But when you’re a graduate student or postdoc struggling to make a career in physics, that quote rarely feels true. Instead, you are usually made to feel like productivity and technical ability are the qualities that will make or break your career.

But Leo Kadanoff was someone who made that quote feel true.

Leo Kadanoff, one of the true giants in theoretical physics during the last half century, passed away just a few days ago. Kadanoff’s work was marked by its depth of thought and its relentless creativity. I’m sure that over the next week many people will be commemorating his life and his career.

But I thought it might be worth telling a brief story about my own memories of Leo Kadanoff, however minor they may be.

During the early part of 2014, I had started playing with some ideas that were well outside my area of expertise (if, indeed, I can be said to have any such area). I thought these projects were pretty cool, but I was tremendously unconfident about them. My lack of confidence was actually pretty justified: I was highly ignorant about the fields to which these projects properly belonged, and I had no reason to think that anyone else would find them interesting. I was also working in the sort-of-sober environment of the Materials Science Division at Argonne National Laboratory, and I was afraid that at any moment my bosses would tell me to shape up and do real science instead of nonsense.

In a moment of insecure hubris (and trust me, that combination of emotions makes sense when you’re a struggling scientist), I wrote an email to Leo Kadanoff. I sent him a draft of a manuscript (which had already been rejected twice without review) and asked him whether he would be willing to give me any comments. The truth is that the work really had no connection to Kadanoff, or to any of his past or present interests. I just knew that he was someone who had wide-ranging interests and a history of creativity, and I wasn’t sure who else to write to.

His reply to me was remarkable. He told me that he found the paper interesting, and that I should come give a seminar and spend a day at the University of Chicago. I quickly took him up on the offer, and he slotted me into his truly remarkable seminar series called “Computations in Science.” (“The title is old,” he said, “inherited from the days when that title would bring in money. We are closer to ‘concepts in science’ or maybe ‘all things considered.’ ”)

When I arrived for the seminar, Kadanoff was the first person to meet me. He had just arrived himself, by bike. Apparently at age 77 he still rode his bike to work every day. We had a very friendly conversation for an hour or so, in which we talked partly about science and partly about life, and in which he gave me a brief guide to theater and music in the city of Chicago. (At one point I mentioned that my wife was about to start medical residency, which is notorious for its long and stressful hours. He sympathized, and said “I am married to a woman who has been remarkably intelligent all of her life, except during her three years of residency.”)

When it came time for the talk to start, I was more than a little nervous. Kadanoff stood in front of the room to introduce me.

“Brian is a lot like David Nelson …” he began.

[And here my eyes got wide. David Nelson is another giant in theoretical physics, well-respected and well-liked by essentially everyone. So I was bracing myself for some outrageous compliment.]

“… he grew up in a military family.”

I don’t think that line was meant as a joke. But somehow it put me in a good mood, and the rest of the day went remarkably well. The seminar was friendly, and the audience was enthusiastic and critical (another combination of emotions that goes very well together in science). In short, it was a beautiful day for me, and I basked in the atmosphere that surrounded Kadanoff in Chicago. It seemed to me a place where creativity and inquisitiveness were valued intensely, and I found it immensely energizing and inspiring.

Scientists love to tell the public about how their work is driven by the joy of discovery and the pleasure of figuring things out. But rarely does it feel so directly true as it did during my visit to University of Chicago.

On the whole, the truth is that I didn’t know Leo Kadanoff that well. My interactions with him didn’t extend much beyond one excellent day, a few emails, and a few times where I was in the audience of his talks. But when Kadanoff was around, I really felt like science and the profession of scientist lived up to their promise.

It’s pretty sad to think that I will probably never get that exact feeling again.

Take a moment, if you like, and listen to Kadanoff talk about his greatest work. It starts with comic books.

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*You can read it here.*

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Specifically, this problem:

The problem was written into this year’s Higher Maths exam in Scotland, and has since been the source of much angst for Scottish high schoolers and many Twitter jokes for everyone else.

As with most word problems, I’m sure that what confounded people was making the translation between a verbal description of the problem and a set of equations. The actual math problem that needs to be solved is pretty standard for a calculus class. It just comes down to finding the minimum of the function (which is one of the things that calculus is absolutely most useful for). In practical terms, that means taking the derivative and setting it equal to zero.

But it turns out that there is a more clever way to solve the problem that doesn’t require you to know any calculus or take any derivatives. It has fewer technical steps (and therefore comes with a smaller chance of screwing up your calculation somewhere along the way), but more steps of logical thinking. And it goes like this.

The problem is essentially asking you to find the path of shortest time for a crocodile moving from one point to another. If the crocodile were just walking on land, this would be easy: the quickest route is always a straight line. The tricky part is that the crocodile has to move partly on land and partly on water, and the water section is slower than the land section.

If you want to know the speed of the crocodile on land and on water, you can pretty much read them off directly from the problem statement. The problem gives you the equation . The quantity in that equation represents the path length for the on-land section, and the quantity is the path length for the water section. (That square root is the length of the hypotenuse of a triangle with side lengths and — apparently the river is 6 meters wide.) Since , this means that the on-land speed is m/s, and the water velocity is m/s (remember that the units of time were given in tenths of a second, and not seconds — not that it matters for the final answer).

That was just interpreting the problem statement. Now comes the clever part.

The trick is to realize that the problem of “find the shortest time path across two areas with different speed” is not new. It’s something that nature does continually whenever light passes from one medium to another:

I’m talking, of course, about Fermat’s principle: any time you see light go from one point to another, you can be confident that it took the shortest time path to get there. And when light goes from one medium to another one where it has a different speed, it bends. (Like in the picture above: light moves slower through the glass, so the light beam bends inward in order to cross through the glass more quickly.)

The bending of light is described by Snell’s law:

,

where and are the speeds in regions and , and and are the angles that the light makes with the surface normal.

Since our crocodile is solving the exact same problem as a light ray, it follows that its motion is described by the *exact same equation*. Which means this:

Here, m/s is the crocodile speed on land, and m/s is its speed in the river. The sine of is , and the sine of is .

So in the end the fastest path for the crocodile is the one that satisfies

.

If you solve that equation (square both sides and rearrange), you’ll get the correct answer: m.

So knowing some basic optics will give you a quick solution to the crocodile problems.

This happy coincidence brings to mind a great Richard Feynman quote: “Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.” It turns out that this particular tapestry had both light rays and crocodiles in it.

By the way, this post has a very cool footnote. It turns out that ants frequently have to solve a version of this same problem: they need to make an efficient trail from the anthill to some food source, but the trail passes over different pieces of terrain that have different walking speeds.

And, as it turns out, ants understand how to follow Fermat’s principle too! (original paper here)

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You are invited to teach a class to a group of highly-motivated high school students. It can be about absolutely any topic, and can last for as little as 5 minutes or as long as 9 hours.

What topic would you choose for your class?

As it happens, this is not just a hypothetical question for me at the moment. In November MIT is hosting its annual MIT Splash event, and the call for volunteer teachers is almost exactly what is written in the quote above. Students, staff, and faculty from all over MIT are invited to teach short courses on a topic of their choosing, and the results are pretty wild.

A few of my favorite courses from last year:

- The History of Video Game Music
- How to Create a Language
- Cryptography for People Without a Computer
- Build a Mini Aeroponic Farm
- Calculating Pi With a Coconut
- Advanced Topics in Murder

So now I would like to turn to you, dear blog readers, for help.

What should I teach about? Please let me know, in the comments, what you think about either of these two questions:

- If you were a high school student, what kind of class would you want to go to?
- If you were in my position, what kind of class would you want to teach?

The two ideas that come to mind immediately are:

- Quantum Mechanics with middle school math

Use Algebra 1 – level math to figure out answers to questions like: What is wave/particle duality? How big is an atom? How do magnets work? What is quantum entanglement?

- The Math Behind Basketball Strategy

Learn about some of the difficult strategic decisions that basketball teams are faced with, and see how they can be described with math. Then solve a few of them yourself!

Imagining yourself as a high school student, which of those two sounds better to you? Any suggestions for alternative ideas or refinements?

**UPDATE:** You can find my courses listed on the MIT Splash catalog here. Thanks for all your helpful comments, everyone! This should be a lot of fun.

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* *

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 , 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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You can read the post here.

As a teaser, I’ll give you my preferred picture of a Cooper pair:

which I think is an upgrade over the typical illustrations.

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This may surprise you, but the answer to the question in the title leads to some profound quantum mechanical phenomena that fall under the umbrella of the Berry or geometric phase, the topic I’ll be addressing here. There are also classical manifestations of the same kind of effects, a prime example being the Foucault pendulum.

Let’s now return to the original question. Ultimately, whether or not one returns to oneself after walking in a closed loop depends on what you are and where you’re walking, as I’ll describe below. The geometric phase is easier to visualize classically, so let me start there. Let’s consider a boy, Raj, who is pictured below:

Now, Raj, for whatever reason, wants to walk in a rectangle (a closed loop). But he has one very strict constraint: he can’t turn/twist his body while he’s walking. So if Raj starts his walk in the top right corner of the rectangle (pictured below) and then walks forward normally, he has to start side-stepping when he reaches the bottom right corner to walk to the left. Similarly, on the left side of the rectangle, he is constrained to walk backwards, and on the top side of the rectangle, he must side-step to the right. The arrow in the diagram below is supposed to indicate the direction which Raj faces as he walks. When Raj returns to the top right corner, he ends up exactly in the same place that he did when he started — not very profound at all! But now let’s consider the case where Raj is not walking on a plane, but walking on the surface of a sphere:

Again, the arrow is supposed to indicate the direction that Raj is facing. This time, Raj starts his trek at the north pole, heads to Quito, Ecuador on the equator, then continues his walk along the equator and heads back up to the north pole. Notice that on this journey, even though Raj obeys the non-twisting constraint, he ends up facing a different direction when he returns to the north pole! Even though he has returned to the same position, something is slightly different. We call this difference anholonomy.

Why did anholonomy result in the spherical case and not the flat rectangle case? Amazingly, it turns out to be related to the hairy ball theorem (don’t ask me how it got its name). Crudely speaking, the hairy ball theorem states that if you have a ball covered with hairs, you can’t comb the hairs straight without leaving at least one little bald spot or tuft. In the image below you can see a little tuft on both the top and bottom of the sphere:

The sphere which Raj traversed had a little bald spot at the north pole, leading to the anoholonomy.

Now before moving onto the Foucault pendulum, I want to explicitly state the items that were critical in obtaining the anholonomy for the spherical case: (i) The object that is transported must have a direction (i.e. be a vector); (ii) The object must be a transported on a surface on which one cannot properly comb hair.

Now, how does the Foucault pendulum get tied up in all this? Well, the Foucault pendulum, swinging in Paris, does not oscillate in the same plane after the earth makes a full 24-hour rotation. This difference in angle between the original and the next-day plane of oscillation is also an anholonomy, except the earth rotates instead of the pendulum taking a walk. Check out this great animation from Wikipedia below:

If the pendulum was at the north (south) pole, the pendulum would come back to itself after a 24-hour rotation of the earth, (). If the pendulum was at the equator, the pendulum would not change its oscillation plane at all (). Now, depending on the latitude in the northern hemisphere where a Foucault pendulum is set up, the pendulum will make an angle between and after the earth makes its daily rotation. The north pole case, in light of the animation above, can be inferred from this image by imagining the earth rotating about its vertical axis:

The concept on anholonomy in the quantum mechanical case can actually be pictured quite similarly to the classical case described above. In quantum mechanics, we describe particles using a wavefunction, which in a very basic sense is also a vector. The vector does not exist in “real space” but what physicists refer to as “Hilbert space”. Nonetheless, the geometrical game I played with classical anholonomy can also be played in this abstract “Hilbert space”. The main difference is that *the anholonomy angle becomes a phase factor in the quantum realm*. The correspondence is as so:

(Classical) (Quantum)

The expression on the right is precisely the Berry/geometric phase. In the quantum case, regarding the two criteria above, we already have (i) a “vector” in the form of the wavefunction — so all we need is (ii) an appropriate surface on which hair cannot be combed straight. It turns out that in quantum mechanics, there are many ways to do this, but the most famous is undoubtedly the case of the Aharonov-Bohm effect.

In the Aharonov-Bohm experiment, one prepares a beam of electrons, splits the beam and passes them on either side of the solenoid, recombining them on the other side. A schematic of this experimental steup is shown below:

In the image, **B** labels the magnetic field and **A** is the vector potential. While many readers are probably familiar with the magnetic field, the vector potential may not be as household a concept. The vector potential was originally thought up by Maxwell, and considered to be a mathematical oddity. He realized that one could obtain **B **by measuring the curl of **A** at each point in space, but **A** was not given any physical meaning.

Now, it doesn’t immediately seem like this experiment would give one a geometric phase, especially considering the fact that the magnetic field *outside* the solenoid is *zero*. But let’s take look at the pattern for the vector potential outside the solenoid (top-down view):

Interestingly, the vector potential outside the solenoid looks like it would have a tuft in the center! Criterion number (ii) may therefore potentially be met. The one question left to be answered is this: can the vector potential actually “rotate” the electron wavefunction (or “vector”)? The answer to that question deserves a post to itself, and perhaps Brian or I can fill that hole in the future, but the answer seems to be emphatically in the affirmative.

The equation describing the relationship between the anholonomy angle and the vector potential is:

where is the rotation (or anholonomy) angle, the integral is over the closed loop of the electron path, and is just a proportionality constant.

The way to think about the equality is as so: is an infinitesimal “step” that the electron takes, much in the way that Raj took steps earlier. At each step the wavefunction is rotated a little compared to the previous step by an amount , dictated by the vector potential. When I add up all the little rotations caused by over the entire path of the electrons, I get the integral around the closed loop.

Now that we have the anholonomy angle, we need to use the classical quantum relation from above. This gives us a phase difference of between the electrons that go to the right and left of the solenoid. Whenever there is a non-zero phase difference, one should always be able to measure it using an interference experiment — and this is indeed the case here.

An experiment consisting of an electron beam fired at double-slit interference setup coupled with a solenoid demonstrates this interference effect most profoundly. On the setup to the left, the usual interference pattern is set up due to the path length difference of the electrons. On the setup to the right, the entire spectrum is shifted because the extra phase factor from the anholonomy angle.

Again, let me emphasize that there is no magnetic field in the region in where the electrons travel. This effect is due purely to the geometric effect of the anholonomy angle, a.k.a. the Berry phase, and the geometric effect arises in relation to the swirly tuft of hair!

So next time you’re taking a long walk, think about how much the earth has rotated while you’ve been walking and whether you really end up where you started — chances are that something’s just a little bit different.

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Got one?

I predict, using my psychic powers, that you were much more likely to have thought of a number that begins with 1, 2, or 3 rather than a number that begins with 7, 8, or 9.

As it turns out, the probability is about four times higher. In fact, the probability of having a particular first digit decreases monotonically with the value the digit (1 is a more common first digit than 2, 2 is more common than 3, and so on). And the odds of you having picked a number that starts with 1 are about seven times higher than the odds of you having picked a number that starts with 9.

This funny happenstance is part of a larger observation called Benford’s law. Broadly speaking, Benford’s law says that the lower counting numbers (like 1, 2, and 3) are disproportionately likely to be the first digit of naturally-occurring numbers.

In this post I’ll talk a little bit about Benford’s law, its quantitative form, and how one can think about it.

But first, as a fun exercise, I decided to see whether Benford’s law holds for the numbers I personally tend to use and care about.

(Here I feel I must pause to acknowledge how deeply, ineluctably nerdy that last sentence reveals me to be.)

So I made a list of the physical constants that I tend to think about — or, at least, of the ones that occurred to me at the moment of making the list. These are presented below in no particular order, and with no particular theme or guarantee for completeness and non-redundancy (i.e., some of the constants on this list can be made by combining others).

After a quick look-over, it’s pretty clear that this table has a lot more numbers starting with 1 than numbers starting with 9. A histogram of first digits in this table looks like this:

Clearly, there are more small digits than large digits. (And somehow I managed to avoid any numbers that start with 4. This is perhaps revealing about me.)

As far as I can tell, there is no really satisfying proof of Benford’s law. But if you want to get some feeling for where it comes from, you can notice that those numbers on my table cover a really wide range of values: ranging in scale from (the Planck length) to (the sun’s mass). (And no doubt they would cover a wider range if I were into astronomy.) So if you wanted to put all those physical constants on a single number line, you would have to do it in logarithmic scale. Like this:

The funny thing about a logarithmic scale, though, is that it distorts the real line, giving more length to numbers beginning with lower integers. For example, here is the same line from above, zoomed in to the interval between 1 and 10:

You can see in this picture that the interval from 1 to 2 is much longer than the interval from 9 to 10. (And, just to remind you, the general rule for logarithmic scales is that the same interval separates any two numbers with the same ratio. So, for example, 1 and 3 are as far from each other as 2 and 6, or 3 and 9, or 500 and 1500.) If you were to choose a set of numbers by randomly throwing darts at a logarithmic scale, you would naturally get more 1’s and 2’s than 8’s and 9’s.

What this implies is that if you want a quantitative form for Benford’s law, you can just compare the lengths of the different intervals on the logarithmic scale. This gives:

,

where is the value of the first digit and is the relative abundance of that digit.

If you have a large enough data set, this quantitative form of Benford’s law tends to come through pretty clearly. For example, if you take all 335 entries from the list of physical constants provided by NIST, then you find that the abundance of different first digits is described by the formula above with pretty good quantitative accuracy:

Now, if you don’t like the image of choosing constants of nature by throwing darts at a logarithmic scale, let me suggest another way to see it: Benford’s law is what you’d get as the result of a *random walk* using multiplicative steps.

In the conventional random walk, the walker steps randomly to the right or left with steps of constant length, and after a long time ends up at a random position on the number line. But imagine instead that the random walker takes steps of constant *multiplicative* value — for example, at each “step” the walker could have his position multiplied by either 2/3 or 3/2. This would correspond to steps that appeared to have constant length on the logarithmic scale. Consequently, after many steps the walker would have a random position on the logarithmic axis, and so would be more likely to end up in one of those wider 1–2 than in the shorter 8–9 bins.

The upshot is that one way to think about Benford’s law is that the numbers we have arise from a process of multiplying many other “randomly chosen” numbers together. This multiplication naturally skews our results toward numbers that begin with low digits.

By the way, for me the notion of “randomly multiplying numbers together” immediately brings to mind the process of doing homework as an undergraduate. This inspired me to grab a random physics book off my shelf (which happened to be Tipler and Llewellyn’s *Modern Physics*, 3rd edition) and check the solutions to the homework problems in the back.

So the next time you find yourself trying to randomly guess answers, remember Benford’s law.

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First of all, don’t panic.

I’m going to try in this post to introduce you to quantum field theory, which is probably the deepest and most intimidating set of ideas in graduate-level theoretical physics. But I’ll try to make this introduction in the gentlest and most palatable way I can think of: with simple-minded pictures and essentially no math.

To set the stage for this first lesson in quantum field theory, let’s imagine, for a moment, that you are a five-year-old child. You, the child, are talking to an adult, who is giving you one of your first lessons in science. Science, says the adult, is mostly a process of figuring out what things are made of. Everything in the world is made from smaller pieces, and it can be exciting to find out what those pieces are and how they work. A car, for example, is made from metal pieces that fit together in specially-designed ways. A mountain is made from layers of rocks that were pushed up from inside the earth. The earth itself is made from layers of rock and liquid metal surrounded by water and air.

This is an intoxicating idea: everything is made from something.

So you, the five-year-old, start asking audacious and annoying questions. For example:

*What are people made of*?

People are made of muscles, bones, and organs.

*Then what are the organs made of?*

Organs are made of cells.

*What are cells made of?*

Cells are made of organelles.

*What are organelles made of?*

Organelles are made of proteins.

*What are proteins made of?*

Proteins are made of amino acids.

*What are amino acids made of?*

Amino acids are made of atoms.

*What are atoms made of?*

Atoms are made of protons, neutron, and electrons.

*What are electrons made of?*

Electrons are made from the electron field.

*What is the electron field made of?*

…

And, sadly, here the game must come to an end, eight levels down. This is the hard limit of our scientific understanding. To the best of our present ability to perceive and to reason, the universe is made from fields and nothing else, and these fields are not made from any smaller components.

But it’s not quite right to say that fields are the *most fundamental* thing that we know of in nature. Because we know something that is in some sense even more basic: we know the rules that these fields have to obey. Our understanding of how to codify these rules came from a series of truly great triumphs in modern physics. And the greatest of these triumphs, as I see it, was quantum mechanics.

In this post I want to try and paint a picture of what it means to have a field that respects the laws of quantum mechanics. In a previous post, I introduced the idea of fields (and, in particular, the all-important electric field) by making an analogy with ripples on a pond or water spraying out from a hose. These images go surprisingly far in allowing one to understand how fields work, but they are ultimately limited in their correctness because the implied rules that govern them are completely classical. In order to *really* understand how nature works at its most basic level, one has to think about a field with quantum rules.

***

The first step in creating a picture of a field is deciding how to imagine what the field is made of. Keep in mind, of course, that the following picture is mostly just an artistic device. The real fundamental fields of nature aren’t really made of physical things (as far as we can tell); physical things are made of *them*. But, as is common in science, the analogy is surprisingly instructive.

So let’s imagine, to start with, a ball at the end of a spring. Like so:

This is the object from which our quantum field will be constructed. Specifically, the field will be composed of an infinite, space-filling array of these ball-and-springs.

To keep things simple, let’s suppose that, for some reason, all the springs are constrained to bob only up and down, without twisting or bending side-to-side. In this case the array of springs can be called, using the jargon of physics, a *scalar field*. The word “scalar” just means a single number, as opposed to a set or an array of multiple numbers. So a *scalar field* is a field whose value at a particular point in space and time is characterized only by a single number. In this case, that number is the height of the ball at the point in question. (You may notice that what I described in the previous post was a *vector field*, since the field at any given point was characterized by a velocity, which has both a magnitude and a direction.)

In the picture above, the array of balls-and-springs is pretty uninteresting: each ball is either stationary or bobs up and down independently of all others. In order to make this array into a *bona fide* field, one needs to introduce some kind of coupling between the balls. So, let’s imagine adding little elastic bands between them:

Now we have something that we can legitimately call a field. (My quantum field theory book calls it a “mattress”.) If you disturb this field – say, by tapping on it at a particular location – then it will set off a wave of ball-and-spring oscillations that propagates across the field. These waves are, in fact, the *particles* of field theory. In other words, when we say that there is a particle in the field, we mean that there is a wave of oscillations propagating across it.

These particles (the oscillations of the field) have a number of properties that are probably familiar from the days when you just thought of particles as little points whizzing through empty space. For example, they have a well-defined propagation velocity, which is related to the weight of each of the balls and the tightness of the springs and elastic bands. This characteristic velocity is our analog of the “speed of light”. (More generally, the properties of the springs and masses define the relationship between the particle’s kinetic energy and its propagation velocity, like the of your high school physics class.) The properties of the springs also define the way in which particles interact with each other. If two particle-waves run into each other, they can scatter off each other in the same way that normal particles do.

(A technical note: the degree to which the particles in our field scatter upon colliding depends on how “ideal” the springs are. If the springs are perfectly described by Hooke’s law, which says that the restoring force acting on a given ball is linearly proportional to the spring’s displacement from equilibrium, then there will be no interaction whatsoever. For a field made of such perfectly Hookean springs, two particle-waves that run into each other will just go right through each other. But if there is any deviation from Hooke’s law, such that the springs get stiffer as they are stretched or compressed, then the particles will scatter off each other when they encounter one another.)

Finally, the particles of our field clearly exhibit “wave-particle duality” in a way that is easy to see without any philosophical hand-wringing. That is, our particles by definition *are* waves, and they can do things like interfere destructively with each other or diffract through a double slit.

All of this is very encouraging, but at this point our fictitious field lacks one very important feature of the real universe: the discreteness of matter. In the real world, all matter comes in discrete units: single electrons, single photons, single quarks, etc. But you may notice that for the spring field drawn above, one can make an excitation with completely arbitrary magnitude, by tapping on the field as gently or as violently as one wants. As a consequence, our (classical) field has no concept of a minimal piece of matter, or a smallest particle, and as such it cannot be a very good analogy to the actual fields of nature.

***

To fix this problem, we need to consider that the individual constituents of the field – the balls mounted on springs – are themselves subject to the laws of quantum mechanics.

A full accounting of the laws of quantum mechanics can take some time, but for the present pictorial discussion, all you really need to know is that a quantum ball on a spring has two rules that it must follow. 1) It can never stop moving, but instead must be in a constant state of bobbing up and down. 2) The amplitude of the bobbing motion can only take certain discrete values.

This quantization of the ball’s oscillation has two important consequences. The first consequence is that, if you want to put energy into the field, you must put in at least one quantum. That is, you must give the field enough energy to kick at least one ball-and-spring into a higher oscillation state. Arbitrarily light disturbances of the field are no longer allowed. Unlike in the classical case, an extremely light tap on the field will produce literally zero propagating waves. The field will simply not accept energies below a certain threshold. Once you tap the field hard enough, however, a particle is created, and this particle can propagate stably through the field.

This discrete unit of energy that the field can accept is what we call the *rest mass energy* of particles in a field. It is the fundamental amount of energy that must be added to the field in order to create a particle. This is, in fact, how to think about Einstein’s famous equation in a field theory context. When we say that a fundamental particle is heavy (large mass ), it means that a lot of energy has to be put into the field in order to create it. A light particle, on the other hand, requires only a little bit of energy.

(By the way, this why physicists build huge particle accelerators whenever they want to study exotic heavy particles. If you want to create something heavy like the Higgs boson, you have to hit the *Higgs field* with a sufficiently large (and sufficiently concentrated) burst of energy to give the field the necessary one quantum of energy.)

The other big implication of imposing quantum rules on the ball-and-spring motion is that it changes pretty dramatically the meaning of empty space. Normally, empty space, or *vacuum*, is defined as the state where no particles are around. For a classical field, that would be the state where all the ball-and-springs are stationary and the field is flat. Something like this:

But in a quantum field, the ball-and-springs can *never* be stationary: they are always moving, even when no one has added enough energy to the field to create a particle. This means that what we call *vacuum* is really a noisy and densely energetic surface:

This random motion (called *vacuum fluctuations*) has a number of fascinating and eminently noticeable influences on the particles that propagate through the vacuum. To name a few, it gives rise to the Casimir effect (an attraction between parallel surfaces, caused by vacuum fluctuations pushing them together) and the Lamb shift (a shift in the energy of atomic orbits, caused by the electron getting buffeted by the vacuum).

In the jargon of field theory, physicists often say that “virtual particles” can briefly and spontaneously appear from the vacuum and then disappear again, even when no one has put enough energy into the field to create a real particle. But what they really mean is that the vacuum itself has random and indelible fluctuations, and sometimes their influence can be felt by the way they kick around real particles.

That, in essence, is a quantum field: the stuff out of which everything is made. It’s a boiling sea of random fluctuations, on top of which you can create quantized propagating waves that we call particles.

I only wish, as a primarily visual thinker, that the usual introduction to quantum field theory didn’t look quite so much like this. Because behind the equations of QFT there really is a tremendous amount of imagination, and a great deal of wonder.

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There’s a good chance that, at some point in your life, someone told you that nature has four fundamental forces: gravity, the strong nuclear force, the weak nuclear force, and the electromagnetic force.

This factoid is true, of course.

But what you probably weren’t told is that, at the scale of just about any natural thing that you are likely to think about, only one of those four forces has any relevance. Gravity, for example, is so obscenely weak that one has to collect planet-sized balls of matter before its effect becomes noticeable. At the other extreme, the strong nuclear force is so strong that it can never go unneutralized over distances larger than a few times the diameter of an atomic nucleus ( meters); any larger object will essentially never notice its existence. Finally, the weak nuclear force is extremely short-ranged, so that it too has effectively no influence over distances larger than meters.

That leaves the electromagnetic force, or, in other words, the Coulomb interaction. This is the familiar law that says that like charges repel each and opposites attract. This law alone dominates the interactions between essentially all objects larger than an atomic nucleus ( meters) and smaller than a planet ( meters). That’s more than twenty powers of ten.

But not only does the “four fundamental forces” meme give a false sense of egalitarianism between the forces, it is also highly misleading for another reason. Namely, in physics forces are not considered to be “fundamental”. They are, instead, byproducts of the objects that really *are* fundamental (to the best of our knowledge): *fields*.

Let me back up a bit. To understand what a force is, one first has to accept the idea that empty space is not really empty.

Empty space, or vacuum, is the stage upon which the pageantry of nature plays out. Just as the setup of the stage in a theater determines what kind of plays can be performed, so too do the properties of the vacuum determine what kind of natural laws we have.

Let me be a little less wishy-washy. As we currently understand it, empty space is filled with a number of all-pervasive, interpenetrating *fields*. To a physicist, these fields are mathematical objects: they are functions that take a particular value (or vector of values) at every point in space. But for the daydreamer (which, of course, includes those same physicists), these fields can be visualized as something like a stretchy fabric, or a fluid. To be concrete with the imagery, let’s say that a field is something like the surface of a pond. When not perturbed, that surface is placid (as long as you don’t look too closely, say, at the molecular level). But when something disturbs the pond, it creates a ripple that propagates stably across the surface.

In the modern view of physics, what we call “particles” are really just ripples across a field. The word *electron*, for example, is what we use to refer to a ripple on the electron field. The photon is just a ripple on the photon field (also called the electromagnetic field). And so on. For each of the elementary particles there is a corresponding field upon which that particle is a ripple. It is these fields (and they alone) that define the properties of the universe: what kind of particles can exist in the universe, and how they interact with each other.

Just to belabor the point a little more: a particle like an electron is not any more “fundamental” or “elementary” than a wave lapping the shore of a beach. It is the sea that is fundamental. If you want to understand where waves come from and how they move, you must first understand the water.

(Aside: If you are encountering for the first time this idea of fields as the fundamental objects of the universe, I hope that it bothers you. I hope that it makes you feel uncomfortable and a little incredulous. That’s certainly the way I felt at first, and for me those feelings were the beginning of my ability to appreciate an idea that has come to feel deeply wondrous, and deeply useful. If you do indeed feel unhappy with this idea, all I can suggest is time and perhaps the wonderful essay about fields (classical and quantum) by the great Freeman Dyson.)

With that pictorial overview, let’s return to the Coulomb interaction. What I want to claim is that the Coulomb force, like any other force, is an emergent property of the field that mediates it (in this case, the electromagnetic field). And so, if the force between two charged particles is to be properly understood, then it must be explained in terms of how the field behaves in the vicinity of those two charged particles. From the resulting picture of a perturbed field one should be able to see, in a quantitative way, that Coulomb’s law emerges.

Let’s start with the basics.

Coulomb’s law, as it is usually written, says that the strength of the force between two charges and separated by a distance is

. (1)

(You can think that the term at the beginning of the equation, , as just a constant conversion factor that turns the nebulous units of “charge” into something that has units of force. But I’ll say a little more this factor in a bit.)

To see how the Coulomb force arises from the properties of a field, it makes sense to first talk about the electric field created by a single charge. For this we can invoke the second piece of canonical knowledge related to point charges, which says that the strength of the electric field around a point charge follows

. (2)

This electric field points radially outward (for positive charges) or inward (or for negative charges) from the position of the charge.

At first sight, this equation looks essentially identical to the previous one: the only difference is that there is only one in it instead of two. In fact, high school classes tend to explain the idea of electric field in a way that makes you question why it needs to exist as a separate concept from force. I think my high school physics class, like many others, literally defined an electric field as “the force that would be felt by a unit charge if it were placed at a particular location.”) In this sense the concept of electric field often sounds subsidiary to the concept of force. But if you’re prepared instead to think about the field as a truly fundamental object, then this second equation becomes quite interesting.

In our analogy, the electromagnetic field is something like a fluid that fills all of space. This fluid exists at every point in space and at every moment in time, but at moments and locations where there are no charges around you can imagine that it is stationary and calm. In the presence of electric charges, however, the fluid starts to move. What we normally call the *electric field strength*, or (perhaps confusingly) just the *electric field*, can be imagined as the local velocity of the fluid at a particular point.

What emerges from equation (2), then, is a picture of how the fluid is moving in the vicinity of a point charge. For a positive charge, for example, it moves radially outward, in such a way that the speed of the fluid falls of as the square of the distance from the center. As it happens, this inverse square law is special: it guarantees that the total amount of fluid flowing across any closed surface containing the point charge is the same, regardless of the shape or size of the surface. What’s more, the fluid flow rate is directly proportional to the charge .

The simple way to see this is by drawing a sphere of radius around the point charge. The flow rate of water across the surface is equal to the product of the “velocity” times the area of the sphere. Since the surface area of a sphere is proportional to , and the velocity is inversely proportional to , the flow rate of water is the same for any sized-sphere and is proportional to the charge inside. This is basically just a restatement of Gauss’s law.

All this is to say that, in our fluid analogy, the (positive) point charge is something like a spout of water or a hose: fluid comes flying out of it in all directions. This fluid is fast moving at the source, and slower as it spreads out. It is not created or destroyed anywhere except at the point charge itself. Conversely, a negative point charge is something like the opposite of a hose: a suction source or drain that pulls water into itself. In both cases, the total flow rate of water (either sprayed out or sucked in) is quantified by what we call the charge, .

This analogy might strike you as perhaps a little too precious, but it turns out to give an intuition that works at a surprisingly quantitative level. In particular, the analogy gets quite good when you ask the question “how much energy is there in the field?”

When fluid is in motion, it has kinetic energy. You may remember from high school physics that an object with mass and velocity has kinetic energy . For a fluid, where different locations have different velocities, you can generalize this formula by integrating over different locations:

. (3)

Here, the symbol means an integral over all different parts of space, and is the density of the fluid.

As it turns out, the expression for the energy stored in an electric field looks almost exactly the same:

. (4)

By comparing the last equations, you can see that the value of the electric field, , *really is* playing the same role as the fluid velocity. That constant in equation (4) is a special constant called the *permittivity of free space*. In our analogy, you can think of it as something related to the “density of fluid” in the electromagnetic field.

You can also notice that the quantity is an *energy density*. It tells you how much energy is stored in the electric field at a particular location. For fluids, energy density is closely related to the concept of pressure: a fluid under high pressure has a lot of energy stored in it. (You can check, if you want, that energy per unit volume and pressure have the same physical units.) We can therefore think of the quantity as something like a *pressure* that builds up in the electromagnetic “fluid”.

Having made a correspondence between *electric field* and *pressure*, the final step toward understanding Coulomb’s law is relatively straightforward. When two spouts of water are brought together, pressure builds up between them, and they are pushed apart. Similarly, when two electric charges are brought near each other, pressure builds up in the electric field between them. This pressure ends up pushing the two charges apart from each other, in the same way that two fire hoses would be pushed away from each other if you fired them toward each other.

With equations for the “field pressure” in-hand, you can even calculate the exact mathematical form of the repulsive force between the two “hoses”. If you want the technical details: you can calculate the pressure at the midplane between the two, and then integrate the pressure over the midplane area. (This is the same procedure that you would follow if you wanted to know the force of a fire hose spraying against a wall. Of course, there are other approaches for doing the calculation.) What comes out of this procedure is exactly what I promised from the beginning:

.

If you want a conceptual picture for the attractive force between opposite charges, you can approach it in a similar way. In particular, when two opposite charges are brought near each other, this is like bringing a hose that emits water close to a strong suction hose. One of the two hoses is furiously emitting fluid, while the other is happily sucking it in, and consequently the pressure between the two of them becomes relatively small. As a result, the two “hoses” end up being pushed together by the larger pressure of the fluid outside.

At this point, we have a conceptual explanation of where electric forces come from. But to close this post, it is perhaps worth making a remark about simplicity in physics. It may strike you that the story I have told here is *not simple*. I started with a very simple equation – Coulomb’s law, which is usually introduced as the simplest quantitative starting point for thinking about electric charges – and I gave it a complicated origin story. This story required me to invoke nebulous, space-filling force fields; to make questionable, convoluted analogies; and to compute multi-dimensional integrals over vector-valued functions. This story also never explained *why* a point charge behaves like a “source” or “sink” of “fluid”; it just *does*. Or, at least, it *needs to*, if the story is to hold together.

You may reasonably feel, then, that the picture I painted is essentially worthless. It is much easier to simply remember equation (1) than to remember how to describe the way that pressure builds up in a space-filling, fluid-like field. And it requires essentially the same number of arbitrary assumptions.

If you feel this way, then probably all I can offer is an apology for wasting your time. But for a physicist, the construction of such “origin stories” is perhaps the very most important part of the profession. It is absolutely integral to physics that its developers never be satisfied with any level of description of reality. To every law or equation or theorem, we must always ask “yes, but why is it that way?” This impertinent questioning, where it succeeds, ultimately always turns one question into another question. But along the way it can rewrite very fundamentally the way we perceive nature. And, when those revisions succeed, they pave the way for significant new insights and discoveries while recapitulating all the results that came before. (For the record, the classical and quantum theories of fields are probably the most successful scientific theories that mankind has yet produced.)

You can also view the question of simplicity another way. In telling this story, *I* have not been particularly simple, but *nature *has been very simple indeed. It has provided an extremely succinct mathematical law and allowed it to govern the universe over more than 20 orders of magnitude in scale. Perhaps the greatest proof of Nature’s simplicity is not that I can write Coulomb’s law in a single line, or that I can give it a particular origin story, but rather that I can think about it in many different ways and derive it through many different avenues, and all of those avenues turn out to be equivalent.

I’ll leave you with the words of Richard Feynman, who expressed this same sentiment very nicely in his Nobel lecture:

The fact that electrodynamics can be written in so many ways … was something I knew, but I have never understood. It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but, with a little mathematical fiddling you can show the relationship. … I don’t know why this is – it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what the reason for this is. I think it is somehow a representation of the simplicity of nature. A thing like the inverse square law is just right to be represented by the solution of Poisson’s equation, which, therefore, is a very different way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what it means, that nature chooses these curious forms, but maybe that is a way of defining simplicity. Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing.

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