(1) Feynman diagrams are always read bottom to top.

(2) String theory seems to eliminate the ultraviolet divergences in any single Feynman diagram, but the sum over Feynman diagrams seems to still diverge. I say “seems to” because it seems hard to find precise theorems about this lore.

]]>(2) Is it necessary that the divergences arise from the hypothesis of continuous space-time? Don’t string theorists claim to be able tame the divergences by replacing point particles with more extended objects?

]]>I concluded with a famous example: the magnetic moment of the electron. With a truly heroic computation, physicists have used QED to compute this quantity up to order If we also take other Standard Model effects into account we get agreement to roughly one part in

However, if we continue adding up terms in this power series, there is no guarantee that the answer converges!

]]>Thanks again, this helps!

]]>For what you want, it would be better to take the Hamiltonian for the theory a 2-dimensional grid—one space dimension and one time dimension. If you do a nonrelativistic version (that is, take the limit) you’d get the ordinary Schrödinger equation for massive particles on a line, discretized, but with a certain amplitude for particles to interact. This interaction can turn 2 particles into 2, 3 into 1, 1 into 3, 4 into 0 or 0 into 4.

It would take a little while to explain it well enough to code it up. But I don’t think it’s very easy to “see” the wavefunction of a multiparticle state! It’ll be a function of many variables: the positions of all the particles in your collection.

(Yes, this is the Fock space business.)

]]>Thanks, your explanation about scattering was crucial for me! I’ve read up a little on scattering in general, and in particle physics in particular, including the quite interesting hardware: bubble and cloud and spark and drift chambers. So, mathematically, it’s all about before-after asymptotics.

I had secretly hoped I could come up with a computer visualization of some low-dimensional toy QFT, where particles flit around in front of me as probability clouds. But apparently, achieving that would start from a different story than the one in your blog post.

]]>I should emphasize that there are lots of different viewpoints on quantum field theory, so when you start learning it you feel like you’re talking to blind men who are experts on elephants: everyone will give you a different story, and it’s all very confusing. After a decade of study things get clearer.

In this post I’m telling just one version of the story: how to use path integrals to perturbatively compute the scattering matrix, also known as the ‘S-matrix’. The S-matrix says: if you shoot in a collection of particles with specified energies and momenta at past infinity, what is the amplitude for a collection of particles with specified energies and momenta to come out at future infinity?

The S-matrix is what people doing particle accelerator experiments care about, so it’s what you typically learn how to compute in a first course in quantum field theory. It is *quite hard* to take the S-matrix and read off the answers to the questions you’re talking about.

Your questions are about the *local* behavior of quantum field theory: e.g, what’s going on in one hypercubic meter of spacetime. (You said ‘cubic meter of space’, but since everything here is relativistic, a spacetime viewpoint would be more appropriate.) The S-matrix is a purely *global* thing: computing it involves integrals over all of Minkowski spacetime. To make matters worse (from your viewpoint), these integrals are easiest to do with the help of a Fourier transform. So, the calculations actually involve integrals over particle energy-momenta

not locations of events

Luckily, there is more to quantum field theory than the S-matrix! There *is* a local story. It can again be studied using path integrals (among several other methods). Now, however, instead of integrating over all of spacetime, we should integrate over a finite 4-volume: for example, a hypercube.

This makes Fourier transforms somewhat less attractive. So, instead of integrating over energy-momenta, one for each edge of a Feynman diagram, we integrate over spacetime points, one for each vertex of our Feynman diagram! These vertices are ‘events’: points where an interaction occurs.

If you do this, you can fix the spacetime locations of the incoming and outgoing particles rather than their momenta. That is, you fix where the particles enter and leave your hypercube. These correspond to the ‘loose ends’ of your Feynman diagram.

Technically, this sort of calcuation is a lot harder than computing an S-matrix, so you’ll never see it in an introductory textbook. Conceptually it is easier.

]]>I thought about this more and realized: my uncertainty comes from not knowing the mathematical structure of the phase space. In particular since the number of particles can apparently vary over time. It this the Fock space thing? My “dicing” remark was inspired by your phrase “external edges are held fixed”…

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