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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Why is there this relationship between quaternions and Pauli matrices?
I've just started studying quantum mechanics, and I've come across this correlation between Pauli matrices ($\sigma_i$) and quaternions which I can't grasp: namely, that $i\sigma_1$, $i\sigma_2$ and $...
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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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"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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Show that any proper homogeneous Lorentz transformation may be expressed as the product of a boost times a rotation
I am trying to read Weinberg's book Gravitation and Cosmology. In which he derives the Lorentz transformation matrix for boost along arbitrary direction, (equations 2.1.20 and 2.1.21):
$$\Lambda^i_{\,\...
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General Lorentz boost for a spinor in Hyperbolic form with Pauli matrices
I am having trouble deriving the general Lorentz boost for a spinor with rapidity $\rho $ using the hyperbolic functions. I know that the matrix: $$\exp [-\rho/2 \mathbf{n}\cdot \mathbf{\sigma }]=\...
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Does all symmetry breaking have corresponding unitary group?
In high energy physics. Symmetry breaking like electroweak's has corresponding $SU(2)\times U(1)$ unitary gauge group broken down to $U(1)$. Does it mean all kinds of symmetry breaking (even low ...
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Total spin on a two-particle system [closed]
How can I know the total spin of a given state quickly? I first encountered this problem when proving the Clebsch-Gordan series.
Consider a two-particle system (to make it easier, let's say identical ...
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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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Connection between particle physics and weight diagrams
I have a hard time combining two topics that are often discussed in physics in a coherent way.
In a lot of Introduction to particle physics-classes one will hear about "multiplets", which ...
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Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?
I've been reading Weinberg's QFT Vol 1. and more specifically section 5.6. I would like to know if my understanding is correct or if I missed something. He starts with the full Lorentz group $\mathrm{...
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How to find Casimir operator eigenvalues of $SU(N)$? [closed]
The $[f1, f2, f3…fn]$ in the image represent the irreducible representations of $SU[n]$. How to find the irreducible representations of $SU[n]$ that conform to the form $[f1, f2...fn]$. Can you give ...
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Is intrinsic spin a quantum or/and a relativistic phenomenon?
Ok, this is my reasoning. I am probably making some wrong assumptions here, pls tell me where I am going wrong.
Spin as a quantum phenomenon:
Quantum phenomena disappear as the Planck constant goes to ...
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Have all the symmetries of the standard model of particle physics been found?
Background
The standard model of particle physics is entirely determined by writing down its Lagrangian or, equivalently, writing down the corresponding system of PDEs.
Every set of PDEs has a ...
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?
To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\...
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Weights and roots of $SU(3)$
I am self-studying group theory from Lie Algebras in Particle Physics by H. Georgi and I am having trouble following some of his arguments. In section 7.2 titled Weights and roots of $SU(3)$ he starts ...
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