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This is admittedly a provocative title coming from a particle physicist, and you might think it tongue-in-cheek. But it’s really not.

We live in a cosmos with quantum physics, relativity, gravity, and a bunch of elementary fields, whose ripples we call elementary particles. These elementary “particles” include objects like electrons, photons, quarks, Higgs bosons, etc. Now if, in ordinary conversation in English, we heard the words “elementary” and “particle” used together, we would probably first imagine that elementary particles are tiny balls, shrunk down to infinitesimal size, making them indivisible and thus elementary — i.e., they’re not made from anything smaller because they’re as small as could be. As mere points, they would be objects whose diameter is zero.

But that’s not what they are. They can’t be.

I’ll tell this story in stages. In my last post, I emphasized that after the Newtonian view of the world was overthrown in the early 1900s, there emerged the quantum physics of the 1920s, which did a very good job of explaining atomic physics and a variety of other phenomena. In atomic physics, the electron is indeed viewed as a particle, though with behavior that is quite unfamiliar. The particle no longer travels on a path through physical space, and instead its behavior — where it is, and where it is going — is described probabilistically, using a wave function that exists in the space of possibilities.

But as soon became clear, 1920s quantum physics forbids the very existence of elementary particles.

In 1920s Quantum Physics, True Particles Do Not Exist

To claim that particles do not exist in 1920s quantum physics might seem, at first, absurd, especially to people who took a class on the subject. Indeed, in my own blog post from last week, I said, without any disclaimers, that “1920s quantum physics treats an electron as a particle with position x and momentum p that are never simultaneously definite.” (Recall that momentum is about motion; in pre-quantum physics, the momentum of an object is its mass m times its speed v.) Unless I was lying to you, my statement would seem to imply that the electron is allowed to have definite position x if its momentum p is indefinite, and vice versa. And indeed, that’s what 1920s quantum physics would imply.

To see why this is only half true, we’re going to examine two different perspectives on how 1920s quantum physics views location and motion — position x and momentum p.

1. There is a perfect symmetry between position and momentum (today’s post)
2. There is a profound asymmetry between position and momentum (next post)

Despite all the symmetry, the asymmetry turns out to be essential, and we’ll see (in the next post) that it implies particles of definite momentum can exist, but particles of definite position cannot… not in 1920s quantum physics, anyway.

The Symmetry Between Location and Motion

The idea of a symmetry between location and motion may seem pretty weird at first. After all, isn’t motion the change in something’s location? Obviously the reverse is not generally true: location is not the change in something’s motion! Instead, the change in an object’s motion is called its “acceleration” (a physics word that includes what in English we’d call acceleration, deceleration and turning.) In what sense are location and motion partners?

The Uncertainty Principle of Werner Heisenberg

In a 19th century reformulation of Newton’s laws of motion that was introduced by William Rowan Hamilton — keeping the same predictions, but rewriting the math in a new way — there is a fundamental symmetry between position x and momentum p. This way of looking at things is carried on into quantum physics, where we find it expressed most succinctly through Heisenberg’s uncertainty principle, which specifically tells us that we cannot know a object’s position and momentum simultaneously.

This might sound vague, but Heisenberg made his principle very precise. Let’s express our uncertainty in the object’s position as Δx. (Heisenberg defined this as the average value of x2 minus the squared average value of x. Less technically, it means that if we think the particle is probably at a position x0, an uncertainty of Δx means that the particle has a 95% chance of being found anywhere between x0-2Δx and x0+2Δx.) Let’s similarly express our uncertainty about the object’s momentum (which, again, is naively its speed times its mass) as Δp. Then in 1920s quantum physics, it is always true that

• Δp Δx > h / (4π)

where h is Planck’s constant, the mascot of all things quantum. In other words, if we know our uncertainty on an object’s position Δx, then the uncertainty on its momentum cannot be smaller than a minimum amount:

• Δp > h / (4π Δx) .

Thus, the better we know an object’s position, implying a smaller Δx, the less we can know about the object’s momentum — and vice versa.

This can be taken to extremes:,

• if we knew an object’s motion perfectly — if Δp is zero — then Δx = h / (4π Δp) = infinity, in which case we have no idea where the particle might be
• if we knew an object’s location perfectly — if Δx is zero — then Δp = h / (4π Δx) = infinity, in which case we have no idea where or how fast the particle might be going.

You see everything is perfectly symmetric: the more I know about the object’s location, the less I can know about its motion, and vice versa.

(Note: My knowledge can always be worse. If I’ve done a sloppy measurement, I could be very uncertain about the object’s location and very uncertain about its location. The uncertainty principle contains a greater-than sign (>), not an equals sign. But I can never be very certain about both at the same time.)

An Object with Known Motion

What does it mean for an object to have zero uncertainty in its position or its motion? Quantum physics of the 1920s asserts that any system is described by a wave function that tells us the probability for where we might find it and what it might be doing. So let’s ask: what form must a wave function take to describe a single particle with perfectly known momentum p?

The physical state corresponding to a single particle with perfectly known momentum P0 , which is often denoted |P0>, has a wave function

times an overall constant which we don’t have to care about. Notice the ; this is a complex number at each position x. I’ve plotted the real and imaginary parts of this function in Fig. 1 below. As you see, both the real (red) and imaginary (blue) parts look like a simple wave, of infinite extent and of constant wavelength and height.

Figure 1: In red and blue, the real and imaginary parts of the wave function describing a particle of known momentum (up to an overall constant). In black is the square of the wave function, showing that the particle has equal probability to be at each possible location.

Now, what do we learn from the wave function about where this object is located? The probability for finding the object at a particular position X is given by the absolute value of the wave function squared. Recall that if I have any complex number z = x + i y, then its absolute value squared |z2| equals |x2|+|y2|. Therefore the probability to be at X is proportional to

(again multiplied by an overall constant.) Notice, as shown by the black line in Fig. 1, this is the same no matter what X is, which means the object has an equal probability to be at any location we choose. And so, we have absolutely no idea of where it is; as far as we’re concerned, its position is completely random.

An Object with Known Location

As symmetry requires, we can do the same for a single object with perfectly known position X0. The corresponding physical state, denoted |X0>, has a wave function

again times an overall constant. Physicists call this a “delta function”, but it’s just an infinitely narrow spike of some sort. I’ve plotted something like it in Figure 2, but you should imagine it being infinitely thin and infinitely high, which obviously I can’t actually draw.

This wave function tells us that the probability that the object is at any point other than X0 is equal to zero. You might think the probability of it being at X0 is infinity squared, but the math is clever and the probability that it is at X0 is exactly 1. So if the particle is in the physical state |X0>, we know exactly where it is: it’s at position X0.

Figure 2: The wave function describing a particle of known position (up to an overall constant). The square of the wave function is in black, showing that the particle has zero probability to be anywhere except at the spike. The real and imaginary parts (in red and blue) are mostly covered by the black line.

What do we know about its motion? Well, we saw in Fig. 1 that to know an object’s momentum perfectly, its wave function should be a spread-out, simple wave with a constant wavelength. This giant spike, however, is as different from nice simple waves as it could possibly be. So |X0> is a state in which the momentum of the particle, and thus its motion, is completely unknown. [To prove this vague argument using math, we would use a Fourier transform; we’ll get more insight into this in a later post.]

So we have two functions, as different from each other as they could possibly be,

• Fig. 1 describing an object with a definite momentum and completely unknown position, and
• Fig. 2 describing an object with definite position and completely unknown momentum.

CAUTION: We might be tempted to think: “oh, Fig. 1 is the wave, and Fig. 2 is the particle”. Indeed the pictures make this very tempting! But no. In both cases, we are looking at the shape of a wave function that describes where an object, perhaps a particle, is located. When people talk about an electron being both wave and particle, they’re not simply referring to the relation between momentum states and position states; there’s more to it than that.

CAUTION 2: Do not identify the wave function with the particle it describes!!! It is not true that each particle has its own wave function. Instead, if there were two particles, there would still be only one wave function, describing the pair of particles. See this post and this one for more discussion of this crucial point.

Objects with More or Less Uncertainty

We can gain some additional intuition for this by stepping back from our extreme |P0> and |X0> states, and looking instead at compromise states that lie somewhere between the extremes. In these states, neither p nor x is precisely known, but the uncertainty of one is as small as it can be given the uncertainty of the other. These special states are often called “Gaussian wave packets”, and they are ideal for showing us how Heisenberg’s uncertainty principle plays out.

In Fig. 3 I’ve shown a wave function for a particle whose position is poorly known but whose momentum is better known. This wave function looks like a trimmed version of the |P0> state of Fig. 1, and indeed the momentum of the particle won’t be too far from P0. The position is clearly centered to the right of the vertical axis, but it has a large probability to be on the left side, too. So in this state, Δp is small and Δx is large.

Figure 3: A wave function similar to that of Fig. 1, describing a particle that has an almost definite momentum and a rather uncertain position.

In Fig. 4 I’ve shown a wave function of a wave packet that has the situation reversed: its position is well known and its momentum is not. It looks like a smeared out version of the |X0> state in Fig. 2, and so the particle is most likely located quite close to X0. We can see the wave function shows some wavelike behavior, however, indicating the particle’s momentum isn’t completely unknown; nevertheless, it differs greatly from the simple wave in Fig. 1, so the momentum is pretty uncertain. So here, Δx is small and Δp is large.

Figure 4: A wave function similar to that of Fig. 2, describing a particle that has an almost definite position and a highly uncertain momentum.

In this way we can interpolate however we like between Figs. 1 and 2, getting whatever uncertainty we want on momentum and position as long as they are consistent with Heisenberg’s uncertainty relation.

Wave functions in the space of possible momenta There’s even another more profound, but somewhat more technical, way to see the symmetry in action; click here if you are interested.

As I’ve emphasized recently (and less recently), the wave function of a system exists in the space of possibilities for that system. So far I’ve been expressing this particle’s wave function as a space of possibilities for the particle’s location — in other words, I’ve been writing it, and depicting it in Figs. 1 and 2, as Ψ(x). Doing so makes it more obvious what the probabilities are for where the particle might be located, but to understand what this function means for what the particle’s motion takes some reasoning.

But I could instead (thanks to the symmetry between position and momentum) write the wave function in the space of possibilities for the particle’s motion! In other words, I can take the state |P0>, in which the particle has definite momentum, and write it either as Ψ(x), shown in Fig. 1, or as Ψ(p), shown in Fig. 1a.

Figure 1a: The wave function of Fig. 1, written in the space of possibilities of momenta instead of the space of possibilities of position; i.e., the horizontal axis show the particle’s momentum p, not its position x as is the case in Figs. 1 and 2. This shows the particle’s momentum is definitely known. Compare this with Fig. 2, showing a different wave function in which the particle’s position is definitely known.

Remarkably, Fig. 1a looks just like Fig. 2 — except for one crucial thing. In Fig. 2, the horizontal axis is the particle’s position. In Fig. 1a, however, the horizontal axis is the particle’s momentum — and so while Fig. 2 shows a wave function for a particle with definite position, Fig. 1a shows a wave function for a particle with definite momentum, the same wave function as in Fig. 1.

We can similarly write the wave function of Fig. 2 in the space of possibilities for the particle’s position, and not surprisingly, the resulting Fig. 2a looks just like Fig. 1, except that its horizontal axis represents p, and so in this case we have no idea what the particle’s momentum is — neither the particle’s speed nor its direction.

Fig. 2a: As in Fig. 1a, the wave function in Fig. 2 written in terms of the particle’s momentum p.

The relationship between Fig. 1 and Fig. 1a is that each is the Fourier transform of the other [where the momentum is related to the inverse wavelength of the wave obtained in the transform.] Similarly, Figs. 2 and 2a are each other’s Fourier transforms.

In short, the wave function for the state |P0> (as a function of position) in Fig. 1 looks just like the wave function for the state |X0> (as a function of momentum) in Fig. 2a, and a similar relation holds for Figs. 2 and 1a. Everything is symmetric!

The Symmetry and the Particle…

So, what’s this all got to do with electrons and other elementary particles? Well, if a “particle” is really and truly a particle, an object of infinitesimal size, then we certainly ought to be able to put it, or at least imagine it, in a position state like |X0>, in which its position is clearly X0 with no uncertainty. Otherwise how could we ever even tell if its size is infinitesimal? (This is admittedly a bit glib, but the rough edges to this argument won’t matter in the end.)

That’s where this symmetry inherent in 1920s quantum physics comes in. We do in fact see states of near-definite momentum — of near-definite motion. We can create them pretty easily, for instance in narrow electron beams, where the electrons have been steered by electric and magnetic fields so they have very precisely defined momentum. Making position states is trickier, but it would seem they must exist, thanks to the symmetry of momentum and position.

But they don’t. And that’s thanks to a crucial asymmetry between location and motion that we’ll explore next time.

As expected, the Musk/Trump administration has aimed its guns at the US university system, deciding that universities that get grants from the federal government’s National Institute of Health will have their “overhead” capped at 15%. Overhead is the money that is used to pay for the unsung things that make scientific research at universities and medical schools possible. It pays for staff that keep the university running — administrators and accountants in business offices, machinists who help build experiments, janitorial staff, and so on — as well as the costs for things like building maintenance and development, laboratory support, electricity and heating, computing clusters, and the like.

I have no doubt that the National Science Foundation, NASA, and other scientific funding agencies will soon follow suit.

As special government employee Elon Musk wrote on X this weekend, “Can you believe that universities with tens of billions in endowments were siphoning off 60% of research award money for ‘overhead’? What a ripoff!”

The actual number is 38%. Overhead of 60% is measured against the research part of the award, not the total award, and so the calculation is 60%/(100%+60%) = 37.5%, not 60%/100%=60%. This math error is a little worrying, since the entire national budget is under Musk’s personal control. And never mind that a good chunk of that money often comes back to research indirectly, or that “siphon”, a loaded word implying deceit, is inappropriate — the overhead rate for each university isn’t a secret.

Is overhead at some universities too high? A lot of scientific researchers feel that it is. One could reasonably require a significant but gradual reduction of the overhead rate over several years, which would cause limited damage to the nation’s research program. But dropping the rate to 15%, and doing so over a weekend, will simply crush budgets at every major academic research institution in the country, leaving every single one with a significant deficit. Here is one estimate of the impact on some of the United States leading universities; I can’t quickly verify these details myself, but the numbers look to be at the right scale. They are small by Musk standards, but they come to something very roughly like $10000, more or less, per student, per year.

Also, once the overhead rate is too low, having faculty doing scientific research actually costs a university money. Every new grant won by a scientist at the university makes the school’s budget deficit worse. Once that line is crossed, a university may have to limit research… possibly telling some fraction of its professors not to apply for grants and to stop doing research.

It is very sad that Mr. Musk considers the world’s finest medical/scientific research program, many decades in the making and of such enormous value to the nation, to be deserving of this level of disruption. While is difficult to ruin our world-leading medical and scientific research powerhouse overnight, this decision (along with the funding freeze/not-freeze/kinda-freeze from two weeks ago) is a good start. Even if this cut is partially reversed, the consequences on health care and medicine in this country, and on science and engineering more widely, will be significant and long-lasting — because if you were one of the world’s best young medical or scientific researchers, someone who easily could get a job in any country around the globe, would you want to work in the US right now? The threat of irrational chaos that could upend your career at any moment is hardly appealing.

When the electron, the first subatomic particle to be identified, was discovered in 1897, it was thought to be a tiny speck with electric charge, moving around on a path governed by the forces of electricity, magnetism and gravity. This was just as one would expect for any small object, given the incredibly successful approach to physics that had been initiated by Galileo and Newton and carried onward into the 19th century.

But this view didn’t last long. Less than 15 years later, physicists learned that an atom has a tiny nucleus with positive electric charge and most of an atom’s mass. This made it clear that something was deeply wrong, because if Newton’s and Maxwell’s laws applied, then all the electrons in an atom should have spiraled into the nucleus in less than a second.

From 1913 to 1925, physicists struggled toward struggled toward a new vision of the electron. They had great breakthroughs and initial successes in the late 1920s. But still, something was off. They did not really find what they were looking for until the end of the 1940s.

Most undergraduates in physics, philosophers who are interested in physics, and general readers mainly learn about quantum physics of the 1920s, that of Heisenberg, Born, Jordan and of Schrödinger. The methods developed at that time, often called “quantum mechanics” for historical reasons, represented the first attempt by physicists to make sense of the atomic, molecular, and subatomic world. Quantum mechanics is all you need to know if you just want to do chemistry, quantum computing, or most atomic physics. It forms the basis of many books about the applications of quantum physics, including those read by most non-experts. The strange puzzles of quantum physics, including the double-slit experiment that I reviewed recently, and many attempts to interpret or alter quantum physics, are often phrased using this 1920s-era approach.

What often seems to be forgotten is that 1920s quantum physics does not agree with data. It’s an approximation, and sometimes a very good one. But it is inconsistent with Einstein’s relativity principle, a cornerstone of the cosmos. This is in contrast to the math and concepts that replaced it, known as relativistic quantum field theory. Importantly, electrons in quantum field theory are very different from the electrons of the 1920s.

And so, when trying to make ultimate conceptual sense of the universe, we should always be careful to test our ideas using quantum field theory, not relying on the physics of the 1920s. Otherwise we risk developing an interpretation which is inconsistent with data, at a huge cost in wasted time. Meanwhile, when we do use the 1920s viewpoint, we should always remember its limitations, and question its implications.

Overview

Before I go into details, here’s an overview.

I have argued strongly in my book and on this blog that calling electrons “particles” is misleading, and one needs to remember this if one wants to understand them. One might instead consider calling them “wavicles“, a term itself from the 1920s that I find appropriate. You may not like this term, and I don’t insist that you adopt it. What’s important is that you understand the conceptual point that the term is intended to convey.

Most crucially, electrons as wavicles is an idea from quantum field theory, not from the 1920s (though a few people, like de Broglie, were on the right track.) In the viewpoint of 1920s quantum physics, electrons are not wavicles. They are particles. Quantum particles.

Before quantum physics, an electron was described as an object with a position and a velocity (or a momentum, which is the electron’s mass times its velocity), moving through the world along a precise path. But in 1920s quantum physics, an electron is described as a particle with a position or a momentum, or some compromise between the two; its path is not definite.

In Schrödinger’s viewpoint [and I emphasize that there are others — his approach is just the most familiar to non-experts], there is a quantum wave function (or more accurately, a quantum state) that tells us the probabilities for the particle’s behavior: where we might find it, and where it might be going.

A wave function must not be identified with the particle itself. No matter how many particles there are, there is only one wave function. Specifically, if there are two electrons, then a single quantum wave function tells us the probabilities for their joint behavior — for the behavior of the system of two electrons. The two electrons are not independent of one another; in quantum physics I can’t say what one’s behavior might be without worrying about what the other is doing. The wave function describes the two electrons, but it is not either one of them.

Then we get to quantum field theory of the late 1940s and beyond. Now we view an electron as a wave — as a ripple in a field, known as the electron field. The whole field, across all of space, has to be described by the wave function, not just the one electron. (In fact, that’s not right either: our wave function has to simultaneously describe all the universe’s fields.) This is very different conceptually from the ’20s; the electron is never an object with a precise position, and instead it is generally spread out.

So it’s really, really important to remember that it is relativistic quantum field theory that universally agrees with experiments, not the quantum physics of the ’20s. If we forget this, we risk drawing wrong conclusions from the latter. Moreover, it becomes impossible to understand what modern particle physicists are talking about, because our description of the physics of “particles” relies on relativistic quantum field theory.

The Electron Over Time

Let me now go into more detail, with hope of giving you some intuition for how things have changed from 1900 to 1925 to 1950.

1900: Electrons Before Quantum Physics A Simple Particle

Pre-quantum physics (such as one learns in a first-year undergraduate course) treats an electron as a particle with a definite position which changes in a definite way over time; it has a definite speed v which represents the rate of the change of its motion. The particle also has definite momentum p equal to its mass m times its speed v. Scientists call this a “classical particle”, because it’s what Isaac Newton himself, the founder of old-school (“classical”) physics would have meant by the word “particle”.

Figure 1: A classical particle (blue dot) moves across across physical space. At the moment shown, it is at position A, and its path takes it to the right with a definite velocity. Two Simple Particles

Two particles are just two of these objects. That’s obvious, right? [Seems as though it ought to be. But as we’ll see, quantum physics says that not only isn’t it obvious, it’s false.]

Figure 2: Two particles, each traveling independently on its own path. Particle 1 moves rapidly to the right and is located at A, while particle 2 moves slowly to the left and is located at B. Two Particles in the “Space of Possibilities”

But now I’m going to do something that may seem unnecessarily complicated — a bit mind-bending for no obvious purpose. I want to describe the motion of these two particles not in the physical space in which they individually move but instead in the space of possibilities for two-particle system, viewed as a whole.

Why? Well, in classical physics, it’s often useful, but it’s also unnecessary. I can tell you where the two particles are in physical space and be done with it. But it quantum physics I cannot. The two particles do not, in general, exist independently. The system must be viewed as a whole. So to understand how quantum physics works, we need to understand the space of possibilities for two classical particles.

This isn’t that hard, even if it’s unfamiliar. Instead of depicting the two particles as two independent dots at two locations A and B along the line shown in Fig. 2, I will instead depict the system by indicating a point in a two-dimensional plane, where

• the horizontal axis depicts where the first particle is located
• the vertical axis depicts where the second particle is located

To make sure that you remember that I am not depicting any one particle but rather the system of two particles, I have drawn what the system is doing at this moment as a star in this two-dimensional space of possibilities. Notice the star is located at A along the horizontal axis and at B along the vertical axis, indicating that one particle is at A and the other is at B.

Figure 3: Within the space of possibilities, the system shown in Fig. 2 is located at the star, where the horizontal axis (the position of particle 1) is at A and the vertical axis (the position of the particle 2) is at B. Over time the star is moving to the right and downward, as shown by the arrow, indicating that in physical space particle 1 moves to the right and the particle 2 to the left, as shown in Fig. 2.

Moreover, in contrast to the two arrows in physical space that I have drawn in Fig. 2, each one indicating the motion of the corresponding particle, I have drawn a single arrow in the space of possibilities, indicating how the system is changing over time. As you can see from Fig. 2,

• the first particle is moving from A to the right in physical space, which corresponds to rightward motion along the horizontal axis of Fig. 3;
• the second particle is moving from B to the left in physical space, which corresponds to downward motion along the vertical axis in Fig. 3;

and so the arrow indicating how the system is changing over time points downward and to the right. It points more to the right than downward, because the motion of the particle at A is faster than the motion of the particle at B.

Why didn’t I bother to make a version of Fig. 3 for the case of just one particle? That’s because for just one particle, physical space and the space of possibilities are the same, so the pictures would be identical.

I suggest you take some time to compare Figs. 2 and 3 until the relationship is clear. It’s an important conceptual step, without which even 1920s quantum physics can’t make sense.

If you’re having trouble with it, try this post, in which I gave another example, a bit more elaborate but with more supporting discussion.

1925: Electrons in 1920s Quantum Physics A Quantum Particle

1920s quantum physics, as one learns in an upper-level undergraduate course, treats an electron as a particle with position x and momentum p that are never simultaneously definite, and both are generally indefinite to a greater or lesser degree. The more definite the position, the less definite the momentum can be, and vice versa; that’s Heisenberg’s uncertainty principle applied to a particle. Since these properties of a particle are indefinite, quantum physics only tells us about their statistical likelihoods. A single electron is described by a wave function (or “state vector”) that gives us the probabilities of it having, at a particular moment in time, a specific location x0 or specific momentum p0. I’ll call this a “quantum particle”.

How can we depict this? For a single particle, it’s easy — so easy that it’s misleading, as we’ll see when we go to two particles. All we have to do is show what the wave function looks like; and the wave function [actually the square of the wave function] tells us about the probability of where we might find the particle. This is indicated in Fig. 4.

Figure 4: A quantum particle corresponding to Fig. 1. The probability of finding the particle at any particular position is given by the square of a wave function, here sketched in red (for wave crests) and blue (for wave troughs). Rather than the particle being at the location A, it may be somewhere (blue dot) near A , but it could be anywhere where the wave function is non-zero. We can’t say exactly where (hence the question mark) without actually measuring, which would change the wave function.

As I mentioned earlier, the case of one particle is special, because the space of possibilities is the same as physical space. That’s potentially misleading. So rather than think too hard about this picture, where there are many potentially misleading elements, let’s go to two particles, where things look much more complicated, but are actually much clearer once you understand them.

Two Quantum Particles

Always remember: it’s not one wave function per particle. It’s one wave function for each isolated system of particles. Two electrons are also described by a single wave function, one that gives us the probability of, say, electron 1 being at location A while electron 2 is simultaneously at location B. That function cannot be expressed in physical space! It can only be expressed in the space of possibilities, because it never tells us the probability of finding the first electron at position 1 independent of what electron 2 is doing.

In other words, there is no analogue of Fig. 2. Quantum physics is too subtle to be squeezed easily into a description in physical space. Instead, all we can look for is a generalization of Fig. 3.

And when we do, we might find something like what is shown in Fig. 5; in contrast to Fig. 4, where the wave function gives us a rough idea of where we may find a single particle, now the wave function gives us a rough idea of what the system of two particles may be doing — and more precisely, it gives us the probability for any one thing that the two particles, collectively, might be doing. Compare this figure to Fig. 2.

Figure 5: The probability of finding the two-particle system at any given point in the space of possibilities is given by the square of a wave function, shown again in red (wave crests) and blue (wave troughs). We don’t know if the positions of the two particles is as indicated by the star (hence the question mark), but the wave function does tell us the probability that this is the case, as well as the probability of all other possibilities.

In Fig. 2, we know what the system is doing; particle 1 is at position A and particle 2 is at position B, and we know how their positions are changing with time. In Fig. 5 we know the wave function and how it is changing with time, but the wave function only gives us probabilities for where the particles might be found — namely that they are near position A and position B, respectively, but exactly can’t be known known until we measure, at which point the wave function will change dramatically, and all information about the particles’ motions will be lost. Nor, even though roughly that they are headed right and left respectively, we can’t know exactly where they are going unless we measure their momenta, again changing the wave function dramatically, and all information about the particles’ positions will be lost.

And again, if this is too hard to follow, try this post, in which I gave another example, a bit more complicated but with more supporting discussion.

1950: Electrons in Modern Quantum Field Theory

1940s-1950s relativistic quantum field theory, as a future particle physicist typically learns in graduate school, treats electrons as wave-like objects — as ripples in the electron field.

[[[NOTA BENE: I wrote “the ElectrON field”, not “the electrIC field”. The electrIC field is something altogether different!!!]

The electron field (like any cosmic field) is found everywhere in physical space.

(Be very careful not to confuse a field, defined in physical space, with a wave function, which is defined on the space of possibilities, a much larger, abstract space. The universe has many fields in its physical space, but only one wave function across the abstract space of all its possibilities.)

In quantum field theory, an electron has a definite mass, but as a ripple, it can be given any shape, and it is always undergoing rapid vibration, even when stationary. It does not have a position x, unlike the particles found in 1920s quantum field theory, though it can (very briefly) be shaped into a rather localized object. It cannot be divided into pieces, even if its shape is very broadly spread out. Nevertheless it is possible to create or destroy electrons one at a time (along with either a positron [the electron’s anti-particle] or an anti-neutrino.) This rather odd object is what I would mean by a “wavicle”; it is a particulate, indivisible, gentle wave.

Meanwhile, there is a wave function for the whole field (really for all the cosmic fields at once), and so that whole notion is vastly more complicated than in 1920s physics. In particular, the space of possibilities, where the wave function is defined, is the space of all possible shapes for the field! This is a gigantic space, because it takes an infinite amount of information to specify a field’s shape. (After all, you have to tell me what the field’s strength is at each point in space, and there are an infinite number of such points.) That means that the space of possibilities now has an infinite number of dimensions! So the wave function is a function of an infinite number of variables, making it completely impossible to draw, generally useless for calculations, and far beyond what any human brain can envision.

It’s almost impossible to figure out how to convey all this in a picture. Below is my best attempt, and it’s not one I’m very proud of. Someday I may think of something better.

Figure 6: In quantum field theory — in contrast to “classical” field theory — we generally do not know the shape of the field (its strength, or “value”, shown on the vertical axis, at each location in physical space, drawn as the horizontal axis.) Instead, the range of possible shapes is described by a wave function, not directly shown. One possible shape for a somewhat localized electron, roughly centered around the position A, is shown (with a question mark to remind you that we do not know the actual shape.) The blue blur is an attempt to vaguely represent a wave function for this single electron that allows for other shapes, but with most of those shapes somewhat resembling the shape shown and thus localized roughly around the position A. [Yeah, this is pretty bad.]

I’ve drawn the single electron in physical space, and indicated one possible shape for the field representing this electron, along with a blur and a question mark to emphasize that we don’t generally know the shape for the field — analogous to the fact that when I drew one electron in Fig. 4, there was a blur and question mark that indicated that we don’t generally know the position of the particle in 1920s quantum physics.

[There actually is a way to draw what a single, isolated particle’s wave function looks like in a space of possibilities, but you have to scramble that space in a clever way, far beyond what I can explain right now. We’ll see it later this year.]

Ugh. Writing about quantum physics, even about non-controversial issues, is really hard. The only thing I can confidently hope to have conveyed here is that there is a very big difference between electrons as they were understood and described in 1920’s quantum physics and electrons as they are described in modern quantum field theory. If we get stuck in the 1920’s, the math and concepts that we apply to puzzles like the double slit experiment and “spooky action at a distance” are never going to be quite right.

As for what’s wrong with Figure 6, there are so many things, some incidental, some fundamental:

• The picture I’ve drawn would be somewhat accurate for a Higgs boson as a ripple in the Higgs field. But an electron is a fermion, not a boson, and trying to draw the ripple without being misleading is kind of impossible.
• The electron field is given by a complex numbers, and in fact more than one, so drawing it as though it has a shape like the one shown in Fig. 6 is an oversimplification.
• At best, Fig. 6 sketches how an electron would look if it didn’t experience any forces. But because electrons are electrically charged and do experience electric and magnetic forces, we can’t just show the electron field without showing the electromagnetic field too; the wave function for an electron deeply involves both. That gets super-complicated.
• The wave function is suggested by a vague blur, but in fact it always has more structure than can be depicted here.
• And there are probably more issues, as I’m sure some readers will point out. Go ahead and do so; it’s better to state all the flaws out loud.

What about two electrons — two ripples in the electron field? This is currently beyond my abilities to sketch. Even ignoring the effects of electric and magnetic forces, describing two electrons in quantum field theory in a picture like Fig. 6 seems truly impossible. For one thing, because electrons are precisely identical in quantum field theory, there are always correlations between the two electrons that cannot be avoided — they can never be independent, in the way that two classical electrons are. (In fact this correlation even affects Fig. 5; I ignored this issue to keep things simpler.) So they really cannot be depicted in physical space. But the space of possibilities is far too enormous for any depiction (unless we do some serious rescrambling — again, something for later in the year, and even then it will only work for bosons.)

And what should you take away from this? Some things about quantum physics can be understood using 1920’s language, but not the nature of electrons and other elementary “particles”. When we try to extract profound lessons from quantum physics without using quantum field theory, we have to be very careful to make sure that those lessons still apply when we try to bring them to the cosmos we really live in — a cosmos for which 1920’s quantum physics proved just as imperfect, though still useful, as the older laws of Newton and Maxwell.

Dylan Curious is an bright and enthusiastic fellow, and he has a great YouTube channel focused on what is happening in AI around the world. But Dylan’s curiosity doesn’t stop there. Having read and enjoyed Waves in an Impossible Sea (twice!), he wanted to learn more… so he and I had a great conversation about humans and the universe for about 90 minutes. Don’t let the slightly odd title deter you; we covered a broad set of interesting topics of relevance to 21st century life, including

• In what sense is all motion relative?
• Why haven’t we already encountered intelligent life from other stars?
• Might we live in a simulation?
• Could the universe have glitches akin to what happens in computer games?
• Should the language of science be reconsidered?
• Are the particles we’re made of really waves?

Dylan is fun to talk to and I’m sure you’ll enjoy our discussion. And follow him, as I do, as a way of keeping up with the fast-changing AI landscape!

Last week, when I wasn’t watching democracy bleed, I was participating in an international virtual workshop, attended by experts from many countries. This meeting of particle experimenters and particle theorists focused on the hypothetical possibility known as “hidden valleys” or “dark sectors”. (As shorthand I’ll refer to them as “HV/DS”). The idea of an HV/DS is that the known elementary particles and forces, which collectively form the Standard Model of particle physics, might be supplemented by additional undiscovered particles that don’t interact with the known forces (other than gravity), but have forces of their own. All sorts of interesting and subtle phenomena, such as this one or this one or this one, might arise if an HV/DS exists in nature.

Of course, according to certain self-appointed guardians of truth, the Standard Model is clearly all there is to be found at the Large Hadron Collider [LHC], all activities at CERN are now just a waste of money, and there’s no point in reading this blog post. Well, I freely admit that it is possible that these individuals have a direct line to God, and are privy to cosmic knowledge that I don’t have. But as far as I know, physics is still an experimental science; our world may be going backwards in many other ways, but I don’t think we should return to Medieval modes of thought, where the opinion of a theorist such as Aristotle was often far more important than actually checking whether that opinion was correct.

According to the methods of modern science, the views of any particular scientist, no matter how vocal, have little value. It doesn’t matter how smart they are; even Nobel Prize-winning theorists have often been wrong. For instance, Murray Gell-Mann said for years that quarks were just a mathematical organizing principle, not actual particles; Martinus Veltman insisted there would be no Higgs boson; Frank Wilczek was confident that supersymmetry would be found at the LHC; and we needn’t rehash all the things that Newton and Einstein were wrong about. In general, theorists who make confident proclamations about nature have a terrible track record, and only get it right very rarely.

The central question for modern science is not about theorists at all. It is this: “What do we know from experiments?”

And when it comes to the possibility of an HV/DS, the answer is “not much… not yet anyway.”

The good news is that we do not need to build another multibillion dollar experimental facility to search for this kind of physics. The existing LHC will do just fine for now; all we need to do is take full advantage of its data. But experimenters and theorists working together must develop the right strategies to search for the relevant clues in the LHC’s vast data sets. That requires completely understanding how an HV/DS might manifest itself, a matter which is far from simple.

Last week’s workshop covered many topics related to these issues. Today I’ll just discuss one: an example of a powerful, novel search strategy used by the ATLAS experiment. (It’s over a year old, but it appeared as my book was coming out, and I was too busy to cover it then.) I’ll explain why it is a good way to look for strong forces in a hidden valley/dark sector, and why it covers ground that, in the long history of particle physics, has never previously been explored.

Jets-of-Jets, and Why They’re Tricky

I already discussed topics relevant to today’s post in this one from 2022, where I wrote about a similar workshop, and you may well find reading that post useful as a complement to this one. There the focus was on something called “semi-visible jets”, and in the process of describing them I also wrote about similar “jets-of-jets”, which are today’s topic. So here is the second figure from that older post, showing ordinary jets from known particles, which are covered in this post, as well as the jets-of-jets and semi-visible jets that might arise from what is known as a “confining HV/DS.”

Figure 1: Left: Ordinary jets of hadrons will form from an ordinary, fast-moving quark; the total energy of the jet is approximately the total energy of the unobserved original quark. Center: A fast-moving hidden quark will make a jet of hidden (or “dark”) hadrons; but these, in turn, may all decay to ordinary quark/anti-quark pairs, each of which leads to a jet of ordinary hadrons. The result is a jet of jets. Right: if only some of the dark hadrons decay, while some do not, then the jet of jets is semi-visible; those that don’t decay (grey dotted arrows) will escape the detector unobserved, while the rest will produce observable particles.

How does a jet-of-jets form? In a hidden valley with a “confining” force (a few examples of which were explored by Kathryn Zurek and myself in our first paper on this subject), some or all of the HV/DS particles are subject to a force that resembles one we are familiar with: the strong nuclear force that binds the known quarks and gluons into protons, neutrons, pions, and other hadrons. By analogy, a confining HV/DS may have “valley quarks” and “valley gluons” (also referred to as “dark quarks” and “dark gluons”) which are bound by their own strong force into dark hadrons.

The analogy often goes further. As shown at the left of Fig. 1, when a high-energy quark or gluon comes flying out of a collision of protons in the LHC, it manifests itself as a spray of hadrons, known as a jet. I’ll call this an “ordinary jet.” Most of the particles in that ordinary jet are ordinary pions, with a few other familiar particles, and they are observed by the LHC detectors. Images of these jets (not photographs, but precise reconstructions of what was observed in the detector) tend to look something like what is shown in Fig. 2. In this picture, the tracks from each jet have been given a particular color. You see that there are quite a lot of tracks in the highest-energy jets, whose tracks are colored green and red. [These tracks are mostly from the electrically charged pions. Electrically neutral pions turn immediately into photons, which are also detected but don’t leave tracks; they and instead are absorbed in the detector’s “calorimeters” (the red and green circular regions.) The energy from all the particles, with and without tracks, is depicted by the dark-green/yellow/dark-red bars drawn onto the calorimeters.]

Figure 2: From ATLAS, a typical proton-proton collision with two energetic ordinary jets (plus a few less energetic ones.) The proton beams are coming in and out of the screen; the collision point is at dead center. From the collision emerge two energetic jets, the narrow groupings of nearly straight tracks shown in bright green and red; these (and other particles that don’t make tracks) leave lots of energy in the “calorimeters”, as shown by the dark green/yellow and dark red rectangles at the outer edges of the detector.

But what happens if a dark quark or dark gluon is produced in that collision? Well, as shown in the center panel of Fig. 1, a spray of dark hadrons results, in the form of a dark jet. The dark hadrons may be of various types; their precise nature depends on the details of the HV/DS. But one thing is certain: because they are hidden (dark), they can’t be affected by any of the Standard Model’s forces: electromagnetic, strong nuclear, or weak nuclear. As a result, dark hadrons interact with an LHC detector even less than neutrinos do, which means they sail right through it. And so there’s no hope of observing these objects unless they transform into something else that we can observe.

Fortunately [in fact this was the main point of my 2006 paper with Zurek], in many HV/DS examples, some or all of the dark particles

• will in fact decay to known, observable particles, and
• will do so fast enough that they can be observed in an LHC detector.

This is what makes the whole subject experimentally interesting.

For today, the main question is whether all or some of the dark hadrons decay faster than a trillionth of a second. If all of them do, then, as depicted in the central panel of Fig. 1, the dark jet of dark hadrons may turn into a jet-of-jets (or into something similar-looking, if a bit more complex to describe.) If only a fraction of the dark hadrons decay, while others pass unobserved through the detector, then the result is a semi-visible jet (or semi-visible jet-of-jets, really), shown in the right panel of Fig. 1.

Cool! Let’s go look through LHC data for jets-of-jets!

The Key Distinction Between Jets and Jets-Of-Jets

Not so fast. There’s a problem.

You see, ordinary jets come in such enormous numbers, and vary so greatly, that it’s not immediately obvious how to distinguish a somewhat unusual ordinary jet from a true jet-of-jets. How can this be done?

Theorists and especially experimenters have been looking into all sorts of complex approaches. Intricate measures of jet-weirdness invented by various physicists are being pumped en masse into machine learning algorithms (the sort of AI that particle physicists have been doing for over a decade). I’m all in favor of sophisticated strategies — go for it!

However, as I’ve emphasized again and again in these workshops, sometimes it’s worth doing the easy thing first. And in this context, the ATLAS experimental collaboration did just that. They used the simplest strategy you can think of — the one already suggested by the left and center panels of Figure 1. They exploit the fact that a jet-of-jets of energy E (or transverse momentum pT) generally has more tracks than an ordinary jet with the same energy E (or pT). [This fact, emphasized in Figs. 19 and 20 of this paper from 2008, follows from properties of confining forces; I’ll explain its origin in my next post on this subject.]

So at first glance, to look for this sign of an HV/DS, all one has to do is look for jets with an unusual number of tracks. Easy!

Well, no. Nothing’s ever quite that simple at the LHC. What complicates the search is that the number of LHC collisions with jets-of-jets might be just a handful — maybe two hundred? forty? a dozen? Making HV/DS particles is a very rare process. The number of LHC collisions with ordinary jets is gigantic by comparison! Collisions that make pairs of ordinary jets with energy above 1 TeV — a significant fraction of the energy of LHC’s proton-proton collisions — number in the many thousands. So this is a needles-in-a-haystack problem, where each of the needles, rather than being shiny metal, looks a lot like an unusual stalk of hay.

For example, look at the event in Fig. 3 (also from ATLAS). There are two spectacular jets, rather wide, with lots of tracks (and lots of energy, as indicated by the yellow rectangles on the detector’s outer regions.) Might this show two jets-of-jets?

Figure 3: As in Fig. 2, but showing an event with two jets that each display an extreme numbers of tracks. This is what a pair of jets-of-jets from an HV/DS might look like. But is that what it is?

Maybe. Or maybe not; more likely this collision produced two really unusual but ordinary jets. How are we to tell the difference?

In fact, we can’t easily tell, not without sophisticated methods. But with a simple strategy, we can tell statistically if the jets-of-jets are there, employing a trick of a sort commonly used at the LHC.

A Efficient, Simple, Broad Experimental Strategy

The key: both the ordinary jets and the jets-of-jets often come in pairs — for analogous reasons. It’s common for a high-energy quark to be made with a high-energy anti-quark going the opposite direction, giving two ordinary jets; and similarly it would be common for a dark quark to be made with a dark anti-quark, making two jets-of-jets. (Gluon pairs are also common, as would be pairs of dark gluons.)

This suggests the following simple strategy:

• Gather all collisions that exhibit two energetic jets (we’ll call them “dijet events”) and that satisfy a certain criterion that I’ll explain in the next section.
• Count the tracks in each jet; let’s call the number of tracks in the two jets n1 and n2.
• Suppose that we consider 75 tracks or more to be unusual — more typical of a jet-of-jets than of an ordinary jet. Then we can separate the events into four classes:
  • Class A: Those events where n1 and n2 are both less than 75;
  • Class B: Those events where n1 < 75 ≤ n2 ;
  • Class C: Those events where n2 < 75 ≤ n1 ;
  • Class D: Those events where n1 and n2 are both 75 or greater.
• Importantly, the two ordinary jets in a typical dijet event form largely independent of one another (with some caveats that we’ll ignore), so we can apply simple probability. If the probability that an ordinary jet has 75 tracks or more is p, then (see Fig. 4 below)
  • the number of events NA in class A is proportional to (1-p)2,
  • the number of events NB in class B and NC in class C are both proportional to p(1-p), and
  • the number of events ND in class D is proportional to p2.

These proportions are just those of the areas of the corresponding regions of the divided square in Fig. 4.

Figure 4: For independently-forming jets that have probability p of being unusual, the relations between NA , NB , NC and ND are exactly those of the areas of a square cut into four pieces, where each side of the square is split into lengths p and 1-p. Knowing the area of regions A and B (or C), one can predict the area of D. The same logic allows prediction of ND from NA , NB , NC.

As suggested by Fig. 4, because the two jets are of the same type, NB ≈ NC (where “≈“ means “approximately equal” — they differ only due to random fluctuations.) Furthermore, because the probability p of having more than 75 tracks in an ordinary jet is really small, we can write a few relations that are approximately true both of the numbers in each class and of the corresponding areas of the square in Fig. 4.

• Ntotal ≈ NA
  • (i.e. almost all the events are in Class A)
• NB / NA ≈ NC / NA ≈ p(1-p) / (1-p)2 = p / (1+p) ≈ p
  • (i.e. the fraction of events in class B or C is nearly p)
• ND / Ntotal ≈ p2 ≈ (NB / NA)2 ≈ NB NC / ( Ntotal )2
  • (i.e therefore by measuring NB , NC , and Ntotal , we can predict the number of events in class D. )

Would you believe this strategy and others like it are actually called the “ABCD method” by experimental particle physicists? That name is more than a little embarrassing. But the method is indeed simple, and whatever we call it, it works. Specifically, it allows us to predict the number ND before we actually count the number of events in class D. And when the count is made, two things may happen:

• If the measured ND is roughly the same as the prediction, we know that most of the events in Class D — the dijet events where both jets have an extreme number of tracks — are probably pairs of unusual ordinary jets, and there’s no sign of anything unexpected.
• If the measured ND is significantly larger than the prediction, then we have discovered a new source of dijet events where both jets have an extreme number of tracks, one that is not expected in the Standard Model. Maybe they are from an HV/DS, or maybe from something else — but that’s a detail to be figured out later, when we’re done drinking all the champagne in France.

[Note: I chose the number 75 for simplicity. The experimenters make their choice in a more complicated way, but this is a detail which doesn’t change the basic logic of the search.]

No similar search for jets-of-jets had ever previously been performed, so I’m sure the experimenters were quite excited when they finally unblinded their results and took a look at the data. But nothing unusual was seen. (If it had been, you would have already heard about it in the press, and France would have run out of bubbly.) Still, even though a null result isn’t nearly as revolutionarily important as a discovery, it is still evolutionarily important, representing an important increase in our knowledge.

What exactly we learn from this null result depends on the individual HV/DS example. Basically, if a specific HV/DS produces a lot of jets-of-jets, and those jets-of-jets have lots of tracks, then it would have been observed, so we can now forget about it. HV/DS models that produce fewer or less active jets-of-jets are still viable. What’s nice about this search is that its elegant simplicity allows a theorist like me to quickly check whether any particular HV/DS is now excluded by this data. That task won’t be so easy for the more sophisticated approaches that are being considered for other search strategies, even though they will be even more powerful, and necessary for some purposes.

One More Criterion in the Strategy

As I began to outline the strategy, I mentioned a criterion that was added when the dijet events were initially selected. Here’s what it is.

Click here for the details

The ATLAS experimenters assumed a simple and common scenario. They imagined that the jets-of-jets are produced when a new particle X with a high mass mX is produced, and then the X immediately decays to two jets-of-jets. Simple examples of what X might be are

• a heavy version of a Z boson made in a collision of a quark and an anti-quark, or
• a heavy version of a Higgs-like boson created in the collision of two gluons.

An example of the former, in which the heavy Z-like particle is called a “Z-prime”, is shown in Fig. 5.

Figure 5: A diagram showing a possible source of HV/DS jets-of-jets events, in which a quark and anti-quark (left) collide, making a Z-like boson of high mass, which subsequently decays (right) to a dark quark and anti-quark.

If the X particle were stationary, then its total energy would be given by Einstein’s formula E=mXc2. If such a particle were subsequently to decay into two jets-of-jets, then the total energy of the two jet-of-jets would then also be E=mXc2 (by energy conservation.) In such a situation, all the events from X particles would have the same total energy, and we could use that to separate possible jets-of-jets events from pairs of ordinary jets, whose energy would be far more random.

Typically, however, the X particle made in a proton-proton collision will not be stationary. Fortunately, a similar strategy can be applied, using something know as the invariant mass of the two jets-of-jets, which will always be mX. [Well, nothing is simple at the LHC; these statements are approximately true, for various reasons we needn’t get into now.]

And so, when carrying out the strategy, the experimenters

• Pick a possible value of mX ;
• Select all dijet events where the two jets together are measured to have an invariant mass approximately equal to mX ;
• Carry out an ABCD search only within that selected set of events, to see if the number of Class D events exceeds the prediction;
• Repeat for a new value of mX .

Missed Opportunity?

I have only one critique of this search, one of omission. It’s rather unfair, since we must give the experimenters considerable credit for doing something that had never been tried before. But here it is: a (temporarily) lost opportunity.

Click here for the details

For very large classes of HV/DS examples, the resulting jets-of-jets not only have many tracks but also have one or more of the following properties that are very unusual in ordinary jets:

• If their dark hadrons very often produce bottom quarks, which travel a tiny but measurable distance before they themselves decay to the hadrons we measure, a large fraction of the many tracks in the jet-of-jets will be “displaced”, meaning that they will not trace back precisely to the location of the proton-proton collision. [This too, is shown in Figure 19-20 of this paper.] Such a thing almost never happens in ordinary jets.
• If their dark hadrons themselves travel a tiny but measurable distance before they decay to ordinary hadrons or other Standard Model particles, then again a large fraction of the many tracks in the jet-of-jets will be displaced.
• If the dark hadrons in the dark jet very often decay to muons, or to bottom quarks and taus (which often subsequently decay to muons), then it will be common for a jet-of-jets to have three or more muons embedded within it. [This is observed in Table II of this paper, though in many HV/DS models the effect is even more dramatic.] While this is certainly not unheard of in ordinary jets, it is not at all typical.

And so, if one were to require not only many tracks but also many displaced tracks and/or several muons in each observed jet, then the fraction p of ordinary jets that would satisfy all these criteria would be substantially lower than it is in ATLAS’s current search, and the expected ND would be much smaller. This would then allow ATLAS to discover an even larger class of HV/DS models, ones whose jets-of-jets are significantly rarer or that produce somewhat fewer tracks, but make up for it with one of these other unusual features.

I hope that the experimenters at ATLAS (or CMS, if they try the same thing) will include these additional strategies the next time this method is attempted. Displaced tracks and embedded muons are very common in HV/DS jets-of-jets, and adding these requirements to the existing search will neither complicate it greatly nor make it more difficult for theorists to interpret. The benefit of much smaller background from ordinary jets, and the possibility of a discovery that the current search would have missed, seems motivation enough to me.

Congrats to ATLAS, and a Look Ahead

Let me conclude with a final congratulations to my ATLAS colleagues. Some physicists seem to think that if the LHC were creating particles not found in the Standard Model, we would know by now. But this search is a clear demonstration that such a viewpoint is wrong. Marked by simplicity and power, and easy to understand and interpret, it has reached deep into uncharted HV/DS territory using a strategy never previously tried — and it had the potential to make a discovery that all previous LHC searches would have missed.

Nor is this the end of the story; many more searches of the wide range of HV/DS models remain to be done. And they must be done; to fail to fully explore the LHC’s giant piles of data would be a travesty, a tremendous waste of a fantastic machine. Until that exploration is complete, using as many innovations as we can muster, the LHC’s day is not over.

Happy New Year! 2025 is the centenary of some very important events in the development of quantum physics — the birth of new insights, of new mathematics, and of great misconceptions. For this reason, I’ve decided that this year I’ll devote more of this blog to quantum fundamentals, and take on some of the tricky issues that I carefully avoided in my recent book.

My focus will be on very basic questions, such as: How does quantum physics work, to the extent we humans understand it? Which of the widely-held and widely-promulgated ideas about quantum weirdness are true? And for those that aren’t, what is the right way to think about them?

I’ll frame some of this discussion in the context of the quantum two-slit experiment, because

• it’s famous,
• it’s often poorly explained
• it’s often poorly understood,
• it highlights (when properly understood) an extraordinarily strange aspect of quantum physics.

Not that I’ll cover this subject all in one post… far from it! It’s going to take quite some time.

The Visualization Problem

We humans often prefer to understand things visually. The problem with explaining quantum physics, aside from the fact that no one understands it 100%, is that all but the ultra-simplest problems are impossible to depict in an image or animation. This forces us to use words instead. Unfortunately, words are inherently misleading. Even when partial visual depictions are possible, they too are almost always misleading. (Math is helpful, but not as much as you’d think; it’s usually subtle and complicated, too.) So communication and clear thinking are big challenges throughout quantum physics.

These difficulties lead to many widespread misconceptions (some of which I myself suffered from when I was a student first learning the subject.) For instance, one of the most prevalent and problematic, common among undergraduates taking courses in chemistry or atomic physics, is the wrong idea that each elementary particle has its own wavefunction — a function which tells us the probability of where it might currently be located. This confusion arises, as much as anything else, from a visualization challenge.

Consider the quantum physics of the three electrons in a lithium atom. If you’ve read anything about quantum physics, you may have been led to believe that that each of the three electrons has a wave function, describing its behavior in three-dimensional space. In other words,

• naively the system would be described by three wave functions, each in three dimensions; each electron’s wave function tells us the probability that it is located at this point or that one;
• but in fact the three electrons are described by one wave function in nine dimensions, telling us simultaneously the overall probability that the first electron is t be found at this point, the second at that point, and the third at some other point.

Unfortunately, drawing something that exists in nine dimensions is impossible! Three wave functions in three dimensions is much easier to draw, and so, as a compromise/approximation that has some merits but is very foncusing, that method of depiction is widely used in images of multiple electrons. Here, for instance, two of the lithium atom’s electrons are depicted as though they have wave functions sharing the yellow region (the “inner shell”), while the third is drawn as though it has a wave function occuping the [somewhat overlapping] blue region (the “next shell”). [The atomic nucleus is shown in red, but far larger than it actually is.] Something similar is done in this image of the electrons in oxygen from a chemistry class.)

Yet the meat of the quantum lithium atom lies in the fact that there’s actually only one wave function for the entire system, not three. Most notably, the Pauli exclusion principle, which is responsible for keeping the electrons from all doing the same things and leads to the shell-like structure, makes sense only because there’s only one wave function for the system. And so, the usual visual depictions of the three electrons in the atom are all inherently misleading.

Yet there’s no visual image that can replace them that is both correct and practical. And that’s a real problem.

That said, it is possible to use visual images for two objects traveling in one dimension, as I did in a recent article that explains what it means for a system of two particles to have only one wave function. But for today, we can set this particular issue aside.

What We Can’t Draw Can Hurt Our Brains

Like most interesting experiments, the underyling quantum physics of the quantum double slit experiment cannot be properly drawn. But depicting it somehow, or at least parts of it, will be crucial in understanding how it works. Most existing images that are made to try to explain it leave out important conceptual points. The challenge for me — not yet solved — is to find a better one.

In this post, I’ll start the process, opening a conversation with readers about what people do and don’t understand about this experiment, about what’s often said about it that is misleading or even wrong, and about why it’s so hard to draw anything that properly represents it. Over the year I expect to come back to the subject occasionally. With luck, I’ll find a way to describe this experiment to my satisfaction, and maybe yours, before the end of the year. I don’t know if I’ll succeed. Even if I do, the end product won’t be short, sweet and simple.

But let’s start at the beginning, with the conventional story of the quantum double-slit experiment. The goal here is not so much to explain the experiment — there are many descriptions of it on the internet — but rather to focus on exactly what we say and think about it. So I encourage you to read slowly and pay very close attention; in this business, every word can matter.

Observing the Two Slits and the Screen

We begin by throwing an ultra-microscopic object — perhaps a photon, or an electron, or a neutrino — toward a wall with two narrow, closely spaced slits cut in it. (The details of how we do this are not very important, although we do need to choose the slits and the distance to the screen with some care.) If the object manages to pass through the wall, then on the other side it continues onward until it hits a phosphorescent screen. Where it strikes the screen, the screen lights up. This is illustrated in Fig. 1, where several such objects are showing being sent outward from the left; a few pass through the slits and cause the screen to light up where they arrive.

Figure 1: Microscopic objects are emitted from a device at left and travel (orange arrows) toward a wall (grey) with two narrow slits in it. Each object that passes through the slits reaches a screen (black) where it causes the screen to light up with an orange flash.

If we do this many times and watch the screen, we’ll see flashes randomly around the screen, something like what is shown in Fig. 2:

Figure 2: (click to animate if necessary): The screen flickers with little dots, one for each object that impacts it.

But now let’s keep a record of where the flashes on the screen appear; that’s shown in Fig. 3, where new flashes are shown in orange and past flashes are shown in blue. When we do this, we’ll see a strange pattern emerge, seen not in each individual flash but over many flashes, growing clearer as the number of flashes increases. This pattern is not simply a copy of the shape of the two slits.

Figure 3 (click to animate if necessary): Same as Fig. 2, except that we record the locations of past flashes, revealing a surprising pattern.

After a couple of thousand flashes, we’ll recognize that the pattern is characteristic of something known as interference (discussed further in Figs. 6-7 below):

Figure 4: The interference pattern that emerges after thousands of objects have passed through the slits.

By the way, there’s nothing hypothetical about this. Performing this experiment is not easy, because both the source of the objects and the screen are delicate and expensive. But I’ve seen it done, and I can confirm that what I’ve told you is exactly what one observes.

Trying to Interpret the Observations

The question is: given what is observed, what is actually happening as these microscopic objects proceed from source through slits to screen? and what can we infer about their basic properties?

We can conclude right away that the objects are not like bullets — not like “particles” in the traditional sense of a localized object that travels upon a definite path. If we fired bullets or threw tiny balls at the slitted wall, the bullets or balls would pass through the two slits and leave two slit-shaped images on the screen behind them, as in Fig. 5.

Figure 5: If balls, bullets or other particle-like objects are thrown at the wall, those that pass through the slits will arrive at the screen in two slit-shaped regions.

Nor are these objects ripples, meaning “waves” of some sort. Caution! Here I mean what scientists and recording engineers mean by “wave”: not a single wave crest such as you’d surf at a beach, but rather something that is typically a series of wave crests and troughs. (Sometimes we call this a “wave set” in ordinary English.)

If each object were like a wave, we’d see no dot-like flashes. Instead each object would leave the interference pattern seen in Fig. 4. This is illustrated in Fig. 6 and explained in Fig. 7. A wave (consisting of multiple crests and troughs) approaches the slits from the left in Fig. 6. After it passes through the slits, a striking pattern appears on the screen, with roughly equally spaced bright and dark regions, the brightest one in the center.

Figure 6: If a rippling pattern — perhaps one of sound waves or of water waves — is sent toward the wall, what appears on the screen will be an interference pattern similar to that of Fig. 4. See Fig. 7 for the explanation. The bright zones on the screen may flicker, but the dark zones will always be dark.

Where does the interference pattern come from? This is clearest if we look at the system from above, as in Fig. 7. The wave is coming in from the left, as a linear set of ripples, with crests in blue-green and troughs in red. The wall (represented in yellow) has two slits, from which emerge two sets of circular ripples. These ripples add and subtract from one another, making a complex, beautiful “interference” pattern. When this pattern reaches the screen at the opposite wall, it creates a pattern on the screen similar to that sketched in Fig. 6, with some areas that actively flicker separated by areas that are always dark.

Fig. 7: The interference pattern created by a linear wave pattern passing through two slits, as depicted from above. The two slits convert the linear ripples to two sets of circular ripples, which cross paths and interfere. When the resulting pattern arrives at the screen at right, some areas flicker, while others between them always remain quiet. A similar pattern of activity and darkness, though with some different details (notably fewer dark and bright areas), is seen in Figs. 3, 4 and 6. Credit: Lookang, with many thanks to Fu-Kwun Hwang and author of Easy Java Simulation = Francisco Esquembre, CC BY-SA 3.0 Creative Commons license via Wikimedia Commons

It’s important to notice that the center of the phosphorescent screen is dark in Fig. 5 and bright in Fig. 6. The difference between particle-like bullets and wave-like ripples is stark.

And yet, whatever objects we’re dealing with in Figs. 2-4, they are clearly neither like the balls of Fig. 5 nor the waves of Fig. 6. Their arrival is marked with individual flashes, and the interference pattern builds up flash by flash; one object alone does not reveal the pattern. Strangely, each object seems to “know” about the pattern. After all, each one, independently, manages to avoid the dark zones and to aim for one of the bright zones.

How can these objects do this? What are they?

What Are These Objects?!

According to the conventional wisdom, Fig. 2 proves that the objects are somewhat like particles. When each object hits the wall, it instantaneously causes a single, tiny, localized flash on the screen, showing that it is itself a single, tiny, point-like object. It’s like a bullet leaving a bullet-hole: localized, sudden, and individual.

According to the conventional wisdom, Figs. 3-4 prove that the objects are somewhat like waves. They leave the same pattern that we would see if ocean swell were passing through two gaps in a harbor’s breakwater, as in Fig. 7. Interference patterns are characteristic only of waves. And because the interference pattern builds up over many independent flashes, occurring at different times, each object seems to “know,” independent of the others, what the interference pattern is. The logical conclusion is that each object interferes with itself, just as the waves of Figs. 6-7 do; otherwise how could each object “know” anything about the pattern? Interfering with oneself is something a wave can do, but a bullet or ball or anything else particle-like certainly cannot.

To review:

• A set of particles going through two slits wouldn’t leave an interference pattern; it would leave the pattern we’d expect of a set of bullets, as in Fig. 5.
• But waves going through two slits wouldn’t leave individual flashes on a screen; each wave would interfere with itself and leave activity all over the screen, with stronger and weaker effects in a predictable interference pattern, as in Figs. 6-7.

It’s as though the object is a wave when it goes through and past the slits, and turns into a particle before it hits the screen. (Note my careful use of the words “as though”; I did not say that’s what actually happens.)

And thus, according to the conventional wisdom, each object going through the slits is… well… depending on who you talk to or read…

• both a wave and a particle, or
• sometimes a wave and sometimes a particle, or
• equally wave and particle, or
• a thing with wave-like properties and with particle-like properties, or
• a particle-like thing described by a probability-wave (also known as a “wave function”), or
• a wave-like thing that can only be absorbed like a particle, or…
• ???

So… which is it?

Or is it any of the above?

Looking More Closely

We could try to explore this further. For instance, we could try to look more closely at what is going on, by asking whether our object is a particle that goes through one slit or is a wave that goes through both.

Figure 8: We might try to investigate further, by adding sensors just behind the slits, to see whether each object goes through one slit (as for a bullet) or goes through both (as for a sound wave). With certain sensors, we will find it goes through only one — but in this case, what appears on the screen will also change! We will see not what is in Fig. 4 but rather what appears in Fig. 9.

But the very process of looking at the object to see what slit it went through changes the interference pattern of Figs. 4 and 6 into the pattern in Fig. 5, shown in Fig. 9, that we’d expect for particles. We find two blobs, one for each slit, and no noticeable interference. It’s as though, by looking at an ocean wave, we turned it into a bullet, whereas when we don’t look at the ocean wave, it remains an ocean wave as it goes through the gaps, and only somehow coalesces into a bullet before it hits (or as it hits) the screen.

Figure 9: If sensors are added to try to see which slit each object passes through (or both), the pattern seen on the screen changes to look more like that of Fig. 5, and no clarity as to the nature of the objects or the process they are undergoing is obtained.

Said another way: it seems we cannot passively look at the objects. Looking at them is an active process, and it changes how they behave.

So this really doesn’t clarify anything. If anything, it muddies the waters further.

What sense can we make of this?

Before we even begin to try to make a coherent understanding out of this diverse set of observations, we’d better double-check that the logic of the conventional wisdom is accurate in the first place. To do that, each of us should read very carefully and think very hard about what has been observed and what has been written about it. For instance, in the list of possible interpretations given above, do the words “particle” and “wave” always mean what we think they do? They have multiple meanings even in English, so are we all thinking and meaning the same thing when we describe something as, say, “sometimes a wave and sometimes a particle”?

If we are very careful about what is observed and what is inferred from what is observed, as well as the details of language used to communicate that information, we may well worry about secret and perhaps unjustified assumptions lurking in the conventional wisdom.

For instance, does the object’s behavior at the screen, as in Fig. 2, really resemble a bullet hitting a wall? Is its interaction with the screen really instantaneous and tiny? Are its effects really localized and sudden?

Exactly how localized and sudden are they?

All we saw at the screen is a flash that is fast by human standards, and localized by human standards. But why would we apply human standards to something that might be smaller than an atom? Should we instead be judging speed and size using atomic standards? Perhaps even the standards of tiny atomic nuclei?

If our objects are among those things usually called “elementary particles” — such as photons, electrons, or neutrinos — then the very naming of these objects as “elementary particles” seems to imply that they are smaller than an atom, and even than an atom’s nucleus. But do the observations shown in Fig. 2 actually give some evidence that this is true? And if not… well, what do they show?

What do we precisely mean by “particle”? By “elementary particle”? By “subatomic particle”?

What actually happened at the slits? at the screen? between them? Can we even say, or know?

These are among the serious questions that face us. Something strange is going on, that’s for sure. But if we can’t first get our language, our logic, and our thinking straight — and as a writer, if I don’t choose and place every single word with great care — we haven’t a hope of collectively making sense of quantum physics. And that’s why this on-and-off discussion will take us all of 2025, at a minimum. Maybe it will take the rest of the decade. This is a challenge for the human mind, both for novices and for experts.

Tonight (January 13th) offers a wonderful opportunity for all of us who love the night sky, and also for science teachers. For those living within the shaded region of Fig. 1, the planet Mars will disappear behind the Moon, somewhere between 9 and 10 pm Eastern (6 and 7 pm Pacific), before reappearing an hour later. Most easily enjoyed with binoculars. (And, umm, without clouds, which will be my own limitation, I believe…)

For everyone else, look up anyway! Mars and the Moon will appear very close together, a lovely pair.

Figure 1: the region of Earth’s surface where Mars will be seen to disappear behind the Moon. Elsewhere Mars and the Moon will appear very close together, itself a beautiful sight. Image from in-the-sky.org.

Why is this Cool?

“Occultations”, in which a planet or star disappears behind our Moon, are always cool. Normally, even though we know that the planets and the Moon move across the sky, we don’t get to actually see the motion. But here we can really watch the Moon close in on Mars — a way to visually experience the Moon’s motion around the Earth. You can see this minute by minute with the naked eye until Mars gets so close that the Moon’s brightness overwhelms it. Binoculars will allow you to see much more. With a small telescope, where you’ll see Mars as a small red disk, you can actually watch it gradually disappear as the Moon crosses in front of it. This takes less than a minute.

A particularly cool thing about this particular occultation is that it is happening at full Moon. Occultations like this can happen at any time of year or month, but when they happen at full Moon, it represents a very special geometry in the sky. In particular, it means that the Sun, Earth, Moon and Mars lie in almost a straight line, as shown (not to scale!!!) in Fig. 2.

• The Moon is full because it is fully lit from our perspective, which means that it must lie almost directly behind the Earth relative to the Sun. [If it were precisely behind it, then it would be in Earth’s shadow, leading to a lunar eclipse; instead it is slightly offset, as it is at most full Moons.]
• And when the Moon covers Mars from our perspective, that must mean Mars lies almost directly behind the Moon relative to the Earth.

So all four objects must lie nearly in a line, a relatively rare coincidence.

Figure 2: (Distances and sizes not to scale!!) For a full Moon to block our sight of Mars, it must be that the Sun, Earth, Moon and Mars lie nearly in a line, so that the night side of the Earth sees the Moon and Mars as both fully lit and in the same location in the sky. This is quite rare. What Does This Occultation Teach Us?

Aside from the two things I’ve already mentioned — that an occultation is an opportunity to see the Moon’s motion, and that an occultation at full Moon implies the geometry of Fig. 2 — what else can we learn from this event, considered both on its own and in the context of others like it?

Distances and Sizes

Let’s start with one very simple thing: Mars is obviously farther from Earth than is the Moon, since it passes behind it. In fact, the Moon has occultations with all the planets, and all of them disappear behind the Moon instead of passing in front of it. This is why it has been understood for millennia that the Moon is closer to Earth than any of the planets.

Less obvious is that the map in Fig. 1 teaches us the size of the Moon. That’s because the width of the band where the Moon-Mars meeting is visible is approximately the diameter of the Moon. Why is that? Simple geometry. I’ve explained this here.

“Oppositions” and Orbital Periods

The moment when Mars is closest to Earth and brightest in the sky is approximately when the Sun, Earth and Mars lie in a straight line, known as “opposition”. Fig. 2 implies that an occultation of a planet at full Moon can only occur at or around that planet’s opposition. And indeed, while today’s occultation occurs on January 13th, Mars’ opposition occurs on January 15th.

Oppositions are very interesting for another reason; you can use them to learn a planet’s year. Mars’ most recent oppositions (and the next ones) are given in Fig. 3. You notice they occur about 25-26 months apart — just a bit more than two years.

Figure 3: A list of Martian oppositions (when Mars lies exactly opposite the Sun from Earth’s perspective, as in Fig. 2) showing they occur a bit more than two years apart. From nakedeyeplanets.com. [The different size and brightness of Mars from one opposition to the next reflects that the planetary orbits are not perfect circles.]

This, in turn, implies something interesting, but not instantly obvious: the time between Martian oppositions tells us that a Martian year is slightly less than two Earth years. Why?

Fig. 4 shows what would happen if (a) a Martian year (the time Mars takes to orbit the Sun) were exactly twice as long as an Earth year, and (b) both orbits were perfect circles around the Sun. Then the time between oppositions would be exactly two Earth years.

Figure 4: If Mars (red) took exactly twice as long to orbit the Sun (orange) as does Earth (blue), then an opposition (top left) would occur every two Earth years (bottom). Because oppositions occur slightly more than 24 months apart, we learn that Mars’ orbit of the Sun — its year — is slightly less than twice Earth’s year. (Yes, that’s right!) Oppositions for Jupiter and Saturn occur more often because their years are even longer.

But neither (a) nor (b) is exactly true. In fact a Martian year is 687 days, slightly less than two Earth years, whereas the time between oppositions is slightly more than two Earth years. Why? It takes a bit of thought, and is explained in detail here (for solar conjuctions rather than oppositions, but the argument is identical.)

The Planets, Sun and Moon are In a Line — Always!

And finally, one more thing about occultations of planets by the Moon: they happen for all the planets, and they actually happen pretty often, though some are much harder to observe than others. Here is a partial list, showing occultations of all planets [except Neptune is not listed for some unknown reason], as well as occultations of a few bright stars, in our current period. Why are these events so common?

Well (although the news media seems not to be aware of it!) the Moon and the planets are always laid out roughly in a (curved) line across the sky, though not all are visible at the same time. Since the Moon crosses the whole sky once a month, the chance of it passing in front of a planet is not particularly small!

Why are they roughly in a line? This is because the Sun and its planets lie roughly in a disk, with the Earth-Moon system also oriented in roughly the same disk. A disk, seen from someone sitting inside it, look like a line that goes across the sky… or rather, a huge circle that goes round the Earth.

To get a sense of how this works, look at Fig. 5. It shows a flat disk, seen from three perspectives (left to right): first head on, then obliquely (where it appears as an ellipse), and finally from the side (where it appears as a line segment.) The closer we come to the disk, the larger it will appear — and thus the longer the line segment will appear in side view. If we actually enter the disk from the side, the line segment will appear to wrap all the way around us, as a circle that we sit within.

Figure 5: A disk, seen from three perspectives: (left) face on, (center) obliquely, and (right) from the side, where it appears as a line segment. The closer we approach the disk the longer, the line segment. If we actually enter the disk, the line segment will wrap all the way around us, and will appear as a circle that surrounds us. Upon the sky, that circle will appear as a curved line (not necessarily overhead) from one horizon to the other, before passing underneath us.

Specifically for the planets, this means the following. Most planetary systems with a single star have the star at the near-center and planets orbiting in near-circles, with all the orbits roughly in a disk around the star. This is shown in Fig. 6. Just as in Fig. 5, when the star and planets are viewed obliquely, their orbits form an ellipse; and when they are viewed from the side, their orbits form a line segment, as a result of which the planets lie in a line. When we enter the planetary disk, so that some planets sit farther from the Sun than we do, then this line becomes a circle that wraps around us. That circle is the ecliptic, and all the planets and the Sun always lie close to it.

Fig. 6: (Left) Planets (colored dots) orbiting a central star (orange) along orbits (black circles) that lie in a plane. (Center) the same system viewed obliquely. (Right) The same system viewed from the side, in which case the planets and the star always lie in a straight line. (See also Fig. 5.) Viewed from one of the inner planets, the other planets and the star would seem to lie on a circle wrapping around the planet, and thus on a line across the night sky.

Reversing the logic, the fact that we observe that the planets and Sun lie on a curved line across the sky teaches us that the planetary orbits lie in a disk. This, too, has been known for millennia, long before humans understood that the planets orbit the Sun, not the Earth.

(This is also true of our galaxy, the Milky Way, in which the Sun is just one of nearly a trillion stars. The fact that the Milky Way always forms a cloudy band across the sky provides evidence that our galaxy is in the shape of a disk, probably somewhat like this one.)

The Mysteries of the Moon

But why does the Moon also lie on the ecliptic? That is, since the Moon orbits the Earth and not the Sun, why does its orbit have to lie in the same disk as the planets all do?

This isn’t obvious at all! (Indeed it was once seen as evidence that the planets and Sun must, like the Moon, all orbit the Earth.) But today we know this orientation of the Moon’s orbit is not inevitable. The moons of the planet Uranus, for instance, don’t follow this pattern; they and Uranus’ rings orbit in the plane of Uranus’ equator, tipped almost perpendicular to the plane of planetary orbits.

Well, the fact that the Moon’s orbit is almost in the same plane as the planets’ orbits — and that of Earth’s equator — is telling us something important about Earth’s history and about how the Moon came to be. The current leading explanation for the Moon’s origin is that the current Earth and Moon were born from the collision of two planets. Those planets would have been traveling in the same plane as all the others, and if they suffered a glancing blow within that plane, then the debris from the collision would also have been mostly in that plane. As the debris coalesced to form the Earth and Moon we know, they would have ended up orbiting each other, and spinning around their axes, in roughly this very same plane. (Note: This is a consequence of the conservation of angular momentum.)

This story potentially explains the orientation of the Moon’s orbit, as well as many other strange things about the Earth-Moon system. But evidence in favor of this explanation is still not overwhelmingly strong, and so we should consider this as an important question that astronomy has yet to fully settle.

So occultations, oppositions, and their near-simultaneous occurrence have a great deal to teach us and our students. Let’s not miss the opportunity!

When it comes to the weak nuclear force and why it is weak, there’s a strange story which floats around. It starts with a true but somewhat misleading statement:

• The weak nuclear force (which is weak because its effects only extend over a short range) has its short range because the particles which mediate the force, the W and Z bosons, have mass [specifically, they have “rest mass”.] This is in contrast to electromagnetic forces which can reach out over great distances; that’s because photons, the particles of light which mediate that force, have no rest mass.

This is misleading because fields mediate forces, not particles; it’s the W and Z fields that are the mediators for the weak nuclear force, just as the electromagnetic field is the mediator for the electromagnetic force. (When people speak of forces as due to exchange of “virtual particles” — which aren’t particles — they’re using fancy math language for a simple idea from first-year undergraduate physics.)

Then things get worse, because it is stated that

• The connection between the W and Z bosons’ rest mass and the short range of the weak nuclear force is that
  • the force is created by the exchange of virtual W and Z bosons, and
  • due to the quantum uncertainty principle, these virtual particles with mass can’t live as long and/or travel as far as virtual photons can, shortening their range.

This is completely off-base. In fact, quantum physics plays no role in why the weak nuclear force is weak and short-range. (It plays a big role in why the strong nuclear force is strong and short-range, but that’s a tale for another day.)

I’ve explained the real story in a new webpage that I’ve added to my site; it has a non-technical explanation, and then some first-year college math for those who want to see it. It’s gotten some preliminary comments that have helped me improve it, but I’m sure it could be even better, and I’d be happy to get your comments, suggestions, questions and critiques if you have any.

[P.S. — if you try but are unable to leave a comment on that page, please leave one here and tell me what went wrong; and if you try but are unable to leave a comment here too for some reason, please send me a message to let me know.]

In a previous post, I showed you that the Standard Model, armed with its special angle θw of approximately 30 degrees, does a pretty good job of predicting a whole host of processes in the Standard Model. I focused attention on the decays of the Z boson, but there were many more processes mentioned in the bonus section of that post.

But the predictions aren’t perfect. They’re not enough to convince a scientist that the Standard Model might be the whole story. So today let’s bring these predictions into better focus.

There are two major issues that we have to correct in order to make more precise predictions using the Standard Model:

• In contrast to what I assumed in the last post, θw isn’t exactly 30 degrees
• Although I ignored them so far, the strong nuclear force makes small but important effects

But before we deal with these, we have to fix something with the experimental measurements themselves.

Knowledge and Uncertainty: At the Center of Science

No one complained — but everyone should have — that when I presented the experimental results in my previous post, I expressed them without the corresponding uncertainties. I did that to keep things simple. But it wasn’t professional. As every well-trained scientist knows, when you are comparing an experimental result to a theoretical prediction, the uncertainties, both experimental and theoretical, are absolutely essential in deciding whether your prediction works or not. So we have to discuss this glaring omission.

Here’s how to read typical experimental uncertainties (see Figure 1). Suppose a particle physicist says that a quantity is measured to be x ± y — for instance, that the top quark mass is measured to be 172.57± 0.29 GeV/c2. Usually (unless explicitly noted) that means that the true value has a 68% chance of lying between x-y and x+y — “within one standard deviation” — and a 95% chance of lying between x-2y and x+2y — “within two standard deviations.” (See Figure 1, where x and y are called and . The chance of the true value being more than two standard deviations away from x is about 5% — about 1/20. That’s not rare! It will happen several times if you make a hundred different measurements. But the chance of being more than three standard deviations away from x is a small fraction of a percent — as long as the cause is purely a statistical fluke — and that is indeed rare. (That said, one has to remember that big differences between prediction and measurement can also be due to an unforseen measurement problem or feature. That won’t be an issue today.)

Figure 1: Experimental uncertainties corresponding to , where is the “central value” and “” is a “standard deviation. W Boson Decays, More Precisely

Let’s first look at W decays, where we don’t have the complication of θw , and see what happens when we account for the effect of the strong nuclear force and the impact of experimental uncertainies.

The strong nuclear force slightly increases the rate for the W boson to decay to any quark/anti-quark pair, by about 3%. This is due to the same effect discussed in the “Understanding the Remaining Discrepancy” and “Strength of a Force” sections of this post… though the effect here is a little smaller (as it decreases at shorter distances and higher energies.) This slightly increases the percentages for quarks and, to compensate, slightly reduces the percentages for the electron, muon and tau (the “leptons”).

In Figure 2 are shown predictions of the Standard Model for the probabilities of the W- boson’s various decays:

• At left are the predictions made in the previous post.
• At center are better predictions that account for the strong nuclear force.

(To do this properly, uncertainties on these predictions should also be provided. But I don’t think that doing so would add anything to this post, other than complications.) These predictions are then compared with the experimental measurements of several quantities, shown at right: certain combinations of these decays that are a little easier to measure are also shown. (The measurements and uncertainties are published by the Particle Data Group here.)

Figure 2: The decay probabilities for W– bosons, showing the percentage of W bosons that decay to certain particles. Predictions are given both before (left) and after (center) accounting for effects of the strong nuclear force. Experimental results are given at right, showing all measurements that can be directly performed.

The predictions and measurements do not perfectly agree. But that’s fine; because of the uncertainties in the measurements, they shouldn’t perfectly agree! All of the differences are less than two standard deviations, except for the probability for decay of a W– to a tau and its anti-neutrino. That deviation is less than three standard deviations — and as I noted, if you have enough measurements, you’ll occasionally get one larger than two standard deviations. We still might wonder if something funny is up with the tau, but we don’t have enough evidence of that yet. Let’s see what the Z boson teaches us later.

In any case, to a physicist’s eye, there is no sign here of any notable disgreement between theory and experiment in these results. Within current uncertainties, the Standard Model correctly predicts the data.

Z Boson Decays, More Precisely

Now let’s do the same for the Z boson, but here we have three steps:

• first, the predictions when we take sin θw = 1/2, as we did in the previous post;
• second, the predictions when we take sin θw = 0.48;
• third, the better predictions when we also include the effect of the strong nuclear force.

And again Figure 3 compares predictions with the data.

Figure 3: The decay probabilities for Z bosons, showing the percentage of Z bosons that decay to certain particles. Predictions are given (left to right) for sin θw = 0.5, for sin θw =0.48, and again sin θw = 0.48 with the effect of strong nuclear force accounted for. Experimental results are given at right, showing all measurements that can be directly performed.

You notice that some of the experimental measurements have extremely small uncertainties! This is especially true of the decays to electrons, to muons, to taus, and (collectively) to the three types of neutrinos. Let’s look at them closely.

If you look at the predictions with sin θw = 1/2 for the electrons, muons and taus, they are in disagreement with the measurements by a lot. For example, in Z decay to muons, the initial prediction differs from the data by 19 standard deviations!! Not even close. For sin θw = 0.48 but without accounting for the strong nuclear force, the disagreement drops to 11 standard deviations; still terrible. But once we account also for the strong nuclear force, the predictions agree with data to within 1 to 2 standard deviations for all three types of particles.

As for the decays to neutrinos, the three predictions differ by 16 standard deviations, 9 standard deviations, and… below 2 standard deviations.

My reaction, when this data came in in the 1990s, was “Wow.” I hope yours is similar. Such close matching of the Standard Model’s predictions with highly precise measurements is a truly stunning sucesss.

Notice that the successful prediction requires three of the Standard Model’s forces: the mixture of the electromagnetic and weak nuclear forces given by the magic angle, with a small effect from the strong nuclear force. Said another way, all of the Standard Model’s particles except the Higgs boson and top quark play a role in Figs. 2 and 3. (The Higgs field, meanwhile, is secretly in the background, giving the W and Z bosons their masses and affecting the Z boson’s interactions with the other particles; and the top quark is hiding in the background too, since it can’t be removed without changing how the Z boson interacts with bottom quarks.) You can’t take any part of the Standard Model out without messing up these predictions completely.

Oh, and by the way, remember how the probability for W decay to a tau and a neutrino in Fig. 2 was off the prediction by more than two standard deviations? Well there’s nothing weird about the tau or the neutrinos in Fig. 3 — predictions and measurements agree just fine — and indeed, no numbers in Z decay differ from predictions by more than two standard deviations. As I said earlier, the expectation is that about one in every twenty measurements should differ from its true value by more than two standard deviations. Since we have over a dozen measurements in Figs. 2 and 3, it’s to be expected that one might well be two standard deviations off.

Asymmetries, Precisely

Let’s do one more case: one of the asymmetries that I mentioned in the bonus section of the previous article. Consider a forward-backward asymmetry shown in Fig. 4. Take all collisions in which an electron strikes a positron (the anti-particle of an electron) and turns into a muon and an anti-muon. Now compare the probability that the muon goes “forward” (roughly the direction that the electron is heading) to the probability that it goes “backward” (roughly the direction that the positron is heading.) If the two probabilities are equal, then the asymmetry would be zero; if the muon always goes to the left, then the asymmetry would be 100%; if always to the right, the asymmetry would be -100%.

Figure 4: In electron-positron collisions that make a muon/anti-muon pair, the forward-backward asymmetry compares the rate for “forward” production (where the muon travels roughly in the same direction as the electron) to “backward” production.

Asymmetries are special because the effect of the strong nuclear force cancels out of them completely, and so they only depend on sin θw. And this particular “leptonic forward-backward” asymmetry is an example with a special feature: if sin θw were exactly 1/2, this asymmetry for lepton production would be predicted to be exactly zero.

But the measured value of this asymmetry, while quite small (less than 2%), is definitely not zero, and so this is another confirmation that sin θw is not exactly 1/2. So let’s instead compare the prediction for this asymmetry using sin θw = 0.48, the choice that worked so well for the Z boson’s decays in Fig. 3, with the data.

In Figure 5, the horizontal axis shows the lepton forward-backward asymmetry. The prediction of 1.8% that one obtains for sin θw = 0.48, widened slightly to cover 1.65% to 2.0%, which is what obtains for sin θw between 0.479 and 0.481, is shown in pink. The four open circles represent four measurements of the asymmetry by the four experiments that were located at the LEP collider; the dashes through the circles show the standard deviations on their measurements. The dark circle shows what one gets when one combines the four experiments’ data together, obtaining an even better statistical estimate: 1.71 ± 0.10%, the uncertainty being indicated both as the dash going through the solid circle and as the yellow band. Since the yellow band extends to just above 1.8%, we see that the data differs from the sin θw = 0.480 prediction (the center of the pink band) by less than one standard deviation… giving precise agreement of the Standard Model with this very small but well-measured asymmetry.

Figure 5: The data from four experiments at the LEP collider (open circles, with uncertainties shown as dashes), and the combination of their results (closed circle) giving an asymmetry of 1.70% with an uncertainty of ±0.10% (yellow bar.) The prediction of the Standard Model for sin θw between 0.479 and 0.481 is shown in pink; its central value of 1.8% is within one standard deviation of the data.

Predictions of other asymmetries show similar success, as do numerous other measurements.

The Big Picture

Successful predictions like these, especially ones in which both theory and experiment are highly precise, explain why particle physicists have such confidence in the Standard Model, despite its clear limitations.

What limitations of the Standard Model am I referring too? They are many, but one of them is simply that the Standard Model does not predict θw . No one can say why θw takes the value that it has, or whether the fact that it is close to 30 degrees is a clue to its origin or a mere coincidence. Instead, of the many measurements, we use a single one (such as one of the asymmetries) to extract its value, and then can predict many other quantities.

One thing I’ve neglected to do is to convey the complexity of the calculations that are needed to compare the Standard Model predictions to data. To carry out these computations much more carefully than I did in Figs. 2, 3 and 5, in order to make them as precise as the measurements, demands specialized knowledge and experience. (As an example of how tricky these computations can be: even defining what one means by sin θw can be ambiguous in precise enough calculations, and so one needs considerable expertise [which I do not have] to define it correctly and use that definition consistently.) So there are actually still more layers of precision that I could go into…!

But I think perhaps I’ve done enough to convince you that the Standard Model is a fortress. Sure, it’s not a finished construction. Yet neither will it be easily overthrown.

Well, gosh… what nice news as 2024 comes to a close… My book has received a ringing endorsement from Ethan Siegel, the science writer and Ph.D. astrophysicist who hosts the well-known, award-winning blog “Starts with a Bang“. Siegel’s one of the most reliable and prolific science writers around — he writes for BigThink and has published in Forbes, among others — and it’s a real honor to read what he’s written about Waves in an Impossible Sea.

His brief review serves as an introduction to an interview that he conducted with me recently, which I think many of you will enjoy. We discussed science — the nature of particles/wavicles, the Higgs force, the fabric (if there is one) of the universe, and the staying power of the idea of supersymmetry among many theoretical physicists — and science writing, including novel approaches to science communication that I used in the book.

If you’re a fan of this blog or of the book, please consider sharing his review on social media (as well as the Wall Street Journal’s opinion.) The book has sold well this year, but I am hoping that in 2025 it will reach an even broader range of people who seek a better understanding of the cosmos, both in the large and in the small.