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
Okay, let me give it a try. Note: I am not a physicist or mathematician, but i have read a bit of differential geometry. So this is probably wrong in subtle ways. Cool? Cool! I will try i) and ii)
For simplicity, imagine our universe being flat like a piece of paper. And do it in a very specific way: we live on a line and the second dimension is time. so if you move along your 1d position universe, this is represented by a curve on the 2d plane.
You can bend this paper-universe, but not fold, rip or tear it. The sides of the paper might be glued together, for example to form a doughnut. i) and ii) together mean, that you can look at a very small part of your bent piece of paper and pretend it to be flat. you can do that simply by pressing down with your finger to straighten it out. In this flat part (Which might be very tiny) you can measure distances and angles with your normal geometrical tools. Most likely you can’t do it everywhere at the same time (e.g. on a doughnut). So for big distances, measuring distances and angles can become complicated, because you need to flatten the parts differently to straighten them out.
Physically this means that for slow objects which are close together, everything looks normal, so you can for example measure distances and speed easily. for very fast objects, the object position changes a lot in a small amount of time which in our 2d paper universe means that it has a large distance and you might get into trouble trying to flatten out the universe enough to be able to measure distances properly.
This has consequences, for example you will have difficulties to compare distances of objects that are fast with distances of objects that are slow. To be able to create a theory, we need a way to compare behaviors at different points (i can observe something at my current speed/position/time. How would it look like at a different speed/position/time?). Luckily, there is a tool to do this and it is called a gauge-group. But it behaves in unintuitive ways.
so the important part here is the existence of these tools because they allow the description in a way that is not depending on where you are, when you are and how fast you are. you can develop a theory at a certain point and the gauge-group will tell you how it will look like somewhere else.
Latest Answers