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<p>Let's discuss about $SU(3)$. I understand that the most important representations (relevant to physics) are the defining and the adjoint. In the defining representation of $SU(3)$; namely $\mathbf{3}$, the Gell-Mann matrices are used to represent the generators
$$
\left[T^{A}\right]_{ij} = \dfrac{1}{2}\lambda^{A},
$$
where $T^A$ are the generators and $\lambda^A$ the Gell-Mann matrices. In adjoint representation, on the other hand, an $\mathbf{8}$, the generators are represented by matrices according to
$$
\left[ T_{i} \right]_{jk} = -if_{ijk},
$$
where $f_{ijk}$ are the structure constants.</p>
<p>My question is this, how can one represent the generators in the $\mathbf{10}$ of $SU(3)$, which corresponds to a symmetric tensor with 3 upper or lower indices (or for that matter how to represent the $\mathbf{6}$ with two symmetric indices). What is the general procedure to represent the generators in an arbitrary representation?</p> | 1,039 | [
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<p>So in the context of a set of notes I am reading about acoustics I get to equation (23) in this <a href="http://www3.nd.edu/~atassi/Teaching/ame%2060639/Notes/fundamentals_w.pdf" rel="nofollow">paper</a>. Basically it comes down to showing that (<strong>note the dots above the a's meaning time derivative!)</strong></p>
<p>$$
f(t) = -a_0c_0\int_{-\infty}^t \dot{a}(t'+\frac{a_0}{c_0})e^{-\frac{c_0}{a_0}(t-t')} dt'\ = -a_0^2\dot{a}(t),.
$$</p>
<p>under the assumption that $\frac{c_oT}{a_0} >>1$. $T$ can be thought of as a characteristic time scale for this problem. For those interested in more than just the math, please see the paper. Now I show this in the following way:</p>
<p>Integrating by parts:
$$
\begin{align}
f(t) = & \\
=& -a_0^2\dot{a}(t+\frac{a_0}{c_0})+a_0c_0\int_{-\infty}^t \ddot{a}(t'+\frac{a_0}{c_0})\frac{a_0}{c_0}e^{-\frac{c_0}{a_0}(t-t')} dt'\\\
&
\end{align}
$$</p>
<p>By expanding $\dot{a}$ as a taylor series we get:</p>
<p>$$
\dot{a}(t+\frac{a_0}{c_0}) = \dot{a}(t) +\frac{a_0}{c_0}\ddot{a}(t)+O\left(\frac{a_0^2}{c_0^2}\right)
$$</p>
<p>Now if we make the order of magnitude estimate:</p>
<p>$$
\frac{a_0}{c_0}\ddot{a}(t) \simeq \frac{a_0}{c_0T}\dot{a}(t)
$$</p>
<p>then I hope it is clear by applying this reasoning to the equation for $f(t)$ that we have shown $f(t) = -a_0^2\dot{a}(t)$ for $\frac{c_oT}{a_0} >>1$. </p>
<p><strong>Now here comes my question:</strong> How can I justify the time derivative operator behaving as $1/T$, where $T$ was just given as the length scale of the problem (which seems so arbitrary)? Is my reasoning above correct? I was hoping someone could put my mind at ease about the "hand wavy" nature of these order of magnitude approximations. </p> | 1,040 | [
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<p>I was wondering with a question for a quite long time, thought to ask here.</p>
<p>I need to know is there any material or element which can block magnetic field? I mean I am searching for such material or element that cannot allow magnetic field though itself?</p>
<p>The practical scenario is, there are two permanent magnets and those are positioned within each other's magnetic field. I want to put something so that both the magnets become free of interference withing themselves.</p>
<p>Hope I could clarify my question.</p>
<p>Can anyone help me of give me some suggestion on this aspect please?</p> | 0 | [
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] |
<p>Why is the gravitation force always attractive? Is there a way to explain this other than the curvature of space time? </p>
<p>PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve with concavity?</p> | 81 | [
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<p>I am wondering what will be the physics to explain how two neutral, chemically nonreactive objects stick. I know that using van der Waals formalism, we can treat neutral body electrodynamic forces and arrive with attractive forces that pull the objects together. </p>
<p>Now, once the objects touch (say a mechanical cantiliver in a MEMS sensor like the one used in an iPhone), what happens to the forces? A quantitative answer or some estimate on how strong the attractive force is for simple cases will be very appreciated.</p>
<p>in response to anna's comment :
Let us consider what happens in vacuum for ultra smooth surfaces, with no residual electrical charge and fully chemically stablized surfaces (example, silicon crystals with stabilised surface bonds). </p> | 1,041 | [
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<p>I have a question about the relation:
$\exp(-i \vec{\sigma} \cdot \hat{n}\phi/2) = \cos(\phi/2) - i \vec{\sigma} \cdot \hat{n} \sin(\phi/2)$.</p>
<p>In my texts, I see $\phi\hat{n}$ always as c-numbers. My question is whether or not this relation can be generalized for $\hat{n}$ being an operator?</p>
<p>If so how exactly would the expression be different?</p>
<p>Thanks.</p> | 1,042 | [
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<p>The second postulate of <a href="http://en.wikipedia.org/wiki/Special_relativity" rel="nofollow">special relativity</a> deals with constancy of light in inertial reference frames. But, how did Einstein came to this conclusion? Did he knew about the <a href="http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment" rel="nofollow">Michelson-Morley experiment</a>? </p> | 1,043 | [
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<p>Does one need to invoke quantum mechanics to explain casimir force or vander waals force. I see that textbooks show derivation of vander waal force with no QM but casimir force is typically described with QM. </p>
<p>Is there a difference between vanderwaal and casimir forces ? Are there distinct examples of these two forces in real life. Is there a way to prove a given force is vanderwaal and not casimir or vice versa. </p> | 1,044 | [
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"<p>Einstein has suggested that light can behave as a waves as well as like a particle i.e, it has d(...TRUNCATED) | 1 | [-0.012009003199636936,0.008917891420423985,0.06659446656703949,-0.025958653539419174,0.048719726502(...TRUNCATED) |
"<p>When studying the de Broglie relations, I have stumbled across the following problem:</p>\n\n<bl(...TRUNCATED) | 2 | [-0.012932011857628822,-0.011209017597138882,0.0012715015327557921,-0.022642482072114944,0.014670946(...TRUNCATED) |
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