New answers tagged group-theory
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Does all symmetry breaking have corresponding unitary group?
In general you can have broken symmetry groups different from $U(n)$. For example the quantum Ising model has a discrete symmetry (enacted by $P=\prod_j \sigma^z_j$) that breaks spontaneously in the ...
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Does all symmetry breaking have corresponding unitary group?
As in this SE post, you can have the Lagrangian with $SO(N)$ symmetry
$${\mathcal{L}} = \frac{1}{2}(\partial_\mu \Phi)^T (\partial^\mu \Phi) - \left(\frac{1}{2}\mu^2 \Phi^T \Phi + \frac{1}{4}\lambda (...
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3
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Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?
Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).\tag{1}$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\...
- 207k
2
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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 am not going to read Weinberg's book with you to your satisfaction, nor should I.
Indeed, the little group of a massive particle (go to its rest frame) is the rotation group SO(3), sharing a Lie ...
- 64.2k
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Have all the symmetries of the standard model of particle physics been found?
My answer is also ‘No’ but by direct construction: over the past few years, various research groups have understood new symmetries of the Standard Model.
Over the past decade, field theorists have ...
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?
It's actually Spin(n) which is the universal and double cover of SO(n) for n>=3.
However, it turns out that Spin(3) is isomorphic to SU(2). This is important because SU(2) is defined via complex 2 ×...
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
$\newcommand{\bl}[1]{\boldsymbol{#1}}
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\newcommand{\mb}[1]{\mathbf {#1}}
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2
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Accepted
$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$
The original question is really almost there. The answer is that we can almost prove $(1)$ directly from $(2)$. All we need in addition is a few facts about the pauli matrices $\sigma$ and the ...
- 14.4k
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
You appear to have two unrelated gaps, the final one being the rotation formula for the vector (triplet) representation, the celebrated Rodrigues rotation formula. $\vec \sigma$ rotates like a vector. ...
- 64.2k
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
Your equation $(*)$ is a bit hand-wavy since it doesn't hold exactly (on the r.h.s. you have a vector in $\mathbb{R}^3$ and on the l.h.s. a vector of Pauli matrices), but it's clear what you mean. I ...
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Accepted
How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
Physically, you can solve the Heisenberg equations of motion. It is equivalent to all the general arguments of the adjoint action, but at least it cuts to the chase and gives your formula explicitly. ...
- 13.1k
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Can you ever obtain a pure rotation from composing Lorentz transformations?
Ever? Always, of course, for infinitesimal boosts.
Review your Wigner rotations but consider the evidently superior 2$\times$2 matrix representation of the spinor map, which protects you from the busy ...
- 64.2k
4
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From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?
It seems the answer is negative: given one arbitrary $\mathrm{SO}(8)$ element, we cannot always find one corresponding $\mathrm{SU}(3)$ element in the sense of the transformation of the coefficients ...
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