Linked Questions

0
votes
0answers
27 views

Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
10
votes
2answers
6k views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
6
votes
2answers
1k views

In quantum mechanics(QM), can we define a high-dimensional “spin” angular momentum other than the ordinary 3D one?

Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get ...
7
votes
2answers
381 views

Exponential of a differential operator

I have a differential operator $L$, $\displaystyle L = i (t\frac{\partial}{\partial z} - z\frac{\partial}{\partial t})$ I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, $...
5
votes
1answer
245 views

Why Must Conserved Currents of Lorentz Symmetry Satisfy the Lorentz Algebra

I've seen it written many times that the commutation relation $[M^{I-},M^{J-}]=0$ is required for Lorentz invariance in the light cone gauge quantisation of the bosonic string. This follows ...
2
votes
2answers
357 views

How to construct generators and Lie Algebra for Lorentz group?

I'm trying to figure out Lorentz group in 2+1. First of all, I'd like to think the special orthgonal group as a combination of rotation and boost in space. Then I construct it as below. First rotation ...
-1
votes
1answer
288 views

Derivation of the Lorentz algebra explicity [closed]

I need the complete proof for commutation relation of the Lorentz group generators. The proof of Lorentz algebra using this commutation relation.
2
votes
1answer
346 views

$SO(N)$ symmetric theory of $N$ real scalar fields, why do charges have correct commutation relations of generators?

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda(\Phi^a \Phi^a)^2.$$...
1
vote
1answer
333 views

Lie algebra of lorentz group

I'm stuck in following calcualtion from sredniki's QFT book.(Its actually in the solution manual) How can i get from $$\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma}) $...
2
votes
2answers
157 views

What does a left-right arrow in a tensor formula mean?

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing ...
1
vote
1answer
72 views

How to interpret the extra indices of the generators of the Lorentz algebra in Peskin & Schroeder?

In Peskin & Schroeder p.39 they introduce the 4x4-matrices $$\left(\mathcal{J}^{\mu\nu}\right)_{\alpha\beta} = i \left(\delta^{\mu}_{\;\alpha} \delta^{\nu}_{\;\beta} - \delta^{\mu}_{\;\beta}\...