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

# Pieter Jan De Smet Calculations

Various topics in mathematics and physics

## Tuesday, September 27, 2016

## 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?

## Friday, July 15, 2016

### Resonance in pseudoscalar Yukawa theory

A post with calculations in pseudoscalar Yukawa theory and plots of cross sections to illustrate a resonance.

## Friday, July 8, 2016

### Scattering in Yukawa theory

I illustrate the cross section of the scattering $e^+ e^- \to e^+ e^-$ in Yukawa theory.

## Thursday, June 30, 2016

### Two-particle scattering at one loop

In his book on quantum field theory, Srednicki performs many calculations in the $\phi^3$ theory in six dimensions, which has Lagrangian
\begin{equation*}
\mathcal{L} = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^2 \phi^2 + \frac{g}{6} \phi^3
\end{equation*}
For example, in chapter 20 Srednicki calculates the two particle scattering amplitude, including all one-loop corrections. In this post I illustrate the result.

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