# 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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### "Linear independency" of Lie Brackets [migrated]

I was watching this eigenchris video. At 21:49, he says: $$[g_i, g_j]=\Sigma_k {f_{ij}}^{k}g_k$$ for $\mathfrak{so}(3)$. Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What ...
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### A reference for the fact that the second cohomology of the full Poincare algebra is zero

S. Weinberg in his book "The quantum theory of fields" vol. I says in page 86 that the full Poincare algebra is not semi-simple but its central charges can be eliminated (as he showed in the ...
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### $Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$

This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
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### How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?

Disclaimer: I'm sure this has been asked 100 times before, but I can't find the question asked or answered quite like this. If there are specific duplicates that could give me a simple satisfactory ...
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### Why are Lorentz transformations singular at $i^0$?

On pg. 16 of Strominger's lectures, it is said Lorentz transformations themselves are not smooth at spatial infinity, because the vector fields that generate them are singular at $i_0$. A boost ...
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### Is the factorization method of Hamiltonian related to the theory of Lie groups?

I was learning about algebraic methods to solve the H atom, when I came across the factorization method. It is mentioned in various textbooks, notes and papers, like the one from Infeld and Hull. I am ...
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### One-Loop beta function for gauge couplings

I am currently doing my homework on Standard Model one-loop correction. When I am reading Quantum Field Theory by Mark Srednicki and Journeys Beyond the Standard Model by Pierre Ramond, I notice some ...
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### Wigner-Eckart theorem in classical physics?

The Wigner-Eckart theorem is a useful result in quantum physics and its many applications. Most presentations of this material in books on QM and online lecture notes seem to be variations on the same ...
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