You might think I’m about to tell a joke. But no, not me. This is serious physics, folks!
Suppose a particle falls into a hole, and, as in a nightmare (or as in a certain 1970s movie featuring light sabers), the walls of the hole start closing in. The particle will just stay there, awaiting the end. But if the same thing happens to a wavicle, the outcome is very different. Like a magician, the wavicle will escape!
Today I’ll explain why.
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ON July 11, 2024
In my role as a teacher and explainer of physics, I have found that the ambiguities and subtleties of language can easily create confusion. This is especially true when well-known English words are reused in scientific contexts, where they may or may not be quite appropriate.
The word “particle”, as used to describe “elementary particles” such as electrons and quarks and photons, is arguably one such word. It risks giving the wrong impression as to what electrons etc. are really like. For this reason, I sometimes replace “particle” with the word “wavicle”, a word from the 1920s that has been getting some traction again in recent years. [I used it in my recent book, where I also emphasized the problems of language in communicating science.]
In today’s post I want to contrast the concepts of particle, wave and wavicle. What characterizes each of these notions? Understanding the answer is crucial for anyone who wants to grasp the workings of our universe.
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ON July 9, 2024
From the structure of the Standard Model of particle physics, one might wonder if the electron, muon and tau, so similar except for their masses, might really be the same object seen in three different guises. Last week, starting with a post for general readers, and then looking at the situation in more detail (Part 1 of this two-part series), I showed one way in which this idea fails to agree with experiment. Today I’ll give you another point of failure, focused on a property of particles known as “spin”.
What is “spin” in physics? It’s related to the word “spin” in English, but with some adjustment. I’ll write a longer article about spin elsewhere, but here’s a brief introduction.
The Earth “spins”, both in the English sense and in the physics sense: it rotates. The same is true of a tennis ball. The rotation can be changed when the ball interacts with a tennis racket or with the ground. It can also be changed when it interacts with another tennis ball. In fact, if two balls strike each other, some of the spin of one ball can be transferred to the other, as in Fig. 1.
Elementary “particles” can have spin too, which can be (in part) changed by or transferred to other objects that they interact with. But this type of spin is somewhat different from ordinary rotation.
For example, an electron “spins”. Always. An interaction can change the direction of an electron’s spin, but it cannot change its overall amount. I’ll tell you what I mean by “amount” in a moment, but the quantity of the amount is a famous number:
where h, Planck’s constant, shows up whenever quantum physics matters. This amount of spin is tiny; if a tennis ball had this amount of spin, it wouldn’t have even rotated once since the universe was born. But for an electron, it’s quite a lot.
Usually, as shorthand, one takes the ℏ to be implicit, and just says that “the electron has spin 1/2”. I’ll do that in what follows.
How can we visualize this spinning? It’s not easy. It’s best not to visualize an electron as a small rotating ball; it’s the wrong picture. Quantum field theory, the modern theory of “particles”, gives a different picture. Just as we should think of light as a wave made of wave-like photons, we should think of the electron as a wave — specifically, a wave in the electron field. [This wave is not to be confused with a wave function, which is something else]. A wave in the electron field can rotate in ways that a little ball cannot. This, however, is hard to draw, and in any case is a story for another day.
The important point is that the spin of an elementary “particle” is not like the simple rotational spinning of a tennis ball, even though it is in the same category. An electron has spin intrinsically, by its very nature; there is no electron without its spin, just as there is no electron without its internal energy and rest mass (as emphasized in my recent book, Chapter 17). That’s certainly not true for a tennis ball, which can have any spin, including none.
These details won’t matter much in what follows, though, so let’s step away from these subtleties and move on.
So far, I’ve suggested two types of spinning:
Both contribute to the surprising way that physicists use the word “spin”.
Suppose we have an object that is made of multiple elementary particles. Its total angular momentum involves combining the intrinsic angular momentum of its elementary particles with the ordinary angular momentum that the particles may have as they move around each other. Physicists now do something unexpected: they refer to the object’s total angular momentum as its spin. They do this even though the object is not elementary, meaning that its spin may potentially combine both intrinsic spinning and ordinary spinning of the objects inside it.
So the real meaning of spin, as particle physicists use the term, is this: it is the total angular momentum of an isolated object — an object that may be elementary, or that may itself be formed from multiple elementary objects. That means that the spin of an object like an atom, made of electrons, protons and neutrons, or of a proton, made of quarks, antiquarks and gluons, may potentially arise from multiple sources.
For example, let’s take a hydrogen atom, made from an electron and a proton. (For starters we’ll treat both of these subatomic particles as though they were elementary. We’ll return to their possible internal structure later.)
I’ll refer in the following to four atomic states, illustrated below in Fig. 2.
These four states are sketched in Figure 2. (Spin-flipped versions of the 2s and 2p states exist but are not shown.)
The fact that the ground state has spin 0, and yet the 2p state has larger spin specifically due to the electron’s motion around the proton, illustrates the main point of this post. If an object is made from multiple constituent objects, nothing can prevent those constituents from moving around one another. That means they can have ordinary (or “orbital”) angular momentum, which then contributes, along with the constituents’ spin, to the combined object’s spin — i.e., to its total angular momentum.
Therefore, an object that is not elementary, and so contains multiple objects inside it, will inevitably have excited states with different amounts of spin. Indeed, the hydrogen atom has excited states of total angular momentum 0, 1, 2, 3, and so on. All atoms exhibit similar behavior.
The same applies for protons, which aren’t elementary either. The proton has spin 1/2, but its excited states have spin 1/2, 3/2, 5/2, 7/2, and so on. The first excited state of the proton, the Delta, has spin 3/2. This is most easily understood as a rearrangement of the spins of the quarks, gluons and anti-quarks that it contains (though the exact rearrangement is not obvious, due to the complexity of a proton; see also chapter 6 of the book.) The next excited state, the p(1440), has spin 1/2 like the proton. But many other excited states have been observed, with spin 3/2, 5/2, 7/2, 9/2, and perhaps even 11/2.
A string, such as one finds in string theory, is another object with internal constituents, which one might call bits-of-string. A string can always be spun faster. That’s why, in the superstring theory that is sometimes touted as a potential “theory-of-everything” (or in less grandiose language, a complete theory of space, fields and particles), there are states of all possible spin — any integer times 1/2. Unfortunately, were this really the theory of our universe, the higher-spin states of the string would probably have far too much mass for us to make them in near-term experiments, putting this prediction of the theory out of reach for now.
But string theory isn’t just useful in this rarified context. It can also be used to describe the physics of “hadrons” — objects made from quarks and gluons, including protons. All indications from experiments and numerical calculations do indeed suggest that hadrons come in all possible spins; this includes the excited states of the proton already mentioned above. (That said, the higher the spin, the harder it is to make the states, making it more and more challenging to observe them.)
None of this is true for electrons, muons or taus, all of which have spin 1/2. No electron-like particle with spin 3/2 has ever been observed.
This argues strongly against the electron, muon and tau all being made from the same object. Atoms, protons and strings all have excited states with the same spin as the ground state, but at roughly the same mass, they also have excited states with more spin. If the muon were an excited state of the electron, we would expect to see an object with spin 3/2 that has a mass comparable to the muon, and certainly below the mass of the tau. Such a state would easily have been observed decades ago, so it doesn’t exist.
Are there loopholes to this logic? Yes. It is possible, in special circumstances, for the excited states with higher spin to have much larger masses than the excited states which share the same spin as the ground state. This is a long story which I won’t try to tell here, but examples arises in the context of extra dimensions, and others in the related context of exotic theories of quark-like and gluon-like objects (with buzzwords such as “AdS/CFT” or “gauge/string duality”).
However, it’s hard to apply the loophole to the muon and tau. In such a scenario, the electron should have many more cousins than just two, and some of the others should have observed by now.
Furthermore, data now confirms that both the tau and muon get their rest mass from the Higgs field; see Fig. 3. For such particles, the loopholes I just mentioned don’t apply.
We must also recall the arguments given in the first part of this series. If muons and taus are excited states of electrons, it should be possible for any sufficiently energetic collision to turn an electron into a muon or tau, and for decays via photons to do the reverse. But these processes are not observed.
In short, the properties of the electron, muon and tau disfavor the idea that they are somehow secretly the same object in three different quantum states. The explanation of their similarities must lie elsewhere.
ON May 15, 2024
A couple of days ago, I posted an article describing how the size of a quantum object, such as a proton or electron, can be measured. This isn’t obvious. For example, scientists say that an electron spreads out and is wave-like, and yet that it has no size. This apparent contradiction needs resolution. While I addressed this puzzle in the book‘s chapter 17, I didn’t do so in detail, and so I wrote this article to fill in the gaps.
Now, in response to a reader’s question, I’ve added a section to the end of the article, entitled “Estimating the Object’s Size From Its Excited States”. There I explain in more detail how one goes from simple measurements, which confirm that a proton’s size isn’t zero, to an actual estimate of a proton’s size. The discussion is a little more technical than the rest of the article; you will probably need first-year physics to follow it. But I hope that some readers will find it useful!
ON May 1, 2024
Quantum physics is certainly confusing.
How can both these things be true?
Well, to clarify this, let’s look at objects that do have an intrinsic size, such as protons and neutrons. How are their sizes actually determined? While this question is addressed in the book’s Chapter 17 (see Figure 40 and surrounding text, and footnote 2), I didn’t go into much detail there.
To supplement what’s in the book, I have written a webpage that outlines two classic methods that are used to measure the intrinsic size of a proton, or of any ultra-microscopic object.
When these same methods are used on electrons, one finds no evidence of any finite size, and so one concludes that their intrinsic size (if any) is too small to measure.
ON April 29, 2024
In recent talks at physics departments about my book, I have emphasized that the elementary “particles” of nature — electrons, photons, quarks and so on — are really little waves (or, to borrow a term that was suggested by Sir Arthur Eddington in the 1920s, “wavicles”.) But this notion inevitably generates confusion. That’s because of another wavy concept that arises in “quantum mechanics” —the quantum physics of the 1920s, taught to every physics student. That concept is Erwin Schrödinger’s famous “wave function”.
It’s natural to guess that wave functions and wavicles are roughly the same. In fact, however, they are generally unrelated.
Before quantum physics came along, field theory was already used to predict the behavior of ordinary waves in ordinary settings. Field theory is useful for sound waves in air, seismic waves in rock, and waves on water.
Quantum field theory, the quantum physics that arose out of the 1940s and 1950s, adds something new: it tells us that waves in quantum fields are made from wavicles, the gentlest possible waves. A photon, for instance, is a wavicle of light — the dimmest possible flash of light.
By contrast, a wave function describes a system of objects operating according to quantum physics. Importantly, it’s not one wave function per object — it’s one wave function per system of interacting objects. That’s true whether the objects in the system are particles in motion, or something as simple as particles that cannot move, or something as complex as fields and their wavicles.
One of the points I like to make, to draw the distinction between these two types of waves in quantum physics, is this:
Daniel Whiteson, the well-known Large Hadron Collider physicist, podcaster and popular science writer, liked this phrasing so much that he quoted the second half on X/Twitter. Immediately there were protests. One person wrote “Everything that has ever hurt anyone was in truth a wave function.” Another posted a video of an unfortunate incident involving the collision between a baseball and a batter, and said: “the wave function of this baseball disagrees.”
It’s completely understandable why there’s widespread muddlement about this. We have two classes of waves floating around in quantum physics, and both of them are inherently confusing. My aim today is to make it clear why a wave function couldn’t hurt a fly, a cat, or even a particle.
Wavicles, such as photons or electrons, are real objects. X-rays are a form of light, and are made of photons — wavicles of light. A strong beam of X-ray photons can hurt you. The photons travel across three-dimensional space carrying energy and momentum; they can strike your body, damage your DNA, and thereby cause you to develop cancer.
The wave function associated with the X-ray beam, however, is not an object. All it does is describe the beam and its possible futures. It tells us what the beam’s energy may be, but it doesn’t have any energy, and cannot inflict the beam’s energy on anything else. The wave function tells us where the beam may go, but itself goes nowhere. Though it describes a beam as it crosses ordinary three-dimensional space, the wave function does not itself exist in three-dimensional space.
In fact, if the X-ray beam is interacting with your body, then the X-ray beam cannot be said to have its own wave function. Instead, there is only one wave function — one that describes the beam of photons, your atoms, and the interactions between your atoms and the photons.
More generally, if a bunch of objects interact with each other,the multiple interacting objects form a single indivisible system, and a single wave function must describe it. The individual objects do not have separate wave functions.
This point is already illustrated by Schrödinger’s famous (albeit unrealistic) example of the cat in a box that is both dead and alive. The box contains a radioactive atom which will, via a quantum process, eventually “decay” [i.e. transform itself into a new type of atom, releasing a subatomic particle in the process], but may or may not have done so yet. If and when the atom does decay, it triggers the poisoning of the cat. The cat’s survival or demise thus depends on a quantum effect, and it becomes a party to a quantum phenomenon.
It would be a mistake to say that “the atom has a wave function” (or even worse, that “the atom is a wave function”) and that this wave function can kill the cat. To do so would miss Schrödinger’s point. Instead, the wave function includes the atom, the killing device, and the cat.
Initially, when the box is closed, the three are independent of one another, and so they have a relatively simple wave function which one may crudely sketch as
This wave function represents our certainty that the atom has not yet decayed, the murder weapon has not been triggered, and the cat is still alive.
But this initial wave function immediately begins evolving into a more complicated form, one which depends on two time-varying complex numbers C and D, with |C|2 + |D|2 = 1:
The wave function is now a sum of two “branches” which describe two distinct possibilities, and assigns them probabilities |C|2 and |D|2, the former gradually decreasing and the latter gradually increasing. [Note these two branches are added together in the wave function; its branches cannot be rearranged into wave functions for the atom, device and cat separately, nor can the two branches ever be separated from one another.]
In no sense has the wave function killed the cat; in one of its branches the cat is dead, but the other branch describes a live cat. And in no sense did the “wave function of the atom” or “of the device” kill the cat, because no such wave functions are well-defined in this interacting system.
Let’s now look at an example, similar to the cat but more concrete, and easier to think about and draw.
Let’s take two particles [not wavicles] A and B. These particles travel only in a one-dimensional line, instead of in three-dimensional space.
Initially, particle B is roughly stationary and particle A comes flying toward it. There are two possible outcomes.
In the second case, we may indeed say that A “hurts” B — at least in the sense of causing B to recoil suddenly.
Before we answer quantum questions, let’s first think about how one might describe this situation in a world without quantum physics. There are several ways of depicting what may happen.
We could describe how the particles move within their one-dimensional universe, using arrows to illustrate their motions over time. In the figure below, I show both the “hi” possibility and the “ow” possibility.
Or, using an animation, we can show the time-dependence more explicitly and more clearly. In the second case, I’ve assumed that B has more mass than A, so it recoils more slowly from the blow than does A.
But we could also describe how the particles move as a system in their two-dimensional space of possibilities. Each point in that space tells us both where A is and where B is; the point’s location along the horizontal axis gives A’s position, and its location along the vertical axis gives B’s position. At each moment, the system is at one point in that space; over time, as A and B change their positions, the location of the system in that two-dimensional space also changes.
The motion of the system for the Hi and Ow cases is shown in Fig. 3. It has exactly the same information as Fig. 1, though depicted differently and somewhat more precisely. Instead of following the two dots that correspond to the two particles as they move in one dimension, we now depict the whole system as a single diamond that tells us where both particles are located.
In the first part of Fig. 3, we see that B’s position is at the center of the space, and remains there, while A’s position goes from the far right to the far left; compare to Fig. 1. In the second part of Fig. 3, A and B collide at the center, following which A moves to positive position, B moves to negative position, and correspondingly, within its space of possibilities, the system as a whole moves down and to the right.
And finally, let’s look at an animation in the two-dimensional space of possibilities. Compare this to Fig. 3, and then to Fig. 2, noting that it has the same information.
In Fig. 4, we see that
Now, what if the system is described using quantum physics? What’s up with the system’s wave function?
As noted, we do not have a wave function for particle A and a separate wave function for particle B, and so we do not have a collision of two wave functions. Instead, we have a single wave function for the A/B system, one which describes the collision of the two particles and the aftermath thereof.
It is impossible to depict the wave function using the one-dimensional universe that the particles live in. The wave function itself only exists in the space of possibilities. So in quantum physics, there are no analogues to Figs. 1 and 2.
Meanwhile, although we can depict the wave function at any one moment in the two dimensional space, we cannot simply use arrows to depict how it changes over time. This is because we cannot view the “Hi” and “Ow” cases as distinct, and as something we can draw in two separate figures, as we did in Figs. 3 and 4. In quantum physics, we have to view both possibilities as described by the same wave function; they are not distinct outcomes.
The only option we have is to do an animation in the two-dimensional space of possibilities, somewhat similar to Fig. 4, but without separating the Hi and Ow outcomes. There’s just one wave function that shows both the “Hi” and “Ow” cases together. The square of this wave function, which gives the probabilities for the system’s possible futures, is sketched in Fig. 5.
[Note that what is shown is merely a sketch! It is not the true wave function, which requires a complete solution of Schrödinger’s wave equation. While the solution is well known, it is tricky to get all the details of the math exactly right, and I haven’t had the time. I’ll try to add the complete and correct solution at a later date.]
Compare Fig. 5 with Fig. 4, recalling that the probability of the Hi case is 30% and the probability of the Ow case is 70%. Both possibilities appear in the wave function, with the branch corresponding to the Ow case carrying larger weight than the branch corresponding to the Hi case.
In contrast to Fig. 4, the key differences are that
Notice that even though particle A has a 70% probability of “hurting” B, the wave function itself does not, and cannot, “hurt” B. It just describes what may happen; it contains both A and B, and describes both the possibility of Hi and Ow. The wave function isn’t a part of the A/B system, and doesn’t participate in its activities. Instead, it exists outside the system, as a means for understanding that system’s behavior.
A system has a wave function, but individual objects in the system do not have wave functions. That’s the key point.
To be fair, it is true that when objects or groups of objects in a system interact weakly enough, we may imagine the system’s full wave function as though it were a simple combination of wave functions for each object or group of objects. That is true of the initial Schrödinger cat wave function, which is a product of separate factors for the atom, device and cat, and is also true of the wave function in Fig. 5 before the collision of A and B. But once significant interactions occur, this is no longer the case, as we see in the later-stage Schrödinger cat wave function and in Fig. 5 after the collision.
A wave function expresses how the overall system moves through the full space of its possibilities, and grows ever more complex when there are many possible paths for a system to take. This is completely unrelated to wavicles, which are objects that move through physical space and create physical phenomena, forming parts of a system that itself is described by a wave function.
As a final comment: I’ve given this simple example because it’s one of the very few that one can draw start to finish.
Wave functions of systems with just one particle are misleading, because they make it easy to imagine that there is one wave function per particle. But with more than one particle, the only wave functions that can easily be depicted are those of two particles moving in one dimension, such as the one I have given you. Such examples offer a unique opportunity to clarify what a wave function is and isn’t, and it’s therefore crucial to appreciate them.
Any wave function more complicated than this becomes impossible to draw. Here are some things to consider.
These difficulties explain why one almost never sees a proper discussion of wave functions of complicated systems, and why wave functions of fields are almost never described and are never depicted.
ON April 25, 2024
One of the most challenging aspects of writing a book or blog about the universe (as physicists currently understand it) is that both writer and reader must confront the concept of fields. The problem isn’t that fields are intrinsically that complicated. It’s that they are an unfamiliar abstraction — and novel abstractions of any sort are always difficult both for a writer to describe and for a reader to grasp.
What I’ll do today is give an explanation of fields that is complementary to the one that appears in the book’s chapters 13 and 14. The book’s approach is slow, methodical, and detailed, but today’s will be more of an overview, brief and relatively shallow, and presented in a different order. You will likely come away with many unanswered questions, but the book should help with that. And if the book and today’s post combined are still not enough, you can ask a question in the comments below, or on the book question page.
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ON March 21, 2024
I have posted my fourth article discussing zero-point energy. (Here are the first, the second, and the third, which covered respectively the zero-point energy of a ball on a spring, a guitar string, and a bosonic field whose particles have zero mass, such as the electromagnetic field.) Today’s article looks at fields whose particles have non-zero mass, such as the Higgs field, and fermionic fields, such as the electron field and quark fields. It presents some simple formulas, and in its final section, shows how one can obtain them using math.
Along the way we’ll encounter the idea of “supersymmetry” and its failed role in the cosmological constant problem. This is a word which (for some good historical reasons) generates a lot of heat. But stay calm; I’m neither promoting it nor bashing it. Supersymmetry is an idea which proves useful as a conceptual tool, whether it is true in nature or not.
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ON February 19, 2024
My two new webpages from earlier this week addressed the zero-point energy for the simple case of a ball on a spring and for the much richer case of a guitar string; the latter served as a warmup to today’s webpage, the third in this series, which explains the zero-point energy of a field of the universe. This subject will lead us head-first into the cosmological constant problem. As before, the article starts with a non-mathematical overview, and then obtains the results stated in the overview using pre-university math (except for one aside.) [As always, please comment if you spot typos or find some of the presentation especially confusing!]
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ON February 16, 2024
In my last post, I introduced a new webpage concerning “zero-point energy”, the core concept that lies at the heart of the hierarchy puzzle. I have now posted the next webpage in the series, which extends the notion of zero-point energy to a slightly more complex system, an ordinary metal string of finite length. It’s a bit schematic, but it serves to teach us some crucial and surprising lessons about the zero-point energy associated with the internal vibrations of physical objects. [Please let me know in the comments if you spot any typos or if you find some of the presentation especially confusing!]
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ON February 14, 2024
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