Friday, February 3, 2017

Comment about particle on a circle

The wave function of a particle on a circle is a solution of the Schrödinger equation $$\label{eq:20170129a} i \frac{\partial \psi}{\partial t} = - \frac{1}{2 m} \frac{\partial^2 \psi}{\partial x^2}$$ with $x \in [0 , 2 \pi]$ and $\hbar = 1$. When \eqref{eq:20170129a} is solved in physics books, it is usually imposed that the wave function should be periodic [1]. I used to be puzzled why one has to impose the periodicity. After all, I thought, only the probability density function $|\psi|^2$ has physical meaning, so one could as well impose that $$\label{eq:20170129b} \psi(2 \pi) = e^{ i \alpha} \psi(0) \quad\text{with}\quad \alpha\in\mathbb{R}$$

Thursday, January 19, 2017

A magnetostatic exercise in 10 dimensions

I calculate the electromagnetic field generated by electrical currents in 10 spacetime dimensions (9 space and 1 time). The set up is as follows: the current flows down the positive $x_1$-axis, hits the origin and then spreads out isotropically in the $x_2 x_3 x_4$ subspace, see figure 1 and 2. I wanted to calculate this because in string theory a similar calculation is needed to obtain the Kalb-Ramond field generated by a string ending on a $D3$-brane [1]

Wednesday, January 11, 2017

A calculation in magnetostatics

I wanted to calculate the magnetic field generated by a current which flows down the positive $z$-axis, hits the origin and then spreads out radially over the $xy$ plane, see figure 1.

Friday, January 6, 2017

Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...] The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result \begin{align} \left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty} \left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\ &\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103} \end{align} This is the commutator of two Lorentz transformations in the light-cone gauge. The commutator should be zero for string theory to be Lorentz invariant. The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up. I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting results that I found.

Monday, December 5, 2016

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 $$\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0$$ 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.

Monday, November 21, 2016

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.

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 $SO(3)$.