I made a big mistake. I told you that protons and neutrons are like bags full of a soup of virtual quark-antiquark pairs - a so-called 'quark condensate'. But I got it exactly backward. In fact, this condensate pervades the vacuum *outside* protons and neutrons! (1/n)

I don't want to formalize any of my work on mathematics. First because, as Emily Riehl notes, formalization tends to impose consensus. And second, because I find it boring. It steals time from creative thought to nail things down with more rigidity than I need or want.

I'm only now learning about "vector meson dominance", a big idea put forth by Sakurai and others around 1960. Just as the pion comes in +, - and neutral forms and has spin 0, the rho meson comes in +, - and neutral forms and has spin 1 - so it's described by a "vector", like the photon. (1/n)

I conjecture that the Standard Model gauge group consists of all isometries of the octonionic projective plane that preserve an octonionic projective line and a complex projective plane intersecting in a complex projective line. Check it out: johncarlosbaez.wordpress.com/2026/03/27/g...

Doesn't meet minimum standards, but it still gets a B? What is this, Harvard?

I'm learning more about Feynman diagrams. 'Virtual particles' are just internal lines in these diagrams. They don't need to obey m² = E²−p² like real particles. But when they do, we say they're 'on shell'. When they're close to on shell, the amplitude of the diagram gets big. (1/n)

Don't want Microsoft to take what you do on GitHub and use it for AI model training? Click "disable" here. You can also click "disable" to turn off Copilot on GitHub. github.com/settings/cop...

The Higgs boson gives elementary particles their mass, but 98% of the visible mass in the Universe (not dark matter) comes from a less famous mechanism: chiral symmetry breaking. This is why protons and neutrons are so much heavier than their quarks! en.wikipedia.org/wiki/Chiral_...

From @johncarlosbaez "I think it's better to figure out the theorems before stating the definitions. That is: figure out the patterns that hold, and turn these into theorems (or at least conjectures), and then notice that these theorems are easiest to state if one makes certain definitions." […]

Pions were a revolutionary discovery in the 1930s — they were some of the first animals in the ‘particle zoo’ — but here I’m explaining them as a way to work toward the math and physics concepts needed for the Standard Model. (1/n) www.youtube.com/watch?v=AC7K...

This circle represents an octave. Each new dot is a perfect fifth above the last. There are always either 2 or 3 sizes of space between notes, listed at the bottom. When there are 2, it's called a "moment of symmetry" and the number in the middle - the number of notes - turns red. (1/2)

The list of happiest countries puts Finland at top, followed by Iceland, Denmark, Costa Rica, Sweden and Norway. Clearly to be happy you need to freeze your ass off in the dark for half the year, and Costa Rica is a lot further north than I thought. data.worldhappiness.report/table

Yeah, a bell curve with mean zero and standard deviation zero.

The proton, 1836 times heavier than the electron, is made of two up quarks and a down, with two of their spins aligned and one pointing the other way. The same quarks with all spins aligned give a new particle, the Δ⁺, that's 2411 times heavier than the electron! (1/n)

"This is probably a good place to admit that I regularly made great demands for computer time at Livermore, with little recognition of the fact that the taxpayers installed that gigantic computer complex there in order to build bombs, not discover resonances." More: mathstodon.xyz/@johncarlosb...

When you add up lots of tiny independent random effects the overall random effect tends to be distributed like a bell curve¹. It's a lot like how river rocks tend to get smoothed down by many small knocks. ¹ Unless those tiny effects are so "long-tailed" their standard deviations are infinite.

In the vacuum, away from those pesky electrical charges and currents, there's a perfect symmetry between the electric and magnetic fields! It's the prototype of some much deeper dualities we see later in physics. (1/n)

Inconvenience stores

Holy moly! There are 40 old Soviet uranium-fueled nuclear reactor cores in orbit! I didn't know that: mastodon.social/@sundogplane... A while back, one exploded over northern Canada and spewed out radioactive material: www.cbc.ca/arts/operati...

Another particle has been found! In fact the Large Hadron Collider has found 80 new particles in addition to the Higgs boson. All of these 80 are 'hadrons', collections of quarks and/or antiquarks held together by the strong force. (1/5) home.cern/news/news/ph...

Maxwell's equations imply conservation of electric charge! Try to visualize each step here. When current shoots outward, the divergence of the electric field must decrease — and this is only explicable by a decrease in charge density!

Until now, the interesting thing about Daniel Biss to me was how two big papers of his, published in Annals of Mathematics and Advances in Mathematics, turned out to be wrong and had to be retracted: en.wikipedia.org/wiki/Daniel_...

Pope Leo asked me to meet him in a private audience at the Vatican today I presented him with documents and evidence detailing decades of abuse inside Opus Dei and urged him to launch a full independent investigation into the group A THREAD 🧵

The physicist Ørsted discovered that electric current makes the magnetic field curl around it. Maxwell realized that a changing electric field does this too. Maxwell's fourth equation says exactly how this works. This completes the theory of electromagnetism! (1/n)

Scott Aaronson's blog once featured a philosopher wringing his hands about the possibility that AI will become superhumanly powerful and completely benevolent. If this happens, we will whine about it. NOTHING can rob us of the ability to whine, unless we are genetically engineered not to.

A changing magnetic field makes the electric field curl around. If you move a magnet through a loop of wire, the resulting electric field will push electrons around the wire — and you'll get an electric current! The third Maxwell equation makes this precise. (1/n)

Unsatisfying bookstore.

Margaret Cavendish was the first woman to attend meetings of the Royal Society - after which they banned that for 278 years. And she wrote an SF novel when Newton was still a teenager! www.atlasobscura.com/articles/one-of-the-earliest-science-fiction-books-was-written-in-the-1600s-by-a-duchess

Like the electric field, the magnetic field 𝐁 is an arrow at each point of space. But the second Maxwell equation says that unlike the electric field, the magnetic field cannot spread outwards or inwards. Its divergence is zero! (1/n)

Have you ever wished you could be in charge of the arXiv? Now you can earn $300,000 / year doing exactly that: jobs.chronicle.com/job/37961678...

The electric field 𝐄 is a field of arrows in space. At each point these arrows 'diverge', or spread out, by an amount equal to the density of electric charge there. The charge density is called ρ. I call this fact the first Maxwell equation, since it was discovered first. (1/n)

Beato isn't really saying AI will fail: despite the denialists who claim it's never good for anything, it's too useful for too many things to simply "fail". He's claiming something more interesting.... (1/n) www.youtube.com/watch?v=YTLn...

We're listing our most fear-inducing functions. While I could flex and say something obscure like "Legendre functions of the second kind" - whatever they are - it's actually hypergeometric functions. They've got subscripts on both sides, lots of arguments, and there are entire books about them.

Four equations tell us everything we need to know about the electric and magnetic field — not counting the the behavior of the matter that they interact with, or the subtleties of quantum mechanics. This is when I started really loving physics.

There's a broken cuneiform tablet from the Old Babylonian period, nearly 4,000 years ago, which preserves a tiny portion of a dialogue between two friends. It feels a bit like the conversations I've been having for the past week, so I wanted to share it.

In the Swedish town of Södertälje, crows are being paid to pick up cigarette butts. When they take one and stick it in a machine, they get rewarded with some peanuts. It's called "Corvid Cleaning": www.samodobrevijesti.com/en/news/wild...

What's the 'quantum metric' discussed in this article? The state space ℂPⁿ has a Kähler structure g + iω (Riemannian metric + i times symplectic structure). Given any map from any manifold to ℂPⁿ, we can pull this back and get a 'quantum metric' on that manifold! Now physicists are measuring it!

No, it's not a pipe dream! For more details on how category theory helps us model complex systems, try the talk, slides and papers here: math.ucr.edu/home/baez/co...

The perspective looks all wrong. You're staring straight at this barmaid, but her reflection in the mirror is way off to right. Even worse, her reflection is facing a guy who doesn't appear in the main view! But in 2000, a researcher showed this perspective is actually possible! (1/n)

Hey! Now you can read about my work on applied category theory here on Quanta magazine! And not just me, lots of folks.... www.quantamagazine.org/can-the-most...

Here's how Hideki Yukawa predicted the mass of the particles that carry the nuclear force. It was a stunning episode in the history of physics, and the start of a new era. (1/9)

@uberwensch.bsky.social - hey, here is someone who actually works on hadron scattering and how quarks manifest as jets. Complicated stuff!

Funny how the huge excitement about counting geometrical things using ideas from string theory is being described now: "aside from a brief resurgence in the 1990s, enumerative geometry seemed to have been set aside for good." Fact is, good hard problems never die.

Another place I want to visit on my trip to New Mexico is called Aztec Ruins - named that before white folks realized the ancestral Puebloans were a very different civilization. It's huge, with ~450 rooms, some 3 stories high! People lived here from about 1110 to 1200 AD. (1/n)

I'm enjoying the novel "There Is No Antimemetics Division". I'll quote some for you: ..... Item #: SCP-3125 Object Class: Keter Special Containment Procedures: SCP-3125 is kept inside Cognitohazard Containment Unit 3125 on the first floor of Site 41. (1/n) en.wikipedia.org/wiki/There_I...

The 'mediant' of two fractions a/b and c/d is (a+c)/(b+d). Beginners sometimes add fractions this way by mistake. But it's important in number theory... ... and I just learned it has a simple geometrical meaning, too!

Collect these figurines based on Hieronynus Bosch's paintings! Then cower in terror as they come alive at night and climb into your bed. Starting at $34.50 here: www.amazon.com/s?k=bosch+fi...

When you slam particles into each other, you can try to describe what happens using a 'scattering matrix', or S-matrix. This says the probability (or really 'amplitude') that certain particles shoot out with certain momenta, given what you shot in. But it has limitations.

In the northern hemisphere, the sun moves clockwise in the sky. In 2014 the Bolivians finally decided to break free of this colonial legacy. So now the clock on their parliament looks like this. I like it. But it must make a tempting target for counter-revolutionaries.

Some say our troubles started when we learned to grow grain: the elites could get paid laborers or slaves to grow lots of grain, store the surplus, use it to feed armies, etc. On the bright side, this is also when cats started hanging out with humans - to eat the mice that ate the grain.