here), a set of equations which was the central focus of my PhD thesis and several resultant papers. When I asked Weinstein for certain mathematical details about how he had arrived at the Seiberg-Witten equations, his vague responses led me to doubt his claims. Though Weinstein proposed to host a more in-depth discussion about GU and the requisite math and physics, no such discussion ever materialized.

These difficulties in communicating with Weinstein is what motivated our response paper. Suffice it to say that it was no easy task, as it required repeatedly watching his YouTube lecture and carefully timestamping its content in order to cite the material. These appear as clickable links in our response paper for those who wish to verify that our transcription of Weinstein's presentation is accurate.

Here's the high-level overview of how GU makes a claim towards a Theory of Everything. Essentially, GU asserts that there is a set of equations in 14 dimensions that are to contain the Einstein equations, Dirac equation, and Yang-Mills equations. Because the Einstein equations describe gravity, the Dirac equation accounts for fermions, and the Yang-Mills equations account for gauge-theories describing the strong and electroweak forces, all fundamental forces and particle types are therefore superficially accounted for. It is our understanding that it is in this very limited and weak sense that GU attempts to position itself as a Theory of Everything.

The most glaring deficiency in Weinstein’s presentation is that it does not incorporate any quantum theory. Establishing a consistent quantum theory of gravity alone has defied the efforts of nearly a century’s worth of vigorous research and is part of what makes formulating a Theory of Everything an enormous challenge. For GU to overlook this obstacle means that it has no possible claim on being a Theory of Everything.

Our findings are that even aside from its status as Theory of Everything, GU contains serious technical gaps both mathematical and physical. In summary:
  • GU introduces a “shiab” operator that overlooks a required complexification step. Omitting this step creates a mathematical error but including it precludes having a physically sensible quantum theory. 
  • The choice of gauge group for GU naively leads to a quantum gauge anomaly, thereby rendering the quantum theory inconsistent. Any straightforward attempt to eliminate this anomaly would make the shiab operator impossible to define, compounding the previous objection. 
  • The setup of GU asserts that it will have supersymmetry. In 14 dimensions, adopting supersymmetry is highly restrictive. It implies that the proposed gauge group of GU cannot be correct and that the theory as stated is incomplete. 
  •  Essential technical details of GU are omitted, leaving many of the central claims unverifiable.

Coincidentally, the night before we posted our response paper, Weinstein announced on Lex Fridman’s podcast that he plans on releasing a paper on GU on April 1st. We look forward to seeing Weinstein's response to the problems we have identified.