Part III: Representations 10. Sums and Products. Frobenius began in ##1896## to generalize Weber’s group characters and soon investigated homomorphisms from finite groups into general linear groups ##GL(V)##, supported by earlier considerations from Dedekind. Representation theory was born, and it developed fast in the following decades. The basic object of interest, however, has…
Part II: Structures 5. Decompositions. Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg algebra ##\mathfrak{H}=\langle X,Y,Z\,:\,…
Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus…
Among the most famous people are often geniuses. It’s hard to tell whether this is the reason for the many anecdotes which are told about them, or whether this is just incidentally true. Doubts are allowed, since most scientists are quite ordinary people. But some of them are cranky and all kind of stories circulate…
Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##, ##z \mapsto \overline{z}## for the complex conjugate, ##\tau## for transposing matrices or vectors, which we interpret as written in a column if given a basis, and ##\dagger## for the combination of conjugation and transposition, the adjoint…
Basics Language first: There is no such thing as the Hilbert space. Hilbert spaces can look rather different, and which one is used in certain cases is by no means self-evident. To refer to Hilbert spaces by a definite article is like saying the moon when talking about Jupiter, or the car on an automotive…
Part 1 Representations Image source: [24] 6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the numbering of the Pauli matrices ##(\text{I 4}), (\ref{Pauli-I})##, the linear combinations of them in the definition to basis vectors ##\mathfrak{B}## of ##\mathfrak{su}(2,\mathbb{C}) \; (\text{I 5}), (\ref{Pauli-II}), (\ref{Pauli-III})##, the embedding of the orthogonal groups ##(\text{I…
Part 2 Differentiation, Spheres and Fiber Bundles Image source: [24] The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir elements or irreps. This already reveals…
Exam situations are always situations of stress. It comes with our endeavor to be as good as possible together with our fear to fail. Some students handle these situations better than others. But there are some tricks I encountered over the years tutoring young students. I’m sure everybody has developed their own ways to get…
All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confused. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind are forgotten. So what are they? Especially when it comes to the adjoint…
Let me start with a counter-question. What is a number? Before you laugh, there is more to this question as one might think. A number can be something we use to count, or more advanced an element of a field like the real numbers. Students might answer that a number is a scalar. This is…
Important Theorems – biased, of course Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in $$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||< \varepsilon_1 \} \times \{y \in \mathbb{R}^m\,\vert \,||y-y_0||< \varepsilon_2\}$$ and ## f: U_1 \times U_2 \rightarrow \mathbb{R}^m ## a function with ##f(x_0,y_0)=0## which is totally…
Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications, since a tensor field includes a variety of instances, e.g. vectors, functions or differential forms. In case of vector fields we additionally get a Lie algebra structure. This is, although formulated in…
Some Topology Whereas the terminology of vector fields, trajectories and flows almost by itself suggests its origins and physical relevance, the general treatment of vector fields, however, require some abstractions. The following might appear to be purely mathematical constructions, and I will restrict myself to a minimum, but they actually occur in modern physics:…
Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section of complex functions (The Pantheon of Derivatives – Part I), the main parts of defining a differentiation process are a norm and a direction. So to extend the differentiation concepts on normed vector spaces seems to be the obvious thing to do. Fréchet Derivative…
Differentiation in a Nutshell I want to gather the various concepts at one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum. There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in…
First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space 1). We further want this action to preserve the given structure, because its structure is exactly what we’re interested in. And this other space here should…