# Positive Definite Cartan Matrices in Quantum Physics

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• fresh_42
In summary, the conversation discusses the observation of a positive definite but non-symmetric Cartan matrix and its potential relationship to symmetry breaks in quantum physics. The concept of Cartan matrices and their structures are also briefly mentioned.
fresh_42
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TL;DR Summary
Cartan Matrices are not symmetric.
As I was looking for an example for a metric tensor that isn't among the usual suspects, I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?

fresh_42 said:
I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?
We'll need a bit more info on what you (think you) mean by "symmetry breaks in quantum physics". Spontaneous symmetry breaking? Pathologies at boundaries between different phases of matter? Something else?

I just wondered whether this asymmetry in the matrix that in the end defines a metric for mathematicians, and which I have been told years ago on PF is therefore essentially responsible that especially simple Lie groups play such a prominent role in QM, has any physical consequences and if, what they are. And please, don't open a distraction with ##U(1)## or the ##A_l## series.

Last edited:
What precisely are you talking about? What are "Cartan matrices"?

\begin{align*} A_l &: &\begin{pmatrix}2&-1&0&&& \cdots &&0 \\ -1&2&-1&0&&\cdots &&0\\ 0&-1&2&-1&0&\cdots&&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdots &\cdot&\cdot\\ 0&0&0&0&0&\cdots&-1&2 \end{pmatrix}\\ \\ \hline &&\\ B_l &: &\begin{pmatrix}2&-1&0&&\cdots&&&0\\ -1&2&-1&0&\cdots&&&0\\ \cdot&\cdot&\cdot&\cdot&\cdots&\cdot&\cdot&\cdot\\ 0&0&0&0&\cdots&-1&2&-2\\ 0&0&0&0&\cdots&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ C_l &: &\begin{pmatrix}2&-1&0&&\cdots&&&0\\ -1&2&-1&&\cdots&&&0\\ 0&-1&2&-1&\cdots&&&0\\ \cdot&\cdot&\cdot&\cdot&\cdots&\cdot&\cdot&\cdot\\ 0&0&0&0&\cdots&-1&2&-1\\ 0&0&0&0&\cdots&0&-2&2 \end{pmatrix}\\ \\ \hline &&\\ D_l &: &\begin{pmatrix}2&-1&0&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&0\\ -1&2&-1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\cdot&\cdot&\cdot&-1&2&-1&0&0\\ 0&0&\cdot&\cdot&\cdot&\cdot&-1&2&-1&-1\\ 0&0&\cdot&\cdot&\cdot&\cdot&0&-1&2&0\\ 0&0&\cdot&\cdot&\cdot&\cdot&0&-1&0&2 \end{pmatrix}\\ \\ \hline && \end{align*}
\begin{align*} E_6 &: &\begin{pmatrix}2&0&-1&0&0&0\\ 0&2&0&-1&0&0\\ -1&0&2&-1&0&0\\ 0&-1&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ 0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ E_7 &: &\begin{pmatrix}2&0&-1&0&0&0&0\\ 0&2&0&-1&0&0&0\\ -1&0&2&-1&0&0&0\\ 0&-1&-1&2&-1&0&0\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&-1\\ 0&0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ E_8 &: &\begin{pmatrix}2&0&-1&0&0&0&0&0\\ 0&2&0&-1&0&0&0&0\\ -1&0&2&-1&0&0&0&0\\ 0&-1&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&0\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&-1\\ 0&0&0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline\\ F_4 &: &\begin{pmatrix}2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ G_2 &: &\begin{pmatrix}2&-1\\ -3&2 \end{pmatrix}\\ \\ \hline && \end{align*}

Someone is good with TeX!

## 1. What is a Positive Definite Cartan Matrix?

A Positive Definite Cartan Matrix is a matrix used in quantum physics to describe the symmetries and interactions of particles in a system. It is a square matrix with positive integer entries that represent the strength of the interactions between different types of particles.

## 2. How is a Positive Definite Cartan Matrix used in quantum physics?

In quantum physics, a Positive Definite Cartan Matrix is used to determine the energy levels and properties of particles in a system. It is also used to study the symmetries and interactions between particles, which can provide insights into the behavior of the system.

## 3. What are the properties of a Positive Definite Cartan Matrix?

A Positive Definite Cartan Matrix has several important properties, including being symmetric, positive definite, and having a determinant of 1. These properties ensure that the matrix accurately describes the interactions between particles in a system.

## 4. How is a Positive Definite Cartan Matrix related to Lie algebras?

In quantum physics, a Positive Definite Cartan Matrix is closely related to Lie algebras, which are mathematical structures used to study symmetries. The entries in the matrix correspond to the structure constants of the Lie algebra, which describe the commutation relations between different elements of the algebra.

## 5. What are some real-world applications of Positive Definite Cartan Matrices in quantum physics?

Positive Definite Cartan Matrices have many applications in quantum physics, including in the study of particle physics, condensed matter physics, and quantum information theory. They are also used in the development of new materials and technologies, such as in quantum computing and quantum cryptography.

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