Albert Einstein is one of the most famous scientists in the history of science. He not only completely changed our understanding of motion and gravity with his theory of relativity, upending 300 years of Newtonian physics, but he also contributed substantially to the quantum revolution. He is responsible for perhaps one of the most famous equations in the world, ee equals em cee squared, which equates energy with mass. His iconic look, meanwhile, with long white hair and a droopy mustache, combined with a German accent, has become the de facto image of a scientist in the modern world.

His theory of relativity has stood the test of time, having been confirmed numerous times with rigorous tests.

But what if I told you that the theory that we know today as Einstein’s General Theory of Relativity is not the theory Einstein was even looking for, and that, although his ideas evolved over his lifetime, he never came to the consensus that most physicists hold today? Moreover, although Newtonian physics and even Einstein’s special theory of relativity contain relativist concepts of observer dependence, so-called general relativity does not, in fact, contain any nontrivial relativity at all. And that may be a big problem.

Einstein was born in 1879 at a time when physicists largely believed physics to be a solved problem. They had Newton’s theory of forces and motion, as well as Maxwell’s theory that explained electromagnetism. There were still some tricky questions about the physics of systems of many particles, like fluids and gases, but the underlying physics was thought to be well-known.

Some discrepancies remained in well-established physics: for example, the orbit of Mercury deviated very slightly from Newton’s prediction.

Also, it was not known how electromagnetic waves propagated through empty space. It was thought that some kind of substance called a luminiferous ether must be responsible. Michelson and Morley, two Americans, attempted to measure the Earth’s motion through this ether by measuring the speed of light in the direction of the orbit and perpendicular to it. They showed no difference.

Einstein seemed an unlikely candidate to change all this. He was a lazy student, and although he excelled in physics and mathematics, he spent two years failing to gain a teaching position in those subjects. Einstein worked in a Swiss patent office, a position he only gained through the help of his friend’s father. He spent a great deal of his time, however, thinking about physics and working out theories about space, time, and matter.

He was deeply engaged with the philosophy of the time. This both helped him in his goals but also hindered him in his ability to understand and embrace the implications of the scientific revolutions he helped initiate.

Einstein was a follower of Ernst Mach, who, besides being known for measuring the speed of sound, formulated Mach’s Principle, which is that all motion is relative to the motion of other objects, including rotation. (“Mach’s Principle” was, in fact, coined by Einstein himself.)

Mach formulated the principle in response to a thought experiment Sir Isaac Newton had introduced, called the bucket argument.

The idea is that if you have a bucket of water on a rope and you spin it, at first, the water sits there while the pail rotates. After a while, however, the water also begins to rotate, and a depression will appear in the surface of the water as the water is pushed to the outer rim of the bucket away from the center. If you are watching the bucket spin, this makes perfect sense, but what if you are somehow sitting on the bucket?

In this case, you should know you are rotating because everything else is spinning around you, including the distant stars. But what if you are in an empty universe with no other matter?

The surface of the water will still be concave, Newton argues, and therefore motion cannot be entirely relative but must be absolute.

Mach came along and argued,

Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotations with respect to the mass of the earth and other celestial bodies. - Ernst Mach, as quoted by L. Bouquiaux in Leibniz, p. 104

In other words, the motion of the water is still relative to all the matter in the universe.

Einstein strongly agreed, and when he published his special theory of relativity at age 26, he demonstrated how this principle would affect the way different observers measure space and time.

He later attempted to drag this idea into his general theory of relativity, which would address the bucket argument properly. This general theory explained not only accelerating and rotating motion but also gravity, and it is what we still use today.

In a talk at Princeton in 1921, after he became famous for this theory thanks to measured light bending around the sun in 1919, he claimed (as he never did in writing),

Whenever we talk about the motion of a body, we always mean by the very concept of motion relative motion…we might as well say ‘the street moves with respect to the car’ as ‘the car moves with respect to the street’ … These conditions are really quite trivial … we can only conceive of motion as relative motion … All this goes without saying and does not need any further discussion. - Quoted by Michel Janssen.

Far from being trivial, however, these points are not even true.

Einstein had a habit of clinging to his earlier philosophical beliefs, it seems, with the public, even when he had conceded them academically. By 1918, Einstein had already accepted the defeat of relativity at the hands of Erich Kretschmann. His own theory did not support his claim that all motion is relative.

The phrase “general relativity” is, in fact, one of those historical oddities in that its name implies something that the theory to which it refers lacks.

Einstein had believed, up until 1918, that general covariance was a property of his theory that made it generally relativistic. General covariance, however, is only the ability to express a theory in any coordinate system (or none at all). Thus, I am free to choose my coordinate systems to be anything, even something ridiculous. Because the equations are expressed in a generally covariant way, this is physically equivalent to saying that the laws of physics are the same for all observers.

In 1917, Erich Kretschmann, a former student of Max Planck and a high school teacher, attacked Einstein’s claims about his theory. In particular, Kretschmann showed that just about any space-time theory could be made generally covariant, and, indeed, by 1923, Newton’s theory was made generally covariant.

While general covariance says that all observers experience the same physical laws, it doesn’t say what those laws are.

In what sense, therefore, is GR generally relativistic?

Einstein argued that his theory was generally relativistic because it had these three elements:

1. General covariance - all observers have the same physical laws.

2. Mach’s Principle - motion of matter is relative to the motion of other matter, with their gravitational fields being the mediator between them.

3. Equivalence Principle - a gravitational field is indistinguishable from accelerated motion.

With great difficulty, Einstein produced, in November 1915, several short papers containing his field equations, and the next year published a review paper laying the entire theory out as well as its implications, including corrections that explained Mercury’s orbit and light bending around the Sun.

Einstein believed he had succeeded.

This turned out to be an illusion.

His theory did not meet condition #2 above. It did not guarantee that all gravitational fields have material sources, and therefore, matter could have motion that has nothing to do with the motion of other matter.

In 1917, he decided to add a cosmological constant to his theory. Many popular science communicators claim this is because Einstein was attached to the idea of a steady-state universe and needed the constant to stop the universe from changing density. That is true, and, if you just read his paper on the topic, you might conclude that is all he’s after, but the reason why he wanted a static universe in the first place isn’t obvious. Many popularizers erroneously conclude that it is because he was stuck in an old philosophical school that wants to have an eternal universe with no beginning.

But that isn’t why: it was, in fact, a flailing attempt to prop up Mach’s Principle.

He wanted a universe that firmly held onto its matter so that it could not become empty.

Almost immediately, however, Dutch physicist Willem de Sitter discovered that you could have an empty universe with a cosmological constant.

Einstein accused him, saying, “You have violated Mach’s principle.”

Thanks to de Sitter, Einstein, however, realized that his cosmological constant could not rescue the principle.

Absolute motion was here to stay, and absoluteness is built into general relativity.

To see why, take a simple example: suppose you have two observers, Alice and Bob, moving in non-uniform motion with respect to one another. Is Alice, Bob, or both moving non-uniformly? If you can answer that question within Einstein’s theory, then Mach’s Principle fails.

Assume a flat spacetime and let Alice be in inertial motion. Bob is experiencing a constant acceleration a.

GR can tell us which one is inertial and which one is accelerated, regardless of the reference frame. Because Alice is in what is called geodesic motion, meaning free fall, and Bob is not, their motion, independently of one another, satisfy different equations. Alice satisfies the geodesic equation. Bob does not.

Physically, each one can carry an accelerometer and determine if they are accelerating regardless of their relative motion, even in a totally empty universe.

GR also shows that Mach is wrong about Newton’s bucket. There is such a thing as absolute rotation, as Newton said, and Einstein’s equations show that and exactly why it is absolute.

Thus, some motion is relative but not all motion. What motion is relative? Straight line, uniform motion.

Two principles down: general covariance is not unique to general relativity and Mach’s Principle fails, but what about the third, the equivalence principle?

Without the other two, this principle alone cannot guarantee relativity.

For this to be true, you would have to be able to attribute any acceleration or non-uniform motion to a gravitational field, and Einstein believed he could prove this.

In 1907, Einstein came up with the thought experiment of a person in an elevator. This person cannot distinguish between acceleration caused by the motion of the elevator and that caused by a gravitational field, he thought. Five years later, he gave the name “equivalence principle” to this idea.

Einstein hoped that this would allow a theory of relativity for non-uniform motion, but it was doomed to fail. If Einstein were correct, then you could have situations like this: Suppose Alice is on an airplane, enjoying nice, smooth motion. She can claim, because her motion is uniform, either that she is moving or that the ground is moving. That is special relativity (or Galilean relativity). Now, a sudden jolt happens, and her drink spills.

Can she claim that a random passing gravitational field caused her drink to spill?

No, she can’t. To do so is to violate general relativity itself. A gravitational field is not just an acceleration. It must produce tidal forces, affect nearby objects, curve spacetime, and change the motion of other freely falling bodies. It cannot simply explain the spill of one drink.

With general relativity, all that relative motion is now a special case. General solutions and states of motion in Einstein’s universe are not relative but absolute.

This explains, for example, why the twins paradox exists. In this paradox, two twins, sometimes named Peter and Paul, start out on Earth, and one takes off in a spaceship, travels near the speed of light for several years, turns around, and comes back. Now the twin that left is younger than the twin that stayed behind.

Richard Feynman put it this way:

This is called a “paradox” only by the people who believe that the principle of relativity means that all motion is relative; they say, “Heh, heh, heh, from the point of view of Paul, can’t we say that Peter was moving and should therefore appear to age more slowly? By symmetry, the only possible result is that both should be the same age when they meet.” But in order for them to come back together and make the comparison, Paul must either stop at the end of the trip and make a comparison of clocks or, more simply, he has to come back, and the one who comes back must be the man who was moving, and he knows this, because he had to turn around. When he turned around, all kinds of unusual things happened in his space ship—the rockets went off, things jammed up against one wall, and so on—while Peter felt nothing.

So the way to state the rule is to say that the man who has felt the accelerations, who has seen things fall against the walls, and so on, is the one who would be the younger; that is the difference between them in an “absolute” sense, and it is certainly correct.

Both of these cases, Alice on the airplane and Peter and Paul, are examples of absolute motion and violate the equivalence principle Einstein hoped to introduce.

The principle Einstein hoped for would read something like:

But that is not true. The equivalence principle allows some gravitational effects to be locally mimicked by acceleration.

Modern physicists use a different definition altogether:

In other words, if Alice is in orbit, in free fall, she can claim (approximately) that her own reference frame is a flat, special relativistic frame, no matter what gravity she is experiencing. It says little about acceleration. This is a far cry from what Einstein was after with his thought experiment.

Einstein was forced to abandon this hope as well.

Thus, all three of the principles that Einstein hoped his theory was built upon in 1915-16 were torn down in short order, such that, by 1920, he ought to have been singing a different tune, but still seemed to cling to his earlier philosophy despite Kretschmann, de Sitter, and others systematically disillusioning him.

Kretschmann went as far as to say,

Einstein’s theory physically satisfies no relativity principle whatsoever…; it is a completely absolute theory in regard to its content.

You might even call general relativity, general absoluteness.

In fact, special relativity is far more relativistic than general.

Despite this, there are still questions about whether the theory of gravity itself contains more relativity than these early pioneers found in Einstein’s theory.

Could GR be overly restrictive?

In special relativity, there actually are different reference frames. They have different physical experiences, but their measurements can be related by certain mappings called Lorentz transformations.

You will often hear people argue that general relativity has diffeomorphic invariance. This is the modern belief system. This means that solutions to Einstein’s equations that can be related to one another by a differentiable mapping are physically equivalent.

Diffeomorphic invariance, however, is not a true physical symmetry, unlike the invariance under Lorentz transformations in special relativity. Rather, it means that there is redundancy in the mathematical representation of the theory.

Yet, GR is really about how motion couples to geometry. Its central field is called the metric tensor, but it can also be called an inertio-gravitational field. Thus, it combines both motion and gravity.

The question is: does the inertio-gravitational field represent a unique reality, or can visibly distinct examples represent physically equivalent realities?

Take an example: suppose you have two gravitational fields that give the same free-fall trajectories. If all you measure is inertial motion, then you can’t distinguish the two fields. If inertial motion is what matters, then all the fields that represent the same set of free-fall trajectories represent the same reality.

Alternatively, you can apply conformal mappings to the gravitational field, where you stretch or compress the metric field by a different amount at each point at the same time. Basically, it means that the volume of spacetime at any given point is irrelevant to physics. All that matters is the causal structure. Under those transformations, light propagation is unchanged. This determines the causal structure of the universe.

This would mean that all inertio-gravitational fields that have the same causal structure but different volumes represent the same physical reality.

What is physically meaningful?

Is it diffeomorphic invariance? Free-fall paths? Causality?

There is an old program from the 1970s that attempted to reconstruct spacetime geometry from the behavior of particles and light. This is the precursor to quantum gravity approaches such as Causal Set Theory, which is based on the causal structure of spacetime. CST argues that conformal mappings don’t change the fundamental causal structure of spacetime and therefore aren’t physically measurable. As long as you know the causal structure, you know just about everything. Several other theories, including Causal Dynamical Triangulation (CDT) and Twistor Theory, use the same idea.

Philosophically, it bears thinking about that Einstein started with some very wrong ideas but came out with the right theory. Nevertheless, we have only just scratched the surface of what is really fundamental to that theory. What he considered to be fundamental, many researchers in the field consider to be emergent. The problem is identifying what is fundamental.

Kretschmann’s Analysis of Covariance, Robert Rynasiewicz, Expanding Worlds of General Relativity, Einstein Studies, vol. 7, 1999.

Janssen, Michel. "'No success like failure...': Einstein's Quest for general relativity, 1907-1920." (2008).

You probably learned about numbers when you were very small. You may have been subjected to worksheets in elementary school asking you to add, subtract, multiply, and divide. Later on, you learned you would replace those numbers with letters when you didn’t know what the numbers were and had to solve for x.

But I bet nobody ever explained to you where numbers come from or what they really are.

After all, when learning what cats, dogs, cows, and horses are as a small child, you can visit a farm and see all of them and point them out. They are real things. In fact, most words you know correspond to real things.

Likewise, if you learned that a cat is white, you could understand that.

A white cat makes sense. But what about white itself?

You can probably visualize white in your head. When I do it, I imagine a splotch of white paint against a black background.

I’m not sure if I can just imagine white. I can imagine white things. I can imagine a blank field of whiteness. But is that what white really is? Am I not simply imagining something white, perhaps something that takes up my whole field of vision, like I have my nose pressed against a white wall?

The same is true of numbers. Numbers used to just be adjectives. You couldn’t just have three. You had to have three apples or three cats.

You can visualize three somethings.

Try, however, to visualize just three by itself.

You can’t.

And you were never visualizing white by itself either.

You can’t.

The reason you can’t is that both concepts, threeness and whiteness, are abstractions. They are adjectives that have been stripped of their nouns and turned into nouns themselves.

This has a lot of advantages.

When I do math, I don’t need to worry about what the numbers refer to. I just want to know how numbers relate to other numbers or transform under certain transformations.

When I start to add units to numbers, which are the nouns these adjectives refer to, I am in the realm of science, not math.

Thus, numbers come from the same place that all adjectives come from: language.

They are a way of adding attributes or properties to nouns.

Indeed, all abstractions are really adjectives. You can have a beautiful vase, but Beauty is abstract. You can have a just person, but Justice is abstract.

This explanation for numbers may be satisfying to some, but it hardly goes far enough to equate them, generically, with abstract concepts. After all, numbers appear to have a reality all their own. They have a logic and relationship with reality that is so fundamental that it demands a new system of thought.

You also know, if you read me regularly, that things are about to get more complicated.

Read more

In 2010, I visited Florida State University in Tallahassee. It was early spring and not too hot yet. The grass was green and the shade inviting as many students lay on the quad, reading or listening to music. My host at the university had a special treat for us as he invited us to the upper floors of the library, where we met with the archivist for one of the strangest and most famous men in the history of physics.

FSU seemed like an unlikely resting place for the papers of one of the fathers of quantum mechanics, who had spent the majority of his career at Cambridge University. There, he held the Lucasian Chair of Mathematics, the same professorship that Sir Isaac Newton had held and which, later, Stephen Hawking would hold. Yet, in his later years, he had longed for warmer climes and landed in the Florida panhandle, bringing a lifetime with him.

Paul Dirac’s books and papers were neatly arranged on shelves and in boxes within a small room with glass doors, and my fellow pilgrims, for that is what we were, were invited to rifle through them while the archivist explained what they did and how it would have been nice if Professor Dirac had put dates on more of his papers.

As I examined some random items, the archivist handed me a thin notebook. It looked like a cheap one. It wasn’t leatherbound, but the sort schoolchildren use. I opened to the first page. There, I saw written in pencil, with a neat script along the lined page, these words:

Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be included to think could not be seriously modified without having all its attractive features spoilt. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated into the new scheme.

I recognized these words as the opening of Paul Dirac’s classic textbook, Principles of Quantum Mechanics, first published in 1930. (I have the 4th edition published in 1967.) These words had not changed from when he first wrote them in that, now, nearly 100-year-old notebook to this day.

Many students, however, never get very far with this subject because the mathematics and the concepts are so confusing and nonintuitive. It is hard to believe that any of them can apply to real life. Classical mechanics is all about billiard balls, rockets, and things we can see and understand. Quantum mechanics is about wavefunctions and operators as much as it is about particles. It seems more like abstract mathematics than real life.

Quantum mechanics, however, isn’t that much harder to understand than classical mechanics. It is just harder to visualize. As Dirac says, it is much more elegant too, and, if you suspend your disbelief for a moment, you too can understand it.

But why bother with this at all? Why do we need quantum mechanics?

We need quantum mechanics because, without it, we could not explain why matter exists at all. A classical atom or molecule, modeled like a little solar system with electrons zipping around a nucleus, would be inherently unstable. It is because atoms are more digital and less analog that we can exist.

Even on philosophical grounds, quantum mechanics is necessary. In classical physics, size is always relative. Something is big only in comparison to something smaller. There is no concept of absolute size. If you were to shrink yourself down smaller and smaller, insects would suddenly become big, then cells, then molecules, and then atoms. Yet you could go further to subatomic particles and so on. This would mean that a classical particle must be big relative to something. And logically, that something must therefore exist. This goes on ad infinitum.

Quantum mechanics, however, introduces the notion of something that is absolutely small. This arises because the means by which we observe anything requires some disturbance to that thing. To observe an ant, we must recognize light scattering off it. The ant is large compared to the light. Therefore, the light makes little disturbance. The same is true for cells. Yet, once we approach the size of molecules, and certainly atoms, light that is sufficiently intense to see them would cause them to shoot away as if shot by a cannonball. We have to use lower power, small wavelength objects like electrons, which is why electron microscopes are a thing, for molecules. Going smaller, there is a finite limit to what is achievable. This is what makes smallness absolute. This is not simply an engineering limit on our abilities but a physical law, a transition from the classical to the quantum encoded by Heisenberg’s uncertainty principle.

While philosophy demands that something like quantum mechanics exist, quantum mechanics also makes demands on philosophy. It tells us that causality, as we know it, is flawed. We can no longer propose observing a closed system evolving as we can in classical physics because our act of observing a small system violates the closed assumption. We can describe undisturbed systems evolving in quantum mechanics, but those equations only relate probabilistically to what we actually observe. Quantum mechanics itself fails to account for how observation affects things.

Perhaps one of the most startling aspects of quantum mechanics is the superposition of states, which can be a feature of undisturbed systems. This is the background for the famous Schroedinger’s cat thought experiment, but the thought experiment itself tends to obfuscate the phenomenon. This principle is fundamental to quantum theory but can be confusing in the general case. Let’s look at a simple case of light polarization.

Did you know that sunglasses use a quantum phenomenon to shade your eyes?

Sunglasses are polarized, which means that they only let light through that lines up with the direction of polarization. There are three kinds of polarization: vertical, horizontal, and circular.

These terms, however, are misleading.

Imagine that you and a friend are holding a rope between the two of you, and one of you starts jiggling their end up and down. Waves, which are aligned in the vertical direction, start to travel along the rope. Likewise, I could do the same in the horizontal direction (ignore gravity for the moment).

Suppose you stop and thread the rope through a barrier with a long vertical slit cut into it. Now, if you start jiggling the rope up and down, the waves pass easily through the slit. If, however, you try jiggling the rope side to side, the waves crash into the barrier, dissipating against it.

I could reverse things and use a horizontal slit, and then the vertical waves could not pass through.

Circular polarization is a bit more complicated, but I imagine we could achieve something similar for that as well.

This is a nice picture of how polarization works, but it is about as accurate as the image of an atom as a mini-solar system. It doesn’t take quantum into account.

Consider that if I cut a diagonal slit into the barrier, neither my vertical nor my horizontal waves could pass through at all. In fact, classical physics tells me that I don’t have three polarizations at all, but an infinite number depending on the angle.

Not so with quantum mechanics.

If I produce some randomly polarized light and pass it through a polarizing filter, such as sunglasses, guess how much will be able to pass through?

If it were like the rope, then we might guess a very small fraction would happen to be polarized in a way that it could pass through.

The real answer, however, is half.

Fully half will randomly select to pass through.

After passing through, moreover, all the light coming through the other side will be polarized in alignment with the filter.

If I place some similar sunglasses behind the first ones but at a right angle, all the light will be blocked out.

As I turn the second set of sunglasses, however, I will see the amount passing through increase according to a formula that is the cosine squared of the angle between the two sunglasses.

If light were only a wave, we might imagine that this makes some sort of sense, but now we recall that light must also be a particle, called a photon.

We know that light is made of photons for many different reasons. One is called the photoelectric effect, which is where, by shining light onto certain kinds of atoms, they emit light particles at a precise frequency. It was this phenomenon that got Albert Einstein the Nobel Prize (not relativity).

Each photon itself has a state of polarization.

How can we then account for the behavior of photons passing through the polarization filter? If each photon has a particle polarization, then surely only those that line up with the filter should pass through, not some weird fraction.

To understand this better, let us therefore set up an experiment where we send only one already polarized photon at a time through the second polarizing filter.

What will happen is that we will never measure a fraction of a photon passing through the filter. Instead, we will either measure a whole photon coming out or none at all. If we send many through, one at a time, we will find the probability that a photon passes through is equal to the cosine squared of the angle.

This experiment doesn’t tell us anything about how the photon decides whether to pass through or not, and, indeed, quantum mechanics cannot tell us that either. Quantum theory, however, can tell us how to interpret the state of the photon before passing through the second filter.

The photon is in a state of superposition when it encounters the second filter placed at an angle to the first, where it is partly vertically polarized with respect to the second filter and partly horizontally polarized.

This is possible in classical physics if you have a wave, but not with a particle. Quantum mechanics allows it to be applied to a single photon.

When the photon meets the filter, the filter performs an observation because it only allows photons with vertical polarization through. Because it is an observation, it forces the photon to choose one of its states in superposition, and the other must vanish.

Although the choice that is made cannot be predicted, the photon, nevertheless, makes it and jumps to one or the other. If the result is vertical, it passes through. If horizontal, it does not.

Here is a diagram of how this works.

The first polarizer vertically polarizes the photon, but that is only in the basis of that polarizer. These are in purple above. A basis is like a coordinate system, but for quantum states. We can shift the basis to that of the second polarizer, which rotates the polarization so that now it is partly vertical and partly horizontal. This is in blue. In the basis of the second polarizer, it is a superposition of “pass” which means the polarization of the photon will jump to match that of the polarizer, and “block,” which means it will jump to be perpendicular.

The photon will pass through the second polarizer with a probability that is the cosine squared of the angle. This is because probabilities in quantum physics are computed by squaring them. In the “pass” case, the original superposition state is projected onto the vertical polarization state and squared. This is in green.

To project a state onto another state is like shining a light on the first state. It is a silhouette or shadow of the original state. The same thing happens when the sun projects your shadow onto the ground. The two-dimensional cutout of your body is projected onto the ground.

The “blocked” state probability is the square of the projection of the original state onto the horizontal polarization state in the basis of the second polarizer. It is equal to the sine squared of the angle. This is in red.

The diagram uses a notation invented by Dirac called bra-ket notation. This is a bra:

And this is a ket

The symbol inside can be anything you want. It is just a symbol to help you keep track of what the bra and ket refer to.

You can combine bra’s and ket’s like this

These are kind of like vectors if you are familiar with those, with the bras being like row vectors and the kets being like column vectors, but the easiest way to think about it is that these are quantum states. To combine two quantum states to get a single number, you have to combine a bra with a ket in that order. If you combine a ket with a bra, you get something called a density matrix, but that is way beyond the scope of this article.

Because this applies to a single photon, the polarization is not an emergent phenomenon from many particles waving together. Rather, it is inherent to the photon, like a bit of information about it. In fact, it is a quantum bit or qubit.

Quantum superposition is sometimes described as the particle being in both states at the same time, as if the cat were alive and dead at the same time in the thought experiment. That is not a good way to describe it.

For one thing, a quantum superposition is a pure quantum state, meaning that it is a single state, not two states at once. It is only our classical intuition that makes us want to interpret it that way.

A better way to look at it might be something like a position on a sphere. The cosine and sine lend themselves easily to spherical coordinates. Classical information only includes states that are at the North or South Pole of the sphere, but quantum physics can wander anywhere on the sphere. We have a new sphere for every qubit of information.

When we make a measurement, however, the state snaps to the North or South pole, and we can never observe it anywhere else.

A change of basis is like rotating the sphere so the North and South poles are somewhere else.

So, you can see in the diagram that after the first polarizer (blue box in the upper left), the state is simply at the North Pole in the basis of that polarizer.

The second polarizer, however, is in a different basis. The green box on the left side shows where the vertical polarization of the second polarizer is in terms of the polarizations of the first polarizer. In other words, we have to express the angle between the two polarizers, not as a number but as a ket, so we can use it in the next step.

The original state of the photon can be expressed in the basis of Polarizer 2 as a combination of the angle and its perpendicular, both as kets. This is the purple box on the upper right side. The ket

refers to the state corresponding to the vertical or “pass” polarization of the second polarizer, the green line on the small sphere at the bottom left. The ket

corresponds to its perpendicular or “block” polarization, the red line on the small sphere at the bottom right.

The original state is expressed as a superposition of these two states. The states are weighted by cosine and sine.

The measurement by the polarizer, which passes or blocks the photon, snaps the photon’s state to a position on the sphere corresponding to the green line (pass) or the perpendicular (block), which points to the opposite side of the sphere. These are the North and South poles in the second polarizer’s basis.

While an ordinary classical bit of information has only two values, a quantum bit of information can range anywhere over the sphere. It is only when a measurement is made that it snaps to one of two possible values, which correspond to a North and South pole in the relevant basis (but not all bases). Superposition is, therefore, precisely the state that is on neither the North nor the South Pole in the relevant basis.

From this perspective, superposition is obvious. It is not somehow in the state of the North Pole and the South Pole at the same time, any more than the location of New York City is a combination of the North and South Poles on Earth. It is a single location at neither pole. Yet, it is a location that cannot survive observation; it is only in the post-analysis that we interpret it as being both states at once. It is only one state, but we cannot observe it. Nature hides superposition from us, and we don’t know why.

The scientist, in particular, must always be prepared to notice the anomaly:

You may know the more popular Isaac Asimov version of the above quote, but when I researched it to find the citation, it turned out that the quote has no known origin with Asimov’s writings. This is despite others searching for it through his extensive bibliography, 4000 short pieces, and 500 books (many of which have not been digitized).

In wasting time on this quote’s provenance, I made a discovery.

Its earliest attribution is the 1987 release of the Fortune program for UNIX systems. Spending several hours, into the following day, searching through as many of Asimov’s works as I could find for the quote, I came up empty. It even appears in a 1997 book by Alan Hale on the discovery of comet Hale-Bopp, which has a foreword by Janet Asimov, Isaac’s second wife and widow, Isaac having passed away in 1992. Hale claimed to have learned of the quote just before discovering the comet in 1995 and thought it apropos to his noticing the faint shimmer of the icy sojourner (simultaneous with amateur astronomer Thomas Bopp).

It is often like that when you start on something. You find yourself in places you never expected, and so I found myself recognizing that discovery, great, world-changing discovery, depends not on luck nor on mysterious emissions from the unconscious but on being prepared to notice when something is off and knowing what to do about it.

That means that the vast majority of the time, the scientist, to make progress, has to be on task.

Yet, that is not what we want to hear. We want the “Eureka!” moment to be the genuine source of discovery, not painstaking grinding. This is perhaps why these moments of discovery get mythologized so much, including the original story.

The Rattray quote and its more pithy, Asimov-attributed form, which contrasts the “that’s funny” experience with the “Eureka!” moment, only includes half of what it takes to make a profound discovery.

Neither situation points to what one must be doing before those experiences to have those great discoveries, although examples abound. The Eureka Phenomenon, where Asimov wrote about it, suggests that people who are banging their heads against a problem can solve it by letting go for a while, sleeping on it, and coming back. The answer will suddenly present itself.

The original Eureka moment about Archimedes discovering the principle of buoyancy comes from a story by Roman architect and writer Vitruvius, writing about 200 years after the fact. It is likely apocryphal or a rewriting of whatever actually happened into a more dramatic legend.

The original story is set in 250 B.C. in the city of Syracuse on the island of Sicily. The king, Hieron II, asked a goldsmith to make him a new gold crown. In the common practice of the time, the king gave the goldsmith a specific amount of unformed gold with the expectation that his gold would be returned to him in crown form.

Upon receipt of the crown, however, Hieron was suspicious that the goldsmith had mixed in some copper or silver with the gold and pocketed the extra precious metal for himself.

The king, however, had a famous relative, the scientist and mathematician Archimedes, and asked him to solve the problem.

The key to determining the purity at the time was measuring the density of the metal, which was a fixed number for an element on the periodic table, such as gold (or other metals it could be mixed with, like silver or copper). If the original density of the gold was known, then the crown should match it.

Archimedes, smart as he was, however, could not immediately work out how to calculate the crown’s density, which would have told him how pure it was, without pounding it flat to calculate its volume using the mathematics of the time, which only worked for regular solids. Calculus didn’t exist yet, which might have helped.

Hieron did not want his new crown destroyed. Find another way, he said to his relative.

Unable to come up with a solution, it is said that Archimedes visited the public baths and, sinking into the water, saw the water level rise. This led him to realise that the amount of water displaced by his body would correspond to his body’s volume. Legend has it he leapt from the bath and ran naked through the town shouting “Eureka!” (I’ve found it!)

The way to calculate the crown’s density, according to the story, is to weigh it and then place it in a jar of water. By precisely measuring the amount of water displaced, the volume of the crown can be calculated and its density compared to that of a pure gold ingot of the same weight. Problem solved. Perhaps the goldsmith is a cheat and gets executed for it.

It’s a nice story about scientific discovery coming from relaxing and letting the mind drift. Like the Asimov quote above, however, it likely never happened or didn’t happen that way.

The problem is that the technical details are implausible.

Modern historians suggest that using water displacement to measure the crown’s density would have been far too inaccurate. Pure gold has a density of 19.3 grams per cubic centimeter, while silver is 10.5 (copper, which is the metal Asimov uses for his version of the story, is 8.92). If the goldsmith only replaced a small fraction of gold, the volume change would have been a few cubic centimeters. With ancient equipment, Derek de Solla Price, an historian who spent a lot of time working on the ancient computer, the Antikythera mechanism, has pointed out, it would have been almost impossible to measure this difference.

It’s possible, of course, that Archimedes didn’t realize the flaw in his idea when he ran through the town and only later realized the measurement would not work.

This is probably the most common result of Eureka moments: unnatural excitement over a false lead.

Many historians, however, think that Archimedes was smarter than that, and he actually did come up with a way to measure the crown’s density, one far better than the dumbed-down method that Vitruvius describes, which Asimov parrots.

Instead of measuring volume directly from water overflowing the container, Archimedes may have invented a hydrostatic balance. It works like this:

1. You weigh the crown in the air.

2. You weigh it again, submerged in water.

3. The crown will lose weight equal to the weight of displaced water.

4. From that, the volume can be determined.

In the ancient world, it was far easier to measure weight on a scale than the volume of water, and this method would likely have worked for a crown. You simply have to submerge a set of scales and weights to measure the underwater weight. This also, of course, depends on buoyancy.

Archimedes developed the principle of buoyancy in his treatise On Floating Bodies, and the hydrostatic weighing approach fits better with the mathematics in that book. It may have been that Archimedes did develop these ideas from the problem his royal relative gave him, but the rest is probably a later embellishment.

I suspect in many cases some of these moments are mythologized after the fact. It is far more dramatic to say, “I had a dream that gave me the answer” than “I sifted through the data and a picture of the answer gradually formed.” Yet I suspect that the latter is the more common course of science.

We all hear about Sir Isaac Newton and the apple, although in this case, the real story, which Newton attested to, is that watching an apple fall caused him to wonder if the apple tugs on the Earth in the same way the Earth tugs on the apple. It would be decades before he turned this into the theory of universal gravitation, and his real discovery was that the inverse square law could be used to reproduce both terrestrial parabolic trajectories as well as Kepler’s laws of planetary motion.

Eureka moments do seem to happen, but I suspect that they result in ideas that are far smaller and ill-formed than the full theories that later result. Storytellers like Isaac Asimov and Vitruvius embellish these stories for their audiences to make them seem more dramatic, yet the original core is far less interesting.

These stories suggest that periods of relaxation do lead to insights, even if they are small, and there are plenty of people who attest to this phenomenon, so there is a core truth there.

Several great examples are out there, such as Kekule discovering the benzene ring in a dream, Poincaré stepping off a bus, Einstein playing the violin, or Charles Darwin taking walks.

One of my favorite examples is about Richard Feynman. Feynman worked on the Manhattan Project during World War II, helping to build the atomic bomb. Feynman had eloped with his first wife, Arline, early in the war after she had been diagnosed with lymphatic tuberculosis and given only 2 years to live. In 1943, Oppenheimer invited him to Los Alamos to work on a secret project where he worked with some of the greatest minds of the time to bring about one of the most terrifying weapons yet developed. Arline passed away during the war.

Following the end of the project and his wife’s death, Feynman became deeply depressed and burnt out. He gave up on “important” physics.

One day he became fascinated by a cafeteria plate wobbling in the air. He spent time calculating its motion for fun.

I went on to work out equations of wobbles. Then I thought about how electron orbits start to move in relativity. Then there's the Dirac Equation in electrodynamics. And then quantum electrodynamics. And before I knew it (it was a very short time) I was “playing” - working, really - with the same old problem that I loved so much, that I had stopped working on when I went to Los Alamos: my thesis-type problems; all those old-fashioned, wonderful things. - “Surely you're joking, Mr. Feynman”, by Richard Feynman

This story makes it seem like Feynman accidentally discovered the mathematics of QED from the wobbling plate, but that isn’t even close to true. The mathematics weren’t related. What happened is that the wobbling plate problem helped to refresh his mind and got him into diagrammatic approaches to doing complicated calculations. These together contributed to his QED work indirectly, but there was no accidental discovery. That is a myth.

If you look at the history of great scientific discoveries, there are a few cases where they were made accidentally, but it took a prepared mind to recognize what they were looking at.

There is the famous discovery of penicillin by Alexander Fleming, who left a contaminated petri dish out overnight. Most researchers would have thrown the plate away, but Fleming noticed that bacteria were not growing near the mold. He recognized that something was killing them. That was the first antibiotic but many people don’t know the story of what happened after.

Fleming made his discovery in 1928 and published it. It was treated as a mild curiosity and ignored. Several attempts were made to purify it. All failed, and it was deemed impossible.

10 years later, in 1937, Howard Florey and Ernst Chain discovered Fleming’s work and put together a team to work on the “Penicillin Project”. The project nearly failed because of ego clashes between the scientists, but after 3 years, they came up with an incredibly inefficient process to produce pure penicillin. Gallons of mould broth (yuck!) were required to produce a few milligrams of antibiotic. The researchers had to use every container available to store all the broth, bathtubs, bedpans, milk churns, and food tins.

Eventually, a fermentation vessel was developed, and production began.

Production for clinical trials was so difficult that there was an immediate shortage, just as World War 2 began. Because the body excretes about 80% of penicillin in the urine, the patient’s urine was collected and recycled into their next doses.

Because of the war, the UK-based team could not find anywhere that would produce their miracle drug and had to turn to a lab in Peoria, Illinois, in the United States to produce it there. A rotting cantaloupe at a Peoria market was discovered that produced 6 times more of the compound than Fleming’s original mold.

After Pearl Harbor, the United States ramped up antibiotics production, and enough was available by 1943 to supply the Allied effort.

Would Penicillin have been discovered if Fleming had not made his accidental observation? I would say probably yes. Consider that 10 years have passed with nobody doing anything about it, and research into chemicals produced by microorganisms like mold was already ramping up. It provided a clue, but it didn’t provide the answer.

Two Bell Labs employees, Arno Penzias and Robert Wilson, in 1964, were trying to eliminate noise in their microwave antenna. They checked the electronics and accounted for atmospheric effects. They even cleaned the antenna, but the noise persisted. It turned out that they had accidentally discovered the Cosmic Microwave Background (CMB).

The story often ends here and fails to mention that Penzias and Wilson, who are usually credited with the CMB discovery, didn’t know what they were looking at. All they concluded was that the source was outside the galaxy. Meanwhile, at Princeton University, Robert Dicke, Jim Peebles, and David Wilkinson had been working on a model for a blast of radiation from the early universe and how it should be detectable in the microwave spectrum. Penzias, fortunately, had a friend, Bernard Burke, at MIT, who made him aware of Peebles’ work. It was only after Penzias invited Dicke to come look at his data that the real discovery was made.

This seems to be a pattern with “accidental” discoveries. They are fortunate but depend strongly on available talent and knowledge. Rontgen discovered X-rays, Spencer, microwave ovens, and Goodyear, vulcanized rubber, but all these accidental discoveries were merely the precursors to long development cycles and more sophisticated reasoning than the popular myths tell. Others had seen what these discoverers had seen before, but didn’t stop to think about what it meant because they lacked the knowledge or connections to make the leap from anomaly to discovery. It takes a lot of persistence after the eureka moment to make the true discovery.

The critical ingredient in great discoveries, therefore, is not the accident occurring. It is having a mind that is prepared to recognize the unexpected and make it known so that the real science can be done. And this is the part that often falls out of the tales later told. These discoverers didn’t just notice the anomaly, but something about it kindled in them (or sometimes someone else) the motivation to see it through over months or years.

Most people ignore anomalies, throw away failed experiments that don’t give the answer being looked for, dismiss odd observations, and return to the task they are supposed to be doing.

People who get curious, follow distractions, investigate nuisances, and treat surprises as clues are the ones who get the gold, but it takes a lot longer than a bath. And sometimes the original discoverer isn’t even the one who benefits, except perhaps in terms of name recognition in the myth that grows up later.

This seems like a personality trait in some. I spent hours looking for the origin of this quote:

The most exciting phrase to hear in science, the one that heralds new discoveries, is not “Eureka!” (I found it!) but “That’s funny …”
— Isaac Asimov

I even remembered reading a book or article where he said this, but I could not find it. I wondered if I had actually read it from Fortune, the UNIX program, and conflated it with some actual work of his over the past 30 years, since I both used Fortune and read Asimov a lot in the ‘90’s.

In my investigation, I did stumble upon Asimov’s 1971 article The Eureka Phenomenon which has nothing to do with the quote. Reading the article for the first time in probably 30 years, I decided to investigate whether the original story, the Eureka story, even happened. I discovered that it probably didn’t. Did this lead me to any great discovery? I think what it led to is a realization that activities that one might consider wasting time, getting off task, going onto tangents, are actually beneficial traits, but only if you are ready to recognize them and follow them up.

Will some discovery emerge from the article you just finished? Who knows? Maybe one of you knows where the quote really comes from and can send me the citation.

* Focus by Gordon Rattray Taylor, April 1965, Science Journal, Volume 1, Number 2, Page 32, Column 1, Associated Iliffe, London, England

Growing up in the 80s and 90s, it felt like string theory was the theory of everything that would eventually describe all matter, forces, and fields.

In an episode of Star Trek: The Next Generation, an alien probe transforms the jittery, fantasy-obsessed engineer, Lieutenant Barclay, into a super-intelligent being. In one scene, Barclay stands on the holodeck before a blackboard, correcting a holographic Albert Einstein about how many dimensions the universe has: is it 26 or 10? Barclay argues, and Einstein agrees, that neither can stomach a 26-dimensional universe. Later, Chief Engineer Geordi LaForge expressed incredulity that Barclay could be debating universal field theory with Albert Einstein.

I couldn’t imagine where those two numbers, 26 and 10, came from at the time. I assumed the show made them up. It did not. It wasn’t explained, but his debate was over whether supersymmetry, or SUSY, is a feature of string theory. Without SUSY, the theory requires 26 spatial dimensions, while with SUSY, you only need 10 (and you get some other benefits as well).

Since the early 90s, when the episode aired, string theory has suffered many setbacks (and some of its greatest achievements). All attempts to confirm its physical predictions have failed. The largest colliders, such as the Large Hadron Collider (LHC) in Switzerland, have never given a hint that supersymmetry or compact dimensions exist, let alone strings.

Respected critics such as Lee Smolin and Peter Woit have argued that string theory is a dead end, an overhyped, overfunded farce. They even suggest that string theory is a beautiful theory about a universe that happens not to be ours.

Such opinions, however, are conjecture. We don’t know if string theory can explain our universe, only that some of the best incarnations seem to fit other universes than our own.

While the end of the 20th century gave us hope that string theory would be the final theory of everything, the early 21st century ushered in the landscape and swampland paradigms. In fact, a recent paper, Strings from Almost Nothing, by Caltech researchers now accepted by the prestigious Physical Review Letters, is the latest in a long program to find the rules by which string theory must arise as a theory of quantum gravity, but not necessarily explain our universe uniquely.

Landscape and swampland, which I will explain in a moment, are two sides of the same coin and are often the targets of criticism. But together they point to something that might make string theory more, not less, compelling:

String theory is a universe factory. In other words, it is a way of describing not only our own universe (we hope) but all possible universes. It is a universal theory of universe construction. String theory may never be narrowed down to a theory that describes our unique universe because it is far more than that. It may, instead, open a porthole into God’s own workshop. This article is about what that workshop looks like according to the latest discoveries and why string theory might be the most likely theory to explain it.

String Theory

The theory arose in the 1960s as a potential explanation for the strong nuclear force but was eventually abandoned in favor of Yang-Mills theory and spontaneous symmetry breaking. The theory proposes that particles are not points but extended, one-dimensional objects that vibrate and, depending on how they vibrate, create different forces or particles.

Early string theorists noticed that from the equations for string theory emerges a spin-2 field that looks a lot like a gravitational field, and they conjectured that string theory might be the high-energy theory of gravity, with Einstein’s General Relativity being the low-energy, Effective Field Theory (EFT) that emerges from it.

String theory originally only included bosons, which are particles with integer spin. Forces are, for the most part, bosons, while matter is made up of fermions, which have spin that is a multiple of a half. Supersymmetry was later added to string theory to encompass both bosons and fermions. Sometimes the earlier string theory is called bosonic string theory, while the latter is fermionic string theory, but the latter contains both.

While point particles trace out lines in time called worldlines, strings trace out two-dimensional objects called worldsheets. These sheets interact in various ways, combining strings and vibrational modes to create different kinds of particles.

Therefore, while the Standard Model of particle physics has many kinds of particles:

String theory only has one kind of particle: strings.

Since strings are tiny loops, at the scales we can probe them, they appear to be point particles, but if we could probe them at very, very high energies, we would find they have a stringy structure.

Strings do not vibrate in ordinary 3D space, however. They need a much higher number of dimensions and specific kinds of high-dimensional spaces that we call compact or compactified spaces.

A good illustration of a compact dimension is to imagine a drinking straw. The straw has one extended dimension down its length, and when you look at the straw from a distance, it appears to be one-dimensional. If you look closely, however, you see another circular dimension. Thus, a tiny insect, like a mite, could walk both along the straw and around it, but a larger insect, such as a beetle, could only walk along the straw.

A field theory defined on a compact dimension is often called a Kaluza-Klein theory after Theodor Kaluza, who tried to combine gravity with electromagnetism using an extra dimension, and Oskar Klein, who suggested compactification as a reason why we couldn’t detect the extra dimension (as well as offering some additional benefits).

Kaluza-Klein theory is historically the idea that forces could all combine geometrically. It was a tantalizing theory of everything for a while, and it got Albert Einstein excited. It was eventually proved that you could not, in fact, combine all forces using compact dimensions. String theory, however, salvaged the Kaluza-Klein compact spaces and proposed that strings were, in fact, vibrating inside these compact dimensions and that this is where matter and forces come from, not the geometries themselves.

To replicate all the forces and matter in a supersymmetric string theory, you only need 6 extra dimensions, while in bosonic string theory, you need 22 extra dimensions.

In superstring theory (supersymmetric or fermionic string theory), the compact spaces are called Calabi-Yau manifolds. Here is a picture of a cross-section of one:

The reason why it needs that number of dimensions is a bit in the weeds, but these numbers are special. (If you really want to know, these are the numbers of dimensions in which the conformal or Weyl anomaly vanishes. These are called “critical” string theories.)

Why does string theory make a good candidate for a theory of quantum gravity?

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About two years ago, I wrote an article on a preprint by a pair of MIT researchers, Lohmiller and Slotine, who claimed to have reproduced quantum mechanics using classical mechanics. Recently, that paper was finally published in a peer-reviewed journal.

The authors claim to reproduce quantum mechanics by allowing branching behavior in classical mechanics, having different classical timelines overlapping and interacting. These branches can still interfere with one another, so the idea is that you don’t need all these crazy zigzag paths that Richard Feynman’s approach requires.

Feynman’s approach is called the path integral because it involves integrating, meaning summing, the quantum phases over all the different paths leading up to a measured particle’s state.

The path integral sums over every path in the universe, meaning that it includes paths that zigzag out to Alpha Centauri, the Andromeda Galaxy, and so on and back to the lab in even the tiniest experiment. Such long paths contribute very little to the integral, but they are there. It would be nice, therefore, if someone could come up with a more intuitive approach where particles follow paths that make more sense.

These guys claim to have achieved exactly that.

In classical mechanics, all trajectories obey something called the principle of least action. This means that there is a function called the action that, when minimized, determines the path an object will take.

Most paths in the path integral violate the principle of least action because their actions exceed the minimal action, like the purple paths in the above image. Those that are minimal, meaning classical paths, contribute the most to the integral, but they are not the only ones.

The authors claim to have done away with Feynman’s extra paths and kept only the minimal necessary set to reproduce quantum mechanics. All of these turn out to be classical paths.

They point out that classical mechanics has branching paths just like quantum mechanics. For example, if you want to know the classical path of a ball from point A to point B, normally there is only one path, but if there are barriers where the ball can bounce or pass through holes, then there will be more than one path that minimizes the action. You can also vary the initial conditions to get different paths as well, something the authors make heavy use of.

The famous double slit experiment is an example of such a multi-path system. There are two classical paths. If they can interfere with one another like waves, then the standard interference pattern will result.

Unfortunately, L&S made a fatal mistake.

Reading the paper, you get the impression that the authors do not normally work in quantum mechanics. The notation is more typical of classical nonlinear dynamics. I’ve worked in both fields, so I recognize this as being more in the language of the applied mathematics/control theory community than physics. This might explain why they run into problems. Looking at their publication history, they have published mainly in nonlinear systems, a fascinating but classical area of physics. They also appear to be mechanical engineers. Engineers are notorious for stumbling into fields they know nothing about and acting like everything is a design issue.

L&S, through a series of mathematical lemmas and a theorem, show that they can reduce the quantum Schrodinger’s equation to the classical Hamilton-Jacobi equation. This is a red flag. (Also, when I wrote my first article about their work, they hadn’t yet made this assumption, although they were getting close.)

The assumption isn’t wrong; it is just a special case. The Hamilton-Jacobi equation is a classical equation that connects classical mechanics to wave mechanics, so it is a natural choice. David Bohm used insights from the Hamilton-Jacobi equation in his Bohmian mechanics in the 1950s. And Edward Nelson used it to derive his stochastic quantum mechanics. Neither of them, however, just assumed that you could get quantum mechanics from the classical equation.

Something extra appears when you transform the Schrodinger equation into the Hamilton-Jacobi equation (using something called the Madelung transform), called the quantum potential, Q. This potential is proportional to the spatial variation of the density of the quantum wavefunction. The equivalence is sometimes called the (Madelung) hydrodynamic form of quantum mechanics. The equations really resemble those of some kind of fluid with a density and a velocity field.

The density variations present in Q are exactly what allow all those crazy zigzag paths in the path integral. It frees reality to deviate from classical mechanics. L&S assume that, along their multi-branching trajectories, the quantum potential is zero, which throws all those crazy paths out from the get-go. This is like assuming your fluid always has constant density. That may cover a lot of fluids, but not all fluids in the universe.

So, they did not show that they can reproduce quantum mechanics from classical branching; they eliminated all quantum mechanics that doesn’t conform to classical branching. It is circular reasoning.

They do manage, however, to demonstrate their model reproduces the wavefunctions for several standard examples, including the double slit. They achieve this in two ways: for some models, like their particle in a box example, the quantum potential is zero anyway. For others, like the quantum harmonic oscillator, they just smuggle it in through their initial conditions, which is clever but makes little sense.

Sabine Hossenfelder refers to this paper as “bullsh*t” on her YouTube channel, and indeed, a detailed takedown of the paper’s main conclusions by Gábor Vattay has appeared on arxiv (it has not been peer reviewed yet). Too bad the peer reviewers didn’t catch this, or, as Hossenfelder suggested, the authors didn’t give ChatGPT/Claude/Grok a pass over it.

To be honest, I’m not sure if the paper is completely useless. It isn’t strictly semi-classical but more of a hybrid. It might have some computational utility in some cases, but it says absolutely nothing about reality other than you can get almost anything published in a peer-reviewed journal these days. I’m not sure I’d want to have a paper become infamous like this even for a few weeks.

After I wrote my last article on the earlier preprint, I did go through the paper in more detail since, at the time, I was working on Hamilton-Jacobi approaches to quantum mechanics. I found it contained some unvalidated assumptions, although it had yet to set the quantum potential to zero, and at the time correctly recognized that it would not be zero for some examples. Strange how it got worse before it was published.

Honestly, my conclusion from spending a few months on the subject is that the HJ equation almost always makes quantum mechanics more complicated than it needs to be. It introduces nonlinearities where Schrodinger has none, and once you start introducing many-bodied dynamics or field theory, the connection to classical hydrodynamics disappears as you end up in higher and higher dimensions, becoming infinite-dimensional with field theory.

Worse, there is an inherent problem when going from Schrodinger to the hydrodynamic form (even if you do it correctly) because Schrodinger’s equation has a symmetry that disappears in the conversion. This is why Nelson’s stochastic quantum mechanics has problems since it depends on stochastic paths that obey the hydrodynamic form.

To understand this problem, imagine you have a particle confined to a one-dimensional ring. In standard quantum mechanics, the wavefunction defined on the ring is periodic, meaning it wraps back around since the ring is a circle. This forces the wavefunction to take on only states that have phases that are integer multiples, so that they all obey the same periodic structure.

Think of it like this. I have a bunch of cars driving around a circular track. For all the cars to cross the finish line on each lap at the same time, they have to be traveling at speeds that are integer multiples of one another, so the first car is just sitting stationary at the finish line, the second car is going around at speed s, the third car at speed 2s, and so on. You even have cars traveling the opposite direction -s, -2s, -3s, and so on. Perhaps this is because the cars are being refueled by some giant cross arm that comes down at a certain rate 1/s. In any case, the cars are synchronized.

This is the kind of restriction enforced on the wavefunction’s states.

When you go to a hydrodynamic form, this restriction disappears from the equations. Suddenly, the cars can go any speed they want and cross the finish line whenever they want. Unless you add it back in artificially, therefore, the hydrodynamic equations are not equivalent to Schrodinger’s because that rule, which is a symmetry, breaks.

Therefore, even though L&S could account for the quantum potential by adding stochastic (random) zigzags to their branching paths, as Nelson does, and that might be an interesting variation, their model would still have a serious issue. Even if L&S managed to get their model to account for the quantum potential correctly, their approach would still fall short of a full quantum theory for the same reason why Nelson’s does. The HJ approach to quantum mechanics just doesn’t work.

Although pre-Socratic philosophers mentioned the soul occasionally in the context of life after death as in the Homeric epics, as wraiths inhabiting the underworld, Plato and Aristotle were the first to provide detailed descriptions of the soul and its properties. Plato believed that the soul encompassed our intellect, which could be freed from the body to contemplate philosophy for eternity.

In The Republic, Socrates asks Glaucon,

Haven’t you realized that our soul is immortal and never destroyed?

Standing in for the reader, no doubt, it continues,

He looked at me [Socrates] with wonder and said: “No, by god, I haven’t. Are you really in a position to assert that?”

This contradicted popular 5th century BC Greek ideas, which had moved on from Homer, that the soul is like smoke and would disperse after death.

In Phaedo, Plato presents his [as Socrates] arguments for the indestructibility of the soul relying on it being more like thought or abstract concepts and less like a physical thing, i.e., immaterial not material.

One of the problems with being a modern person and reading anything from centuries or milliennia ago is that we imagine these texts as addressing fundamentally important problems that we have today, when in fact they may simply be trying to address problems of the time that we don’t understand. In other words, Plato is trying to solve 5th-4th century BC problems like is the soul material, like smoke, like everyone believes or immaterial, like a thought, as he believed. Plato won that argument and hardly anyone now believes the soul is a material thing (attempts to weigh it notwithstanding).

The questions Plato left open then became problems for Aristotle to solve.

Aristotle contradicted his teacher, arguing that the soul vanishes with the body. Aristotle went much further than Plato provided a more complete description of its nature and function.

The Greek concept of the soul includes not only the vital essence of a person but everything that distinguishes living from nonliving, animate from inanimate. Our word “animal” comes from the Latin word anima, which means soul, while our word “psychology” comes from the Greek words psyche and logos, meaning the word or study of the soul. Thus, there is some notion that the soul is both the animating force of life and also the force behind the mind. It also means that all living things, by Greek reasoning, have souls.

Modern materialist philosophy tells us that life force and the mind should be attributed to physical processes such as energy, metabolism, and neural firings, but such a mechanistic understanding of reality is a relatively recent outgrowth of 18th-century rationalism. Philosophers of the Enlightenment such as David Hume and Thomas Hobbes wanted to replace all the old, mystical belief systems with a mechanistic one based on Newtonian physics and scientific reasoning. It is no surprise that mechanical automatons were hugely popular in the 18th century and that philosophers of the time ran towards a theory that life itself was no more than a more sophisticated automaton.

Such mechanistic descriptions, however, are insufficient to explain the existence of life or the mind. They provide only the immediate “how”, not the bigger picture “why” that metaphysics demands. Hence, why the Greeks are still relevant even now.

Aristotle’s theory of the soul rests on his own philosophical school which became popular in the Middle Ages and is having a renaissance now. This school is called hylomorphism, hylo meaning matter and morphism meaning form: the theory of form and matter. While the world is made of matter, form distinguishes one lump of matter from another. This form gives an object not just its shape but its essence, “the essential whatness” of its being. Thus, a table is a lump of wood, but it also has something called “tableness,” which makes it not a mere lump of wood but a table.

If you have been steeped in postmodernism (I almost wrote “modern” postmodernism before realising how silly that sounds), you might find hylomorphism quaint. A table is only a table because we call it a table. Isn’t that true? Reality is matter, and only the human mind gives it form.

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How many of you know the difference between quantum and classical physics? Raise your hands.

If I were to call on you to define it, what would you answer?

The old-school answer might be that classical physics is the limit as Planck’s constant goes to zero. But that isn’t really an answer. It’s just a mathematical statement.

Another might be that it is when Heisenberg’s uncertainty vanishes (which is a consequence of the first answer). In this case, it is possible to know the exact position and velocity of a particle at the same time.

That answer is closer to the truth, but perhaps isn’t the fullness of the truth.

A third answer is that, whereas in classical physics you can predict the past and future of a system precisely, provided you know its exact state, in quantum mechanics, you can at best predict probabilities.

This is getting closer, but is still missing an ingredient. I could say the same thing about a classical stochastic process, and I wouldn’t get quantum mechanics.

So what is the answer? What is my answer?

My answer is that in quantum mechanics, there exists an entity called a wavefunction. This is a wave-like object having both amplitude and phase. Wavefunctions combine in ways that let you add them together to represent new wavefunctions, which are called superpositions. These wavefunctions evolve according to an equation called Schrodinger’s equation that evolves the wavefunction forward in time. The probability that you will find a particle in a particular state when you measure it is equal to the square norm of the wavefunction (which is the wavefunction times its complex conjugate).

I like this answer not because it is the most complete (after all, I have just established that wavefunctions exist ontologically, which is disputed, and I have completely ignored other mathematically formalisms such as the Heisenberg picture and even time-independent Schroedinger’s) and not because it is the easiest to understand, but because it is the most complete answer that is easy to understand.

There are a few important points here as to why I think that: (1) quantum mechanics is weird, and one of the easiest ways to understand why it is weird is to recognize that wavefunctions are not probabilities but rather like square roots of probabilities. They contain not only amplitude, which is what determines the probability, but also phase information. Therefore, they combine like waves, interfering with one another as well as constructively adding together. This is exactly why you can have effects like entanglement, where one particle appears to affect another distant particle instantaneously. This only happens because wavefunctions are waves and not probabilities. (2) Squared wavefunctions are probabilities. This is important and has a name: the Born rule, after quantum mechanics pioneer Max Born. Without this rule, there is no quantum mechanics. And finally (3) Schroedinger’s equation evolves wavefunctions forward in time, so wavefunctions are not static objects, but they move and change.

Classical mechanics, on the other hand, has no wavefunctions. It represents particles purely as having position, velocity, and so on in a definite way, but more importantly, even when particles are represented probabilistically, as in Brownian motion, for example, where they are jostled around by molecules, those probabilities do not combine like waves. There is no weirdness.

Potentially, if we understood quantum mechanics better and where it comes from, we could come up with a shorter and more intuitive answer, but I’m not so sure. Most of the various interpretations that we are aware of, when they are explained to the layperson, ignore the phase information in the wavefunction entirely. The Many Worlds interpretation allows multiple realities to coexist, but this is just ordinary probability. It doesn’t explain how those probabilities got there in the first place! It was through the summation of complex numbers, not classical probabilities.

I guess I’m trying to hammer home here that quantum mechanics is not just classical mechanics with probabilities. That is classical stochastic mechanics. Because of the phase information in the wavefunction, quantum mechanics becomes something far weirder. We can only understand this if we think of all reality in terms of waves.

Quantum theory translates between waves and classical probabilities in a straightforward mechanism called decoherence. You can think of decoherence as when many, many waves interact with one another, their phase coherence (how their peaks and troughs line up) is destroyed. The result is that phase information becomes less and less important, and all that matters is the different probable outcomes.

This is how we go from Schrodinger’s cat to a dead or alive cat.

Decoherence is necessary for all objective interpretations of quantum mechanics, including the Many Worlds. Ultimately, decoherence is what “separates” the worlds from one another, since before that, their phase coherences keep them locked together. Technically, the worlds aren’t separate after decoherence either; they are just phase incoherent, so they can no longer interact.

Decoherence, however, can only remove phase coherences. It cannot tell you what the actual outcome is going to be. In other words, it can tell you that the cat is alive or dead, but not which one it will be.

It is what happens after (or during) decoherence that tells us what quantum mechanics is actually doing.

While some believe we split into many universes, others believe that conscious observation somehow causes decoherence to transition to a single decohered state: alive or dead, up or down, true or false.

Those who believed in the observation-triggered collapse, however, have a problem. They can not explain precisely how or why this would happen.

Enter: objective collapse theory.

The goal of objective collapse theory is to explain this mechanism. Starting in 1986, a model was introduced by Ghirardi, Rimini, and Weber that showed how quantum theory could be modified so that, not only would decoherence happen over time, but the wavefunction itself would spontaneously, without even needing to be observed, collapse all on its own, like a house of cards in a gust of wind.

Their first model included these things called jump operators, which meant that collapse would happen very suddenly with these random, “localization” events called “hits”. (Localization is another word for wavefunction collapse, especially when it occurs for position.)

This means that at random times, according to a probability distribution, the wavefunction would just collapse, whether anyone was looking at it or not.

Later on, in 1990, GRW adapted their model to another kind of spontaneous collapse theory called Continuous Spontaneous Localization (CSL). In this case, the collapse didn’t happen suddenly and randomly but over time. So instead of the house of cards falling over from a sudden gust of wind, it fell down by a thousand little puffs of air.

Think for a moment how things might be different if either of these mechanisms proved true: we would finally know how classical physics emerges from quantum. Our need to find interpretations for that part, at least, would be at an end. Perhaps the wavefunction, or its field theoretic equivalent, really is the fundamental reality, but it is an unstable reality, whereas classical physics is its stable state.

I, for one, like objective collapse theories because they remove the hokiness of believing that conscious minds somehow create the universe with all its attendant philosophical difficulties. It also removes the existential discomfort that the Many Worlds interpretation causes many of us, believing that there are a multitude of copies of ourselves running around in parallel worlds.

Instead, there would be one world: this one, one past, one future, and one you. Unique.

What is happening at the quantum level, although it ultimately determines reality, has limits.

The downside to GRW and what is now called CSL is that both have to modify quantum mechanics by introducing randomness to force the wavefunction to collapse over time (either probabilistically from “hits” or continuously from noise).

This is a serious claim: Quantum mechanics as-is isn’t good enough.

I took this problem head-on in the last few months, although I got there by a circuitous route. My goal: show that quantum mechanics is good enough by itself to collapse the wavefunction.

For a long time, as I have frequently written about here, I have been looking for a theory that would explain quantum mechanics itself. I believed that a five-dimensional universe would, in fact, explain exactly where it comes from.

I became fixated on the idea that classical physics in compact, curled-up dimensions could lead to quantum theory.

And I failed. Instead of reproducing quantum mechanics, I derived an equation that would later prove to me that I had had it backwards the whole time. Quantum mechanics didn’t come from classical. It was the other way around.

In the process, I stumbled into a whole area of quantum physics that I had never encountered before, but which I eventually realized could explain the CSL model.

There is a subfield, going back at least to the 1960s when a lot of the best work on quantum mechanics and field theory was done, called open quantum systems. In physics, a closed system means that your model includes everything; nothing is getting in, and nothing is getting out. A system is open when your model does not include everything. This implies, of course, that the things getting in and getting out of your open system are models, but whatever is causing them to get in and out is not being modeled.

Take a model of a greenhouse as an example. You can model the conditions inside a greenhouse without modeling the nuclear fusion taking place in the sun, nor the ambient temperatures impinging on the exterior of the glass, nor the exterior air currents that may affect your ventilation. All these external influences can be abstracted away.

In other words, you don’t have to model the full water cycle to know that rivers flow downhill.

Quantum systems are no different. Why model 1000 molecules if you only care about one? Instead, abstract the 1000 molecules into some external force.

This is exactly how models of Brownian motion work, too. We don’t model all the individual molecules knocking a bit of pollen around in a petri dish. All we care about is how the collective energy of the molecules influences the pollen.

Albert Einstein explained this in one of his 1905 annus mirabilis papers. His main result in that paper (which is often forgotten as it appeared alongside relativity and the photoelectric effect) was something called a fluctuation-dissipation relation or theorem. Such theorems are essential to understand how macroscopic objects interact with lots and lots of randomly moving microscopic objects.

Random motion in macroscopic objects includes these two opposing forces: fluctuation, which is random noise traveling from the environment to the object, and dissipation, which is energy leaking from the object into the noisy environment. These two together determine the random motion of the object, ensuring that the object doesn’t continually gain energy and acceleration from noise, but also never stops from dissipation. Rather, the two balance each other out.

Open quantum systems likewise have noise and dissipation as their primary opposing elements, but calculating them is a lot more complicated.

This all started around 1963 when Richard Feynman and a student named Frank Vernon, who was working on a PhD in electrical engineering and physics. (He passed away in 2002 but is sometimes confused with a professor of the same name at UCSD.) Together, they used Feynman’s prior research into quantum theory to develop a theory of open quantum systems, where complicated quantum environments could be abstracted away into expressions for fluctuation and dissipation.

This was pretty monumental in itself, but nothing compared to what would become of it later.

There are two kinds of open systems: those that are in equilibrium with their environments and those that are not.

If a macroscopic object is in equilibrium with its environment, that means that where it ends up is pretty much the same as where it started, at least in terms of its probable location. If I have a piece of pollen in a petri dish, I can take a snapshot of where it is now and work either forward or backwards in time to determine where it is likely to be and where it is likely to have been. The probability for where it has been any time in the past, however, will match the probability for the future. That is what equilibrium means. Time doesn’t matter. Nothing is going anywhere in a definite sense.

The pollen isn’t expected to change because of what’s happening.

But if I replace the pollen with something fragile, a mini-house of cards, then I would expect at some point the house of cards would collapse. Since I don’t expect houses of cards to magically assemble themselves, if I work backwards in time, if I see a house of cards now in a noisy environment, I would expect that house of cards to still be there any time in the past. Once it has collapsed, however, I expect that it will remain collapsed forever.

This is an example of a non-equilibrium system. In fact, it is not only not in equilibrium, but it is also an irreversible process. The house of cards cannot reassemble itself.

In physics, equilibrium processes are comparatively easy to understand. There isn’t anything dynamic about them. It is all statistical. I did my PhD thesis on equilibrium statistics, and I can tell you that it is a lot easier to model something that doesn’t change over time than something that fundamentally does.

Perhaps the most confusing thing about modeling quantum systems is how they work in time. Open quantum systems do not evolve from one time to the next the way that classical systems or closed quantum systems do. They evolve along a contour that goes from the infinite past up to the point in time you care about and then back to the infinite past.

In other words, they evolve along an infinitely long, closed time-loop, which sounds like an interesting plot device for a science fiction novel, but is actually critical to understand how open quantum systems differ fundamentally from other kinds.

In open quantum systems, we don’t often work directly with the wavefunction but with an object constructed from wavefunctions called a density matrix.

A density matrix is like a probability distribution except that it includes all the phase information that quantum wavefunctions have as well as probabilities.

Because a density matrix is formed from two wavefunctions, the way you evolve the matrix as a whole forward in time is you actually evolve the left side of the matrix forward in time and the right side backward in time.

Yes, the density matrix lives on two time lines, a forward line and a backward line, straddling them like Colossus of Rhodes (which didn’t actually straddle the harbor, but that’s the expression).

In any closed quantum system, you can treat the timelines as separate entities, but in an open quantum system, you are getting rid of part of the matrix corresponding to your environment and only retaining the part you care about.

The forward side of the system and its backward side interact with the same environment. Because the environment is the same for the forward and backward sides, the timelines become stuck together so that the same environment interacts.

Feynman and Vernon dealt with this issue by making two copies of their system, a forward copy and a backward copy. They then enforced the copies to coincide with each other at the time they were interested in. This forced them to form a closed loop. The forward side comes from the infinite past up to the time of interest, and the backward side goes from that time of interest to the infinite past.

If this makes your head spin as it does mine, imagine the system as being like a time traveler in a universe where the rules of time travel are very strict. In order to retain their identity, a time traveler who travels forward in time must return to their exact place of origin in the past where they left. In other words, their path through time must form a closed loop. If their path does not follow this closed loop, then not only will the time traveler be split in two, but the universe will be split as well! The universe is analogous to the environment. If you want there to be only one universe, your time traveler has to return to the past.

Although Feynman and Vernon had laid the foundation, it took Soviet physicist Leonid Veniaminovich Keldysh to make the conceptual shift and build the house. All Feynman and Vernon had really done was explain how systems decohere when interacting with an environment. They had related this to noise and dissipation. Keldysh invented an entire nonequilibrium field theory. Instead of asking what happens to the density matrix, he asked what any physicist wants to know: what happens to the things we want to measure?

That then leads to what is called the Keldysh contour that the time traveler travels along.

If you were to look at the notation used in Keldysh theory, you would probably want to immediately turn back to your trusty quantum field theory textbook and never grace Keldysh’s door again, but if you do take the time, you will see that Keldysh does for quantum systems what Feynman had earlier done for ordinary quantum field theory. He made it understandable using diagrams.

I won’t go into his diagram method, but it is similar to Feynman’s in some ways.

Thanks to Keldysh’s work, I was able to start in earnest trying to understand how a quantum field could, thanks to a compact 5th dimension, collapse itself.

One of the things that happens when you have a tiny curled-up dimension is that you can take one field defined over all five dimensions with one particle mass and turn it easily into a lot of fields in four dimensions with a ladder of increasing particle masses. (This is called the Kaluza-Klein tower or KK tower.) If the original field interacts with itself, then all the resulting fields will interact with each other, and this is the first step in applying the Keldysh theory.

The lowest mass field in the KK tower has the same mass as the original field. I treated this as my system of interest. The rest of the tower I treated as my environment.

Thanks to the Keldysh contour theory, I was able to calculate the noise and dissipation for this kind of theory, and from that, pretty easily produce the same type of equation that is used in the CSL theory. So far, so good.

I then started to look at whether I would get good collapse from that.

I quickly found out that if the compact dimension was just a bunch of vacuum fields, meaning nothing in there, then collapse could not happen. The whole idea here, after all, is that the KK tower fields need to scramble the wavefunction in the observable field and cause decoherence. They have to force it to decohere. It turns out, however, that in a vacuum that can’t happen because the lightest KK tower will be too heavy (mass-wise), and if that one is too heavy, then the other tower fields are much too heavy.

If, on the other hand, the KK tower fields have some excitations in them, which could be primordial particles or something else, then those excitations can help scramble the wavefunction of the observable field. That’s interesting, of course, because that fits well with Kaluza-Klein dark matter theories, which propose that the universe is filled with such excitations.

It was in this moment that I realized that I had reproduced the CSL theory (under a whole heap of well-justified approximations, of course).

And that’s really all there is to it. My paper is now out on my website and has been submitted to a journal.

https://andersenuniverse.com/wp-content/uploads/2026/04/emergent_wavefunction_collapse_andersen_28apr2026.pdf

Ultimately, what it shows is that a compact 5th dimension can cause a quantum field theory to collapse its observable part while remaining an ordinary quantum field theory. No modifications to Schrodinger’s equation, no information destruction at collapse. It’s all emergent. The lost information just escapes into the compact dimension where it is irretrievable.

This has made me wonder if, philosophically, my chasing after a way to derive quantum mechanics from classical all these years wasn’t a fool’s errand, a kind of philosophical irritation with quantum theory that, I feel, I have resolved by showing that it is the other way around.

While the movie is primarily focused on the buddy relationship between biologist Ryland Grace and his alien counterpart Rocky, physics plays a big role, too. After all, the only way that either Rocky or Ryland made it to Tau Ceti within their lifetimes was by using the fantastic property of astrophage to collect enormous quantities of energy and release it with near-perfect efficiency to cross vast depths of space in short periods of time.

Although the movie only vaguely describes how astrophage works, the book provides a more detailed explanation. It turns out, however, that this explanation is deeply flawed. Yet, I have figured out a better way that astrophage could, within the realm of known physics, exist with its fantastic properties, although it would change some of the plot points of the story, and even then, it is highly speculative.

Astrophage means “star killer” in Latin, and it is a tiny black, dot-like cell that maintains an internal temperature of 96.415 degrees Celcius, just below the boiling point of water at 1 atmosphere of pressure. This temperature is constant no matter where it is, even inside a star, which means that when exposed to temperatures higher than its internal temperature, it absorbs that energy. This is how it destroys stars. It simply absorbs the energy. When it has to release energy, it does so at a particular infrared frequency, which it uses to travel vast interplanetary distances to its breeding grounds, which, in our case, is the CO2-rich atmosphere of Venus.

The astrophage reproduces by seeking out the spectral frequency of CO2 (at 4.26 and 18.31 micrometers). In the book, a Russian researcher, Dimitri Komorov, reports that an astrophage can absorb 1.5 MegaJoules of energy without heating up, becoming 17 nanograms heavier as a result. This kind of direct mass-energy conversion is extremely efficient, approaching that of matter-antimatter reactions.

To collect this much energy, in the book, a large proportion of the Sahara is covered with black panels which absorb enough heat from the sun to feed the astrophage, producing the 2 million kg required to get the Hail Mary and astronauts to Tau Ceti. On Rocky’s planet of Erid, meanwhile, the planet itself is hot enough that they just have to dump the astrophage into their ocean, which is already above 96.415 degrees.

This makes astrophage like any other biological system. It must capture energy, store it, and release it for useful work. Yet, all of these would have to work at a scale and efficiency never seen in the biological world.

How does astrophage capture energy?

Ordinary photosynthesis is very inefficient, with 1-2% of solar energy being stored as chemical energy inside a plant. Moreover, plants generally only absorb a few frequencies of light. Astrophage, on the other hand, is described as completely black and impenetrable to all wavelengths of light, suggesting it absorbs across a very broad spectrum, likely into the gamma rays.

It would need to not only absorb this energy with near-perfect efficiency but also transport it with close to 100% efficiency from end-to-end. Thus, the transition from photon energy to storage would need to have only minimal losses.

How does astrophage store energy?

You may have heard that ATP is the main energy storage medium of biological organisms. These store about 30 kJ/mol, which is nowhere near what astrophage stores.

Even hydrogen, which is the highest-density chemical fuel, wouldn’t come close at 120 MegaJoules/kg. Metallic hydrogen, such as might be found at the core of the planet Jupiter, might come closer to astrophage but would require enormous pressure to store and would not be stable. Chemistry hits a hard wall when it comes to energy storage because of electron energy levels. Chemical energy comes from rearranging electrons in orbitals, which is measured in electronVolts per bond. Astrophage, on the other hand, stores energy in the thousands to millions of electronVolts per particle, which suggests either nuclear physics or high-energy electromagnetic storage.

Astrophage’s storage capacity approaches that of uranium fission or higher, but there is no indication in the book that it requires heavy elements at all.

The book suggests that astrophage stores energy using “neutrinos and a form of pair production”. Neutrinos form from colliding protons together and are able to absorb energy while contained within their astrophage. These neutrinos then release energy at the Petrova line frequency.

This is where I get lost. Andy Weir is a first-rate storyteller, but for whatever reason, he has used dubious physics as the centerpiece of his two most popular novels: Project Hail Mary and The Martian. (I admit that I haven’t read his other novel, Artemis, yet.)

In The Martian, he places a lot of emphasis on Martian storms threatening to blow the MAV over as the reason why the crew needs to leave the surface in a hurry, abandoning Mark Watney on the surface. NASA (the real one, not the one in the book/movie) pointed out at the time that the Martian atmosphere is so thin that a 60 mph wind there would have the force of a 5 mph wind on Earth. Now, I can imagine other great reasons why the crew might have to leave in a hurry. That their getaway vessel is at risk of tipping over (and apparently has no way to stow itself in case of storms) seems awfully weak, given the clearly enormous amount of money and effort spent on the series of Mars missions. The fact that it is also physically unlikely makes it a bit confusing.

In Project Hail Mary, the problem with astrophage is that Weir wants it to be composed only of simple elements. If astrophage needed heavy elements to reproduce, it would be too easy to stop. Heavy elements are rare and tend to be buried in the crusts of rocky planets, produced from ancient supernovae. If all it needs to reproduce is CO2 and hydrogen, then it is much harder to stop. Nevertheless, Weir needs some explanation for how it can store so much energy that each microscopic astrophage can literally propel itself from the Sun to Venus and back again.

Neutrinos seem like a good answer until you actually look at how neutrinos behave.

The idea is, of course, clever in some ways. Neutrinos don’t interact, so they can carry huge amounts of energy without damaging the host organism. Neutrinos only interact via the weak force, which is related to radioactive decay. This means no heating, no structural damage, and no radiation.

Neutrinos can carry sufficient energy in the MeV per particle regime, so that’s another point in their favor.

And there is no need for electromagnetic confinement that you would need with antimatter, because neutrinos have no charge and again don’t interact.

The downside is a big one, however: you can’t confine neutrinos. There is no mirror, trap, or container for them. Neutrino detectors on Earth, like Ice Cube at the South Pole, depend on capturing just a few interactions out of the countless numbers streaming from the Sun.

Suppose you could store them. You run into another problem. You can’t get the energy back because they don’t interact with anything. Neutrinos are hugely reluctant to give up their energy.

Andy Weir waves away the containment issue (maybe physics we don’t know about?) and tries to solve the energy release issue by assuming that neutrinos are their own antiparticle. When a massive particle is its own antiparticle, we call it a Majorana fermion after a guy named Ettore Majorana, who, by the way, disappeared without a trace (likely drowned in a lake in Switzerland).

This turns out not to work either because neutrinos would have just as much trouble interacting with their antiparticles as they would with other matter. Thus, the whole advantage of neutrinos (being safe for biological lifeforms because they don’t interact) makes them impossible as a medium for extracting useful energy. Neutrinos will simply take their energy and spread it through the cosmos for the most part, never being absorbed. Astrophage would have to rely entirely on the weak force to extract energy.

This means that neutrinos have none of the required attributes to be an energy storage and propulsion mechanism, other than the ability to carry a great deal of energy per particle. They cannot be contained, you cannot recover energy from them with the required efficiency, their interaction strength is weak, and there is absolutely no physical experiment that remotely resembles what astrophage is supposedly doing with them.

Mind you, I am open to all kinds of stretches in science fiction. I’m a big Star Trek fan after all, but the problem goes deeper than can be addressed by saying “maybe there is physics we don’t know about in play”. Rather, the book contains an explanation that is based on a contradiction: astrophage wants neutrinos that both don’t interact with things (so they can have a lot of energy but not damage the biology) and do interact with things (so they can be contained and their energy released on demand). These cannot both be true at the same time.

So, how do we, within the intersection of science fiction and fact, rectify this problem? My theory is that astrophage, if one wanted to create it, would not use neutrinos. It also would not use uranium or plutonium, nor would it make antimatter. There are lesser-known mechanisms that work better.

Let’s look, first, at the what astrophage’s requirements are: it needs to store a lot of energy in a stable configuration where it does not randomly explode (although it does explode in the book and movie, killing the science officers intended for the mission [why would you have your main and backup science officers anywhere near each other if this mission was so important, Stratt?], the explosion is triggered accidentally by the science team). It also needs to release the energy slowly in the infrared, not gamma rays, which might be more typical of uncontrolled reactions.

Thus, astrophage sits in a regime that does not occur naturally:

• Nuclear scale energy density

• Chemical-like stability and reversibility

• Low-radiation release

• Biological self-repair

Only one thing that I could find in the literature could come anywhere near satisfying all of these:

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With the death of Paul Ehrlich, author of the 1968 bestseller The Population Bomb, we have seen firsthand how alarmism based on science can lead to government overreach.

Even 50 years later, Ehrlich was still sounding the alarm, despite all his predictions failing. The New York Times claimed his predictions were merely “premature,” but no. As the libertarian Reason Magazine astutely points out, he was just plain wrong, dubbing him “History’s Wrongest Man”. The world, far from declining, showed that both population and standard of living could increase through innovation.

Thanks to Ehrlich and people like him, billions of US dollars flowed into controlling, not so much our own population (although there was some of that), but the population of nations where people were deemed to be less valuable. (Indeed, the population control and eugenics movements are connected.) Through the Office of Population set up in USAID in 1966, the United States government funded contraception, sterilization, and abortion in the developing world. Speaking to the St. Louis Post-Dispatch in 1977, its founder, Dr. Reimert Thorolf Ravenholt, argued that

If sterilization services were made readily available world-wide… about one quarter of the world’s… fertile women… would accept sterilization.

The article summarized this as a fertility target, although Ravenholt later claimed this was an uptake estimate and not a target. In testimony to the US Congress later that year, he claimed to have been misrepresented, and the US Government denied having sterilization targets.

That said, Ravenholt clearly believed in the false premise that there was a fixed-size pie when he said, “resources divided by population equals well-being… we’re trying to lower the denominator.” This included exporting sterilization programs by training foreign doctors. All of this was ostensibly in support of US commercial interests, not any humanitarian reason. These policies led to the sterilization of millions of women across the developing world.

Although there is no evidence that Ehrlich directly influenced China’s disastrous one-child policy, Song Jian and his colleagues in the 1970s applied systems engineering and control theory models to argue that China could not hit certain economic targets without strict fertility controls. For this work, they relied heavily on The Limits of Growth (1972) by the Club of Rome, a global think tank founded in 1968, at the same time Ehrlich was reaping the rewards of population panic. This work led to Soviet-style top-down planning and potentially millions of deaths of female babies in a silent holocaust.

The Limits of Growth is still cited as predicting catastrophic collapse by left-leaning newspapers like The Guardian, but it actually suggested several possible outcomes depending on hidden factors. Its system dynamics model looked at interactions between population, industrial output, food production, pollution, and nonrenewable resources to create several possible futures. The most famous of these is the business-as-usual scenario, and subsequent studies by Graham Turner in 2008 and 2014 and by Gaya Herrington in 2020 show that the global population is following that model quite well. Both show growth slowing and potential decline later this century.

The reality, however, is that even The Limits of Growth failed to model scarcity accurately. So-called “fixed” resources have increased because of discoveries, fluctuating prices, and technological substitution. Technological efficiency was underestimated, and the effects of the computer revolution were underappreciated. Collapse predictions failed to account for human ingenuity.

Even in recent years, this Malthusian argument that there is a fixed-size pie and that human growth is going to use it all up sells books like Jared Diamond’s Collapse. Although the book, intended for popular consumption, popularized many of the systemic reasons societies collapse, its commitment to environmental determinism, that the environment is the primary constraint on society, is misguided. Much of the scholarship in the book is overly simplified, and historical claims, such as Easter Island society falling apart because of deforestation, are contested.

Remarkably, he makes this environmental argument in one book, while in his Pulitzer Prize-winning book, Guns, Germs, and Steel, about the conquest of the Americas, he puts two critical human innovations, guns and steel, in its title as major causes! It is not as if these technologies just grew on trees.

Although there are many older people now who have no grandchildren because they chose not to reproduce in the face of this panic, who may now regret their decision as the world has shown it can increase population and standard of living at the same time, that was at least their own choice. If people choose to install solar panels, drive electric cars, buy energy-efficient appliances, or any of the many other ways that may (or may not) reduce dependency on fossil fuels, that is their choice, and they are free to make it. When it comes to government coercion, alarmism is no excuse.

Yet, calls to government overreach have proliferated on the left (while outright denial has become the norm on the right). And many of the same kinds of arguments are being made for controlling CO2 that were made for controlling population back in the 1970s, based on the assumption that the world was heading towards collapse. Now as then, we underestimate human ingenuity at our peril.

That is not to say that there aren’t good reasons to control pollution. Only an idiot (or someone who stood to benefit) would point to a factory dumping waste into a river and claim that it is their sovereign right. The problem comes when governments make laws to control people based on perceived future catastrophes.

Pollution isn’t something that we have to worry about happening sometime in the future, after all. It is something we can observe affecting people in the here and now.

And let’s be clear, the models show catastrophic effects from climate change if things continue as they are. But is that not what models showed back in the 1970s from population increase? Catastrophe?

Could it be that we are just selling ourselves short? It turns out to be complicated.

But let’s look at the two and compare them:

Firstly, both the population scare of the 1970s and the climate scare of today rely heavily on models that predict nonlinear growth. That nonlinearity fed the urgency of the problem because it was seen as exponential, and by the time the problem became obvious, it would be too late. A linear problem is easy to solve linearly, so the solution can be delayed as long as it can be scaled, but an exponential problem has to be nipped in the bud. Thus, it could justify major outlays of spending and draconian measures to solve a perceived but not realized problem.

These models are necessarily simplifying complex systems. They depend heavily on assumptions, and worst-case scenarios are often the ones that make the news, so people perceive the problem as being worse than it likely is. This is definitely true of both.

Both are based on the Malthusian intuition: growth in demand outstripping supply capacity. In the 1970s, this was about food and resources. Now it is about atmospheric capacity to absorb CO2.

Both debates have been used to justify coercive measures such as China’s one-child policy in the ‘70’s and more recently heavy-handed government quotas, like for EVs, and narrow incentives, like for producing biofuels.

The more existential the risk appears, the more likely governments will deploy strong interventions. Widespread famine was the risk with population growth. Climate change includes sea-level rise, desertification, extreme heat waves, water shortages, and potentially other extreme weather. A lot of the effects are debatable, and certainly not as clear-cut as a shortage of food, but clearly, warming cannot go on forever with only mild effects.

There are some differences. Resource scarcity was mitigated by technological substitution, discovery, and efficiency, but CO2 accumulation is just basic physics.

The effects of population alarmism, such as widespread famines in the 70s and 80s didn’t happen. Food production increased dramatically instead, in what is dubbed the Green Revolution.

Climate projections, on the other hand, in terms of temperatures, sea level, and heat events, have been largely on track. We aren’t seeing predictions just not happen.

In the case of the population crisis, innovation decisively solved the problem, but it isn’t clear in the climate crisis that innovation is happening fast enough. Renewables, storage, nuclear, carbon capture, and carbon-neutral fuels have to replace an entrenched global infrastructure dependent on fossil fuels. There are also feedback loops. Embracing some technologies can backfire, causing more problems than they solve, which is why government incentives are so dangerous. Meanwhile, CO2 emissions will persist for centuries. Even zero emissions won’t immediately solve the problem.

CO2 is a global phenomenon, whereas population growth was always local, even in a global economy. China’s CO2 output will affect the future weather in the United States, no matter what policies are put in place. The crisis requires global cooperation.

This suggests that any solutions to the climate crisis will at best offset it and mitigate its risks because the underlying system is constrained by the thermodynamics of the global climate. Innovation does not have the time or capacity to completely offset it, like the population crisis.

Another point is that whereas population control focused on deeply personal decisions, such as whether to have children and how many, emissions control is almost exclusively focused on sources of energy. I think most people don’t care if their car runs on fossil fuel or some carbon-neutral energy source as long as it works the way they like at a reasonable price.

Perhaps the biggest difference between the population crisis and climate change is the role of the free market. Food as a commodity is subject to market forces. As food becomes scarce, the market innovates to create new, cheaper sources of food. This helps undercut the expensive sources of food and grow market share. For hungry populations and food producers, this is a win-win.

Groceries may seem expensive now, but in 1968, people on a median income spent about 15% of their income on food, whereas in 2023, it was only about 10%. (And by comparison, in 1903, when records started, it was a whopping 42%.) Thus, food is significantly more affordable now for the average American. Mostly, this abundance came from technologies such as mechanized farming, transport, and refrigeration, as well as better agricultural practices.

Markets are good at overcoming resource constraint problems and outpace centralized planning. Food production increases from the green revolution solved the population problem long before sterilization programs would ever have made a dent.

The problem with fossil fuel usage is that there is no resource constraint, at least not until fossil fuels start running out. Emitting CO2 (like emitting any other pollutant) is underpriced as long as it doesn’t affect the emitter in the here and now. Therefore, markets will not work to mitigate it until it is priced in. That is why most economists have argued in favor of carbon taxes and cap-and-trade policies. These would effectively nudge the market in the right direction without more coercive measures.

In fact, command-and-control style regulation is far less likely to help mitigate CO2 emissions than economic policies that enforce a cost on emissions or an incentive for lowering emissions. If governments, like the state government of California, mandate quotas for cars, phase-outs, and bans, automakers will just work to meet the target. There is no incentive to exceed it. But an emissions tax is broad, and businesses can juggle those taxes amid different aspects of the business. Thus, they can optimize the policy in a way that is best for themselves without centralized planners guessing what path they ought to take.

Regulators will never know a firm’s business better than the firm, so their guesses as to what is best will always fall short. Rigid targets will just encourage gaming and metric optimization at the expense of real benefits.

A good example of where a coercive climate policy backfired is biofuel mandates. The U.S. renewable fuel standard and similar EU polices required ethanol/biodiesel blending. These policies backfired as more and more farmers started growing corn and soy for fuel instead of leaving land fallow as grassland or growing food crops. Land that could have been used for food was being burned up to meet a policy. Fertilizer runoff increased as well, ironically causing more pollution.

The net emissions benefit, meanwhile, was small or even negative when land use was taken into account.

This kind of policy has all the hallmarks of top-down planning:

1. The government picked a technology to support instead of letting the market choose.

2. It created a market of compliance rather than innovation (meet this target and get a benefit; exceeding it or innovating around it gets no benefit).

3. It chose a path that seemed beneficial on its face, but actually failed to meet its intended goal.

Other examples include Germany’s nuclear phase-out and focus on renewables, which led to more coal burning for stability, high electricity prices, and slow emissions reductions. Also, California’s Zero Emission Vehicle mandate of the ‘90s, which forced automakers to produce a certain quota of ZEVs. They did so. Nobody bought them.

Now, let’s look at how the market has been used to solve pollution problems.

The United States established, in 1990, a cap-and-trade scheme for acid rain (sulfur dioxide). This essentially told businesses that there was a cap on sulfur dioxide emissions, but they could trade surpluses to other businesses so they could produce more. This created a robust trading system that reduced emissions faster and more cheaply than expected. Compliance costs were lower than predicted, and there was significant innovation in scrubbers and fuels.

This is the kind of policy that has all the hallmarks of an innovative, market-based solution:

1. Firms, not the government, picked the technologies to support their own businesses.

2. It created a market of innovation as firms sought to decrease their emissions.

3. Firms, not the government, chose the path that was beneficial. The government only chose the overall goal.

Other examples are the British Columbia carbon tax and the (most recent) EU cap-and-trade system.

Without the government, the market would have done nothing about the problem, so laissez-faire is not an option, but neither should the government decide how to solve the problem.

The main downside to these schemes is that they are not deployed globally. This means that businesses that are subject to, for example, cap-and-trade and carbon tax schemes may have issues competing with imports and certainly exporting. Nevertheless, such schemes could be applied to imported goods as well. There are some fragmented international mechanisms for cap-and-trade, but the level of international governance required for a global cap-and-trade system is probably beyond the human species at the moment. At best, there can be voluntary agreements.

In addition, because climate change has become a political football, firms can’t depend on the rules and regulations being the same from decade to decade. This makes it hard to invest in innovation when you don’t know if your product is going to be worth anything in ten years because the collective will of the people has changed its mind.

Quotas, narrow compliance targets, and other such schemes are a mistake that tends not only to be overly coercive but also often fails to have its intended effect. Broad incentives will trump narrow coercion every time, but it takes long-term commitment and global cooperation to make it work.

So, it seems that the population bomb never went off, but the climate crisis still can. How bad that will actually be is up for debate, but as with any pollutant, it is best not to find out what CO2 unchecked will do for the next century.

You’d be forgiven for thinking that I am an expert in philosophy. I am only a humble physicist. And while physics can instruct us on the merits of one philosophy over another, it cannot necessarily point us to the right one. It is rather the reverse. Every physicist, deep down, has a philosopher struggling to get out because, without philosophy, we are merely wranglers of equations, slingers of predictions, and collectors of measurement data. Without philosophy, physics is just shutting up and calculating.

Now, I love a good calculation, but at the end of the day, to understand what that calculation means, I need philosophy.

The philosophy I choose can have vast repercussions on how I interpret findings, what theories I find most appealing, and what direction I want my research to take.

In this way, physics is not that different from theology. But whereas theologians have great respect for philosophy, physicists have a tendency to treat it like a waste of time. This is even though all of the most fundamental concepts in physics: time, space, matter, causation, probability, measurement, observation, and even the concept that physics exists at all are based in philosophy.

For centuries, Western civilization has been based upon the philosophy of the Greeks: Plato and Aristotle. Medieval theology drew life from them. Concepts like substance, potential, essence, form, and ideals formed the backbone of how theologians understood God, the soul, heaven, hell, and humanity’s place in the cosmos. Scientists, likewise, understood time, space, and matter similarly. Objects had substance and form. Even as Galileo and Newton swept aside Aristotle’s physics, they kept his understanding of the universe as made up of objects that had essential natures. These objects related to one another, Newton taught, through forces, all the while an impersonal, absolute time moves everything forward.

In fact, classical physics, as we now call everything before the dawn of quantum mechanics, fits perfectly well into this philosophical framework. As theology marched forward with Calvin and his double predestination, where an absolutely sovereign God chooses who goes to heaven and who goes to hell. Laplace argued that because all the laws of physics are deterministic, the future is likewise determined, and one could, with perfect knowledge, predict all future outcomes.

All this was to come crashing down with the invention of quantum mechanics. Suddenly, it became clear that what underlaid the foundations of the cosmos was not necessarily order and clear direction but indeterminism. Suddenly, physics needed a new philosophy.

Niels Bohr, one of the fathers of quantum mechanics, was deeply fond of the philosophy of Kant. Kant argued against the idea that we could ever really know things. He believed that we could know some things. (It would take Heidegger and finally Wittgenstein to argue we couldn’t know anything.) Therefore, Bohr believed that quantum theory was just a representation of how particles interacted with measuring apparatus. Purely symbolic but useful.

Others, like Heisenberg, wanted to reapply Aristotle, arguing that wavefunctions were similar to Aristotelian potentiality. Aristotle had this idea about potential things transmuting into actual things, and to Heisenberg, this was very similar to what was happening in quantum mechanics. A wavefunction represented potential things, and when a measurement was made that potentiality was converted into the actual reality, the measurement made. Nevertheless, Heisenberg kept the idea of substances meaning objects moving through time and space. The only difference now was that those objects embodied potentiality before measurement and actuality afterward. They still retained their identity. When the wavefunction has not yet been measured, the particle does not exist. It only potentially exists.

This is not unlike a young person growing up. They are full of potential; no one knows what they are going to be. As they grow, they become more and more actualized (if they are lucky), and that potential becomes reality.

Albert Einstein opposed the interpretations that Bohr and Heisenberg were proposing. He believed that quantum mechanics was fundamentally incomplete and that a truer theory would be discovered that would accord better with how he viewed the universe. Einstein saw the universe as largely fixed, like a book that had already been written. We just happened to be reading it one page at a time. Einstein’s concept of God was as a Being who had written this book all out, including all the laws governing its pages. This is not that different from ancient and medieval philosophy, such as the timeless, changeless God of the Roman philosopher Boethius and the medieval friar Saint Thomas Aquinas, who stood outside time, watching all of history simultaneously. This God never changed His immutable mind. He had a feeling well up in his impassible heart. God does not suffer. This is what the church has believed since almost the beginning.

In Aristotelian terms, God cannot have potential. He is entirely actualized.

Life would be simple if quantum mechanics were all we had to contend with. But quantum mechanics is merely a special case of quantum field theory. In quantum field theory, the particles that are so important to quantum mechanics are mere excitations of the field, like a toddler jumping on a springy bed. They are here today and gone in the blink of an eye. In a way, fields are another kind of potentiality. Sometimes particles emerge from fields that we can detect. When we can’t, we can still detect fields influencing particles that we can detect, like magnetic fields drawing iron filings to them.

The particle is like the actualization of the field, but the field is not nothing either. Fields have many, many particle interactions inside them. Most of these are virtual interactions, meaning that the particles aren’t quite real. Yet, they still have a measurable effect.

In 1948, a way of understanding field theory was introduced by a Ph.D. student who had suspended his studies during WWII to help with the Manhattan Project.

He wanted, in a way, to turn field theory, which was very abstract at the time, into something more tangible, more like classical physics, where you could picture what was going on. We can picture things like cannonballs rounding the Earth even if we have never seen it, like this one drawn by Sir Isaac Newton himself:

But how can we make sense of an infinite-dimensional summation?

His solution was to turn those summations into diagrams that picture what’s going on in the field theory, the sort of thing that might happen in a particle accelerator or even a nuclear explosion.

This is one of the diagrams that he published in 1949:

It shows two electrons, the straight lines, exchanging a virtual photon, the squiggly line, a common interaction that causes the electrons to, for example, repel one another.

Here are three more examples of these diagrams. These are all from a process, like the one above, called Compton scattering:

What’s interesting about these diagrams is that all the external lines are the same, yet each diagram is different. In fact, the external lines aren’t the most important part. Unlike the diagram for the trajectory of a cannonball, they don’t go anywhere specific or come from anywhere we care about. The placement of each intersection of lines is, likewise, unimportant. Like subway maps, these diagrams don’t necessarily tell you anything about where particles go in physical space and time, nor do they tell you where they come from. What they do tell you is the stops they make along the way. For example, one diagram, on the left, tells you that two electrons exchanged a photon and continued on their way. The middle one says that they exchanged two photons. The right one says they exchanged a photon, which turned into an electron-positron pair, which annihilated and turned back into a photon.

In this sense, these diagrams do not describe objects because objects are not stable in field theory. They transmute into other things and back again. Instead, they describe events. They tell you what happened.

That Ph.D. student, whose name was Richard Feynman, went on to make many more discoveries, but perhaps none so philosophically profound as this pictorial way of representing what field theory represents: the primacy of events over objects.

Events, after all, are part of the fabric of spacetime in a way objects aren’t. Events are what happen while objects are impermanent and in flux.

A philosophy that better represents field theory is not one of potential and actualization of things, but one where events are the primary content of the universe, and objects are merely strings of such events that maintain some similarity to one another. Consider that we only perceive the electron when we observe it, and to observe something, it has to, in field theory parlance, scatter off of something. In other words, we never see the lines in the diagrams. We only see the vertices.

Process philosophers such as Alfred North Whitehead argue that this is a better way of understanding the world and, indeed, God as well. Events are fundamental. Particles don’t exist with events happening to them. Rather, events are all there is.

In his book Process and Reality, Whitehead argues that the universe is made up of events called actual occasions. These are momentary events of becoming. Thus, all things are always becoming, not being, even you and me. Every event integrates influences from the past and actualizes one of many possibilities. These possibilities now become part of the past. Each of these events is an example of the universe experiencing itself and producing novelty. For Whitehead, the future has not been written yet, and we are the genuine authors of our own experience, given the chance to write our own stories.

Whitehead’s slogan was that the world was composed of “drops of experience” and is considered a panexperientialist. All matter is capable of some kind of experience, even atoms. (This might be considered a generalization of panpsychism, the idea that all matter has some experience of consciousness. Experience is somewhat ill-defined but may not be what we think of as conscious experience.)

Thus, atoms and we are not beings but becoming as patterns of experience, and if field theory is any guide, this might fit best with modern physics.

One of the realities that quantum physicists must ignore most of the time is that all our descriptions of quantum reality focus on possibilities and their probabilities for becoming real. Nature, however, has a poorly understood mechanism for transforming possibility, embodied by the quantum wavefunction, into actuality. Depending on what this mechanism is, process philosophy might be a very good or a very poor description of reality.

Essential to process philosophy is this concept of events, which turn possibility into actuality. In the case of human beings, we call this free will. Reality is indeterminant. The future is being written by matter all the way from electrons up to human beings and beyond.

If, however, no actualizing is happening and the universe is entirely deterministic and predictable, then process philosophy is a poor fit. There is no experience happening, no actualization. Rather, the universe is simply unfolding in a way that Einstein understood well.

This is why process theory is much friendlier to interpretations of quantum theory where wavefunctions collapse. Recall that wavefunctions represent what is possible. Therefore, the collapse of the wavefunction into a single reality is, in essence, a conversion of possibility into actuality. Many physicists over the years have proposed that collapse happens when human beings observe particles, but that need not be the case. After all, the only reason we can observe particles is by using detectors, which are large, complicated objects made of many, many particles. Perhaps it is these and not human minds that cause the collapse.

A theory called the Ghirardi-Rimini-Weber (GRW) theory, for example, proposes a physical process by which wavefunctions spontaneously collapse, especially when interacting with macroscopic systems. Thus, the universe is constantly experiencing wavefunction collapses. One formulation, developed by Roderich Tumulka and others, explains this in terms of flash ontology. Fundamental entities in the universe are these “flashes,” and each flash is an event in space and time. The wavefunction determines the probabilities of these flashes. Thus, reality is a network of these flashes all connected, and what we think of as matter is simply patterns of flashes, like dashes and dots over a telegraph wire.

While in the flash ontology, the flashes are purely physical, in a Whiteheadian metaphysics, each flash contains experience and creative self-determination. Thus, the flashes are not merely collapsing wavefunctions but reality inventing itself to experience itself.

What kind of God invents a universe like this?

John B. Cobb Jr. extended Whitehead’s metaphysics into process theology, where God does not control events deterministically. Rather, God offers possibilities or “lures” towards greater value. Human beings have free will to either move towards these lures or away from them through the self-determining nature of events. Each event inherits influences from its past, considers available possibilities, and determines its own final form. This self-determination requires not human beings to exist, but rather some patterns of events are humans becoming. Events link to future events in an ever-connected network of creativity.

Thus, human free will is not an exception to the determinism of nature but a higher form of the same freedom that all matter enjoys.

Unlike in classical theology, God in process theology is seen as actively participating in the universe’s unfolding. Because of the freedom conferred upon all events and all matter, God may not know the final form that each event will choose. God knows as much as is possible to know, but has given the universe the ability to surprise him. Divine power does not determine outcomes but rather persuades. God’s purpose for the universe is emergent rather than imposed.

According to Cobb, God is dipolar, having a primordial pole that generates the possibilities of the universe and a consequent pole that allows him to experience the world along with us. The future is genuinely open, and God may change with the world.

Ancient and medieval theologians, steeped in Plato and Aristotle, believed that God had to be unchanging (immutable) and unaffected by emotions (impassible) as well as outside of time (timeless) because otherwise, if he could change or be affected in any way, he would be less than perfect. In fact, these ideas come straight from the pagan Greeks rather than from any Biblical doctrine. Process theologians, however, reject this on its face, pointing to many instances in the Bible where God changes his mind or expresses emotion (e.g., Genesis 6:6, Exodus 32:11). For process philosophy, all things are becoming, and thus, to be perfect, a being must become perfectly. This is God’s nature, then: God relates to all creation perfectly by responding to all creation perfectly. God truly suffers, shares in the consequences of human actions, and works persuasively towards his greater good. He has the greatest possible capacity for relationships. While God does have unchanging aspects such as his character, his experience does change.

Some process theologians like Cobb steer clear of making definite statements about the nature of Jesus Christ. Although they reject Aristotelian concepts of substance, which makes the Trinity or the nature of Christ difficult to support in its classic form, they seem influenced by modernist, liberal theologies in downplaying the true nature of Christ as divine and of his resurrection as a genuine miracle. They instead focus on drawing “meaning” from these stories symbolically.

None of this is necessarily a consequence of process philosophy itself, but rather a direction that these theologians have chosen, which I think is wrong. Other process theologians, however, like David Ray Griffin, have argued that miracles are entirely consistent with process philosophy and argued for the resurrection as a genuine event, while Marjorie Suchocki has argued for the real divine presence in Jesus and not merely a symbolic presence.

Fundamentally, the Trinity, I think, has to be interpreted relationally, as love shared in the midst of God. Christ’s nature, then, flows from that relationship as the second person of the Trinity. Meanwhile, C. S. Lewis has this to say about miracles:

It is therefore inaccurate to define a miracle as something that breaks the laws of Nature. It doesn’t…If God annihilates or creates or deflects a unit of matter He has created a new situation at that point. Immediately all Nature domiciles this new situation, makes it at home in her realm, adapts all other events to it. (CS Lewis in Miracles, p. 94)

Thus, miracles are part of God’s own creative freedom. Any moment allows for the possibility of a miracle, even ones as shocking as the incarnation and the resurrection.

As we discover more about the universe, it seems increasingly clear that change is the only constant. Quantum field theory and quantum mechanics, in general, suggest that events that actualize possibilities into reality are a natural way to understand the universe. Will our understanding of God keep up or remain in the Middle Ages? It seems that we have a difficult tightrope to walk between updating our philosophy without throwing out everything in favor of a lukewarm scientism.

Last year, Anthropic, the maker of Claude, added “extended thinking” in Claude’s 3.7 release. The idea is that, by letting Claude mull things over longer, it would be able to solve “complex physics problems”.

https://www.rdworldonline.com/anthropic-brings-extended-thinking-to-claude-which-can-solves-complex-physics-problems-with-96-5-accuracy/

The physics problems, however, that Claude was able to solve were not research problems. They were “graduate level” benchmarks.

This article talks about the drawbacks:

https://www.rdworldonline.com/eureka-2-0-ai-is-beginning-to-ace-grad-level-science-but-can-you-trust-it/

Problems that are constructed to test knowledge are inherently different than research problems. Because research is novel, Claude can no longer rely on reasoning already presented in existing research and textbooks.

I notice that when using Claude (I have been using Sonnet 4.5), I have to remind it multiple times of the assumptions I’m making that differ from standard theory. It often falls back into standard reasoning, although it appears to “learn” not to do it as often once it has been reminded a few times. It is sometimes hard to tell if it is trying to smuggle these assumptions in other ways.

I am working on emergent quantum mechanics right now, so I keep having to remind Claude not to assume quantum mechanics to prove that quantum mechanics emerges. After I admonished it a few times, it did it less often.

The biggest problems with Claude are that:

1. It makes claims without proof frequently. It will simply make a bunch of superficial “checks” and then give itself marks. Often, these checks are actually circular reasoning or just another way of calculating the same result that has already calculated. It will give me 10-15 steps to try to establish some result, none of which is fleshed out that well. The superficial checks often hide a very big conceptual error, which leads to the next issue.

2. It fails to establish results that are essential to the claims it is making. Oddly enough, using the Stanford Agentic Peer Reviewer is pretty good at finding these issues, and I can go back and feed Claude its review. Unfortunately, this tends to cause Claude to drop the reasoning it used previously completely and find a different track. This is not unlike a human researcher, but it shouldn’t need a peer review to do this. It needs to proceed more carefully. In this sense, it reminds me of a very smart, enthusiastic 1st year grad student who just doesn’t have the experience to know how to do things the right way. For example, in my case, Claude assumed it could apply ideas from thermal field theory, which relies on a compact time dimension, to a thermal field theory with a compact spatial dimension. The Peer Reviewer pointed out that this is not ok and doesn’t work. Claude, when confronted with this information, admitted it didn’t work and came up with a completely different idea.

3. The other annoying thing it does is play fast and loose with units. It loves to say one quantity is “similar” to another without establishing equality. This may be ok for brainstorming sessions, but it has to be done very carefully in research because often the constants and things you leave out matter, e.g., you have to establish dimensional equivalence. You can ask it to add units back in, and it will establish dimensional equivalence, but not necessarily with good reasoning. It will just make use of whatever constants are available, often providing more than one approach with completely different reasoning.

4. It has trouble tying its ideas back to specific papers (although it is capable of looking at papers, even ones it wasn’t trained on), and it is aware of what the main papers and books are. It can’t tell me that this formula can be found in this paper. It can’t tell me that this chain of reasoning is available in this other paper. This means I have to go scouring through the literature myself, trying to figure out where a particular formula or idea comes from to make sure it wasn’t hallucinated. Unfortunately, this is essential to doing good research. You have to establish your ideas in the literature.

5. Although it has trouble straying from conventional thinking, unlike ChatGPT, it sometimes offers some very unconventional ideas that are usually conceptually flawed. For example, it came up with a very novel, but very wrong interpretation of quantum spin.

Best aspects of Claude

1. It can figure out its mistakes and backtrack, finding a new way to go. I think this is because it has some kind of math server in the backend that it uses to carry out calculations. This at least means it gets its math right. This is a great example of neurosymbolic cooperation.

2. It brings a lot of knowledge together and makes suggestions that I would never have thought of.

3. It is good at discussing the advantages and disadvantages of existing models, which is great for introductions. In this case, it often can point to specific papers.

4. It can answer questions about established physics very well, even for very niche topics. This has always been an issue with lit reviews. It’s hard to find commentary on papers that have been published. Not that they don’t exist. They are often buried in papers about alternative ideas.

Based on my experience over the last couple of months using Claude to do my research. Here is my guide:

How best to use Claude to do physics research:

Don’ts:

• Don’t have Claude do your research for you. It will do it badly. You have to lead it along. Think of it as a 1st year grad student. Bright and knowledgeable but also deeply inexperienced.

• Don’t ask Claude open-ended questions, hoping it will establish your results for you. It will try to do what you ask by making a bunch of unfounded assumptions. It is very much a people pleaser.

• Don’t ask Claude to write your paper for you. It will write perfectly good academese, but you will look like an idiot when it fails to establish its claims and makes basic conceptual errors.

Dos

• Do ask Claude to help with lit reviews, list established results in a particular area, and explain their advantages and disadvantages.

• Do ask Claude to make suggestions about models or existing paradigms that might help solve your problem.

• Do ask Claude to generate code to simulate or calculate results if you need it.

Most importantly, recognize that a good researcher proceeds carefully from established results to novel results, step by step. Every claim has to be supported and fleshed out. You have to know what needs to be established, and you have to couch all your work in the existing literature. That is why experience matters. Intuitive leaps are essential to pick a direction, but then you have to do the work. Claude can be a secretary and assistant, but not a co-researcher.

I also think this is a warning to people who want to do physics research but don’t have the math skills. Claude will give you enough rope to hang yourself with. You need to have the math skills to recognize when Claude is giving you BS.

Niels Bohr, one of the fathers of quantum mechanics, upon being confronted with the equations for the new quantum mechanics, argued that they could only be symbolic and not represent anything real. Part of his reasoning was that Schrodinger’s equation equated momentum with an imaginary quantity, meaning a number times the square root of negative one.

Bohr’s opinion is surprising given that in classical electrodynamics, one can represent waves very conveniently as complex numbers, meaning a real number added to an imaginary one. This is because waves, which are solutions to Maxwell’s equations, consist of three quantities in relation to one another: amplitude, frequency, and phase. If you represent a complex number in polar coordinates within the complex plane, the magnitude is the amplitude, while the phase appears as the direction. Frequency is simply how quickly that direction moves in angle as it sweeps around the plane, like a line on a radar screen.

If you want to compare two waves, it is easy to do so by looking at how they move in the complex plane in this way. Thus, one should not be confused by the word “imaginary” when talking about these numbers. They are anything but.

In fact, any kind of digital signal processor will, typically, convert signals it receives into imaginary numbers. It does this by sampling the signal it receives twice for every wavelength. The first sample is the real part, and the second sample is the imaginary. Thus, the physically measurable, an electromagnetic wave, is converted into imaginary numbers.

Was Bohr ignorant of these facts?

Not at all.

Read more

You might be forgiven for dismissing brain lateralism, the concept that either the left and right hemispheres of the brain have different functions or worse that there are left-brained and right-brained people, as pseudoscience. Indeed, there is little evidence for the pop-culture concept of the right brain as artistic and poetic and the left brain as scientific and logical. Yet, thanks to decades of research, especially on people who have had one or the other side damaged, it has become clear that the left and right brains do have subtle yet importantly distinct roles.

As psychiatrist and author Iain McGilchrist showed in his masterful work The Master and His Emissary, the left brain is primarily for breaking the world apart, reducing it to its components, and understanding mechanism. The right brain is holistic; it integrates what the left brain discovers and derives meaning from it. As he says: “the right hemisphere helps us understand the world, the left only to manipulate.”

The idea that there are left and right-brained people is unfortunately a fallacy. There is no hemispherical distinction between those in STEM fields and those in the humanities for instance. In fact, there are left and right-brained people, but they are either brain-damaged or have psychiatric conditions that are essentially as bad. Schizophrenics, for example, are profoundly lacking in right-brain functions, unable to integrate or find meaning from the world around them. Sentences become strings of words without overall meaning. Meanwhile, damage to the left side primarily affects speech and use of the right hand but not our overall understanding of the world and our sense of self.

Quantum mechanics, you may think, is very much a left-brained activity, the ultimate reductionism of the universe to its smallest components. Yet, its mysterious nature would not worry us were it not for our right-hemispheric desire to understand what it means. We would happily “shut up and calculate”, and that has been, more or less, what scientists have done for the past 100 years. Nevermind the 1000s of papers on quantum interpretation. Interpretations don’t win Nobel Prizes.

McGilchrist, ever the psychiatrist, offers a diagnosis for society as a whole that I think applies especially to the physics community: society has, over the years, become increasingly dominated by left-brain thinking. The tendency to replace the human element with abstract rules perhaps started in Greece with the ancient philosophers and was perfected in Rome with its devotion to law and order. In the modern era, it has increasingly been visible in our architecture, with all straight lines and abstract shapes. A society bloated with bureaucracy, obsessed with control, and prone to paranoia is exactly what we see in many modern societies. This is the left-brained utopia: like Camazotz in A Wrinkle in Time. Everyone is doing the same thing at the same time. Total conformity.

Some in the physics community believe that interpretation should serve calculation, that our job is not to explain the universe but to merely weave mathematical models. Worse are those who believe their job is merely to publish papers, serve on committees, apply for grants, and organise conferences and symposia. They serve the bureaucracy of physics rather than the goal.

This article isn’t an indictment of them. We all have to feed that beast after all. What I’d rather talk about is what I believe is the right approach to quantum mechanics, one that follows McGilchrist’s argument that the right brain the master, the one that truly “groks” the theory, while the left is merely the emissary, going out and gathering details, like how to solve Schrödinger’s equation, and processing them for the right.

McGilchrist’s argument is that scientific thinking is far more “mystical” than we would like to believe. Intuition and imagination are what move science forward, not analysis. People who are very analytical and like everything laid out one step after another, who either don’t want or aren’t able to make intuitive leaps, tend to get frustrated and leave research for something more comfortable, like insurance, accounting, or information technology. I have seen this happen at least three times, where very smart people left research for the more mundane industries. They never seemed, I thought, to recognise either why they were leaving, at least they didn’t express it, perhaps because that felt like admitting failure.

McGilchrist mentions several examples:

After much cogitation, Kekulé seized the shape of the benzene ring, the foundation of organic chemistry, when the image of a snake biting its tail arose from the embers of his fire; Poincaré, having spent 15 days trying to disprove Fuchsian functions, suddenly saw their reality, as, after a cup of black coffee, “ideas rose in crowds—I felt them collide until pairs interlocked”; later their relation to non-Euclidean geometry occurred to him at the moment he put his foot on a bus, though he was in the middle of a completely unrelated conversation (“on my return to Caen, for conscience’s sake I verified the result at my leisure”). The structure of the periodic table of the elements came to Mendeleyev in a dream.

I myself saw in a flash of insight (perhaps a bit less dramatic than the stories above) that quantum field theory could be explained by classical field theory in five dimensions at the age of 28. I have been working on this idea for 17 years, and only now, at age 45, has my left brain managed to work it out.

These stories of profound intuition yielding insights about the natural world might suggest that the right brain is mainly good for intuition, but meaning, integration, and purpose all happen on that side of the brain.

How then can we understand something like quantum mechanics, which has no clear, unified interpretation from the right side of the brain?

I think it is important to recognize that interpreting certain weird quantum effects using some popular framework like Copenhagen, Many Worlds, or Superdeterminism is not right-brain thinking. This is like applying a popular methodology for interpreting a work of art or literature. It is like reading a book on theology to understand who God is. That is still left-brain work.

Music cannot be appreciated by understanding the tricks composers use to trigger certain emotional responses. One must feel the emotions by integrating those techniques.

Thus, what the right brain allows us to do is to turn an objective understanding of an experience into a subjective experience of it. It transforms the explanation of the miracle into the miracle itself.

This is why, in physics (and all of science), it is important to experience what Newton called the “experimentum crucis,” or crucial experiments. These are experiments which demonstrate a fact about nature not only to the mind but to the heart. You have to move beyond the equations to the natural world itself. This is the difference between reading music and hearing it.

When a real experiment is impossible or infeasible, a thought experiment can be a good substitute. Albert Einstein is known for his thought experiments, such as the warping of triangle geometry on a rotating disk in space, leading to curved geometry, or the falling elevator, which led to the development of his principle of equivalence and ultimately general relativity, a clear indication of right-brain thinking to develop the intuition needed to “send out” the left brain to manage the details.

Even the armchair physicist or curious onlooker can participate in a thought experiment without necessarily needing to visit a physics lab or purchase equipment to perform their own real experiments.

Thus, the thought or gendanken experiment should not be thought of as a mere pedagogical tool, designed to explain something already known through analytical means, but rather a means of engaging the right brain in understanding, intuitively, what the left brain knows only by rote.

The most famous of these is Schrödinger’s cat, in which a cat is placed in a box and, because of the superposition of two quantum states, becomes entangled with quantum particles and is both alive and dead at the same time.

I use this thought experiment pedagogically sometimes, but honestly, it isn’t a very good thought experiment. Schrödinger didn’t intend it to be one. Rather, it was created in 1935 after conversations with Einstein, the notorious quantum theory sceptic, to point out the absurdity of quantum mechanics. Since then, however, most physicists embraced the experiment as a critical demarcation between “intuitive” classical thinking and “nonintuitive” quantum thinking.

In fact, I’m not sure that this thought experiment tells you much about quantum mechanics. All it does is replace the word “particle” with “cat” to make a kind of gedanken joke. Yes, one particle can inherit the superposition of states from another particle.

Another, slightly more interesting and manifestly more serious thought experiment is Wigner’s friend thought experiment, developed by Eugene Wigner in the 1960s.

This is actually a variant of Schrödinger’s cat, but it clears up some of the ambiguity in using a cat as a macroscopic quantum state.

In it, Wigner’s friend is in a room and performing an experiment that involves observing the outcome of some quantum superposition. This can be as simple as the observation of a single photon emitted at a half-silvered mirror and seeing whether it is reflected (outcome A) or passes through to a detector (outcome B). Without observing the photon, it remains in superposition with both A and B being equally likely. This is a pure quantum state, so it is not A or B but more like A and B until observed. Wigner’s friend is completely isolated in the lab from Wigner, who is outside.

When Wigner’s friend performs the experiment, she observes a definite outcome A or B. Now the photon is in one or the other state. The question is: what does Wigner observe?

The reason why this thought experiment is so important is that it explores the concept of observation and whether it is objective and universal or subjective and personal. When Wigner’s friend observes the photon and the photon enters state A or B instead of A and B, does Wigner still consider the friend to be in a state A and B as well, because the observation has been made in the closed-off lab, or does Wigner now consider the friend in a state A or B?

Whether Wigner knows the outcome is irrelevant because in quantum mechanics, a quantum probability, A and B, is very different, being a pure state, than a classical probability A or B, being a mixed state, even if you don’t know what the outcome is.

To put it in other words, before observation, the photon is 100% in the state A and B. After the observation, the photon is 50% state A and 50% state B. These are manifestly different states.

If observation is objective and universal, then by observing the outcome, the state should change for both Wigner’s friend and Wigner. If observation is subjective and personal, then the state will be different for Wigner’s friend and Wigner. Wigner’s friend will observe the mixed state A or B, while for Wigner, both the photon and his friend will be in the pure state A and B until he goes into the lab and finds the answer.

Realistically, this experiment cannot be performed because it is impossible to isolate a person in a lab sufficiently so that they remain in a pure quantum state. Their state would almost immediately decohere. This experiment has been sort of performed using surrogate particles, and the results suggest that observation is subjective, not objective, but that hardly conforms to the original intent, which is to understand if consciousness itself affects quantum outcomes.

Different interpretations of quantum theory give different interpretations of this experiment, but that isn’t really the point. The point is that the thought experiment should give some intuitive understanding of how quantum mechanics works and what is at stake when one interprets it one way or another.

I find this thought experiment slightly better than Schrödinger’s cat, but again, I’m not sure if it answers any questions about quantum theory. If anything, it just raises more questions.

Indeed, many early thought experiments in quantum theory seemed designed to befuddle more than help. Therefore, let’s look at some more recent ones.

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Bohmian mechanics, or as it is more formally known, Bohm-de Broglie Pilot Wave theory, is an interpretation of quantum mechanics developed first by de Broglie in the 1920s and rediscovered and strengthened by David Bohm in the 1950s.

The theory proposes to explain the quantum measurement problem: why is it that we never observe the wave function but only single particles?

The standard explanation for this phenomenon is that when we observe the wavefunction, it collapses into a single, localized particle. What this means is that the particle essentially has no distinct reality apart from the wavefunction until it is observed.

How observation causes this to happen is a mystery, and many proponents of collapse theory want to get away from it to a more objective mechanism for collapse, such as the Penrose interpretation (which I’ll be writing about in a separate article).

De Broglie and Bohm sought to explain the phenomenon by proposing that the wavefunction simply guides the particle to where it is supposed to be, kind of like a ghost. The particle itself is what we measure, and it is always there, whether we measure it or not.

Bohm’s theory, however, has a massive flaw in it, one so basic as to make it untenable as a quantum interpretation. I will get to this flaw below, but first, I want to talk about the implications of Bohm’s theory and some other downsides to it.

Bohm’s formulation actually eliminates the type of action at a distance that Einstein called “spooky” in a letter to Max Born in 1947.

Einstein’s objection was about how the wavefunction “knows” that the particle has been detected, so that it knows not to make the particle appear anywhere else. In Bohm’s formulation, the particle is always only in one place, so when it is detected, it simply ceases to be guided by the wavefunction (or the wavefunction decoheres into the macroscopic structure of the detector).

Bohm’s theory has a downside in that the wavefunction can guide as many particles as you like, and it does so instantaneously, allowing one particle to influence another also instantaneously. Although this does not allow for information to be sent faster than light, it does allow for action at a distance. This is what most physicists and science writers refer to as spooky action at a distance now (not knowing the original context of the quote). It is definitely spookier than what Einstein objected to, which is more correctly encapsulated in the Einstein-Podolsky-Rosen (EPR) paradox.

The EPR paradox simply says that if you have two particles created together, as they move apart, they will maintain a single wavefunction. When one is measured, that measurement will instantaneously affect how the other particle is measured.

Bohm’s mechanics accepted nonlocality as the price to have definite particles (realism).

This, however, is not the massive flaw. It is a philosophical choice. A bad one, in my opinion, but still a viable choice.

Bohm’s theory has another problem. It is a theory about quantum mechanics, and so it addresses trajectories of a single particle quite well. While it makes sense to talk about the trajectory of a point particle moving much slower than light as a velocity, as you add more particles, you have to talk about how that entire configuration moves as a whole. You can’t separate them. That means that instead of having a velocity field in 3D, as you have with one particle, and that resembles a kind of fluid flow, you now have a velocity field in 3N-D for N particles. This is much higher-dimensional and very abstract. This is the price you pay for giving up locality. Because particles are not localized, they cannot be treated as free agents. They have to be treated like a group, even when they are far apart. This is one way the resemblance to classical mechanics breaks down. You can deal with this, and people have, but it becomes cumbersome.

When it is made relativistic, incorporating Einstein’s theory, and used to address things that matter to high-energy physics, such as Quantum Field Theory (QFT), it becomes massively difficult to deal with. Now, instead of having a group of particles, you have a field, which is an extended, infinite-dimensional thing. Again, giving up locality has brought on the need to deal with the entire field as a whole. The fluid analogy remains, but now it becomes infinite-dimensional. Also, in relativity, you have to define the field’s trajectory, which means doing some time slicing. You can do it again here, but the resemblance to classical field theory is gone. You can describe how an electric or magnetic field evolves with time by describing how each point in it evolves separately from all others, because you have locality. That is what keeps the number of dimensions finite. Without locality, everything starts to blow up.

Bringing in gravity is where things really start to break down. Einstein’s theory of gravity doesn’t have a fixed notion of time, so it doesn’t have a concept of trajectory built in. You have to choose your time direction. It also doesn’t have a concept of simultaneity, which means it is exceptionally difficult to define time across the entire universe. Yet, because Bohm’s theory lacks locality, you must do exactly that. The Wheeler-DeWitt equation, which is like the Schrödinger equation for gravity, doesn’t even have time in it. The universe is effectively static. This is likely because time doesn’t really exist as a physical thing in general relativity. Global time is a choice you make, not a thing you measure. Only local time is measurable with clocks.

This is where the conflict between Bohm’s mechanics and physics becomes almost insurmountable. Yet you can do it, and this, even this, is not the fatal flaw. After all, we can’t figure out gravity and quantum theory together anyway, so why put the burden on Bohm to solve it?

No, the fatal flaw actually appears with just one particle moving slowly, the most basic of quantum mechanics.

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By many standards, people would consider me wildly successful. I have a high-paying job. I’ve never been laid off. I earned my Ph.D. at a young age and have several publications in high-ranking physics journals as well as a monograph. I have a wonderful family. I’ve been married for 26 years and have three children. I’ve never even had a serious breakup, never run in with the law, and always had enough to eat.

It would perhaps surprise you to learn, however, that for a long time I was unhappy with my life. In my mind, my failures overshadowed my successes, and my shortcomings, my better qualities.

I compared myself not to my peers but to people I read about who seemed to have earned the successes I lacked and to have the qualities I failed to develop.

All this was through a keen awareness that I had a lot to drive others to envy. Yet, I regarded it as of little importance.

Because I yearned to be someone I was not, to have a life I did not have, I could never be happy. In fact, for a period in my early 30s, I was in despair.

I had been an anxious but also arrogant young man in my late teens and early 20s, full of plans for my future success. I had built a life on imagining my future self: independent, successful, and rising above my peers.

My life unfolded rather differently than I had imagined it would, and, as it did so, I began to feel like I had failed.

I didn’t realize it then, but according to Kierkegaard, I had simply exchanged an unconscious despair for a conscious one.

Kierkegaard was a Danish philosopher, perhaps the greatest to come from the small nation of my ancestors. Born in Copenhagen, where incidentally my own father was born, in 1813, Kierkegaard was the son of a businessman. In the 1840s, he began a writing career that led to a short but prolific career until he died in 1855 at age 42.

Mainly writing under pseudonyms, Kierkegaard explored the nature of life, death, God, the soul, and how to live a good life. A devout Christian, he nevertheless criticized the official Christianity of 19th-century Denmark and developed his own ideas about how to get closer to God and live out one’s faith. He is considered one of the early existentialists and is sometimes called the father of existentialism.

Kierkegaard had a great deal to say about what makes people unhappy or “despair,” which he wrote about in his work Sickness unto Death, as well as other works.

Kierkegaard begins his analysis of the human condition by proposing that we are all full of anxiety and that it is our anxiety that leads us into despair.

Anxiety comes from our freedom. Because human beings are free to act as they choose, to do evil or to do good, to kill or to spare life, we are anxious. It is as if we stand at the edge of a cliff and, looking down, recognize that we have the freedom to hurl ourselves into the abyss. Deep down, we are afraid that we want to.

When we try to become someone we are not for the sake of others, smiling just right, laughing just right, saying all the right things, wearing clothes that we think others will like, we do so out of anxiety. We have the freedom to do otherwise, but out of anxiety, we do not. We instead despair that we are not the people that others want us to be, and we hide our true selves behind masks.

Most of us, indeed, live a lie, quietly, fearfully, and in deep despair that we are not that lie.

When we live a lie, when we do wrong, when we try to distract ourselves with addictions to forget the voids in our lives, our despair, he says, we become even more anxious. For now, we have given up our freedom. We are trapped by our choices.

Sometimes, however, this sense of being trapped can lead us back to our Creator, but I’ll get to that later.

Despair ultimately comes from fear in the face of the eternal, and he says there are two kinds.

The first kind is ignorance of the eternal self. This is the person who is mired in everyday secular minutia or pleasure seeking, who loses themselves in addictions, alcohol, drugs, and porn, or buries themselves so deeply in work, status climbing, money making, and various kinds of maxxing (looksmaxxing, statusmaxxing, etc.) that they don’t even know where their despair is coming from.

As he says in Sickness Unto Death,

[W]hen the ambitious man whose slogan is "Either Caesar or nothing" does not get to be Caesar, he despairs over it. But this also means something else: precisely because he did not get to be Caesar, he now cannot bear to be himself. Consequently he does not despair because he did not get to be Caesar but despairs over himself because he did not get to be Caesar....

This was me in my 20s and early 30s.

Meanwhile, deep down, the eternal self is calling out, wanting to be seen.

Kierkegaard says this is the most dangerous form of despair, but not the most painful. Because of our ignorance of despair, our feeling that there is just one more distraction, one more life hack, one more miracle cure self-help book or video that will set our life back on track and make us become that elusive “person we ought to be,” keeps us going, even though we feel empty and alone. This person unconsciously wishes that their higher, eternal nature would take a hike.

When we become conscious of our despair, that is when the wheels fall off, but that is also when we approach our liberation.

Kierkegaard says that despair intensifies with consciousness and can take on two forms: despair of the will to be oneself and the will to not be oneself.

The second one is the more common. Now conscious of our despair, we will not to be ourselves. This can mean wanting to hide oneself, one’s true self, or worse, not wanting to exist at all, or even worse (according to the philosopher) wanting to be someone else entirely.

I often think of this as someone like George Bailey in It’s a Wonderful Life. Bailey, feeling like he had failed all his life, despairs and wants to end his life, finally wishing to the angel Clarence that he had never been born at all. Deep down, he wishes he could have been like one of his friends or even his brother, attending college, traveling the world, and being financially successful.

Unlike the person who seeks to distract themselves with work or addictions, this person, despairing, simply wants to be someone else. He or she defies his or her Creator, like George, saying, “You made a mistake when You made me.”

I became conscious of my despair sometime in my mid-30s when I realized that I was already middle-aged, no longer young.

The other kind of conscious despair, of the will to be oneself, occurs when we become all too conscious of our despair and we recognize that our despair is a conscious act. It is something we are doing to ourselves. It is our choice.

As he says,

in order to despair to will to be oneself, there must be consciousness of an infinite self. This infinite self, however, is really only the most abstract form, the most abstract possibility of the self. And this is the self that a person in despair wills to be, severing the self from any relation to a power that has established it, or severing it from the idea that there is such a power.

Recognizing despair is our choice, we recognize the infinite self but want to own it whole and clear. This person defies their Creator as well, but differently, saying, “I am not yours. I am mine.”

Such a person, aware of their eternal self, nevertheless seeks to own it and to become a kind of abstract, bodiless spirit or being of pure thought.

From this form of despair, we get new age beliefs and practices, modern paganism and the occult, but also to some extent atheism and humanism.

The only way out, then, he says, is to recognize that the eternal and the corporeal self come from the Creator. Only by turning away from living lies, doing wrong, and resting in the One from whom all things came, can the individual be at rest, completely free, yet not in despair.

As the philosopher lays out, the way out is a complex process of both heightening consciousness from the earthly and mundane to the infinite and eternal, and also recognizing one’s own dependence on the Creator.

Yet, now, supposing we have turned towards that One. We find a new source of despair.

Kierkegaard calls this “despair before God”.

At this point, and only at this point, when one stands before the Creator, not necessarily in death, but in the consciousness that one always stands before the Creator, can one become the eternal self.

I joined a church at age 35, but it wasn’t until I had been going regularly for a couple of years that I became aware that I was still in despair and continued for several years after that.

This despair arises because we are not right with our Creator; we are using our freedom, out of anxiety, to do wrong.

When we are not so aware of this, when we are in unconscious despair, we might dismiss our behavior as normal, certainly not illegal. These are modern times after all.

It is hard to make that argument when you feel you are walking on holy ground.

This is, however, the final stage of despair. Having awareness of one’s eternal self, awareness of one’s form of despair, awareness that one stands before the Creator, you can reach out to take the proffered Hand, receive forgiveness, and begin to change your life.

This is the only way out of unhappiness, to become your true self, take off the mask, and live joyously.

It is a simple road, but psychologically requires a lot of maturity and a little bit of faith to leave despair and that other person you wished you were behind. The alternative however is to remain defiant yet unhappy, searching forever for the secret to becoming someone else, of transcending it all, when what you really need already stands before you.

The culprit: sea ice melting from global warming.

At least, that’s what social media is saying.

Scientists, on the other hand, are puzzled by the conflicting data. In some models, the cause is sea ice melting in the Arctic. But other models show confusing predictions about what is happening. A well-cited review article says:

In model experiments, sea ice loss in the Atlantic sector appears to cause a weaker vortex, whereas a stronger vortex is found in response to sea ice loss in the Pacific sector

What is clear is that the polar vortex has been weakening for decades, and we know sea ice has been melting too. Shouldn’t the two be correlated?

Almost certainly, but that doesn’t establish a causal relationship between the two, and the models seem to bear out that the answer is much more complex than we would like.

Even further complicating matters is that scientists aren’t sure whether polar vortex weakening is caused by global warming at all.

That same review article mentions two studies from 2017 that attribute the weakening to “internal variability,” which causes Sudden Stratospheric Warming (SSW), the immediate cause of the weakening.

The reason is that models show that, while the vortex does respond to climate change, the response is small compared to normal variability in the strength of the vortex.

How do we know that the vortex wouldn’t have weakened mostly without global warming? Perhaps the effect of warming is actually negligible.

This is the problem with a lot of studies of climate change. The climate has always changed. How do we separate the effects caused or exacerbated by global warming from those that are unrelated?

It’s not enough to show a causal relationship between warming and a particular effect. What if that causal relationship is weaker than observed and the overarching cause is something unrelated to warming?

Indeed, more model-based studies are showing that global warming and sea ice melting may not be the culprit at all. Some models show sea ice loss strengthening the polar vortex later this century.

So what’s going on?

In this article, I want to explore the research that has been done on the polar vortex, go beyond what news outlets will tell you, and get to the heart of why this question is so complex and currently unanswered.

The polar vortex starts in autumn as the Arctic cools. It is a stable westerly wind high in the stratosphere driven by the rotation of the Earth. It is symmetric and typically circular.

The stratosphere is the layer of the atmosphere directly above the layer in which we live. It is where jet airliners fly.

The vortex is usually stable. Some graphics depict it as a kind of corral that keeps all the nasty Arctic cold in the Arctic and away from us. That is not quite true. It’s more the other way around. The polar vortex surrounds the cold air in the Arctic that forms in winter.

Sometimes the troposphere, where we live, sends warm air upward into the stratosphere, causing it to warm up.

These SSWs happen about every other year. Sixteen such events were observed between 1998 and 2024, although the criteria for identifying them have evolved somewhat.

Most models show, however, that while the polar vortex will weaken with climate change, that does not lead to an increase in SSW events. One model looked at 1000 years of pre-industrial SSWs and showed that they work on 60 to 90-year cycles, “which is associated with long-term variability in the amplitude of the quasibiennial oscillation (QBO)”.

The QBO is a 28 to 29-month oscillation in tropical stratospheric winds. It has both a positive and a negative phase.

During the positive phase, the northern United States in winter becomes cooler than normal, while the Upper Midwest, Rockies, and California are wetter than normal. The South and East become drier than normal.

During the negative phase, the lower 48 of the United States is likely to be cooler than normal in winter, especially in Texas, where I currently live. Meanwhile, the Rockies, plains, and Ohio Valley are drier than normal, while the South is wetter.

If the QBO is the main driver of SSW events, then polar vortex weakening, even if driven by climate change, might not be causing cold snaps in the United States.

And while the northern hemisphere vortex is expected to weaken, its southern hemispheric counterpart is expected to strengthen with climate change.

Another source of cold snaps is caused not by SSW or polar vortex disruptions, but by a stretching of the polar vortex into an oval shape. The 2021 Texas cold snap that caused so much misery, for example, can be linked to such a stretching event in February, which followed an SSW in January.

Sea ice loss seems to modulate the stretching, changing the timing of the effects and also causing them. Sea ice loss causes snowfall to increase at high latitudes because there is more exposed ocean to evaporate into the air. Indeed, snow increase seems to be a better predictor of stretching events than sea ice loss.

Indeed, unlike SSWs, stretching events have much more of a close link to climate change.

As the sea ice melts and snow increases, the exchange of heat between the troposphere and the stratosphere changes as well.

What this means is that sometimes cold snaps are likely to increase because of climate change. In particular, ocean warming in the Arctic causes sea ice loss, which may not raise sea levels (ice floating in the water can’t), but it can cause more cold snaps in winter.

The relationship between sea ice, snow, and polar vortex stretching events is somewhat complex, so I won’t go into it. Also, the debate over how to separate the signal from the noise in the climate continues. While some researchers are very confident in the link between climate change and polar vortex changes, others are more cautious because of the potential for cyclical changes. Models sometimes disagree on findings.

That there are negative impacts from climate change is clear, particularly changes in the Arctic, which is warming far more rapidly than the rest of the globe. What the ultimate result of those impacts will be is hard to see. Eventually, there may be no Arctic sea ice at all, which could cause other changes to accelerate.

As I discuss in the bonus content below, we have had success in coming together to agree to stop climate change issues before, with the banning of CFCs. Unfortunately, climate change is a much, much bigger problem. One cannot simply undo the Industrial Revolution, and the promises that “new” technologies like electric cars will sort that all out tend to sweep away serious issues. I think it is important, however, to maintain a nuanced view about what these effects are and the unintended consequences of some heavy-handed attempts to solve climate change.

As individuals, we can at most saner voices heard above the shrill voices of those seeking merely to gain attention or vent their emotions.

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A couple of years ago, it was revealed that China had had a breakthrough in submarine detection. The success depends on a major refinement of a relatively old quantum technology known as Superconducting Quantum Interference Devices (SQUIDs).

In a recent article in the National Security Journal, quantum magnetometers, enhanced with AI, are enabling China to develop anti-submarine networks that render current and older submarine technology exposed and the ocean “transparent” by 2050.

It’s hard to tell how much of this is overconfidence on the part of the Chinese and how much is just propaganda.

While it is true that the era where a submarine could simply “run quiet” and evade detection is probably over, underwater stealth technology is by no means falling behind.

Many of you are familiar with old submarine movies like The Hunt for Red October, where acoustics was the main way to detect submarines. Although there was a bunch of interesting and potentially classified technology mentioned in that breakout novel, Tom Clancy mentions another submarine detection mechanism in the first of his Jack Ryan series of novels, Sum of All Fears,

Sticking out the tail of the converted Lockheed Electra airliner was a sensitive device called a magnetic anomaly detector. It reported on variations in the earth’s magnetic field, such as those caused by the metallic mass of a submarine.

-Sum of All Fears, Tom Clancy, 1991

Emphasis added.

The magnetic anomaly detection (MAD) device is not a single device but belongs to a class of devices that detect submarines based on their magnetic signature. They are a kind of metal detector for submarines that can be used to find them even when they are running silent.

While these devices have been around for a while, new devices that use quantum superconducting technology, making them extremely sensitive, are changing the game. These devices can detect variations in electromagnetic energy 100 billion times weaker than the energy it takes to move a compass needle.

These technologies are rapidly replacing sonar as the most effective way to find subs. Not only are they now much smaller than they used to be, but they can be mounted on drones, which can spread out and scour the ocean far faster and more reactively than networks of sonar buoys.

Nevertheless, submarines are developing better stealth technology, and I wouldn’t be surprised if that did not include active stealth (deliberately faking magnetic signatures) as well, so I think it’s too early to say that submarines as a strategic asset are over.

The quantum phenomenon behind magnetic signature detection is perhaps one of the strangest in the quantum world: the Aharonov-Bohm effect.

In classical mechanics, if you send a charged particle through space, its motion will depend on the electric and magnetic fields through which it passes. The electric potential and magnetic vector potential have no effect at all. Indeed, these potentials were thought not to be detectable at all at one time, meaning that they could be regarded as mere mathematical conveniences and unphysical.

Even more importantly, electric and magnetic fields elsewhere in the universe do not classically affect charged particle motion at all.

All that changes with the Aharonov-Bohm effect.

Take an experiment where I fire electrons out of a cathode (similar to old-style TVs) at a barrier containing two slits. In between the slits, right on the barrier, I have wound a thin, dense, long coil of wire called a solenoid. Inside the solenoid is a strong magnetic field. (Solenoids are used in many kinds of technology because they can cause a freely moving bar of metal to shoot out one end when a current is applied in one direction and suck it back in when the current is reversed. This is handy for valves, locks, and strikers.)

The magnetic field around the solenoid falls off quickly far from it, and it can even be shielded so that it falls off even more quickly to the point where the magnetic field at the slits is effectively zero.

The barrier should ensure that, following the classical path through either slit, the electrons will not encounter the solenoid.

When this experiment is performed, however, the magnetic flux in the solenoid changes the outcome. Magnetic flux is essentially the sum of the magnetic vector potential around a closed loop. This is non-zero around the solenoid.

We know that the magnetic vector potential induces the magnetic field (and if it varies with time, the electric field). The relationship between the magnetic vector potential and the magnetic field follows the right-hand rule. You can think of the vector potential as being like the axis of rotation, while the magnetic field is the rotation itself. If you point your thumb in the direction of the vector potential with your right hand, your fingers will curl in the direction of the magnetic field.

If you have the case here, however, where the magnetic field is in a straight line, the rule is reversed. Point your thumb in the direction of the magnetic field, and your fingers will curl in the direction of the vector potential. (See Feynman lectures, Figure 15-6 here.)

Along both paths the electron takes, the magnetic field is zero outside the solenoid, but the vector potential is not zero. Instead, it surrounds the solenoid.

When the electrons encounter the vector potential in the form of magnetic flux, their phases change, which changes the interference pattern on the screen. It shifts the entire interference pattern by a constant amount (See Feynman lectures, Figure 15-8.).

Some physicists have regarded this as evidence that the electrons are somehow “entering” the solenoid before proceeding to the slit or being influenced by it in a kind of action at a distance. This is merely a philosophical choice, however, and I don’t think it’s a good one.

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Suppose you took away the Sun. How long would it take for us to notice and for the Earth to fly off into interstellar space?

According to Isaac Newton’s theory of gravity, you would notice instantly because in his theory gravity is instantaneous. This “action at a distance” bothered Newton considerably saying that it was

so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.

— Isaac Newton, Letters to Bentley, 1692/3

Nevertheless, the theory was so successful in describing planetary orbits that it was tolerated for nearly 250 years until Albert Einstein rewrote the gravitational theory completely.

The problem with action at a distance is that it can reverse cause and effect. Although we might see the Sun’s disappearance as being the cause of the Earth flying off into space, visiting aliens, traveling at an appreciable fraction of the speed of light between the Earth and Sun, with Sun behind them and Earth ahead of them, would see the Earth flying off into space first and then the Sun disappear.

That is a consequence of Einstein’s 1905 theory of special relativity which introduced the concept of the speed of light as an absolute speed limit. He spent the next 10 years developing his general theory of relativity, first for accelerating reference frames, and then for gravity. His final theory, published in 1915, showed that if the Sun were to disappear, we would not notice at all for 8 minutes.

This ensures that cause and effect remain in the correct order for all observers, human and alien alike.

Einstein had solved the 250 year old problem of action at a distance and shown that all phenomena propagating through space, including gravity, obey the speed of light. Ten years later, however, Werner Heisenberg, Erwin Schrödinger, and Niels Bohr, among others, develop the theory of quantum mechanics, which explains the strange phenomena being observed in experiments with individual particles.

Heisenberg developed the mathematics for the theory first, a version that has disappeared from the popular consciousness called the Heisenberg Picture. In this picture, particles are modeled as having complex, multifaceted properties rather than the simple ones that Isaac Newton had developed.

When Newton developed his mechanics, he assumed a great many things about objects and what was important about them. For example, when trying to understand how objects, whether they be planets or baseballs, move, he determined that their position, velocity, and mass where the most important and that their behavior was largely affected by something he called forces which applied accelerations to their positions.

All of this he laid out with a mathematical precision that had previously been thought to be impossible.

Position could simply be represented by three numbers, indicating where the particle was in the up-down, left-right, and back-forth directions. Velocity likewise would simply be the particle speed in these three dimensions. Mass was an intrinsic property, meanwhile. When velocity and mass where multiplied, you got another quantity called momentum, which is the amount of inertia that the particle has.

Heisenberg determined that if you took these simple properties like position and momentum and replaced them with mathematical objects called operators, you could model the particles quantum mechanical behavior.

Schrödinger offered a different approach. He decided that instead of replacing the familiar Newtonian concepts of position and momentum, he simply had to invent a new kind of physical object, one based on waves rather than compact bodies.

Of course, scientists had been modeling waves for centuries, water waves, electromagnetic waves, and so on. The beauty of the wave approach was that you didn’t need Heisenberg’s complicated operator formalism to be a property of the particle. Rather, all that machinery was sucked into the wave function and the equations that determined how the wave function evolved in time.

Wave function became the standard way to look at quantum particles and continues to this day.

About 10 years after their introduction, however, Einstein gave a short talk where he pointed out a problem with wave function. By this time Einstein was in his fifties and nobody had seen much new from him since his papers on Bose-Einstein Condensates 10 years before. He was still famous but he was old news. The problem with the wave function was that, as far as Einstein could tell, it reintroduced action at a distance.

Einstein showed that, according to quantum theory, when a particle emerges from an emitter, whether it be an electron or a photon, quantum mechanics models it as an expanding wave. This is the wave function. Yet, when it strikes a detector, the particle only appears at a single point. Einstein asked, therefore, how the rest of the detector knows, instantly, that the particle has been detected somewhere.

He called this “spooky action at a distance” in a letter to Max Born in 1947.

In quantum mechanics parlance, the wave function “collapses” when it interacts with some macroscopic matter like a detector and this happens everywhere all at once. Even if the detector is light years wide, the particle will still only appear at one point.

Most physicists shrugged this off and many actually said that that was exactly what happened and that was ok as long as information wasn’t transmitted instaneously.

Others, such as David Bohm and Hugh Everett later came up with alternative explanations that avoided the problem to some degree. Bohm proposed that there really was a single particle and it was just being guided by the wave function which was never directly detected. Everett proposed that the wavefunction never collapsed at all but rather the detector itself maintained the wavefunction in itself. Bryce DeWitt used this interpretation as the basis for the Many Worlds interpretation of quantum mechanics.

One wonders, however, if this would have been a real problem if Heisenberg’s picture had prevailed over Schrödinger’s wave mechanics. The two are mathematically equivalent but philosophically very different. One proposes a new kind of entity, the wavefunction, while the other proposes a new kind of property, the operator. Both “collapse” when detected but there is a subtle difference.

Heisenberg explained to Einstein back in 1925 when the two met. Einstein had criticized Heisenberg’s theory at a talk and the two had gone off together on a long walk. On that walk, Heisenberg had convinced Einstein that his picture was similar to Einstein’s own theory of special relativity. In essence, the collapse of the particle’s properties was not an intrinsic reality to the particle but an observer effect, meaning that the experimenter was only getting to see one aspect of the particle in the measurement but the rest of it was “still there”.

Later Heisenberg abandoned this interpretation theory in favor of what would later become the Copenhagen interpretation of wavefunction collapse but still it is important to understand that the wavefunction itself is never observed, and there is good reason to believe that it does not exist, at least in the form we think it does.

You might ask where Schrödinger even got his wavefunction from. It was not invented from nothing but came from an earlier theory called Hamilton-Jacobi theory. The idea behind Hamilton-Jacobi or HJ theory is to connect classical mechanics with the theory of waves.

What came out of that development is the idea that there is an equivalence between the trajectories of bodies in motion, such as are governed by Newton’s laws, and waves in motion, specifically wave fronts in motion.

Imagine an explosion, perhaps a star going supernova. It sends light out as a massive wavefront in all directions. We know that light is made up of many, many photons all following trajectories. It turns out that we can find a mathematical equivalence to how an individual photon propagates through space and how a wavefront as a whole propagates.

This was where Schrödinger got his insight. He looked at quantum experiments, saw that the individual photons or electrons behaved like waves even when they were alone, and concluded that they were behaving according to the HJ law. The trouble was that he realized this was not exactly true. There was a random component to the particle trajectories as well that seemed proportional to a constant, called Planck’s constant, which had to be included. Thus, he modified the HJ law so that it became a wave with random trajectories in it and not just classical trajectories.

Now let me ask a question: if the HJ law shows an equivalence between classical trajectories, anything from photons to planets to baseballs, and wave fronts, does that mean that all these objects are classically waves?

Certainly not!

All it means is that wavefronts are equivalent to the trajectories of many, many particles.

Likewise, in the quantum mechanics realm, we can conclude that even though we can model particle trajectories with a wavefunction, that does not mean those particles are waves. All it means is that there is an equivalence between the trajectory of the particle and the wave function.

Thus, the action at a distance Einstein was seeing was simply a byproduct of how the wavefunction was being modeled as a real thing, as if we decided to model the trajectory of the planet Jupiter as a wave and then screamed “action at a distance” when it showed up at a particular location.

The wavefunction, in fact, is nothing more than a useful tool but does not exist.

Unfortunately, while this argument is straightforward for classical physics, it becomes quite difficult to make in quantum physics. This is why, for example, David Bohm had to model the wavefunction as a guiding field rather than doing away with it all together. It is, unfortunately, impossible to eliminate action at a distance because particle motion appears to be influenced by things it could have no contact with under the ordinary rules of special relativity. In other words, action at a distance seems to be baked into any model of quantum theory that assumes that particles actually exist all the time as particles (what we call hidden variable theories.)

While we can get away from Einstein’s specific objection, we cannot get away from the overall sense that quantum theory is violating the speed of light or time traveling or something of that nature. The only alternatives are (1) to assume that the universe somehow “comes into being” when we observe it and therefore no action at a distance is needed because our perceptions are simply creating reality (which is very Zen but not very satisfying to a hardcore realist like myself) or (2) assume that everything in the universe is pre-determined and it just “looks like” action at a distance is happening (which feels to me like creationist arguments that God put dinosaur bones in the Earth to test our faith).

I do wonder, however, if Heisenberg had the right idea in 1925 and it all comes down to relative perspective. Reality may be deeply multifaceted and yet we, for whatever reason, can only see a small part of it and, when a particle strikes a detector there is no collapse, no worlds split, all we see is what anyone in our position would see.

Whereas the playing field of special relativity is space and time, the playing field of quantum mechanics is the Hilbert space. This is an infinite dimensional space of possibilities. Sometimes you can break the Hilbert space down into discrete intervals or even finite numbers of possibilities but fundamentally this is the space in which particles live. In the Heisenberg picture, particle properties such as position, momentum, spin, and the like are evolving operators that act on a kind of fixed Hilbert space position. In the Schrödinger wave mechanics, particle properties are extracted by means of fixed operators on an evolving Hilbert space position. (That’s why they are equivalent.)

If we, fundamentally, live not only in space and time, but also in a Hilbert space of possibility, then there is no reason to suppose that Hilbert space position simply collapses into a familiar classical realm, nor does it make sense that it becomes “walled off” when we observe something. Rather, it seems like when we make a measurement it is like looking at Saturn’s rings. Depending on how Saturn is tilted towards us we might see a very thin line or we might see a big oval shape. We may not understand all the dynamics involved in why we see what we see in the quantum realm, unlike with Saturn, but likewise, we might simply be incapable of seeing the infinite number of dimensions all at once. Instead, we are treated, when we observe something, to merely a small subset of the dimensions that are there.

You can interpret this how you like. Lev Vaidman calls this Many Worlds but without splitting. I don’t know that there are many worlds. It could simply be one world where we are perceiving Hilbert spaces from a particular angle (where I use the term “angle” as a very loose analogy to viewing a particular point in an infinite dimensional space.)

In this case, quantum reality requires no action at all distance at all. Instead, we are simply perceiving an infinite dimensional reality through a very narrow lens and interpreting it as communicating across vast distances when, in fact, those distances are also part of the Hilbert spaces. All of reality is, indeed, merely a matter of perspective.

If relativity of Hilbert spaces is the correct way to talk about quantum mechanics, however, there must be a Hilbert space relativity theory, and indeed there is.

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The title may seem like clickbait. Surely, the speed of light was measured centuries ago. The speed of light is exactly 299,792,458 meters per second. In fact, the meter itself has been defined using the speed of light since 1983, so the speed is exact. The meter is the distance light travels in a vacuum in 1/299,792,458 seconds.

The problem has been that ever since we first became aware that light travels at a finite speed, we not been able to prove that light actually travels at this speed.

In fact, Veritasium posted a video about this about 5 years ago, now with up to 23 million views at the time of writing, where he explains the whole problem in detail.

The problem with this video is that it is now out of date. It’s not impossible. We now have a way to measure the speed of light. The only problem is that we lack the technology (and perhaps the will) to do so.

In this article, I will explain exactly what the problem is, why it is so hard to solve, and how it could be solved using a little-known byproduct of Einstein’s theory of special relativity called the Sagnac effect.

So what is the problem with measuring the speed of light?

The main issue is that we use light to measure the speed of everything else. Suppose you want to know how fast a bullet fired from a gun travels. You set up a high-speed camera and place a ruler behind where the bullet is to pass by (or you can hit a steel plate a known distance away), as close as you can. For example, if your camera is super high speed, like the Shimadzu HPV-X3 at 20 million frames a second, you can make a very accurate measurement.

But the speed of a bullet is nothing compared to the speed of light.

If you fire a beam of light at a target, you can’t use a camera to measure it because light would have to propagate to the camera.

Ah ha, but you don’t need a camera because you can use the light itself to trigger the start and stop of a clock. That clock can then be used to measure the speed of light.

If you fire a laser beam over a distance of 10 km, you can start a clock when the laser beam fires and stop it when it hits a detector 10 km away.

But how do you know when to stop the clock?

The obvious way that scientists have done for centuries is to reflect the light off a mirror and measure when the light returns to where it started. This ensures you can use the same clock for both the start and end of the light’s journey.

There is a problem with this approach, however. Suppose I want to use this method to measure the speed of an airplane. Maybe I attach a balloon to a string. When the airplane hits the balloon, it starts a clock. Now, the airplane flies until it reaches some point, turns around, and comes back. When the airplane returns, it hits another balloon, and the clock stops.

Now, I can use the time between the two balloons popping to measure the average round-trip speed of the airplane. Any variations in the airplane speed are, of course, not measured, but also any major difference between the airplane’s outbound speed and its inbound speed is averaged out.

Suppose I measure that the airplane’s average speed is 1000 kph. I could infer that the airplane’s average outbound speed and inbound speed are both 1000 kph, but there is a problem with that. If there is a constant wind of 100 kph that is a headwind outbound and a tailwind inbound, then the airplane’s actual speed could be 900 kph outbound and 1100 inbound. With the setup I have, I would have no way to measure that.

There is an inherent problem with round-trip measurements.

The solution seems to be to use two clocks, one at the start and one at the end. After all, we can find good ways of synchronizing clocks now, don’t we? Doesn’t the world depend on having atomic clocks that all work in lock step?

It turns out, however, that clock synchronization mechanisms all depend on knowing the speed of light since most rely on sending a beam or electric pulse along a wire from one clock to the other. That is what we are trying to measure.

Einstein addressed this problem in his 1905 paper, which introduced the world to special relativity. Einstein’s annus mirabilis papers were five papers all published in 1905 that changed physics and the world forever, including his discoveries of the equivalence of energy and mass, E=mc², the photoelectric effect, which is the basis for lasers and solar panels and won him the Nobel prize, and others. This was one of those papers, and all of this work was done while the 20-something German was working in a Swiss patent office.

Some of those patent applications were undoubtedly on clock synchronization mechanism which were important for making the trains run on time.

Einstein noticed this issue with clock synchronization and therefore developed, independently of Henri Poincaré, the Poincaré-Einstein Synchronization Convention (PESC), which simply assumes, without proof or evidence, that the one-way speed of light is equal to the two-way or average speed of light. Indeed, Einstein said of this convention that it

is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own free will.

A statement like that would make a peer-reviewer’s hair stand on end now, but at the time, Max Planck, who was the editor and reviewer for these papers, published them without criticism.

Later critics argued that you could, in fact, synchronize two clocks provided you start them out together and slowly move one clock from the starting point to the finish line. The problem with this approach is that, according to Einstein’s special relativity, the moving clock will slow down relative to the stationary clock, and, by the time it gets to the finish line, it will be ever so slightly out of sync. Because the degree to which special relativity unsynchronizes clocks is dependent on the speed of light itself, there is no way to know by how much.

In fact, if the one-way speed of light were significantly different from the average two-way speed, many laws of physics would have to change, but none of those changes would be measurable using current technology.

Because of this problem, some physicists have assumed that theories that have different one-way speeds of light and those where they are the same are equivalent theories, sort of like how different interpretations of quantum mechanics, like wave function collapse and Many Worlds, are equivalent. This turns out not to be the case, however. We can measure the one-way speed of light. The problem is that no one has succeeded in doing so because the technology does not exist.

PESC is, in fact, an axiom upon which all of special relativity and, by extension, general relativity depends. It is what allows us to view time and space as not absolute but relative. In particular, two events which appear to be simultaneous to one observer will not appear to be simultaneous to another observer in motion relative to the first observer.

For example, if lightning strikes the front and back of a moving train at the same time relative to an observer on the train, an observer standing next to the train on the platform will see the lightning at the back strike before the lightning at the front.

This suggests that time and space are intertwined and that simultaneity really is relative.

Einstein took this to mean that the universe is, in fact, a single four-dimensional block with the past and future already in existence.

The famous philosopher Karl Popper recalls in his autobiography arguing with Einstein at Princeton about this in 1950.

I tried to persuade him to give up his determinism, which amounted to the view that the world was a four-dimensional Parmenidean block universe in which change was a human illusion, or very nearly so. (He agreed that this had been his view, and while discussing it I called him “Parmenides.”)

Parmenides, a pre-Socratic philosopher born around 540 BC who profoundly influenced Plato, is widely believed to have argued that all reality was determined and unchangeable. Popper believed that this viewpoint of reality was unacceptable, and I agree.

In the 1990s, the prolific Italian physicist Franco Selleri, in his paper “Noninvariant One-Way Velocity of Light”, looked at what would happen if you jettisoned PESC and assumed that, instead, special relativity is a broken symmetry, in the sense that it is only true up to a point but not absolutely.

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