Science blog of a physics theorist Feed

A scientific brain teaser for readers: here’s a wave function evolving over time, presented in the three different representations that I described in a post earlier this week. [Each animation runs for a short time, goes blank, and then repeats.] Can you interpret what is happening here?

The explanation — and the reasons why this example is particularly useful, informative, and interesting (I promise!) — is coming soon [it will be posted here tomorrow morning Boston time, Friday Feb 21st.]

[Note added on Thursday: I give this example in every quantum mechanics class I teach. No matter how many times I have said, with examples, that a wave function exists in the space of possibilities, not in physical space, it happens every time that 90%-95% thinks this shows two particles. It does not. And that’s why I always give this example.]

What is a wave function in quantum physics?

Such a question generates long and loud debates among philosophers of physics (and more limited debate among most physicists, who tend to prefer to make predictions using wave functions rather than wondering what they are.) I have a foot in both camps, even though I have no real credentials among the former set. But no matter; today I won’t try to answer my own question in any profound way. We can debate the deeper meaning of wave functions another time.

Instead I just want to address the question practically: what is this function for, in what sense does it wave, and how does it sit in the wider context of physics?

Schrödinger’s Picture of the World

Quantum physics was born slowly, in stages, beginning around 1900. The most famous stage is that of 1925, when Heisenberg, along with Born and Jordan, developed one approach, using matrices, and Schrödinger developed another, using a “wave function”. Both methods could predict details of atomic physics and other systems, and Schrödinger soon showed the two approaches were equivalent mathematically. Nevertheless, he (and many others) felt his approach was more intuitive. This is why the wave function approach is emphasized — probably over-emphasized — in many books on quantum physics.

Suppose we want to investigate a physical system, such as a set of interacting subatomic objects that together make up a water molecule. Westart by imagining the system as being in some kind of initial physical state, and then we ask how that state changes over time. In standard first-year undergraduate physics, using the methods of the 17th-19th century, we would view the initial physical state as consisting of the locations and motions of all the objects at some initial time. Armed with that information, we could then calculate precisely what the system would do in the future.

But experimental data on atomic physics revealed that this older method simply doesn’t agree with nature. Some other approach was needed.

In 1920s quantum physics in the style of Schrödinger, the state of the system is (under-)specified by an unfamiliar object: a function on the space of possibilities for the system. This function gives us a complex number for each possibility, whose square tells us the probability for that particular possibility. More precisely, if we measure the system carefully, Schrödinger’s function at the time of the measurement tells us the probability of our measurements giving one outcome versus another.

For instance, suppose the system consists of two particles, and let’s call the possible position of the first particle x1 and that of the second x2. Then Schrödinger’s function will take the form Ψ(x1,x2) — a function giving us a complex number for each of the possible locations of the two particles. (As I’ve emphasized repeatedly, even though we have a system of two particles, there is only one wave function; I’ve given you a couple of examples of what such functions are like here and here.)

If we want to know the probability of finding the first particle at some definite position X1 and the second at a definite position X2 — assuming we do the measurements right now — that probability is proportional to the quantity |Ψ(X1,X2)|2, i.e. the square of the function when the first particle is at X1 and the second is at X2.

If we choose not to make a measurement right away, Schrödinger’s equation tells us how the function changes with time; if the function was initially Ψ(x1,x2; t=0) = Ψ(x1,x2), then after a time T it will have a new form Ψ(x1,x2; t=T) which we can calculate from that equation. If we then measure the positions of the particles, the probabilities for various measurement outcomes will be given by the square of the updated function, |Ψ(x1,x2; t=T)|2.

Schrödinger’s function is usually called a “wave function”. But this comes with a caveat: it’s not always actually a wave…see below. So it is more accurate to call it a “state function.”

Wave Functions Are Not Things

Probably thanks to advanced chemistry classes, in which pictures of atoms are often drawn that suggest that each electron has its own wave function, it is a common error to think that every particle has a wave function, and that wave functions are physical objects that travel through ordinary space and carry energy and momentum from one place to another, much like sound waves and ocean waves do. But this is wrong, in a profound, crucial sense.

If the electrons and atomic nuclei that make up atoms are like characters in a 19th century novel, the wave function is like an omniscient narrator. No matter how many characters appear in the plot, there is only one such narrator. That narrator is not a character in the story. Instead the narrator plays the role of storyteller, with insight into all the characters’ minds and motivations, able to give us many perspectives on what is going on — but with absolutely no ability to change the story by, say, personally entering into a scene and interposing itself between two characters to prevent them from fighting. The narrator exists outside and beyond the story, all-knowing yet powerless.

A wave function describes the objects in a system, giving us information about all the locations, speeds, energies and other properties that they might have, as well as about how they influence one another as they move around in our familiar three-dimensional space. The system’s objects, of which there can be as many as we like, can do interesting things, such as clumping together to form more complex objects such as atoms. As they move around, they can do damage to these clumps; for instance, they can ionize atoms and break apart biological DNA molecules. The system’s wave function, by contrast, does not travel in three-dimensional space and has neither momentum nor energy nor location. It cannot form clumps of objects, nor can it damage them. It is not an object in the way that electrons , photons and neutrinos are objects. Nor is it a field like the electric field, the Higgs field, and the electron field, which exist in three dimensions and whose waves do have momentum, energy, speed, etc. Most important, each system has one, and only one, wave function, no matter how many objects are in the system.

[One might argue that a wave function narrator is less omniscient, thanks to quantum physics, than in a typical novel; but then again, that might depend on the author, no? I leave this to you to debate.]

I wrote the article “Why a Wave Function Can’t Hurt You” to emphasize these crucial points. If you’re still finding this confusing, I encourage you to read that article.

Some Facts About Wave Functions

Here are a few interesting facts about wave functions. I’ll state them mostly without explanation here, though I may go into more details sometime in the future.

• It is widely implied in books and articles that wave functions emerged for the first time in quantum physics — that they were completely absent from pre-quantum physics. But this is not true; wave functions first appeared in the 1830s.
  
  In the “Hamilton-Jacobi” reformulation of Newton’s laws, the evolution of a non-quantum system is described by a wave function (“Hamilton’s characteristic function”) that is a function on the space of possibilities and satisfies a wave equation quite similar to Schrödinger’s equation. However, in contrast to Schrödinger’s function, Hamilton’s function is a real number, not a complex number, at each point in the space of possibilities, and it cannot be interpreted in terms of probabilities. In very simple situations, Hamilton’s function is the argument (or phase) of Schrödinger’s function, but more generally the two functions can be very different.
  
  
• Wave functions are essential in Schrödinger’s approach to quantum physics. But in other approaches, including Heisenberg’s and the later method of Feynman, wave functions and wave equations do not directly appear. (The situation in pre-quantum physics is completely analogous; the wave function of Hamilton appears neither in Newton’s formulation of the laws of motion nor in the reformulation known as the “action principle” of Maupertuis.)
  
  This is an indication that one should be cautious ascribing any fundamental reality to this function, although some serious scientists and philosophers still do so.
  
  
• The relevant space of possibilities of which the wave function is a function is only half as big as you might guess. For instance, in our example of two particles above, even though the function specifies the probabilities for the various possible locations and motions of the objects in the system, it is actually only a function of either the possible locations or the possible motions (more specifically, the particles’ momenta.) If we write it as a function of the possible locations, then the probabilities for the objects’ motions are figured out through a nontrivial mathematical procedure, and vice versa.
  
  The fact that the wave function can only give half the information explicitly, no matter how we write it down, is related to why it is impossible to know objects’ positions and motions precisely at the same time.
  
  
• For objects moving around in a continuous physical space like the space of the room that you are sitting in, waves are a natural phenomenon, and Schrödinger’s function and the equation that governs it are typical of waves. But in many interesting systems, objects do not actually move, and there’s nothing wavy about the function, which is best referred to as a “state function”. As an example, suppose our system consists of two atoms trapped in a crystal, so that they cannot move, but each has a “spin” that can point up or down only. Then
  • the space of possibilities is just the four possible arrangements of the spins: up-up, up-down, down-up, down-down;
  • the wave state function doesn’t look like a wave, and is instead merely a discrete set of four complex numbers, one for each of the four arrangements;
  • the square of the each of these four complex numbers gives us the probabilities for finding the two spins in each of the four possible arrangements;
  • and Schrödinger’s equation for how the state function changes with time is not a wave equation but instead a 4 x 4 matrix equation.

The space of possibilities for two trapped atoms, each with spin that can be up or down, consists only of the above four physical states; Schrödinger’s state function provides a single complex number for each one, and is in no sense wave-like.

• So although the term “wave function” suggests that waves are an intrinsic part of quantum physics, they actually are not. For the design and operation of quantum computers, one often just needs state functions made of a finite set of complex numbers, as in the example I’ve just given you.
  

• Another case where a state function isn’t a wave function in the sense you might imagine is in quantum field theory, widely used both in particle physics and in the study of many materials, such as metals and superconductors. In this context, the state function shows wavelike behavior but not for particle positions, in contrast to 1920s quantum physics. More on this soon.
  
  
• For particle physics, we need relativistic quantum field theory, which incorporates Einstein’s special relativity (with its cosmic speed limit and weird behavior of space and time). But in a theory with special relativity, there’s no unique or universal notion of time. Unfortunately, Schrödinger’s approach requires a wave function defined at an initial moment in time, and his equation tells us how the function changes from the initial time to any later time. This is problematic. Because my definition of time will differ from yours if you are moving relative to me, my form of the wave function will differ from yours, too. This makes the wave function a relative quantity (like speed), not an intrinsic one (like electric charge or rest mass). That means that, as for any relative quantitiy, if we ever want to change perspective from one observer to another, we may have to recalculate the wave function — an unpleasant task if it is complicated.
  
  Despite this, the wave function approach could still be used. But it is far more common for physicists to choose other approaches, such as Feynman’s, which are more directly compatible with Einstein’s relativity.

If you’re of a certain age, you know Alan Alda from his wonderful acting in television shows and in movies. But you may not know of his long-standing interest in science communication and his podcast Clear and Vivid (named for the characteristics that he feels all communication should have.)

Alda and I had a great conversation about the idea that we are made of waves, and what it means for our relationship to the universe. A slimmed-down version of that discussion is now available on his podcast. I hope you enjoy it!

Separately, as promised: to my last post, which covered various ways of depicting and interpreting wave functions, I’ve added explanations of the two quantum wave functions that I placed at the end. Tomorrow I’ll take a step back and consider wave functions from a larger point of view, taking a brief look at what they are (and aren’t), what’s “wavy” (and not) about them, and at their roles in contexts ranging from pre-quantum physics of the 19th century to quantum field theory of the 21st.

Before we knew about quantum physics, humans thought that if we had a system of two small objects, we could always know where they were located — the first at some position x1, the second at some position x2. And after Isaac Newton’s breakthroughs in the late 17th century, we believed that by combining this information with knowledge of the objects’ motions and the forces acting upon them, we could calculate where they would be in the future.

But in our quantum world, this turns out not to be the case. Instead, in Erwin Schrödinger’s 1925 view of quantum physics, our system of two objects has a wave function which, for every possible x1 and x2 that the objects could have, gives us a complex number Ψ(x1, x2). The absolute-value-squared of that number, |Ψ(x1, x2)|2, is proportional to the probability for finding the first object at position x1 and the second at position x2 — if we actually choose to measure their positions right away. If instead we wait, the wave function will change over time, following Schrödinger’s wave equation. The updated wave function’s square will again tell us the probabilities, at that later time, for finding the objects at those particular positions.

The set of all possible object locations x1 and x2 is what I am calling the “space of possibilities” (also known as the “configuration space”), and the wave function Ψ(x1, x2) is a function on that space of possibilities. In fact, the wave function for any system is a function on the space of that system’s possibilities: for any possible arrangement X of the system, the wave function will give us a complex number Ψ(X).

Drawing a wave function can be tricky. I’ve done it in different ways in different contexts. Interpreting a drawing of a wave function can also be tricky. But it’s helpful to learn how to do it. So in today’s post, I’ll give you three different approaches to depicting the wave function for one of the simplest physical systems: a single object moving along a line. In coming weeks, I’ll give you more examples that you can try to interpret. Once you can read a wave function correctly, then you know your understanding of quantum physics has a good foundation.

For now, everything I’ll do today is in the language of 1920s quantum physics, Schrödinger style. But soon we’ll put this same strategy to work on quantum field theory, the modern language of particle physics — and then many things will change. Familiarity with the more commonly discussed 1920s methods will help you appreciate the differences.

Complex Numbers

Before we start drawing pictures, let me remind you of a couple of facts from pre-university math about complex numbers. The fundamental imaginary number is the square root of minus one,

which we can multiply by any real number to get another imaginary number, such as 4i or -17i. A complex number is the sum of a real number and an imaginary number, such as 6 + 4i or 11 – 17i.

More abstractly, a complex number w always takes the form u + i v, where u and v are real numbers. We call u the “real part” of w and we call v the “imaginary part” of w. And just as we can draw a real number using the real number line, we can draw a complex number using a plane, consisting of the real number line combined with the imaginary number line; in Fig. 1 the complex number w is shown as a red dot, with the real part u and imaginary part v marked along the real and imaginary axes.

Figure 1: Two ways of representing the complex number w, either as u + i v or as |w|eiφ .

Fig. 1 shows another way of representing w. The line from the origin to w has length |w|, the absolute value of w, with |w|2 = u2 + v2 by the Pythagorean theorem. Defining φ as the angle between this line and the real axis, and using the following facts

• u = |w| cos φ
• v = |w| sin φ
• eiφ = cos φ + i sin φ

we may write w = |w|eiφ , which indeed equals u + i v .

Terminology: φ is called the “argument” or “phase” of w, and in math is written φ = arg(w).

One Object in One Dimension

We’ll focus today only on a single object moving around on a one-dimensional line. Let’s put the object in a “Gaussian wave-packet state” of the sort I discussed in this post’s Figs. 3 and 4 and this one’s Figs. 6 and 7. In such a state, neither the object’s position nor its momentum [a measure of its motion] is completely definite, but the uncertainty is minimized in the following sense: the product of the uncertainty in the position and the uncertainty in the momentum is as small as Heisenberg’s uncertainty principle allows.

We’ll start with a state in which the uncertainty on the position is large while the uncertainty on the momentum is small, shown below (and shown also in Fig. 3 of this post and Fig. 6 of this post.) To depict this wave function, I am showing its real part Re[Ψ(x)] in red and its imaginary part Im[Ψ(x)] in blue. In addition, I have drawn in black the square of the wave function:

• |Ψ(x)|2 = (Re[Ψ(x)])2 + (Im[Ψ(x)])2

[Note for advanced readers: I have not normalized the wave function.]

Figure 1: For an object in a simple Gaussian wave packet state with near-definite momentum, a depiction of the wave function for that state, showing its real and imaginary parts in red and blue, and its absolute-value squared in black.

But as wave functions become more complicated, this way of doing things isn’t so convenient. Instead, it is sometimes useful to represent the wave function in a different way, in which we plot |Ψ(x)| as a curve whose color reflects the value of φ = arg[Ψ(x)] , the argument of Ψ(x). In Fig. 2, I show the same wave function as in Fig. 1, depicted in this new way.

Figure 2: The same wave function as in Fig. 1; the curve is the absolute value of the wave function, colored according to its argument.

As φ cycles from 0 to π/4 to π/2 to 3π/4 and back to 2π (the same as φ = 0), the color cycles from red to yellow-green to cyan to blue-purple and back to red.

Compare Figs. 1 and 2; its the same information, depicted differently. That the wave function is actually waving is clear in Fig. 1, where the real and imaginary parts have the shape of waves. But it is also represented in Fig. 2, where the cycling through the colors tells us the same thing. In both cases, the waving tells us that the object’s momentum is non-zero, and the steadiness of that waving tells us that the object’s momentum is nearly definite.

Finally, if I’m willing to give up the information about the real and imaginary parts of the wave function, and just want to show the probabilities that are proportional to its squared absolute value, I can sometimes depict the state in a third way. I pick a few spots where the object might be located, and draw the object there using grayscale shading, so that it is black where the probability is large and becomes progressively lighter gray where the probability is smaller, as in Fig. 3.

Figure 3: The same wave function in Figs. 1 and 2, here showing only the probabilities for the object’s location; the darker the grey, the more likely the object is to be found at that location.

Again, compare Fig. 3 to Figs. 1 and 2; they all represent information about the same wave function, although there’s no way to read off the object’s momentum using Fig. 3, so we know where it might be but not where it is going. (One could add arrows to indicate motion, but that only works when the uncertainty in the momentum is small.)

Although this third method is quite intuitive when it works, it often can’t be used (at least, not as I’ve described it here.) It’s often useful when we have just one object to worry about, or if we have multiple objects that are independent of one another. But if they are not independent — if they are correlated, as in a “superposition” [more about that concept soon] — then this technique usually does not work, because you can’t draw where object number 1 is likely to be located without already knowing where object number 2 is located, and vice versa. We’ve already seen examples of such correlations in this post, and we’ll see more in future.

So now we have three representations of the same wave function — or really, two representations of the wave function’s real and imaginary parts, and two representations of its square — which we can potentially mix and match. Each has its merits.

How the Wave Function Changes Over Time

This particular wave function, which has almost definite momentum, does indeed evolve by moving at a nearly constant speed (as one would expect for something with near-definite momentum). It spreads out, but very slowly, because its speed is only slightly uncertain. Here is its evolution using all three representations. (The first was also shown in this post’s Fig. 6.)

I hope that gives your intuition some things to hold onto as we head into more complex situations.

Two More Examples

Below are two simple wave functions for a single object. They differ somewhat from the one we’ve been using in the rest of this post. What do they describe, and how will they evolve with time? Can you guess? I’ll give the full answer tomorrow as an addendum to this post.

Two different wave functions; in each case the curve represents the absolute value |Ψ(x)| and the color represents arg[Ψ(x)], as in Fig. 2. What does each wave function say about the object’s location and momentum, and how will each of them change with time?

Pioneer Works is “an artist and scientist-led cultural center in Red Hook, Brooklyn that fosters innovative thinking through the visual and performing arts, technology, music, and science.” It’s a cool place: if you’re in the New York area, check them out! Among many other activities, they host a series called “Picture This,” in which scientists ruminate over scientific images that they particularly like. My own contribution to this series has just come out, in which I expound upon the importance and meaning of this graph from the CMS experimental collaboration at the Large Hadron Collider [LHC]. (The ATLAS experimental collaboration at the LHC has made essentially identical images.)

The point of the article is to emphasize the relation between the spikes seen in this graph and the images of musical frequencies that one might see in a recording studio (as in this image from this paper). The similarity is not an accident.

Each of the two biggest spikes is a sign of an elementary “particle”; the Z boson is the left-most spike, and the Higgs boson is the central spike. What is spiking is the probability of creating such a particle as a function of the energy of some sort of physical process (specifically, a collision of objects that are found inside protons), plotted along the horizontal axis. But energy E is related to the mass m of the “particle” (via E=mc2) and it is simultaneously related to the frequency f of the vibration of the “particle” (via the Planck-Einstein equation E = hf)… and so this really is a plot of frequencies, with spikes reflecting cosmic resonances analogous to the resonances of musical instruments. [If you find this interesting and would like more details, it was a major topic in my book.]

The title of the article refers to the fact that the Z boson and Higgs boson frequencies are out of tune, in the sense that if you slowed down their frequencies and turned them into sound, they’d be dissonant, and not very nice to listen to. The same goes for all the other frequencies of the elementary “particles”; they’re not at all in tune. We don’t know why, because we really have no idea where any of these frequencies come from. The Higgs field has a major role to play in this story, but so do other important aspects of the universe that remain completely mysterious. And so this image, which shows astonishingly good agreement between theoretical predictions (colored regions) and LHC data (black dots), also reveals how much we still don’t understand about the cosmos.

[An immediate continuation of Part 1, which you should definitely read first; today’s post is not stand-alone.]

The Asymmetry Between Location and Motion

We are in the middle of trying to figure out if the electron (or other similar object) could possibly be of infinitesimal size, to match the naive meaning of the words “elementary particle.” In the last post, I described how 1920’s quantum physics would envision an electron (or other object) in a state |P0> of definite momentum or a state |X0> of definite position (shown in Figs. 1 and 2 from last time.)

If it is meaningful to say that “an electron is really is an object whose diameter is zero”, we would naturally expect to be able to put it into a state in which its position is clearly defined and located at some specific point X0 — namely, we should be able to put it into the state |X0>. But do such states actually exist?

Symmetry and Asymmetry

In Part 1 we saw all sorts of symmetry between momentum and position:

• the symmetry between x and p in the Heisenberg uncertainty principle,
• the symmetry between the states |X0> and |P0>,
• the symmetry seen in their wave functions as functions of x and p shown in Figs. 1 and 2 (and see also 1a and 2a, in the side discussion, for more symmetry.)

This symmetry would seem to imply that if we could put any object, including an elementary particle, in the state |P0>, we ought to be able to put it into a state |X0>, too.

But this logic won’t follow, because in fact there’s an even more important asymmetry. The states |X0> and |P0> differ crucially. The difference lies in their energy.

Who cares about energy?

There are a couple of reasons we should care, closely related. First, just as there is a relationship between position and momentum, there is a relationship between time and energy: energy is deeply related to how wave functions evolve over time. Second, energy has its limits, and we’re going to see them violated.

Energy and How Wave Functions Change Over Time

In 1920s quantum physics, the evolution of our particle’s wave function depends on how much energy it has… or, if its energy is not definite, on the various possible energies that it may have.

Definite Momentum and Energy: Simplicity

This change with time is simple for the state |P0>, because this state, with definite momentum, also has definite energy. It therefore evolves in a very simple way: it keeps its shape, but moves with a constant speed.

Figure 5: In the state |P0>, shown in Fig. 1 of Part 1, the particle has definite momentum and energy and moves steadily at constant speed; the particle’s position is completely unknown at all times.

How much energy does it have? Well, in 1920s quantum physics, just as in pre-1900 physics, the motion-energy E of an isolated particle of definite momentum p is

• E = p2/2m

where m is the particle’s mass. Where does this formula come from? In first-year university physics, we learn that a particle’s momentum is mv and that its motion-energy is mv2/2 = (mv)2/2m = p2/2m; so in fact this is a familiar formula from centuries ago.

Less Definite Momentum and Energy: Greater Complexity

What about the compromise states mentioned in Part 1, the ones that lie somewhere between the extreme states |X0> and |P0>, in which the particle has neither definite position nor definite momentum? These “Gaussian wave packets” appeared in Fig. 3 and 4 of Part 1. The state of Fig. 3 has less definite momentum than the |P0> state, but unlike the latter, it has a rough location, albeit broadly spread out. How does it evolve?

As seen in Fig. 6, the wave still moves to the left, like the |P0> state. But this motion is now seen not only in the red and blue waves which represent the wave function itself but also in the probability for where to find the particle’s position, shown in the black curve. Our knowledge of the position is poor, but we can clearly see that the particle’s most likely position moves steadily to the left.

Figure 6: In a state with less definite momentum than |P0>, as shown in Fig. 3 of Part 1, the particle has less definite momentum and energy, but its position is roughly known, and its most likely position moves fairly steadily at near-constant speed. If we watched the wave function for a long time, it would slowly spread out.

What happens if the particle’s position is better known and the momentum is becoming quite uncertain? We saw what a wave function for such a particle looks like in Fig. 4 of Part 1, where the position is becoming quite well known, but nowhere as precisely as in the |X0> state. How does this wave function evolve over time? This is shown in Fig. 7.

Figure 7: In a state with better known position, shown in Fig. 4 of Part 1, the particle’s position is initially well known but becomes less and less certain over time, as its indefinite momentum and energy causes it to move away from its initial position at a variety of possible speeds.

We see the wave function still indicates the particle is moving to the left. But the wave function spreads out rapidly, meaning that our knowledge of its position is quickly decreasing over time. In fact, if you look at the right edge of the wave function, it has barely moved at all, so the particle might be moving slowly. But the left edge has disappeared out of view, indicating that the particle might be moving very rapidly. Thus the particle’s momentum is indeed very uncertain, and we see this in the evolution of the state.

This uncertainty in the momentum means that we have increased uncertainty in the particle’s motion-energy. If it is moving slowly, its motion-energy is low, while if it is moving rapidly, its motion-energy is much higher. If we measure its motion-energy, we might find it anywhere in between. This is why its evolution is so much more complex than that seen in Fig. 5 and even Fig. 6.

Near-Definite Position: Breakdown

What happens as we make the particle’s position better and better known, approaching the state |X0> that we want to put our electron in to see if it can really be thought of as a true particle within the methods of 1920s quantum physics?

Well, look at Fig. 8, which shows the time-evolution of a state almost as narrow as |X0> .

Figure 8: the time-evolution of a state almost as narrow as |X0>.

Now we can’t even say if the particle is going to the left or to the right! It may be moving extremely rapidly, disappearing off the edges of the image, or it may remain where it was initially, hardly moving at all. Our knowledge of its momentum is basically nil, as the uncertainty principle would lead us to expect. But there’s more. Even though our knowledge of the particle’s position is initially excellent, it rapidly degrades, and we quickly know nothing about it.

We are seeing the profound asymmetry between position and momentum:

• a particle of definite momentum can retain that momentum for a long time,
• a particle of definite position immediately becomes one whose position is completely unknown.

Worse, the particle’s speed is completely unknown, which means it can be extremely high! How high can it go? Well, the closer we make the initial wave function to that of the state |X0>, the faster the particle can potentially move away from its initial position — until it potentially does so in excess of the cosmic speed limit c (often referred to as the “speed of light”)!

That’s definitely bad. Once our particle has the possibility of reaching light speed, we need Einstein’s relativity. But the original quantum methods of Heisenberg-Born-Jordan and Schrödinger do not account for the cosmic speed limit. And so we learn: in the 1920s quantum physics taught in undergraduate university physics classes, a state of definite position simply does not exist.

Isn’t it Relatively Easy to Resolve the Problem?

But can’t we just add relativity to 1920s quantum physics, and then this problem will take care of itself?

You might think so. In 1928, Dirac found a way to combine Einstein’s relativity with Schrödinger’s wave equation for electrons. In this case, instead of the motion-energy of a particle being E = p2/2m, Dirac’s equation focuses on the total energy of the particle. Written in terms of the particle’s rest mass m [which is the type of mass that doesn’t change with speed], that total energy satisfies the equation

For stationary particles, which have p=0, this equation reduces to E=mc2, as it must.

This does indeed take care of the cosmic speed limit; our particle no longer breaks it. But there’s no cosmic momentum limit; even though v has a maximum, p does not. In Einstein’s relativity, the relation between momentum and speed isn’t p=mv anymore. Instead it is

which gives the old formula when v is much less than c, but becomes infinite as v approaches c.

Not that there’s anything wrong with that; momentum can be as large as one wants. The problem is that, as you can see for the formula for energy above, when p goes to infinity, so does E. And while that, too, is allowed, it causes a severe crisis, which I’ll get to in a moment.

Actually, we could have guessed from the start that the energy of a particle in a state of definite position |X0> would be arbitrarily large. The smaller is the position uncertainty Δx, the larger is the momentum uncertainty Δp; and once we have no idea what the particle’s momentum is, we may find that it is huge — which in turn means its energy can be huge too.

Notice the asymmetry. A particle with very small Δp must have very large Δx, but having an unknown location does not affect an isolated particle’s energy. But a particle with very small Δx must have very large Δp, which inevitably means very large energy.

The Particle(s) Crisis

So let’s try to put an isolated electron into a state |X0>, knowing that the total energy of the electron has some probability of being exceedingly high. In particular, it may be much, much larger — tens, or thousands, or trillions of times larger — than mc2 [where again m means the “rest mass” or “invariant mass” of the particle — the version of mass that does not change with speed.]

The problem that cannot be avoided first arises once the energy reaches 3mc2 . We’re trying to make a single electron at a definite location. But how can we be sure that 3mc2 worth of energy won’t be used by nature in another way? Why can’t nature use it to make not only an electron but also a second electron and a positron? [Positrons are the anti-particles of electrons.] If stationary, each of the three particles would require mc2 for its existence.

If electrons (not just the electron we’re working with, but electrons in general) didn’t ever interact with anything, and were just incredibly boring, inert objects, then we could keep this from happening. But not only would this be dull, it simply isn’t true in nature. Electrons do interact with electromagnetic fields, and with other things too. As a result, we can’t stop nature from using those interactions and Einstein’s relativity to turn 3mc2 of energy into three slow particles — two electrons and a positron — instead of one fast particle!

For the state |X0> with Δx = 0 and Δp = infinity, there’s no limit to the energy; it could be 3mc2, 11mc2, 13253mc2, 9336572361mc2. As many electron/positron pairs as we like can join our electron. The |X0> state we have ended up with isn’t at all like the one we were aiming for; it’s certainly not going to be a single particle with a definite location.

Our relativistic version of 1920s quantum physics simply cannot handle this proliferation. As I’ve emphasized, an isolated physical system has only one wave function, no matter how many particles it has, and that wave function exists in the space of possibilities. How big is that space of possibilities here?

Normally, if we have N particles moving in d dimensions of physical space, then the space of possibilities has N-times-d dimensions. (In examples that I’ve given in this post and this one, I had two particles moving in one dimension, so the space of possibilities was 2×1=2 dimensional.) But here, N isn’t fixed. Our state |X0> might have one particle, three, seventy one, nine thousand and thirteen, and so on. And if these particles are moving in our familiar three dimensions of physical space, then the space of possibilities is 3 dimensional if there is one particle, 9 dimensional if there are three particles, 213 dimensional if there are seventy-one particles — or said better, since all of these values of N are possible, our wave function has to simultaneously exist in all of these dimensional spaces at the same time, and tell us the probability of being in one of these spaces compared to the others.

Still worse, we have neglected the fact that electrons can emit photons — particles of light. Many of them are easily emitted. So on top of everything else, we need to include arbitrary numbers of photons in our |X0> state as well.

Good Heavens. Everything is completely out of control.

How Small Can An Electron Be (In 1920s Quantum Physics?)

How small are we actually able to make an electron’s wave function before the language of the 1920s completely falls apart? Well, for the wave function describing the electron to make sense,

• its motion-energy must be below mc2, which means that
• p has to be small compared to mc , which means that
• Δp has to be small compared to mc , which means that
• by Heisenberg’s uncertainty principle, Δx has to be large compared to h/(4πmc)

This distance (up to the factor of 1/4π) is known as a particle’s Compton wavelength, and it is about 10-13 meters for an electron. That’s about 1/1000 of the distance across an atom, but 100 times the diameter of a small atomic nucleus. Therefore, 1920s quantum physics can describe electrons whose wave functions allow them to range across atoms, but cannot describe an electron restricted to a region the size of an atomic nucleus, or of a proton or neutron, whose size is 10-15 meters. It certainly can’t handle an electron restricted to a point!

Let me reiterate: an electron cannot be restricted to a region the size of a proton and still be described by “quantum mechanics”.

As for neutrinos, it’s much worse; since their masses are much smaller, they can’t even be described in regions whose diameter is that of a thousand atoms!

The Solution: Relativistic Quantum Field Theory

It took scientists two decades (arguably more) to figure out how to get around this problem. But with benefit of hindsight, we can say that it’s time for us to give up on the quantum physics of the 1920s, and its image of an electron as a dot — as an infinitesimally small object. It just doesn’t work.

Instead, we now turn to relativistic quantum field theory, which can indeed handle all this complexity. It does so by no longer viewing particles as fundamental objects with infinitesimal size, position x and momentum p, and instead seeing them as ripples in fields. A quantum field can have any number of ripples — i.e. as many particles and anti-particles as you want, and more generally an indefinite number. Along the way, quantum field theory explains why every electron is exactly the same as every other. There is no longer symmetry between x and p, no reason to worry about why states of definite momentum exist and those of definite position do not, and no reason to imagine that “particles” [which I personally think are more reasonably called “wavicles“, since they behave much more like waves than particles] have a definite, unchanging shape.

The space of possibilities is now the space of possible shapes for the field, which always has an infinite number of dimensions — and indeed the wave function of a field (or of multiple fields) is a function of an infinite number of variables (really a function of a function [or of multiple functions], called a “functional”).

Don’t get me wrong; quantum field theory doesn’t do all this in a simple way. As physicists tried to cope with the difficult math of quantum field theory, they faced many serious challenges, including apparent infinities everywhere and lots of consistency requirements that needed to be understood. Nevertheless, over the past seven decades, they solved the vast majority of these problems. As they did so, field theory turned out to agree so well with data that it has become the universal modern language for describing the bricks and mortar of the universe.

Yet this is not the end of the story. Even within quantum field theory, we can still find ways to define what we mean by the “size” of a particle, though doing so requires a new approach. Armed with this definition, we do now have clear evidence that electrons are much smaller than protons. And so we can ask again: can an elementary “particle” [wavicle] have zero size?

We’ll return to this question in later posts.

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.

This week I’ll be at the University of Michigan in Ann Arbor, and I’ll be giving a public talk for a general audience at 4 pm on Thursday, December 5th. If you are in the area, please attend! And if you know someone at the University of Michigan or in the Ann Arbor area who might be interested, please let them know. (For physicists: I’ll also be giving an expert-level seminar at the Physics Department the following day.)

Here are the logistical details:

The Quantum Cosmos and Our Place Within It

Thursday, December 5, 2024, 4:00-5:00 PM ; Rackham Graduate School , 4th Floor Amphitheatre

Click to enlarge map

When we step outside to contemplate the night sky, we often imagine ourselves isolated and adrift in a vast cavern of empty space—but is it so? Modern physics views the universe as more full than empty. Over the past century, this unfamiliar idea has emerged from a surprising partnership of exotic concepts: quantum physics and Einstein’s relativity. In this talk I’ll illustrate how this partnership provides the foundation for every aspect of human experience, including the existence of subatomic particles (and the effect of the so-called “Higgs field”), the peaceful nature of our journey through the cosmos, and the solidity of the ground beneath our feet.

Particle physicists describe how elementary particles behave using a set of equations called their “Standard Model.” How did they become so confident that a set of math formulas, ones that can be compactly summarized on a coffee cup, can describe so much of nature?

My previous “Celebrations of the Standard Model” (you can find the full set here) have included the stories of how we know the strengths of the forces, the number of types (“flavors” and “colors”) and the electric charges of the quarks, and the structures of protons and neutrons, among others. Along the way I explained how W bosons, the electrically charged particles involved in the weak nuclear force, quickly decay (i.e. disintegrate into other particles). But I haven’t yet explained how their cousin, the electrically-neutral Z boson, decays. That story brings us to a central feature of the Standard Model.

Here’s the big picture. There’s a super-important number that plays a central role in the Standard Model. It’s a sort of angle (in a sense that will become clearer in Figs. 2 and 3 below), and is called θw or θweak. Through the action of the Higgs field on the particles, this one number determines many things, including

• the relative masses of the W and Z bosons
• the relative lifetimes of the W and Z bosons
• the relative probabilities for Z bosons to decay to one type of particle versus another
• the relative rates to produce different types of particles in scattering of electrons and positrons at very high energies
• the relative rates for processes involving scattering neutrinos off atoms at very low energies
• asymmetries in weak nuclear processes (ones that would be symmetric in corresponding electromagnetic processes)

and many others.

This is an enormously ambitious claim! When I began my graduate studies in 1988, we didn’t know if all these predictions would work out. But as the data from experiments came in during the 1990s and beyond, it became clear that every single one of them matched the data quite well. There were — and still are — no exceptions. And that’s why particle physicists became convinced that the Standard Model’s equations are by far the best they’ve ever found.

As an illustration, Fig. 1 shows, as a function of sin θw, the probabilities for Z bosons to decay to each type of particle and its anti-particle. Each individual probability is indicated by the size of the gap between one line and the line above. The total probability always adds up to 1, but the individual probabilities depend on the value of θw. (For instance, the width of the gap marked “muon” indicates the probability for a Z to decay to a muon and an anti-muon; it is about 5% at sin θw = 0, about 3% at sin θw = 1/2, and over 15% at sin θw = 1.)

Figure 1: In the Standard Model, once sin θw is known, the probabilities for a Z boson to decay to other particles and their anti-particles are predicted by the sizes of the gaps at that value of sin θw. Other measurements (see Fig. 3) imply sin θw is approximately 1/2 , and thus predict the Z decay probabilities to be those found in the green window. As Fig. 5 will show, data agrees with these predictions.

As we’ll see in Fig. 3, the W and Z boson masses imply (if the Standard Model is valid) that sin θw is about 1/2. Using that measurement, we can then predict that all the various decay probabilities should be given within the green rectangle (if the Standard Model is valid). These predictions, made in the mid-1980s, proved correct in the 1990s; see Fig. 5 below.

This is what I’ll sketch in today’s post. In future posts I’ll go further, showing how this works with high precision.

The Most Important Angle in Particle Physics

Angles are a common feature of geometry and nature: 90 degree angles of right-angle triangles, the 60 degree angles of equilateral triangles, the 104.5 degree angle between the two hydrogen-oxygen bonds in a water molecule, and so many more. But some angles, more abstract, turn out to be even more important. Today I’ll tell you about θw , which is a shade less than 30 degrees (π/6 radians).

Note: This angle is often called “the Weinberg angle”, based on Steven Weinberg’s 1967 version of the Standard Model, but it should be called the “weak-mixing angle”, as it was first introduced seven years earlier by Sheldon Glashow, before the idea of the Higgs field.

This is the angle that lies at the heart of the Standard Model: the smallest angle of the right-angle triangle shown in Fig. 2. Two of its sides represent the strengths g1 and g2 of two of nature’s elementary forces: the weak-hypercharge force and the weak-isospin force. According to the Standard Model, the machinations of the Higgs field transform them into more familar forces: the electromagnetic force and the weak nuclear force. (The Standard Model is often charaterized by the code SU(3)xSU(2)xU(1); weak-isospin and weak-hypercharge are the SU(2) and U(1) parts, while SU(3) gives us the strong nuclear force).

Figure 2: The electroweak right-triangle, showing the angle θw. The lengths of two of its sides are proprtional to the strengths g1 and g2 of the “U(1)” weak-hypercharge force and the “SU(2)” weak-isospin force.

To keep things especially simple today, let’s just say θw = 30 degrees, so that sin θw = 1/2. In a later post, we’ll see the angle is closer to 28.7 degrees, and this makes a difference when we’re being precise.

The Magic Angle and the W and Z Bosons

The Higgs field gives masses to the W and Z bosons, and the structure of the Standard Model predicts a very simple relation, given by the electroweak triangle as shown at the top of Fig. 3:

This has the consequence shown at the top of Fig. 3, rewritten as a prediction

If sin θw = 1/2 , this quantity is predicted to be 1/4 = 0.250. Measurements (mW = 80.4 GeV/c2 and mZ = 91.2 GeV/c2) show it to be 0.223. Agreement isn’t perfect, indicating that the angle isn’t exactly 30 degrees. But it is close enough for today’s post.

Where does this formula for the W and Z masses come from? Click here for details:

Central to the Standard Model is the so-called “Higgs field”, which has been switched on since very early in the Big Bang. The Higgs field is responsible for the masses of all the known elementary particles, but in general, though we understand why the masses aren’t zero, we can’t predict their values. However, there’s one interesting exception. The ratio of the W and Z bosons’ masses is predicted.

Before the Higgs field switched on, here’s how the weak-isospin and weak-hypercharge forces were organized: there were

• 3 weak isospin fields, called W+, W– and W0, whose particles (of the same names) had zero rest mass
• 1 weak-hypercharge field, usually called, X, whose particle (of the same name) had zero rest mass

After the Higgs field switched on by an amount v, however, these four fields were reorganized, leaving

• One, called the electromagnetic field, with particles called “photons” with zero rest mass.
• One, called Z0 or just Z, now has particles (of the same names) with rest mass mZ
• Two, still called W+ and W– , have particles (of the same names) with rest mass mW

Central to this story is θw. In particular, the relationship between the photon and Z and the original W0 and X involves this angle. The picture below depicts this relation, given also as an equation

Figure 3: A photon is mostly an X with a small amount of W0, while a Z is mostly a W0 with a small amount of X. The proportions are determined by θw .

The W+ and W– bosons get their masses through their interaction, via the weak-isospin force, with the Higgs field. The Z boson gets its mass in a similar way, but because the Z is a mixture of W0 and X, both the weak-isospin and weak-hypercharge forces play a role. And thus mZ depends both on g1 and g2, while mW depends only on g2. Thus

where v is the “value” or strength of the switched-on Higgs field, and in the last step I have used the electroweak triangle of Fig. 2.

Figure 3: Predictions (*before accounting for small quantum corrections) in the Standard Model with sin θw = 1/2, compared with experiments. (Top) A simple prediction for the ratio of W and Z boson masses agrees quite well with experiment. (Bottom) The prediction for the ratio of W and Z boson lifetimes also agrees very well with experiment.

A slightly more complex relation relates the W boson’s lifetime tW and the Z boson’s lifetime tZ (this is the average time between when the particle is created and when it decays.) This is shown at the bottom of Fig. 3.

This is a slightly odd-looking formula; while 81 = 92 is a simple number, 86 is a weird one. Where does it come from? We’ll see in just a moment. In any case, as seen in Fig. 3, agreement between theoretical prediction and experiment is excellent.

If the Standard Model were wrong, there would be absolutely no reason for these two predictions to be even close. So this is a step in the right direction. But it is hardly the only one. Let’s check the detailed predictions in Figure 1.

W and Z Decay Probabilities

Here’s what the Standard Model has to say about how W and Z bosons can decay.

W Decays

In this earlier post, I explained that W– bosons can decay (oversimplifying slightly) in five ways:

• to an electron and a corresponding anti-neutrino
• to a muon and a corresponding anti-neutrino
• to a tau and a corresponding anti-neutrino
• to a down quark and an up anti-quark
• to a strange quark and a charm anti-quark

(A decay to a bottom quark and top anti-quark is forbidden by the rule of decreasing rest mass; the top quark’s rest mass is larger than the W’s, so no such decay can occur.)

These modes have simple probabilities, according to the Standard Model, and they don’t depend on sin θw (except through small quantum corrections which we’re ignoring here). The first three have probability 1/9. Moreover, remembering each quark comes in three varieties (called “colors”), each color of quark also occurs with probability 1/9. Altogether the predictions for the probabilities are as shown in Fig. 4, along with measurements, which agree well with the predictions. When quantum corrections (such as those discussed in this post, around its own Fig. 4) are included, agreement is much better; but that’s beyond today’s scope.

Figure 4: The W boson decay probabilities as predicted (*before accounting for small quantum corrections) by the Standard Model; these are independent of sin θw . The predictions agree well with experiment.

Because the W+ and W- are each others’ anti-particles, W+ decay probabilities are the same as those for W–, except with all particles switched with their anti-particles.

Z Decays

Unlike W decays, Z decays are complicated and depend on sin θw. If sin θw = 1/2, the Standard Model predicts that the probability for a Z boson to decay to a particle/anti-particle pair, where the particle has electric charge Q and weak-isospin-component T = +1 or -1 [technically, isospin’s third component, times 2], is proportional to

• 2 (Q/2-T)2 + 2(Q/2)2 = 2+2TQ+Q^2

where I used T2 = 1 in the final expression. The fact that this answer is built from a sum of two different terms, only one of which involves T (weak-isospin), is a sign of the Higgs field’s effects, which typically marry two different types of fields in the Standard Model, only one of which has weak-isospin, to create the more familiar ones.

This implies the relative decay probabilities (remembering quarks come in three “colors”) are

• For electrons, muons and taus (Q=-1, T=-1): 1
• For each of the three neutrinos (Q=0, T=1): 2
• For down-type quarks (Q=-1/3,T=-1) : 13/9 * 3 = 13/3
• For up-type quarks (Q=2/3,T=1): 10/9 * 3 = 10/3

These are shown at left in Fig. 5.

Figure 5: The Z boson decay probabilities as predicted (*before accounting for small quantum corrections) by the Standard Model at sin θw = 1/2 (see Fig. 1), and compared to experiment. The three neutrino decays cannot be measured separately, so only their sum is shown. Of the quarks, only the bottom and charm decays can be separately measured, so the others are greyed out. But the total decay to quarks can also be measured, meaning three of the five quark predictions can be checked directly.

The sum of all those numbers (remembering that there are three down-type quarks and three up-type quarks, but again top quarks can’t appear due to the rule of decreasing rest mass) is:

• 1 + 1 + 1 + 2 + 2 + 2 + 13/3 + 13/3 + 13/3 + 10/3 + 10/3 = 86/3.

And that’s where the 86 seen in the lifetime ratio (Fig. 3) comes from.

To get predictions for the actual decay probabilities (rather than just the relative probabilities), we should divide each relative probability by 86/3, so that the sum of all the probabilities together is 1. This gives us

• For electrons, muons and taus (Q=-1, I=-1): 3/86
• For each of the three neutrinos (Q=0, I=1): 6/86
• For down-type quarks (Q=-1/3,I=-1) : 13/86
• For up-type quarks (Q=2/3,I=1): 10/86

as shown on the right-hand side of Fig. 5; these are the same as those of Fig. 1 at sin θw = 1/2. Measured values are also shown in Fig. 5 for electrons, muons, taus, the combination of all three neutrinos, the bottom quark, the charm quark, and (implicitly) the sum of all five quarks. Again, they agree well with the predictions.

This is already pretty impressive. The Standard Model and its Higgs field predict that just a single angle links a mass ratio, a lifetime ratio, and the decay probabilities of the Z boson. If the Standard Model were significantly off base, some or all of the predictions would fail badly.

However, this is only the beginning. So if you’re not yet convinced, consider reading the bonus section below, which gives four additional classes of examples, or stay tune for the next post in this series, where we’ll look at how things improve with a more precise value of sin θw.

Bonus: Other Predictions of the Standard Model

Many other processes involving the weak nuclear force depend in some way on sin θw. Here are a few examples.

High-Energy Electron-Positron Collisions (click for details)

In this post I discussed the ratio of the rates for two important processes in collisions of electrons and positrons:

• electron + positron ⟶ any quark + its anti-quark
• electron + positron ⟶ muon + anti-muon

This ratio is simple at low energy (E << mZ c2), because it involves mainly electromagnetic effects, and thus depends only on the electric charges of the particles that can be produced.

Figure 6: The ratio of the rates for quark/anti-quark production versus muon/anti-muon production in high-energy electron-positron scattering depends on sin θw.

But at high energy (E >> mZ c2) , the prediction changes, because both electromagnetic and weak nuclear forces play a role. In fact, they “interfere”, meaning that one must combine their effects in a quantum way before calculating probabilities.

[What this cryptic quantum verbiage really means is this. At low energy, if Sf is the complex number representing the effect of the photon field on this process, then the rate for the process is |Sf|2. But here we have to include both Sf and SZ, where the latter is the effect of the Z field. The total rate is not |Sf|2 + |SZ|2 , a sum of two separate probabilities. Instead it is |Sf+SZ|2 , which could be larger or smaller than |Sf|2 + |SZ|2 — a sign of interference.]

In the Standard Model, the answer depends on sin θw. The LEP 2 collider measured this ratio at energies up to 209 GeV, well above mZ c2. If we assume sin θw is approximately 1/2, data agrees with predictions. [In fact, the ratio can be calculated as a function of energy, and thus made more precise; data agrees with these more precise predictions, too.]

Low-Energy Neutrino-Nucleus Collisions (click for details)

When electrons scatter off protons and neutrons, they do so via the electromagnetic force. For electron-proton collisions, this is not surprising, since both protons and electrons carry electric charge. But it’s also true for neutrons, because even though the neutron is electrically neutral, the quarks inside it are not.

By contrast, neutrinos are electrically neutral, and so they will only scatter off protons and neutrons (and their quarks) through the weak nuclear force. More precisely, they do so through the W and Z fields (via so-called “virtual W and Z particles” [which aren’t particles.]) Oversimplifying, if one can

• obtain beams of muon neutrinos, and
• scatter them off deuterons (nuclei of heavy hydrogen, which have one proton and one neutron), or off something that similarly has equal numbers of protons and neutrons,

then simple predictions can be made for the two processes shown at the top of Fig. 7, in which the nucleus shatters (turning into multiple “hadrons” [particles made from quarks, antiquarks and gluons]) and either a neutrino or a muon emerges from the collision. (The latter can be directly observed; the former can be inferred from the non-appearance of any muon.) Analogous predictions can be made for the anti-neutrino beams, as shown at the bottom of Fig. 7.

Figure 7: The ratios of the rates for these four neutrino/deuteron or anti-neutrino/deuteron scattering processes depend only on sin θw in the Standard Model.

The ratios of these four processes are predicted to depend, in a certain approximation, only on sin θw. Data agrees with these predictions for sin θw approximately 1/2.

More complex and detailed predictions are also possible, and these work too.

Asymmetries in Electron-Positron Collisions (click for details)

There are a number of asymmetric effects that come from the fact that the weak nuclear force is

• not “parity-invariant”, (i.e. not the same when viewed in a mirror), and
• not “charge-conjugation invariant” (i.e. not the same when all electric charges are flipped)

though it is almost symmetric under doing both, i.e. putting the world in a mirror and flipping electric charge. No such asymmetries are seen in electromagnetism, which is symmetric under both parity and charge-conjugation separately. But when the weak interactions play a role, asymmetries appear, and they all depend, yet again, on sin θw.

Two classes of asymmetries of great interest are:

• “Left-Right Asymmetry” (Fig. 8): The rate for electron-positron collisions to make Z bosons in collisions with positrons depends on which way the electrons are “spinning” (i.e. whether they carry angular momentum along or opposite to their direction of motion.)
• “Forward-Backward Asymmetry” (Fig. 9): The rate for electron-positron collisions to make particle-antiparticle pairs depends on whether the particles are moving roughly in the same direction as the electrons or in the same direction as the positrons.

Figure 8: The left-right asymmetry for Z boson production, whereby electrons “polarized” to spin one way do not produce Z’s at the same rate as electrons polarized the other way. Figure 9: The forward-backward asymmetry for bottom quark production; the rate for the process at left is not the same as the rate for the process at right, due to the weak nuclear force.

As with the high-energy electron-positron scattering discussed above, interference between effects of the electromagnetic and Z fields, and the Z boson’s mass, causes these asymmetries to change with energy. They are particularly simple, though, both when E = mZ c2 and when E >> mZ c2.

A number of these asymmetries are measurable. Measurements of the left-right asymmetry was made at the Stanford Linear Accelerator Center (SLAC) at their Linear Collider (SLC), while I was a graduate student there. Meanwhile, measurements of the forward-backward asymmetries were made at LEP and LEP 2. All of these measurements agreed well with the Standard Model’s predictions.

A Host of Processes at the Large Hadron Collider (click for details)

Fig. 10 shows predictions (gray bands) for total rates of over seventy processes in the proton-proton collisions at the Large Hadron Collider. Also shown are measurements (colored squares) made at the CMS experiment . (A similar plot is available from the ATLAS experiment.) Many of these predictions, which are complicated as they must account for the proton’s internal structure, depend on sin θw .

Figure 10: Rates for the production of various particles at the Large Hadron Collider, as measured by the CMS detector collaboration. Grey bands are theoretical predictions; color bands are experimental measurements, with experimental uncertainties shown as vertical bars; colored bars with hatching above are upper limits for cases where the process has not yet been observed. (In many cases, agreement is so close that the grey bands are hard to see.)

While minor discrepancies between data and theory appear, they are of the sort that one would expect in a large number of experimental measurements. Despite the rates varying by more than a billion from most common to least common, there is not a single major discrepancy between prediction and data.

Many more measurements than just these seventy are performed at the Large Hadron Collider, not least because there are many more details in a process than just its total rate.

A Fortress

What I’ve shown you today is just a first step, and one can do better. When we look closely, especially at certain asymmetries described in the bonus section, we see that sin θw = 1/2 (i.e. θw = 30 degrees) isn’t a good enough approximation. (In particular, if sin θw were exactly 1/2, then the left-right asymmetry in Z production would be zero, and the forward-backward asymmetry for muon and tau production would also be zero. That rough prediction isn’t true; the asymmetries are small, only about 15%, but they are clearly not zero.)

So to really be convinced of the Standard Model’s validity, we need to be more precise about what sin θw is. That’s what we’ll do next time.

Nevertheless, you can already see that the Standard Model, with its Higgs field and its special triangle, works exceedingly well in predicting how particles behave in a wide range of circumstances. Over the past few decades, as it has passed one experimental test after another, it has become a fortress, extremely difficult to shake and virtually impossible to imagine tearing down. We know it can’t be the full story because there are so many questions it doesn’t answer or address. Someday it will fail, or at least require additions. But within its sphere of influence, it rules more powerfully than any theoretical idea known to our species.

Just a brief note, in a very busy period, to alert those in the Providence, RI area that I’ll be giving a colloquium talk at the Brown University Physics Department on Monday November 18th at 4pm. Such talks are open to the public, but are geared toward people who’ve had at least one full year of physics somewhere in their education. The title is “Exploring The Foundations of our Quantum Cosmos”. Here’s a summary of what I intend to talk about:

The discovery of the Higgs boson in 2012 marked a major milestone in our understanding of the universe, and a watershed for particle physics as a discipline. What’s known about particles and fields now forms a nearly complete short story, an astonishing, counterintuitive tale of relativity and quantum physics. But it sits within a larger narrative that is riddled with unanswered questions, suggesting numerous avenues of future research into the nature of spacetime and its many fields. I’ll discuss both the science and the challenges of accurately conveying its lessons to other scientists, to students, and to the wider public.

If you’re curious to know what my book is about and why it’s called “Waves in an Impossible Sea”, then watching this video is currently the quickest and most direct way to find out from me personally. It’s a public talk that I gave to a general audience at Harvard, part of the Harvard Bookstore science book series.

My intent in writing the book was to illuminate central aspects of the cosmos — and of how we humans fit into it — that are often glossed over by scientists and science writers, at least in the books and videos I’ve come across. So if you watch the lecture, I think there’s a good chance that you’ll learn something about the world that you didn’t know, perhaps about the empty space that forms the fabric of the universe, or perhaps about what “quantum” in “quantum physics” really means and why it matters so much to you and me.

The video contains 35 minutes of me presenting, plus some Q&A at the end. Feel free to ask questions of your own in the comments below, or on my book-questions page; I’ll do my best to answer them.