Recommended reads #1

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

$\displaystyle \begin{cases} \frac{\partial^2}{{\partial t}^2}A - c^2\frac{\partial^2}{{\partial x}^2}A = \gamma B Z \\ \frac{\partial^2}{{\partial t}^2}B - c^2\frac{\partial^2}{{\partial x}^2}B = \gamma A Z \\ \frac{\partial^2}{{\partial t}^2}Z - c^2\frac{\partial^2}{{\partial x}^2}Z = - 4\pi^2 \nu^2 Z + \gamma A B \end{cases} \ \ \ \ \$

As initial conditions we choose ${Z(0,x)=0, \partial_t Z(0,x) = 0}$ , i.e. there is no particle everywhere. Then we choose for ${A}$ and ${B}$ (whose equations at early times are basically just wave equations, since ${Z=0}$ ) two wavepackets traveling toward each other, i.e. given by

$\displaystyle W_{\pm}(t,x)=e^{-(x \pm ct)^2} \cos(2\pi (x\pm ct))$

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
]

Almost Originality

a mathematical journal

Recommended reads #1

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