# 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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### Most general proper Lorentz transformations do not form a group?

How to prove that most general Lorentz transformations with relative velocities in arbitrary directions do not form a group? If the Lorentz transformations always satisfy, $$\eta=\Lambda^T\eta\Lambda$$...
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### How do you observe “silent” quantum vibrations?

In the theory of quantum vibrations (aka phonons) it is useful to divide up the vibrational normal modes of a crystal based on their representation within the symmetry group of the crystal. The ...
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### Three blocks and the representation of $S_3$

I've been studying chapter 1 of the famous group "Lie Algebras in Particles Physics" by Georgi. I am rather confused by section 1.16. The claim is the following. Consider a system of three ...
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### Why are Casimir operators required to be Hermitian?

What is the physical significance of requiring Casimir operators (of e.g. Poincare group or the conformal group) to be Hermitian? What breaks down if we do not impose this condition? EDIT: To be ...
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### Mass as a coupling and mass as a Casimir operator

In Poincare group, we consider mass as a Casimir of the group. Hence it is a constant in various frames (I do not mean old fashion Lorentz transformation). But, in the quantum field theory mass is the ...
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### What does a broken symmetry mean for the Lagrangian?

I am a little confused about symmetry breaking - in particular, what I see to be too different interpretations of it. First, what I have seen taken to be the definition of a broken symmetry - we start ...
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### Do the spinor transformation matrices form a matrix representation of the corresponding Lorentz group?

Suppose $\Psi$ is a Dirac spinor, then let the transformation matrix $S$ be defined as usual: $\Psi'=S(\Lambda)\Psi$, where $\Lambda$ is the Lorentz transformation matrix. Then the questions is: for ...
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### Question in Derivation of Lie algebra

In Weinberg's QFT Volume 1 Chapter2, he "derives" the Lie algebra from the Lie group as follows [...] a connected Lie group [...is a...] group of transformations 𝑇($\theta$) that are ...
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### Why don't $\rm SO(10)/Spin(10)$ or larger GUTs permit electron-pair decay?

$\rm Spin(10)$ unifies all left-handed fermions and anti-fermions, and all right-handed fermions and anti-fermions. And its Pati-Salam subgroup unifies quarks with leptons (SU(4)) and complements SU(...