Here is the paragraph:
>If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations: (i) Spacetime is a **pseudo-Riemannian manifold** *M*, endowed with a **metric tensor** and governed by geometrical laws. (ii) Over *M* is a **vector bundle** *X* with a **non-abelian gauge group** *G*. (iii) **Fermions** are sections of **(Ŝ +⊗VR)⊕(Ŝ ⊗VR¯)(Ŝ+⊗VR)⊕(Ŝ⊗VR¯)**. ***R*** and ***R*****¯** are not **isomorphic**; their failure to be **isomorphic** explains why the light fermions are light and presumably has its origins in representation difference Δ in some underlying theory. All of this must be supplemented with the understanding that the geometrical laws obeyed by the **metric tensor**, the **gauge fields**, and the **fermions** are to be interpreted in quantum mechanical terms.
>
>Edward Witten, “Physics and Geometry”
According to Eric Weinstein (who I know is a controversial figure, but let’s leave that aside for now), this is the most beautiful and important paragraph written in the English language. You can watch him talk about it [here](https://youtu.be/vdW9XDBuxjU?t=3079) or take a deep dive into his [Wiki](https://theportal.wiki/wiki/Graph,_Wall,_Tome).
Could someone (1) literally translate the paragraph so a layman can grasp the gist of it, switching the specific jargon **in bold** with simplified plain English translations? Just assume I have no formal education in math or physics, so feel free to edit the flow of the paragraph for clarity’s sake. For example, something like:
>If one wants to summarize our knowledge of physics in the briefest possible terms, there are three really fundamental observations: (i) Spacetime is a ~~pseudo-Riemannian manifold~~ ***flexible*** ***3-dimension space*** *M*, endowed with a ~~metric tensor~~ **composite list of contingent quantities** and governed by geometrical laws… etc.
And (2) briefly explain the importance of this paragraph in the big picture of physics?
In: Physics
PhD student in theoretical physics here. This paragraph describes what the whole universe really is and what everything in the universe is fundamentally made of. It touches on a lot of very advanced ideas in mathematics, but I’ll do my best to unpack it. Warning: It’ll be long.
**(i)** Age-old question: What does the whole universe look like if you stand “outside” of it? Is it like a flat plane that extends in all directions forever? Is it like a ball, where if you walk in one direction for a while, you’ll return to your original position? Or is it like a large twisted pretzel with holes? In mathematics, we call these different shapes, flat or curved, **manifolds**.
To describe a “pretzel” or a “flat plane”, we need to say where the shape is “rounded” or “curved”, like a blind man touching an elephant, telling his friend where the round belly is and where the trunk is sticking out. We call this description a **metric tensor** on the manifold.
What this point tells us is that our universe has all sorts of bumps and troughs here and there, and everything in the universe—the Sun, the Moon, galaxies—move around according to where the bumps and troughs are, like a small ant on a pretzel trying to walk in a “straight line” but inadvertently walking in circles. In other words, the Earth goes in circles around the Sun (and stars around galaxies) because it follows **geometrical laws** of “how to walk on an irregular bumpy manifold”.
It turns out that clocks run at different speeds depending on where they’re placed in the universe, whether it’s at the bottom of a trough or on flat ground. Physicists figured out that that’s because space isn’t the only thing getting curved and twisted in our universe; time is too. Normal run-of-the-mill manifolds don’t do the job anymore. This is when **pseudo-Riemannian manifolds** come in to describe our universe, where both time and space are twisted with each other, into curved **spacetime**. Don’t even try to imagine what that looks like; we can’t either. All this is described in more detail in this [Introduction to General Relativity](https://en.wikipedia.org/wiki/Introduction_to_general_relativity).
**(ii)** Another age-old question: Magnets, how do they work? Also, what makes rocks stick together and not fall apart? Physicists have figured out that these forces are due to these things called **fields**, which are invisible but permeate everywhere in the entire universe. Take the electromagnetic field as an example. It is everywhere. Closer to a magnet, where the field is stronger, a second magnet gets attracted to the first one very quickly; farther from the magnet, where the field is weaker, the second magnet barely moves at all.
Remember from (i) that our universe is a curved shape called a manifold? Well, the electromagnetic field (and other fields for other forces) is everywhere on this manifold. In mathematics, we call that a **fibre bundle over the manifold**. (Pardon the strange name. If you try to imagine little hairs on a pretzel indicating how strong the field is at each point on the pretzel, it ends up looking like a bundle of fibres.)
Now we need to know how the electromagnetic field makes the magnet move around, how it makes your microwave cook your food, and how light comes out of a lightning. All of this is described by a very special kind of theory called **gauge theory**. What you need to know is that every gauge theory comes with a mathematical structure called a **group**. The simpler the group, the simpler the the interaction between the corresponding force and matter is, and vice versa. The simplest group is an **Abelian group**, which is what describes the electromagnetic field.
The electromagnetic field also keeps rocks together and strong. But if you chop a rock into little pieces, you get to the atom, made of neutrons, protons, and electrons. Neutrons and protons are in turn made of even smaller particles called quarks. What keeps the quarks together in a neutron or a proton is the strong nuclear force, caused by the **gluon field**. Neutrons and protons can also turn into each other, emitting radiation that makes some cancer treatments possible. This is due to the weak nuclear force, caused by the **W and Z boson fields**.
These fields are also described by gauge theories, so they’re called **gauge fields**, but their interaction with matter is so complicated that their corresponding groups are **non-Abelian groups**, which is as complicated as you can get when it comes to groups.
**(iii)** One final age-old question: What’s stuff made of? What’s the smallest bit of matter? As I said before, chopping things down to the smallest pieces gives you electrons and quarks. These particles (and a few others) all belong to a class of particles called **fermions**. The special thing about fermions is that every fermion has a left-handed and a right-handed version, just like gloves—they look alike, but are mirror reflections of each other.
The gluon field doesn’t care whether the fermion is left or right-handed. It doesn’t discriminate. (Be like gluon.) You might think the electromagnetic field and the W and Z boson fields are good guys too, but no, they like to interact more with the left-handed fermions than the right-handed ones. This is what the formula means.
The correct formula is (Ŝ+⊗VR)⊕(Ŝ−⊗VR̃). (Unfortunately, Reddit doesn’t support subscripts.) It basically says the left-handed fermion Ŝ+ interacts with all the gauge fields in this way (VR), while the right-handed fermion Ŝ− interacts with the gauge fields in a different way (VR̃). It is important that VR and VR̃ are **not isomorphic**, which is just a snobby way of saying “they’re different”.
We know that the electron, along with another fermion called the neutrino, is lighter than the quarks. We don’t yet know why that is the case, but every new idea physicists have come up with over the years involves the difference between the way gauge fields interact with left-handed and right-handed fermions (**representation difference**). If one of those ideas turns out to be correct, then it is the **underlying theory** that we all dream about.
Finally, if all of the above is not complicated enough, God plays a joke on us by making everything **quantum mechanical**. In short, this means that the same particle can be in several different places at the same time, and particles are randomly popping in and out of existence everywhere, all the time.
**Personal thoughts:** The paragraph is certainly very concise, mostly because it takes entire textbooks to truly describe the mathematical terms in each of the three bullet points. It is also very beautiful, because it tells you in precise mathematical terms what you’d need if your job were to create another universe. However, it’s like describing an elaborate wedding cake by listing all its ingredients.
“The most beautiful paragraph in physics” is a bit of a stretch. Sure, it encapsulates what our universe is, deep down. But it doesn’t capture any of the emergent phenomena—how these particles come together into complex atoms and molecules, how atoms form beautiful crystals and rocks, how rocks form planets, how stars and dust form galaxies, how too much stuff makes a black hole… Physics is a vast subject with countless interesting questions to study, and “what is the fundamental structure of the universe” is just one of them.
**TL;DR:** Our universe has curved spacetime in which matter, made of very small particles called fermions, interacts through forces controlled by gauge fields. The interaction is different for left-handed and right-handed fermions. The paragraph is beautiful because it tells us what blueprint God had in mind when creating our universe.
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