Tuesday, October 11, 2016

The isotropic harmonic oscillator

While studying Lie-algebras I read that the three-dimensional harmonic oscillator has an $SU(3)$ symmetry. I found this very unexpected; I thought it was ''obvious'' that the symmetry is only $SO(3)$.

Thursday, October 6, 2016

$F_4$ tensor products

The product of two irreducible representations of a simple Lie algebra can be decomposed into irreducible components. There are various techniques to calculate this decomposition, see for example chapter XIV in [1]. However, the decomposition can also be calculated by brute force.

Tuesday, September 27, 2016

Weight diagrams of $G_2$

This post contains pictures of weight diagrams of irreducible representations of the Lie algebra $G_2$.

Monday, September 19, 2016

The 7-dimensional representation of $G_2$

The exceptional Lie algebra $G_2$ has an irreducible representation of dimension 7. In this post I calculate the matrices of this irrep.

Saturday, September 10, 2016

Commutation relations in $G_2$

In this post I calculate the structure constants of the exceptional Lie algebra $G_2$. I assume the reader is familiar with Lie algebras, for example at the level of chapter 9 in [1].

Sunday, July 31, 2016

Decay rate of the muon

The muon is a heavy cousin of the electron and decays into an electron and two neutrinos \begin{equation*} \mu \to e + \nu_{\mu} + \bar{\nu}_e \end{equation*} The decay rate of the muon is calculated in section 10.2 in Griffiths [1]. To calculate the decay rate $\Gamma$ one needs to calculate a 6-dimensional integral coming from 3 particles times 3 momentum integrals with momentum conservation. The calculation of this integral in Griffiths is quite lengthy and I do not have much insight about it. I have questions like
  • Could I have calculated the integral in a different order than the one in Griffiths?
  • Could one still calculate the result analytically if more particles were produced in the decay?
  • Is there a faster way to obtain the result?
I therefore decided to calculate $\Gamma$ in a different way. My calculation is motivated by [2]. I set the mass of the electron $e$, of the muon neutrino $\nu_{\mu}$ and of the electron anti neutrino $\bar{\nu}_e$ to zero.

Friday, July 15, 2016