>>14152684>>141545552. Wigner's theorem. Why particle physics? is a relevant question since 1. didn't seem to require much particle shit. Wigner's theorem is the statement that a particle with spin has degrees of freedom if it's massive, but 2 if it's massless. This is important because it tells you how to build a theory on the spot. Say you imagine a massive spin-0 particle: a scalar field is sufficient to describe this, because it has 1 degree of freedom. So you write down the simplest Lorentz-invariant (translation invariance is usually trivial, and other symmetries can be considered later) Lagrangian with only a scalar field you can imagine. This turns out to give you the Klein-Gordon equation. The nonrelativistic limit would give the Schrodinger equation. That's trivial, so what about higher spin, say massless spin-1? That requires something with at least 2 degrees of freedom, so you need a bigger Lorentz-invariant object. So you'll then say let's try a 4-vector, and you write down some simple terms that look Lorentz invariant, and look at what you get. In that case, the simplest thing you can get is actually the sourceless Maxwell Lagrangian, and to remove the extra degrees of freedom in that 4-vector (which is your potential of course) you have gauge invariance. So what we've done is waved a wand and obtained Maxwell's equations from nothing but symmetry and the existence of a particle with certain spin and mass properties. Repeating this for spin-2 gives the graviton, and its gauge conditions are essentially general relativity (invariance under smooth coordinate changes, basically) although with the caveat that this is perturbative, whereas general relativity is actually non-perturbative.
Massive spin-1 would give you something similar to the massless spin-1 case (the Proca Lagrangian), except without the gauge invariance you get in electromagnetism, and these particles would be W/Z bosons, etc. (cont)
