# Anticommutators of the spin-1 representation

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• Tony3
In summary, the conversation discusses the Pauli matrices of the spin-1 representation and their properties. It is mentioned that the matrices ##T_{1}, T_{2},## and ##T_{3}## are given by specific equations, and the question is posed about the relationship between these matrices. After performing calculations, it is found that the anticommutator of ##T_{1}## and ##T_{2}## is equal to ##i\begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}##, and similar results are found for the other combinations of matrices. The conversation concludes by
Tony3
TL;DR Summary
I need to find a closed form of the anticommutators of the Pauli matrices in the spin-1 representation.
The Pauli matrices of the spin-1 representation are given by: ##T_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}##, ##T_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}## and ##T_{3}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}##. I need to find what ##\left\{T_{i},T_{j}\right\}## is equal to.
Doing some calculations, I found that ##\left\{T_{1},T_{2}\right\}=i\begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}##, ##\left\{T_{1},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}##,##\left\{T_{2},T_{3}\right\}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & i \\ 0 & -i & 0 \end{pmatrix}##.
Is there a general relation that I can derive from these special relations? I think that I am close, but I can't quite see it.

What else than the explicit matrices do you expect to get? They are as closed from as you can make it.

## 1. What is the spin-1 representation in quantum mechanics?

The spin-1 representation in quantum mechanics refers to the mathematical description of the intrinsic angular momentum of a particle. It is a fundamental property of particles and is often denoted by the symbol "s". In the case of spin-1 particles, there are three possible spin states: +1, 0, and -1.

## 2. How do anticommutators relate to the spin-1 representation?

Anticommutators are mathematical operations that involve the multiplication of two operators in a specific order, followed by the subtraction of the same two operators in the reverse order. In the case of the spin-1 representation, anticommutators are used to describe the behavior of spin operators and their commutation relations.

## 3. What are the physical implications of anticommutators of the spin-1 representation?

The physical implications of anticommutators of the spin-1 representation are related to the measurement of spin states. They determine the probabilities of obtaining certain spin values when a measurement is performed on a spin-1 particle. Additionally, anticommutators also play a crucial role in the formulation of quantum field theories.

## 4. How do anticommutators differ from commutators?

Commutators and anticommutators are both mathematical operations involving the multiplication and subtraction of operators. The main difference between them is the order in which the operators are multiplied and subtracted. In commutators, the operators are multiplied in the same order, while in anticommutators, the operators are multiplied in opposite orders.

## 5. Can anticommutators be used to describe other spin representations?

Yes, anticommutators can be used to describe other spin representations, such as spin-1/2 or spin-3/2. However, the specific mathematical expressions and physical implications will differ for each spin representation. The use of anticommutators is not limited to the spin-1 representation and can be applied to various quantum mechanical systems.

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