# Essays

• Demystifying the QCD Vacuum – Part 1 – The Standard Story
After being confused for several weeks about various aspects of the QCD vacuum, I now finally feel confident to write down what I understand. The topic itself isn’t complicated. However, a big obstacle is that there are too many contradictory “explanations” out there. In addition, many steps that are far from obvious are usually treated […]
• Making Sense of Particle Physics Research
There are two things I learned recently that helped me a lot as a young researcher in understanding what is currently going on in modern particle physics. These things not only help me to understand the work of others better, but also allow me to formulate better what kind of research I’m currently doing and […]
• When do you understand?
Over the years I’ve had many discussions with fellow students about the question: when do you understand something? Usually, I’ve taken the strong position that is summarized by this famous Vonnegut quote: “any scientist who couldn’t explain to an eight-year-old what he was doing was a charlatan.” In other words: you’ve only understood a given […]
• Larger Symmetries
“Further progress lies in the direction of making our equations invariant under wider and still wider transformations.” These prophetic lines were written in 1930 by P. A. M. Dirac in his famous book “The Principles of Quantum Mechanics”. In the following centuries, tremendous progress was made exactly as he predicted. Weak interactions were described perfectly […]
• Physics Model Fits in Mathematica
This shouldn’t be hard. We have some physics model and want to find values for the model parameters that yield some experimental measured values. However, there are several small things that aren’t obvious and it took me quite some time to make things work. I wasn’t able to find a good explanation how such a […]
• Resources that helped me understand Grand Unified Theories
I recently finished my master's thesis on dark matter in Grand Unified Theories. Here are some resources that I found particularly helpful.
• What are Quantum Numbers?
For quite some time I didn’t really understand what quantum numbers are. For example, why do we use the words “red”, “blue” and “green” for the charges of the strong interaction? Why does a gluon carry “red anti-green + green anti-red” color? From the group theoretical perspective these things actually make a lot of sense […]
• Solving the Renormalization Group Equations for the Gauge Couplings
We have already discussed why the gauge couplings depend on the energy scale and how we can compute the renormalization group equations (RGEs) that describe how the couplings change with energy. In this post we talk about how we can solve the RGEs. The Standard Model RGEs To solve the RGEs, we need boundary conditions. […]
• Derivation of the Renormalization Group Equations for the Gauge couplings
In this post I discussed why the gauge couplings depend on the energy scale. Here I discuss how we can compute this change with energy in practice. This is another post from the category “I wished this kind of post had existed when I started”. In addition to the general formulas, I discuss two examples […]
• Renormalization Group Flow
The standard model contains three gauge couplings, which are very different in strength. This is not really a problem of the standard model, because we can simply put these measured values in by hand. However, Grand Unified Theories (GUTs) provide a beautiful explanation for this difference in strength. A simple group $G_{GUT}$ implies that we […]
• Classification of all Simple Lie Groups
Simple Lie groups are important, because they are in some sense the building block we can use to build up all Lie groups. Or formulated differently: simple Lie groups are the atoms of Lie theory. They are especially important in theories that unify the fundamental forces, because of the gauge group of the theory is […]
• Write down what you learn – or – Why I wrote a textbook for students as a student
I’ve written a textbook during my master studies. People who hear about this usually ask me: “How did this happen?” The truth is that my book is simply a collection of things that I wrote down for myself. Whenever I understand something, I write it down. It has happened too often to me that something […]
• One Thing You Must Understand About Studying Physics
As a beginner student it’s really easy to feel overwhelmed and stupid, because every page in the textbook you’re reading makes the question mark above your head bigger. It took me quite some time to realize that most textbooks aren’t written for the reader, but for the author. My realization started with a little sentence […]
• Vectors, Forms, p-Vectors, p-Forms and Tensors
This is a topic that can cause quite a bit confusion, so here is a short post I can come back to whenever I get confused. Lets start with the definition of a vector. A vector is… uhmm … I guess you have a rough idea of what a vector is. Otherwise this is stuff […]
• My Book “Physics From Symmetry” has been published!
Update 10.4.16: Almost one year after its publication it’s time for a small recap. On the downside, the book has made me neither rich nor famous so far ;). However there are some things that make me particularly happy: Rutwig Campoamor Stursberg has published a summary and review, in which he writes about Physics from […]
• What’s so special about the adjoint representation of a Lie group?
A representation is a map that maps each element of the set of abstract groups element to a matrix that acts on a vector space (see this post). The problem here is that at the beginning this can be quite confusing: If we can study the representation of any group on any vector space, where […]
• How is a Lie Algebra able to describe a Group?
If you understand the idea Lie Group= Manifold, you can easily understand one of the most curious facts of Lie theory: The Lie algebra $\frak{g}$, which is defined as the tangent space at the identity  $T_eG$, is able to tell us almost everything about a given Lie group $G$. The connection between Lie algebra elements […]
• Short Introduction to and Motivation for Representation Theory
What may seem at a first glance like just another mathematical gimmick of group theory, is of incredible importance in physics. One can consider the Poincaré group (the set of all transformations that leave the speed of light constant) and use the framework of representation theory to construct the irreducible representations of this group. (The […]
• Lie Group Theory – A Completely Naive Introduction
In this posts we discuss how continuous symmetries can be described mathematically. Many important features of such symmetries can be described using something simple, called Lie algebras. In the second part of this post you will see why a new object, called Lie bracket, is the defining feature of a Lie algebra. As an aside: […]
• Origin of the term Delta FUNCTION
Mathematicians get, at least, goosebumps if you say it. Most physicists are aware of the fact that it is somehow incorrect to say. Nevertheless, the word is so common among physicists: Delta function. The object that extracts the value of a function at parameter value zero $\tilde \delta(f(x)) = f(0)$ is not a function, […]
• One Electron and The Egg
In the preface to my book I wrote: “To me, the most beautiful thing in physics is when something incomprehensible, suddenly becomes comprehensible, because of a deep explanation.” Here is one such example, although most experts would argue not a good one. More about that later. We know there are electrons. Many, many electrons. They […]
• Motivation for the Group Theory Axioms
Numbers measure size, groups measure symmetry – M.A. Armstrong: Groups and Symmetry Group theory is the mathematical tool one uses in order to work with symmetries. Because symmetries are defined as invariance under transformations, one defines a group as a collection of transformations. Let’s get started with two easy examples to get a feel for what we want to […]
• Why Group Theory?
Group theory is the branch of mathematics one uses to work with symmetries.  A symmetry of an object is a transformation that leaves the object unchanged. The word object is chosen purposefully, because it is very vague. There is one branch of mathematics that deals with all kinds of symmetries, any kind of object can […]

• Larger Symmetries
“Further progress lies in the direction of making our equations invariant under wider and still wider transformations.” These prophetic lines were written in 1930 by P. A. M. Dirac in his famous book “The Principles of Quantum Mechanics”. In the following centuries, tremendous progress was made exactly as he predicted. Weak interactions were described perfectly […]
• What are Quantum Numbers?
For quite some time I didn’t really understand what quantum numbers are. For example, why do we use the words “red”, “blue” and “green” for the charges of the strong interaction? Why does a gluon carry “red anti-green + green anti-red” color? From the group theoretical perspective these things actually make a lot of sense […]
• Classification of all Simple Lie Groups
Simple Lie groups are important, because they are in some sense the building block we can use to build up all Lie groups. Or formulated differently: simple Lie groups are the atoms of Lie theory. They are especially important in theories that unify the fundamental forces, because of the gauge group of the theory is […]
• What’s so special about the adjoint representation of a Lie group?
A representation is a map that maps each element of the set of abstract groups element to a matrix that acts on a vector space (see this post). The problem here is that at the beginning this can be quite confusing: If we can study the representation of any group on any vector space, where […]
• How is a Lie Algebra able to describe a Group?
If you understand the idea Lie Group= Manifold, you can easily understand one of the most curious facts of Lie theory: The Lie algebra $\frak{g}$, which is defined as the tangent space at the identity  $T_eG$, is able to tell us almost everything about a given Lie group $G$. The connection between Lie algebra elements […]
• Short Introduction to and Motivation for Representation Theory
What may seem at a first glance like just another mathematical gimmick of group theory, is of incredible importance in physics. One can consider the Poincaré group (the set of all transformations that leave the speed of light constant) and use the framework of representation theory to construct the irreducible representations of this group. (The […]
• Lie Group Theory – A Completely Naive Introduction
In this posts we discuss how continuous symmetries can be described mathematically. Many important features of such symmetries can be described using something simple, called Lie algebras. In the second part of this post you will see why a new object, called Lie bracket, is the defining feature of a Lie algebra. As an aside: […]
• Motivation for the Group Theory Axioms
Numbers measure size, groups measure symmetry – M.A. Armstrong: Groups and Symmetry Group theory is the mathematical tool one uses in order to work with symmetries. Because symmetries are defined as invariance under transformations, one defines a group as a collection of transformations. Let’s get started with two easy examples to get a feel for what we want to […]
• Why Group Theory?
Group theory is the branch of mathematics one uses to work with symmetries.  A symmetry of an object is a transformation that leaves the object unchanged. The word object is chosen purposefully, because it is very vague. There is one branch of mathematics that deals with all kinds of symmetries, any kind of object can […]

• Larger Symmetries
“Further progress lies in the direction of making our equations invariant under wider and still wider transformations.” These prophetic lines were written in 1930 by P. A. M. Dirac in his famous book “The Principles of Quantum Mechanics”. In the following centuries, tremendous progress was made exactly as he predicted. Weak interactions were described perfectly […]
• Resources that helped me understand Grand Unified Theories
I recently finished my master's thesis on dark matter in Grand Unified Theories. Here are some resources that I found particularly helpful.
• Solving the Renormalization Group Equations for the Gauge Couplings
We have already discussed why the gauge couplings depend on the energy scale and how we can compute the renormalization group equations (RGEs) that describe how the couplings change with energy. In this post we talk about how we can solve the RGEs. The Standard Model RGEs To solve the RGEs, we need boundary conditions. […]
• Derivation of the Renormalization Group Equations for the Gauge couplings
In this post I discussed why the gauge couplings depend on the energy scale. Here I discuss how we can compute this change with energy in practice. This is another post from the category “I wished this kind of post had existed when I started”. In addition to the general formulas, I discuss two examples […]
• Renormalization Group Flow
The standard model contains three gauge couplings, which are very different in strength. This is not really a problem of the standard model, because we can simply put these measured values in by hand. However, Grand Unified Theories (GUTs) provide a beautiful explanation for this difference in strength. A simple group $G_{GUT}$ implies that we […]
• One Electron and The Egg
In the preface to my book I wrote: “To me, the most beautiful thing in physics is when something incomprehensible, suddenly becomes comprehensible, because of a deep explanation.” Here is one such example, although most experts would argue not a good one. More about that later. We know there are electrons. Many, many electrons. They […]

• Demystifying the QCD Vacuum – Part 1 – The Standard Story
After being confused for several weeks about various aspects of the QCD vacuum, I now finally feel confident to write down what I understand. The topic itself isn’t complicated. However, a big obstacle is that there are too many contradictory “explanations” out there. In addition, many steps that are far from obvious are usually treated […]
• Making Sense of Particle Physics Research
There are two things I learned recently that helped me a lot as a young researcher in understanding what is currently going on in modern particle physics. These things not only help me to understand the work of others better, but also allow me to formulate better what kind of research I’m currently doing and […]
• When do you understand?
Over the years I’ve had many discussions with fellow students about the question: when do you understand something? Usually, I’ve taken the strong position that is summarized by this famous Vonnegut quote: “any scientist who couldn’t explain to an eight-year-old what he was doing was a charlatan.” In other words: you’ve only understood a given […]
• Physics Model Fits in Mathematica
This shouldn’t be hard. We have some physics model and want to find values for the model parameters that yield some experimental measured values. However, there are several small things that aren’t obvious and it took me quite some time to make things work. I wasn’t able to find a good explanation how such a […]
• Write down what you learn – or – Why I wrote a textbook for students as a student
I’ve written a textbook during my master studies. People who hear about this usually ask me: “How did this happen?” The truth is that my book is simply a collection of things that I wrote down for myself. Whenever I understand something, I write it down. It has happened too often to me that something […]
• One Thing You Must Understand About Studying Physics
As a beginner student it’s really easy to feel overwhelmed and stupid, because every page in the textbook you’re reading makes the question mark above your head bigger. It took me quite some time to realize that most textbooks aren’t written for the reader, but for the author. My realization started with a little sentence […]
• My Book “Physics From Symmetry” has been published!
Update 10.4.16: Almost one year after its publication it’s time for a small recap. On the downside, the book has made me neither rich nor famous so far ;). However there are some things that make me particularly happy: Rutwig Campoamor Stursberg has published a summary and review, in which he writes about Physics from […]

• Vectors, Forms, p-Vectors, p-Forms and Tensors
This is a topic that can cause quite a bit confusion, so here is a short post I can come back to whenever I get confused. Lets start with the definition of a vector. A vector is… uhmm … I guess you have a rough idea of what a vector is. Otherwise this is stuff […]
• Origin of the term Delta FUNCTION
Mathematicians get, at least, goosebumps if you say it. Most physicists are aware of the fact that it is somehow incorrect to say. Nevertheless, the word is so common among physicists: Delta function. The object that extracts the value of a function at parameter value zero $\tilde \delta(f(x)) = f(0)$ is not a function, […]