This is a very short digression: I found Prof. Matt Strassler’s blog very intriguing. Here are a couple of links:

Strassler – Fields and their particles (with math)

Strassler – How the Higgs field works (with math)

It attemps (and imho succeds at ) an introduction to the basic concepts of the Standard Model and the mechanism by which particles acquire mass through the Higgs field. It’s presented in a very intuitive way, while preserving enough of the mathematical structure. All the math is actually pretty elementary and there are some white lies here and there, which I think is a good thing. When you’re introducing someone to a subject, you want to sweep the dust under the carpet at first.

All in all, this much quality in science popularization is very rare I think.

I fed one of the naive models of field-particle interactions in the posts above into Mathematica’s NDSolve routine. The model (a system of PDEs) describes the interaction of two massless fields with a massive particle-field: when two quanta of the fields bump into each other, a particle is created as a vibration in the particle-field. The system is the following set of wave equations with additional cross terms that account for the interaction

As initial conditions we choose , i.e. there is no particle everywhere. Then we choose for and (whose equations at early times are basically just wave equations, since ) two wavepackets traveling toward each other, i.e. given by

Here is an animation depicting what happens

It can be interpreted as the creation of a particle (the blue graph).

For those of you that are curious and want to fiddle with the code, I’ve included it here:

\[Gamma] = 0.85; (* interaction parameter *) m = 2; (* (mass of the particle) x c^2 / (h bar) *) \[Nu] = 1; (* frequency *) \[Mu] = 1; (* wavenumber *) G[z_] := Exp[-z^2] Cos[2 \[Pi] z]; (* wavepacket definition *) Sol1 = NDSolve[{ (* NDSolve routine *) { (* system of PDEs above *) D[Z[t, x], t,t] - (\[Nu]/\[Mu])^2 D[Z[t, x], x, x] == -4 \[Pi]^2 m^2 Z[t,x] + \[Gamma] A[t, x]*B[t, x], D[A[t, x], t,t] - (\[Nu]/\[Mu])^2 D[A[t, x], x, x] == \[Gamma] B[t, x]*Z[t, x], D[B[t, x], t,t] - (\[Nu]/\[Mu])^2 D[B[t, x], x, x] == \[Gamma] A[t, x]*Z[t, x] }, Z[0, x] == 0, Derivative[1, 0][Z][0, x] == 0, (* initial values *) A[0, x] == N[G[-\[Mu] (x + 4)]], Derivative[1, 0][A][0, x] == D[G[\[Nu] s - \[Mu] (x + 4)], s] /. s -> 0, B[0, x] == N[G[\[Mu] (x - 4)]], Derivative[1, 0][B][0, x] == D[G[\[Nu] s + \[Mu] (x - 4)], s] /. s -> 0}, (* initial values of A,B are wavepackets traveling in opposite directions *) {Z, A, B}, (* variables *) {t, 0, 20}, {x, -20, 20} (* range *) ]; Animate[ Plot[ Evaluate[{20*Z[t, x], A[t, x], B[t, x]} /. Sol1], {x, -8, 8}, PlotRange -> {{-8, 8}, {-1.5, 1.5}}, PlotPoints -> 50], {t, 0, 12 }, AnimationRate -> 1 ]