# Math Is the Art of Stating Things Clearly

Why do we use math?

In physics we describe everything, from the smallest of particles to the largest of galaxies, with the language of mathematics. Why should that one field be able to describe so much? And why don’t we use something else?

The truth is, this is a trick question. Mathematics isn’t a language like English or French, where we can choose whichever translation we want. We use mathematics because it is, almost by definition, the best choice. That is because mathematics is the art of stating things clearly.

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter. The bartender stops them, pours two beers, and says “You guys should know your limits.”

That was an (old) joke about infinite series of numbers. You probably learned in high school that if you add up one plus a half plus a quarter…you eventually get two. To be a bit more precise:

$\sum_{i=0}^\infty \frac{1}{2^i} = 1+\frac{1}{2}+\frac{1}{4}+\ldots=2$

We say that this infinite sum limits to two.

But what does it actually mean for an infinite sum to limit to a number? What does it mean to sum infinitely many numbers, let alone infinitely many beers ordered by infinitely many mathematicians?

You’re asking these questions because I haven’t yet stated the problem clearly. Those of you who’ve learned a bit more mathematics (maybe in high school, maybe in college) will know another way of stating it.

You know how to sum a finite set of beers. You start with one beer, then one and a half, then one and three-quarters. Sum $N$ beers, and you get

$\sum_{i=0}^N \frac{1}{2^i}$

What does it mean for the sum to limit to two?

Let’s say you just wanted to get close to two. You want to get $\epsilon$ close, where epsilon is the Greek letter we use for really small numbers.

For every $\epsilon>0$ you choose, no matter how small, I can pick a (finite!) $N$ and get at least that close. That means that, with higher and higher $N$, I can get as close to two as a I want.

As it turns out, that’s what it means for a sum to limit to two. It’s saying the same thing, but more clearly, without sneaking in confusing claims about infinity.

These sort of proofs, with $\epsilon$ (and usually another variable, $\delta$) form what mathematicians view as the foundations of calculus. They’re immortalized in story and song.

And they’re not even the clearest way of stating things! Go down that road, and you find more mathematics: definitions of numbers, foundations of logic, rabbit holes upon rabbit holes, all from the effort to state things clearly.

That’s why I’m not surprised that physicists use mathematics. We have to. We need clarity, if we want to understand the world. And mathematicians, they’re the people who spend their lives trying to state things clearly.

# Congratulations to James Peebles, Michel Mayor, and Didier Queloz!

The 2019 Physics Nobel Prize was announced this week, awarded to James Peebles for work in cosmology and to Michel Mayor and Didier Queloz for the first observation of an exoplanet.

Peebles introduced quantitative methods to cosmology. He figured out how to use the Cosmic Microwave Background (light left over from the Big Bang) to understand how matter is distributed in our universe, including the presence of still-mysterious dark matter and dark energy. Mayor and Queloz were the first team to observe a planet outside of our solar system (an “exoplanet”), in 1995. By careful measurement of the spectrum of light coming from a star they were able to find a slight wobble, caused by a Jupiter-esque planet in orbit around it. Their discovery opened the floodgates of observation. Astronomers found many more planets than expected, showing that, far from a rare occurrence, exoplanets are quite common.

It’s a bit strange that this Nobel was awarded to two very different types of research. This isn’t the first time the prize was divided between two different discoveries, but all of the cases I can remember involve discoveries in closely related topics. This one didn’t, and I’m curious about the Nobel committee’s logic. It might have been that neither discovery “merited a Nobel” on its own, but I don’t think we’re supposed to think of shared Nobels as “lesser” than non-shared ones. It would make sense if the Nobel committee thought they had a lot of important results to “get through” and grouped them together to get through them faster, but if anything I have the impression it’s the opposite: that at least in physics, it’s getting harder and harder to find genuinely important discoveries that haven’t been acknowledged. Overall, this seems like a very weird pairing, and the Nobel committee’s citation “for contributions to our understanding of the evolution of the universe and Earth’s place in the cosmos” is a pretty loose justification.

Back in 2015, I did a poll asking how much physics background you guys had. Now four years and many new readers later, I’d like to revisit the question. I’ll explain the categories below the poll:

Amplitudeologist: You have published a paper about scattering amplitudes in quantum field theories, or expect to publish one within the next year or so.

Physics (or related field) PhD: You have a PhD in physics, or in a field with related background such as astronomy or some parts of mathematics.

Physics (or related field) Grad Student: You are a graduate student in physics or a related field. Specifically, you are either a PhD student, or a Master’s student in a research-focused program.

Undergrad or Lower: You are currently an undergraduate student (studying for a Bachelor’s degree) or are in an earlier stage of education (for example a high school student).

Physics Autodidact: Included by popular demand from the last poll: while you don’t have a physics PhD, you have taught yourself about the subject extensively beyond your formal schooling.

Other Academic: You work in Academia, but not in physics or a closely related field.

Other Technical Profession: You work in a technical profession, such as engineering, medicine, or STEM teaching.

None of the Above: Something else.

If you fit more than one category, pick the first that matches you: for example, if you are an undergrad with a published paper in Amplitudes, list yourself as an Amplitudeologist (also, well done!)

# Things I’d Like to Know More About

This is an accountability post, of sorts.

As a kid, I wanted to know everything. Eventually, I realized this was a little unrealistic. Doomed to know some things and not others, I picked physics as a kind of triage. Other fields I could learn as an outsider: not well enough to compete with the experts, but enough to at least appreciate what they were doing. After watching a few string theory documentaries, I realized this wasn’t the case for physics: if I was going to ever understand what those string theorists were up to, I would have to go to grad school in string theory.

Over time, this goal lost focus. I’ve become a very specialized creature, an “amplitudeologist”. I didn’t have time or energy for my old questions. In an irony that will surprise no-one, a career as a physicist doesn’t leave much time for curiosity about physics.

One of the great things about this blog is how you guys remind me of those old questions, bringing me out of my overspecialized comfort zone. In that spirit, in this post I’m going to list a few things in physics that I really want to understand better. The idea is to make a public commitment: within a year, I want to understand one of these topics at least well enough to write a decent blog post on it.

Wilsonian Quantum Field Theory:

When you first learn quantum field theory as a physicist, you learn how unsightly infinite results get covered up via an ad-hoc-looking process called renormalization. Eventually you learn a more modern perspective, that these infinite results show up because we’re ignorant of the complete theory at high energies. You learn that you can think of theories at a particular scale, and characterize them by what happens when you “zoom” in and out, in an approach codified by the physicist Kenneth Wilson.

While I understand the basics of Wilson’s approach, the courses I took in grad school skipped the deeper implications. This includes the idea of theories that are defined at all energies, “flowing” from an otherwise scale-invariant theory perturbed with extra pieces. Other physicists are much more comfortable thinking in these terms, and the topic is important for quite a few deep questions, including what it means to properly define a theory and where laws of nature “live”. If I’m going to have an informed opinion on any of those topics, I’ll need to go back and learn the Wilsonian approach properly.

Wormholes:

If you’re a fan of science fiction, you probably know that wormholes are the most realistic option for faster-than-light travel, something that is at least allowed by the equations of general relativity. “Most realistic” isn’t the same as “realistic”, though. Opening a wormhole and keeping it stable requires some kind of “exotic matter”, and that matter needs to violate a set of restrictions, called “energy conditions”, that normal matter obeys. Some of these energy conditions are just conjectures, some we even know how to violate, while others are proven to hold for certain types of theories. Some energy conditions don’t rule out wormholes, but instead restrict their usefulness: you can have non-traversable wormholes (basically, two inescapable black holes that happen to meet in the middle), or traversable wormholes where the distance through the wormhole is always longer than the distance outside.

I’ve seen a few talks on this topic, but I’m still confused about the big picture: which conditions have been proven, what assumptions were needed, and what do they all imply? I haven’t found a publicly-accessible account that covers everything. I owe it to myself as a kid, not to mention everyone who’s a kid now, to get a satisfactory answer.

Quantum Foundations:

Quantum Foundations is a field that many physicists think is a waste of time. It deals with the questions that troubled Einstein and Bohr, questions about what quantum mechanics really means, or why the rules of quantum mechanics are the way they are. These tend to be quite philosophical questions, where it’s hard to tell if people are making progress or just arguing in circles.

I’m more optimistic about philosophy than most physicists, at least when it’s pursued with enough analytic rigor. I’d like to at least understand the leading arguments for different interpretations, what the constraints on interpretations are and the main loopholes. That way, if I end up concluding the field is a waste of time at least I’d be making an informed decision.

You might notice a change on the site this week: the ads are gone!

When I started this blog back in 2012, it was just a class project. I didn’t want to spend money on it, so I chose WordPress.com’s free hosting option. A consequence of that option is that WordPress got to post ads. These were pretty mild to begin with, I think most of the early posts didn’t even have ads. It seemed like a reasonable deal.

Over the years, WordPress has quietly been adding more ads, and worse ones. I mostly hadn’t noticed: I use an adblocker. For those who don’t, though, the blog began to look increasingly unprofessional, plastered with the kind of shitty, borderline-scam ads that fill certain parts of the internet. Thanks to everyone who let me know this was happening, I don’t think I would have noticed otherwise. To clarify, I never made any money from these ads, all of the revenue went to WordPress.

As of this week I’ve switched the site to a paid hosting plan. The move is long overdue: the plan is actually pretty cheap, and is the sort of thing I could have easily afforded by myself. As it happens I don’t have to afford it by myself: the grant that funds me, a Marie Curie Individual Fellowship, also funds outreach activities. I already had a message thanking them on my About page, but somehow I hadn’t considered actually using their funding here.

The site’s new plan also comes with a free domain, so you can now reach this site with a new simpler address: 4gravitons.com. The old 4gravitons.wordpress.com address should still work as well, if there are any glitches please let me know!

# Pi Day Alternatives

On Pi Day, fans of the number pi gather to recite its digits and eat pies. It is the most famous of numerical holidays, but not the only one. Have you heard of the holidays for other famous numbers?

Tau Day: Celebrated on June 28. Observed by sitting around gloating about how much more rational one is than everyone else, then getting treated with high-energy tau leptons for terminal pedantry.

Canadian Modular Pi Day: Celebrated on February 3. Observed by confusing your American friends.

e Day: Celebrated on February 7. Observed in middle school classrooms, explaining the wonders of exponential functions and eating foods like eggs and eclairs. Once the students leave, drop tabs of ecstasy instead.

Golden Ratio Day: Celebrated on January 6. Rub crystals on pyramids and write vaguely threatening handwritten letters to every physicist you’ve heard of.

Euler Gamma Day: Celebrated on May 7 by dropping on the floor and twitching.

Riemann Zeta Daze: The first year, forget about it. The second, celebrate on January 6. The next year, January 2. After that, celebrate on New Year’s Day earlier and earlier in the morning each year until you can’t tell the difference any more.

# Valentine’s Day Physics Poem 2019

It’s that time of year again! Time for me to dig in to my files and bring you yet another of my old physics poems.

Plagued with Divergences

“The whole scheme of local field theory is plagued with divergences”

Is divergence ever really unexpected?

If you asked a computer, what would it tell you?

You’d hear a whirring first, lungs and heart of the machine beating faster and faster.

And you’d dismiss it.
You knew this wasn’t going to be an easy interaction.
It doesn’t mean you’re going to diverge.

And perhaps it would try to warn you, write it there on the page.
It might even notice, its built-in instincts telling you, by the book,
“This will diverge.”

But instincts lie, and builders cheat.
And it doesn’t mean you’re going to diverge.

Now, you do everything the slow way,
Numerically.
Dismiss your instincts and force yourself through
Piece by piece.

And now, you can’t stop hearing the whir
The machine’s beating heart
Even when it should be at rest

And step by step, it tries to minimize its errors
And step by step, the errors grow

And exhausted, in the end, you see splashed across the screen
Something bigger than it should ever have been.

But sometimes things feel big and strange.
That’s just the way of the big wide world.
And it doesn’t mean you’re going to diverge.

You could have seen the signs,
Power-counted, seen what could overwhelm.
And you could have regulated, with an epsilon of flexibility.

But this one, this time, was supposed to be
Needed to be
Physical Truth
And truth doesn’t diverge

So you keep going,
Wheezing breath and painstaking calculation,
And every little thing blowing up

It’s not like there’s a better way to live.