Historical question: how to reconcile quantum with special relativity? The standard model solved this.
Present question: how to reconcile quantum with general relativity, i.e. gravitation?
Gravity special because it is so weak. The whole earth is pulling you down, but the electrostatic forces in the chair you're sitting in keeps you up.
Standard model was hard to figure out because the distance scale was very small, and things behave differently there. The standard model works, but only through great cleverness. The same problem faces us with gravitation; how does spacetime curvature behave at very, very small distance scales?
Consider 3d gravitation (2 spatial, one temporal). Curvature is very special there; we can think of it as tiny cutout cones, which is "flat" in a certain sense still. So local curvature vanishes, its cosmology and quantization is pretty well understood, etc.
But our universe is (at least!) 4d. Quantization here is hard. If we try to impose local flatness on such a model, one dimension becomes trivial, and we're back to where we started. In 4d spacetime, we pretty much inevitably get gravitational radiation from any moving massive object, which wouldn't happen in flat spacetime.
[sudden digression into alluding to string theory] non-straightness of strings causes difficulty; he's promising to discuss a situation with only "straight" strings. Negative curvature ("surplus of space" around a particle's worldline) must be dealt with as well as positive curvature ("deficit of space").
To calculate the deficit angle, consider a thickening of the string... oh boy I don't understand what he's talking about now. Something something 8πgρ. The angle has something to do with the "tension" of the string, and the mass of the particle.
Talking about intersections of lines flying through space. I didn't realize he was talking about particles that were one-dimensional even instantaneously. In this picture, colliding strings lead to discontinuities, and new vertices, and new edges in the graph.
Unfortunately, it's totally wrong.
Okay, now: SO(3,1) = SL(2,C) (Lorentz group?) Holonomy on the cycle around the "force string" tracing behind the wake of two colliding lines is complicated and depends on the two original particles: it comes out Q-1PQP-1 and this doesn't satisfy some funny condition on the trace, making it not really a string.
These problems are fixed if we allow two new strings to be born from every collision.
The equations that pop out of trying to accommodate these things are quadratic, and it's hard to tell whether there are even solutions. The velocity of all bits should be less than the speed of light.
This causes further problems, which might be resolved by having more resultant intermediary strings.
The whole business can't be quantized. No PCT symmetry.
Maybe we should think about discrete subgroups of the Poincaré group? Crystal groups?
[thought to myself: curvature is visualized as "deficit" or "surplus" of space itself compared to euclidean space, but as far as I remember, usually represented mathematically as a scalar field on a manifold, which amounts to a homeomorphism to euclidean space. What's the relationship between these perspectives?]
At some point I saw quickly skipped past a formula that looked like the
az + b ------ cz + dfrom modular forms...
The biggest two foundational things I didn't understand: one, is there a background of space against which these strings are moving? two, why do they have the dimension they do?