[mfn]

Yeah I wasn't sure how to set things up there - if I kept a single space dimension + a time dimension, then I'd have to explain the negative sign on one of the terms, and probably also talk about the Einstein summation convention to keep things clean. Whereas with a single time dimension, it's not really 'spacetime' as you pointed out.

What motivated this post was that I wanted to give a concrete example of what it really means for some symmetry to 'dictate' the structure of a physical theory, but do so in the simplest way possible - i.e. not deal with spinors, gamma matrices, quantum fields - and the rest of the actual machinery of the standard model. The core idea is so profound that I felt like there has to be a way to get a taste of it across in a way that's accessible.

Turned out to be a lot harder than I thought - I had to skip quite a few steps in the post to keep it from becoming too long, but I'm hoping the model still conveys the essence of how a symmetry + action principle can 'predict' particles.

[nyc111]

I think what you did is standard. But what I question is, if we can solve our problem without assuming spacetime, why do we need the abstraction called spacetime? Spacetime looks like a historical quirk that physicists feel obligated to carry.

For instance, bending of the light experiment is not done in spacetime but in space and time.

[kryptiskt]

But you can't rip them apart in our physical theories, in relativity different observers will have different ideas about space and time, but will agree about certain invariants if you take both space and time into account. Then a few years later, Minkowski formulated special relativity very elegantly using a four dimensional space-time. That view is basic to general relativity, where the foundation is the spacetime metric and the energy-momentum tensor.