# Questions tagged [group-theory]

Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.

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### Relation between the commutator of commutators in Dirac algebra

In an attemption to obtain the curvature tensor related to the spin connection of the fermionic fields I came across this expression with the commutator of the gamma matrices commutators. My question ...
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### Precise definitions for higher spin operators

I am trying to understand the matrices and vectors presented in this section https://en.wikipedia.org/wiki/Spin_(physics)#Spin_projection_quantum_number_and_multiplicity I am looking for a reference ...
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### From the point of view of physics, why is it useful to know the irreps of rotation group?

In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
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### Yet more gauge group nonsense: $D3$? $Q8$? $Z8$?

This'll probably make me look like a total idgit, but I have a new question in the same vein as mine about $SU(4)$, but this time without any guesses. I've looked a bit into groups, and it looks like ...
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### What does it mean for the $\textbf{B}$-field (Hypercharge) to be in the 0 representation within the SM?

I was reading through the wikipedia page for the mathematical formulation of the standard model and I noticed that it listed the representations of the vector bosons under the SM gauge groups as being ...
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### Is every unitary representation of the Poincare group a direct sum of Wigner's irreducible representations?

Is Wigner's classification of unitary irreducible representations of the Poincare group  sufficient for constructing all unitary representations of the Poincare group by taking direct sums? The ...
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### Deriving an identity with rotation generators [closed]

I am trying to justify the following identity on page 68 of Osborn's notes on group theory: $$e^{-i\pi J_{3}}J_{2}e^{i\pi J_{3}} = -J_{2}.$$ Here, the $J_{i}$ are the typical angular momentum ...
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### Extracting $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$, when $S_i$ are higher spin matrices?

This is a cross-post of a question that I posted on the Math SE, that did not get any answers there. It is fundamentally a mathematics question, but it pertains to spin matrices, which many Physics SE ...
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### Orthogonality relations for matrix elements of irreducible representations

I am reading Howard Georgi's "Lie Algebras in Particle Physics" and have a question concerning the presented orthogonality relations for matrix elements of irreducible representations. To ...
### The irreducible representation of rank $n$ spinor in 3D
I was reading Ref. 1, where it is asserted that the irreducible resolution of a rank $n$ spinor can be written as totally symmetric spinors. \Psi^{n}=\Psi^{\{n\}}+\zeta \Psi^{\{n-2\}}+\zeta \zeta \...