On Killing spinors in general dimensions

The following property is true in four spacetime dimensions [1] [2]

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and there is a spinor $\psi$ such that \begin{equation}\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0 \end{equation} and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}

This property (property A from now on) can for example be used to obtain the metric and electromagnetic field of the Israel-Wilson-Perjés (IWP) black holes. Because I wanted to generalize the IWP black holes to higher dimensions, I wanted to find the generalization of property A in higher dimensions.

Extremal black holes via Killing spinors

The usual way to obtain the metric and electromagnetic field of a charged black hole in general relativity is to make a spherically symmetric ansatz, insert this ansatz into the Einstein-Maxwell equations and then solve the resulting set of non-linear ordinary differential equations. In this post I explain an alternative method that uses Killing spinors. This method can be used for extremal black holes. These are black holes with charge equal to the mass.

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 $SO(3)$.

$F_4$ 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.

Weight diagrams of $G_2$

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

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.

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].

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.

Resonance in pseudoscalar Yukawa theory

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

Scattering in Yukawa theory

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

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.

Loop correction to the 3-point vertex

In chapter 18 in Srednicki, the loop correction to the 3-point vertex in $\phi^3$ theory in six dimensions is calculated. In this post, I give comments on its numerical calculation in Mathematica.

Loop correction to the propagator in $\phi^3$ theory

The loop correction to the propagator in the $\phi^3$ theory in six dimensions is given by the Feynman diagrams

Cross section in $\phi^3$ theory

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*} The $\phi^3$ theory in six dimensions is a nice theory to explain many aspects of quantum field theory, because it is a renormalizable theory with only scalar fields. Of course the theory is not realistic because it has six dimensions and the vacuum is not stable, but it is instructive to see some aspects of quantum field theory explained without the extra complications coming from spinors or gauge fields. I also find the $\phi^3$ less cumbersome to calculate with than the more familiar $\phi^4$ theory in four dimensions.

One of the first calculations one can do is to calculate the amplitude of the scattering $\phi\phi \to \phi\phi$ at tree level.

Compton Scattering

I complain about the amplitude of Compton scattering, which is the scattering of a photon by an electron.

Properties of a charged rotating sphere in Maxwell-Chern-Simons theory

I calculate some properties of a charged rotating sphere in five dimensional Maxwell-Chern-Simons theory.

Average of a magnetic field in D dimensions

I solve problem 5.57 in Griffiths, Introduction to Electrodynamics. This problem asks to calculate the average of a magnetic field over a ball; I solve it in \( D \ge 3 \) space dimensions.

A troublesome integral

I calculate an integral that I need when solving problem 5.57 in Griffiths, Introduction to Electrodynamics. I generalized the exercise to \( D \) dimensions.

Electromagnetic properties of a charged rotating sphere

I calculate some properties of the electromagnetic field of a charged rotating sphere.

The Weyl tensor of the five-dimensional Myers-Perry metric

In this post I write the Weyl tensor of the Myers-Perry black hole as a sum of squares.

Charged rotating sphere in Maxwell-Chern-Simons theory

I calculate the electromagnetic field of a charged rotating sphere in five dimensional Maxwell-Chern-Simons theory.

Charged rotating sphere in five dimensions

I calculate the electromagnetic field of a charged rotating sphere in 4 + 1 dimensions. Griffiths calculates the magnetic field of a charged rotating sphere in 3 + 1 dimensions in example 5.11 in his book. In this post, I perform a similar calculation but in 4+1 dimensions.

The Weyl tensor of the D-dimensional Reissner–Nordström metric

In this post I write the Weyl tensor of the Reissner–Nordström black hole as a sum of squares of the electromagnetic field tensor.