# Samuel Beckett’s Guide to Particles and Antiparticles

I was 12 years old when I first encountered this quote by Samuel Beckett:

“Every word is like an unnecessary stain on silence and nothingness.”

That quote impressed me quite a bit at the time. It appeared to my young self to be simultaneously profound, important, and impossible to understand. Now, nineteen years later, I’m still not sure I understand what Beckett meant by that short sentence. But I nonetheless find that its dark Zen has worked itself into me indelibly.

The Beckett quote comes to mind in particular as I sit down to write again about quantum field theory (QFT). QFT, to recap, is the science of describing particles, the most basic building blocks of matter. QFT concerns itself with how particles move, how they interact with each other, how they arise from nothingness, and how they disappear into nothingness again. As a framing idea or motif for QFT, I can’t resist presenting an adaptation of Beckett’s words as they might apply to the idea of particles and fields:

“Every particle is an unnecessary defect in a smooth and featureless field.”

Of course, it is not my intention to depress anyone with existential philosophy. But in this post I want to introduce, in a pictorial way, the idea of particles as defects. The discussion will allow me to draw some fun pictures, and also to touch on some deeper questions in physics like “what is the difference between matter and antimatter?”, “what is meant by rest mass energy?”, “what are fermions and bosons?”, and “why does the universe have matter instead of nothing?”

***

Let’s start by imagining that you have screwed up your zipper.

A properly functioning zipper, in the pictorial land of this blog post, looks like this:

But let’s say that your zipper has become dysfunctional, perhaps because of an overly hasty zip, and now looks more like this:

This zipper is in a fairly unhappy state. There is zipping a defect right in the middle of it: the two teeth above the letter “B” have gotten twisted around each other, and now all the zipper parts in the neighborhood of that pair are bending and bulging with stress. You could relieve all that stress, with a little work, by pulling the two B teeth back around each other.

But maybe you don’t want to fix it. You could, instead, push the two teeth labeled “A” around each other, and similarly for the two “C” teeth. Then the zipper would end up in a state like this:

Now you may notice that your zipping error looks not like one defect, but like two defects that have become separated from each other. The first defect is a spot where two upper teeth are wedged between an adjacent pair of lower teeth (this is centered more or less around the number “1”). The second defect is a spot where two lower teeth are wedged between two upper teeth (“2”).

You can continue the process of moving the defects away from each other, if you want. Just keep braiding the teeth on the outside of each defect around each other. After a long while of this process, you might end up with something that looks like this:

In this picture the two types of defects have been moved so far from each other that you can sort of forget that they came from the same place. You can now describe them independently, if you want, in terms of how hard it is to move them around and how much stress they create in the zipper. If you ever bring them back together again, though, the two defects will eliminate each other, and the zipper will be healed.

My contention in this post is that what we call particles and antiparticles are something like those zipper defects. Empty space (the vacuum) is like an unbroken zipper, with all the teeth sewed up in their proper arrangement. In this sense, empty space can be called “smooth”, or “featureless”, but it cannot really be said to have nothing in it. The zipper is in it, and with the zipper comes the potential for creating pairs of equal and opposite defects that can move about as independent objects. The potential for defects, and all that comes with them, is present in the zipper itself.

Like the zipper, the quantum fields that pervade all of space encode within themselves the potential for particles and antiparticles, and dictate the rules of how they behave. Creating those particles and antiparticles may be difficult, just as moving two teeth around each other in the zipper can be difficult, and such creation results in lots of “stress” in the field. The total amount of stress created in the field is the analog of the rest mass energy of a particle (as defined by Einstein’s famous $$E = mc^2$$, which says that a particle with large mass takes a lot of energy to create). Once created, the particles and antiparticles can move away from each other as independent objects, but if they ever come back together all of their energy is released, and the field is healed.

Since the point of this post is to be “picture book”, let me offer a couple more visual analogies for particles and antiparticles.  While the zipper example is more or less my own invention, the following examples come from actual field theory.

Imagine now a long line of freely-swinging pendulums, all affixed to a central axle.  And let’s say that you tie the ends of adjacent pendulums together with elastic bands. Perhaps something like this:

In its rest state, this field will have all its pendulums pointing downward. But consider what would happen if someone were to grab one of the pendulums in the middle of the line and flip it around the axle. This process would create two defects, or “kinks”, in the line of pendulums. One defect is a 360 degree clockwise flip around the axis, and the other is a 360 degree counterclockwise flip. Something like this:

As with the zipper, each of these kinks represents a sort of frustrating state for the field. The universe would prefer for all those pendulums to be pulled downward with gravity, but when there is a kink in the line this is impossible. Consequently, there is a large (“rest mass”) energy associated with each kink, and this energy can only be released when two opposite kinks are brought together.

(By the way, the defects in this “line of pendulums” example are an example of what we call solitons.  Their motion is described by the so-called Sine-Gordon equation.  You can go on YouTube and watch a number of videos of people playing with these kinds of things.)

In case you’re starting to worry that these kinds of particle-antiparticle images are only possible in one dimension, let me assuage your fears by offering one more example, this time in two dimensions. Consider a field that is made up of arrows pointing in the 2D plane. These arrows have no preferred direction that they like to point in, but each arrow likes to point in the same direction as its neighbors. In other words, there is an energetic cost to having neighboring arrows point in different directions. Consequently, the lowest energy arrangement for the field looks something like this:

If an individual arrow is wiggled slightly out of alignment with its neighbors, the situation can be righted easily by nudging it back into place. But it is possible to make big defects in the field that cannot be fixed without a painful, large-scale rearrangement. Like this vortex:

Or this configuration, which is called an antivortex:

The reason for the name antivortex is that a vortex and antivortex are in a very exact sense opposite partners to each other, meaning that they are created from the vacuum in pairs and they can destroy each other when brought together. Like this:

(A wonky note: this “field of arrows” is what one calls a vector field, as opposed to a scalar field. What I have described is known as the XY model. It will make an appearance in my next post as well.)

***

Now that you have some pictures, let me use them as a backdrop for some deeper and more general ideas about QFT.

The first important idea that you should remember is that in a quantum field, nothing is ever allowed to be at rest. All the pieces that make up the field are continuously jittering back and forth: the teeth of the zipper are rattling around and occasionally twisting over each other; the pendulums are swinging back and forth, and on rare occasion swinging all the way over the axle; the arrows are shivering and occasionally making spontaneous vortex-antivortex pairs. In this way the vacuum is never quiet. In fact, it is completely correct to say that from the vacuum there are always spontaneously arising particle-antiparticle pairs, although these usually annihilate each other quickly after appearing. (Which is not to say that they never make their presence felt.)

If you read the previous post, you might also notice a big difference between the way I talk about fields here and the way I talked about them before. The previous post employed much more pastoral language, going on about gentle “ripples” in an infinite quantum “mattress”. But this post uses the harsher imagery of “unhappy defects” that cannot find rest. (Perhaps you should have expected this, since the last post was “A children’s picture book”, while this one is “Samuel Beckett’s Guide”.) But the two types of language were actually chosen to reflect a fundamental dichotomy of the fields of nature.

In particular, the previous post was really a description of what we call bosonic fields (named after the great Indian physicist Satyendra Bose). A bosonic field houses quantized ripples that we call particles, but it admits no concept of antiparticles. All excitations of a bosonic field are essentially the same as all others, and these excitations can blend with each other and overlap and interfere and, generally speaking, happily coincide at the same place and the same time. Equivalently, one can say that bosonic particles are the same as bosonic antiparticles.  In a bosonic field with many excitations, all particles are one merry slosh and there is literally no way of saying how many of them you have. In the language of physics, we say that for bosonic fields the particle number is “not conserved”.

The pictures presented in this post, however, were of fermionic fields (after the Italian Enrico Fermi). The particles of fermionic fields — fermions — are very different objects than bosons. For one thing, there is no ambiguity about their number: if you want to know how many you have, you just need to count how many “kinks” or “vortices” there are in your field (their number is conserved). Fermions also don’t share space well with each other – there is really no way to put two kinks or two vortices on top of each other, since they each have hard “cores”. These properties of fermions, together, imply that they are much more suitable for making solid, tangible matter than bosons are. You don’t have to worry about a bunch of fermions constantly changing their number or collapsing into a big heap. Consequently, it is only fermions that make up atoms (electrons, protons, and neutrons are all fermions), and it is only fermions that typically get referred to as matter.

Of course, bosonic fields still play an important role in nature.  But they appear mostly in the form of so-called force carriers. Specifically, bosons are usually seen only when they mediate interactions between fermions.  This mediation is basically a process in which some fermion slaps the sloshy sea of a bosonic field, and thereby sets a wave in motion that ends up hitting another fermion. It is in this way that our fermionic atoms get held together (or pushed apart), and fermionic matter abides.

Finally, you might be bothered by the idea that particles and antiparticles are always created together, and are therefore seemingly always on the verge of destruction. It is true, of course, that a single particle by itself is perfectly stable. But if every particle is necessarily created together with the agent of its own destruction, an antiparticle, then why isn’t any given piece of matter subject to being annihilated at any moment? Why do the solid, matter-y things that we see around us persist for so long? Why isn’t the world plagued by randomly-occurring atomic blasts?

In other words, where are all the anti-particles?

The best I can say about this question is that it is one of the biggest puzzles of modern physics. (It is often, boringly, called the “baryon asymmetry” problem; I might have called it the “random atomic bombs” problem.) To use the language of this post, we somehow ended up in a universe, or at least a neighborhood of the universe, where there are more “kinks” than “antikinks”, or more “vortices” than “antivortices”.  This observation brings up a rabbit hole of deep questions. For example, does it imply that there is some asymmetry between matter and antimatter that we don’t understand? Or are we simply lucky enough to live in a suburb of the universe where one type of matter predominates over the other? How unlikely would that have to be before it seems too unlikely to swallow?

And, for that matter, are we even allowed to use the fact of our own existence as evidence for a physical law? After all, if matter were equally common as antimatter, then no one would be around to ask the question.

And perhaps Samuel Beckett would have preferred it that way.

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About Brian Skinner

Brian Skinner is a physicist who specializes in the theory of strongly-interacting many-body systems. He is currently a postdoctoral researcher at MIT, and he blogs at Gravity and Levity

### Comments

1. Spin-glass models of magnetic materials are sort of like this right? Each domain is a sort of locally happy neighborhood, but the overall material has a lot of frustration?

There was a very interesting paper in Axelrod’s complexity of cooperation book that did a metaphor mapping in the other direction, to politics. The authors used spin-glass models to predict alignment in WW II from pre-war data.

A question that bothers me is that while it is easiest to understand such frustrated systems as perturbations to unperturbed systems, does that actually imply that the universe evolved in some way from an unperturbed system? What’s wrong with pictures of the universe that *start* with a baryonic imbalance, other than that they are not as satisfyingly simple as symmetric models?

• Yes, the arrow model is very much a model of magnetism as well. It’s going to appear next time also, in a very different guise.
That’s one of the great things about field theory. Any reasonable simply field that you write down is bound to be a good description of many different types of problems, since in the end all that usually matters is whether it has the right symmetries.

There’s actually nothing wrong with the idea that the universe somehow started with more particles than antiparticles, and then had no mechanism to get rid of them. That might even be the right answer (that there was some asymmetry present in/before the big bang itself), but it isn’t very satisfying because it’s the sort of answer that shuts down further questioning.

As for WWII: I haven’t seen that explanation in particular, but these (and similar) magnetism-inspired models are quite canonical and tend to get used and abused in all sorts of ways. (Like a recent claim that they can explain hipsterism: http://www.washingtonpost.com/news/storyline/wp/2014/11/11/the-mathematician-who-proved-why-hipsters-all-look-alike/ ) In my opinion, though, such adaptations rarely have any predictive power. The person who wants to model something just tweaks the properties of the field until what they want comes out of it.

2. Also, this is *by far* the best explanation I’ve ever read of particles/antiparticles via a metaphor. I was never happy with them just popping out of symmetries in Dirac’s equations. The zipper metaphor is brilliant, and probably a viable basis for a whole pop-science book called “The Unzipped Universe”, complete with the right kind of joke.

• Thanks for the kind words. I admit to taking no small amount of glee in being able to write the sentence “Let’s start by imagining that you have screwed up your zipper.”

3. Pliny says:

Taleb wrote “it looks like the secret of life is antifragility.”

This article implies that antifragility is the secret of *matter* itself — stressors applied to a Tohu wa-bohu[1] quantum field.

(No, I’m not trying to back into a vindication of Judeo-Christian creation narratives, but the metaphor is too rich to pass up. And I’m sure some theologian has already gone there.)

4. In physics, we like the words “protected” or “conserved” to express the idea of non-fragility.

• Carl says:

“Conserved” means robust, in the language of Taleb. An example of an anti-fragile measure would be entropy, which always increases or stays the same.

5. Aaron Thornton says:

Thanks again Brian.
As I sit and look at the animated gif of the vector field, watching the vortex and anti-vortex distortions appear and disappear in an endless dance, I can’t help project this to a grander and more abstract scale: Life as the vortex and death as the anti-vortex.

• juan says:

Or how about death as vortex, and life as anti-vortex?

• Aaron Thornton says:

I was actually thinking life as matter and death as anti matter.
Either way… you can run, but you can’t hide.

6. Kevin says:

These posts have been excellent. Great to see Ribbonfarm evolve and fracture in all sorts of new directions.

Two questions come to mind:

1) What relationship exists between QFT and massive molar movements, such as the formation of stars or the shuffling of tectonic plates? How might you use fermionic and bosonic fields to explain the stratification of the earth into layers of increasing density (lightest atoms in the exosphere, heaviest atoms deep in the inner core)?

2) What metaphysical implications exist pertaining to QFT? Do any modern philosophers (of which you might be aware), touch on metaphysics in a way which is specifically compatible with QFT?

Thanks for the posts, looking forward to learning more in the future.

7. Hi Kevin,

Thanks! I’m glad you’re enjoying them. There should be two more from me in the next couple months.

1) The short answer is that there really isn’t one, but I can’t restrain myself from offering a longer, more philosophical answer.
In physics we tend to have a philosophy that is expressed by the phrase “more is different” (and was laid down by Philip Anderson, probably the greatest physicist that you never heard of). What that phrase expresses is the idea that every level of reality emerges from the level below it, but in an unpredictable way that must be studied in its own right. Thus, even though QFT is the “most basic” theory of nature that we have, it cannot, even in principle, tell you anything about how nature works at the scale of human beings (or larger). QFT can successfully describe fundamental particles, but when a bunch of those particles get together they create a whole other type of science called Chemistry, and the rules of Chemistry must be studied in their own right and cannot be extracted just from knowing QFT. Similarly, simply knowing Chemistry cannot teach you the rules of biology. And simply knowing biology cannot teach you the principles of psychology, and so forth. It is in that sense that I cannot, just by knowing things about QFT, say anything about what happens at the level of stars and planets. Maybe some people have tried, but I very much doubt that anything can be learned that way.

2) I have to admit that I don’t know much about metaphysics, but the truth is that quantum mechanics by itself has all sorts of unresolved philosophical issues. The biggest one is “what is a wave function?” — we don’t know whether the wave function (which is the most basic object in quantum mechanics) represents something real in the universe, or only represents our knowledge about something in the universe. (see here, for example, https://www.sciencenews.org/blog/context/physicists-debate-whether-quantum-math-real-atoms .) In the face of such profound ignorance, it’s hard to build a consistent philosophy.

8. The vortex is depicted as a counterclockwise swirl of arrows; can you depict the antivortex as a clockwise one? I know you didn’t, but the asymmetry disturbs me. If you put a clockwise swirl next to a counterclockwise swirl, I would think the winding number of the vector as you walk around both swirls would vanish, and I’m thinking it’s this winding number that describes the “topological charge”.

• Hi John,

The vortex and the anti-vortex as I drew them actually do have opposite topological charge, in the sense that walking a closed loop around the vortex/antivortex produces a +/-360 degree rotation of the spin.

The XY “arrow” field certainly does admit both clockwise and counterclockwise vortices, and clockwise and counterclockwise antivortices. (i.e., you could just take the mirror image of all the pictures I drew.) But a clockwise vortex is not the antiparticle of a counterclockwise vortex, in the sense that you can’t conjure the pair of them out of the vacuum by making only local rearrangements of the field.

• Thanks! That makes it sound like there are two kinds of charge, “clockwise vortex number” and “counterclockwise vortex number”. And since this is all about the topology of vector fields (as far as I can tell), that makes it sound like there are two ways to compute a number from a vector field in some region of the plane. I’m puzzled because I only know one, the “index” of the vector field, as in page 5 here:

http://web.stanford.edu/~amwright/PoincareHopf.pdf

This “index” is the winding number of the vector field as we go around the boundary of the region: if the region is topologically a disk and the vector field doesn’t vanish on the boundary, it gives us a continuous map from the circle to itself.

If you’re planning to talk about this more later, I can wait.

• No, your original intuition is correct. There is only one type of charge in this problem: the total winding number. There isn’t a separate conservation of clockwise and counterclockwise vortex number.

Here’s a better answer that I should have given to your first question. In the pictures I drew above, my elementary defects were the the vortex, with charge +1, and the antivortex with charge -1. But I can change what my elementary defects look like by making a gauge transformation. If I rotate all spins by 180 degrees, then I just change the signs of the two defects. But if I rotate all spins by 90 degrees, then the vortex looks like an electric charge, with field lines pointing radially (either inward or outward). The antivortex still looks qualitatively the same, buts its axes are redefined.

So in some descriptions, the fundamental defects of the XY model don’t contain vortices at all, but only radial charges and “antivortices” of the sort that I drew. (This is the way it’s presented, for example, in Kardar’s book on the statistical mechanics of fields.)

In that sense, the basic symmetry between clockwise vortices and counterclockwise vortices is a gauge symmetry. What you want your +1 defects and -1 defects to look like is your choice.

• Thanks! My sense of order is restored.

9. Michael says:

You mentioned in your post about how difficult it is to find real connections between levels of “understanding.” Like how the quantum level and the relativity level both are proven to exist, but it’s difficult to find the actual relationship between the two. Would you say that the same difficulties lie between superstring theory and quantum physics?

10. Unfortunately I know almost literally nothing about superstring theory. So I’m probably not qualified to answer your question!