Use of Gell-Mann matrices as the SU(3) basis for gluon states?

[James1238765]

The 8 gluon fields of SU(3) can be represented (generated) by the 8 Gel-Mann matrices:

$$ \lambda_1 =
\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} , \lambda_2 =
\begin{bmatrix}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} , \lambda_3 =
\begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{bmatrix} $$
$$\lambda_4 =
\begin{bmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{bmatrix}, \lambda_5 =
\begin{bmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0
\end{bmatrix} , \lambda_6 =
\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix} $$
$$\lambda_7 =
\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0
\end{bmatrix} , \lambda_8 =
\begin{bmatrix}
\frac{1}{\sqrt3} & 0 & 0 \\
0 & \frac{1}{\sqrt3} & 0 \\
0 & 0 & -\frac{2}{\sqrt3}
\end{bmatrix}
$$

While I have seen many derivations for the Gel-Mann matrices, I have not seen a demonstration of the basic usage of these matrices.

Suppose we have a "red gluon". Is this to be represented by the column vector
$$ \vec{red} = \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}?$$

Then, if we would like to calculate the inverse of this "red gluon", do we multiply this column vector representation against one of the Gel-mann matrices, such as $\lambda_1 \vec {red}$:

$$
\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} =
\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}$$

What does $\lambda_1 \vec {red} = \vec {green}$ mean in this representation?

 

[PeroK]

While I have seen many derivations for the Gel-Mann matrices, I have not seen a demonstration of the basic usage of these matrices.

That's because you haven't bought a textbook, but are just trying to eat graduate-level topics like physics is box of Christmas chocolates.

 

[James1238765]

@PeroK What does the $\lambda_1 \vec{red} = \vec{green}$ mean using the Gel-Mann matrices representation?

 

[PeterDonis]

While I have seen many derivations for the Gel-Mann matrices, I have not seen a demonstration of the basic usage of these matrices.

As @PeroK has said, this is a graduate level topic. That means that an "I" level thread can't really address it properly; you would need an "A" level thread, but then you would need the background knowledge to be able to understand an "A" level discussion. And in any event, since this is a complicated topic, a properly thorough treatment of it is well beyond the scope of a PF discussion; as @PeroK has said, you really need to learn this from a textbook and take the time to do it properly.

That said, there are fairly straightforward answers that can be given to some of your questions, although they probably won't satisfy you. That's because, as above, satisfying you would amount to giving you a course in this topic from a textbook, and that's not what PF is for. That's something you will need to do yourself.

Suppose we have a "red gluon".

There is no such thing. Gluons don't have single colors.

Is this to be represented by the column vector
$$ \vec{red} = \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}?$$

No. The column vector you represent is a vector in the color basis, so the three basis vectors correspond to "red", "green", and "blue", but the color basis does not describe the colors of gluons, since, as above, gluons don't have single colors.

Then, if we would like to calculate the inverse of this "red gluon"

As above, there is no such thing as a "red gluon". Also, I don't know what you mean by "inverse".

What does $\lambda_1 \vec {red} = \vec {green}$ mean in this representation?

It means that $\lambda_1$ represents something that transforms a "red" object into a "green" object. At least, that's a reasonable rough heuristic description. But to make physical sense of it, you need to know:

(1) What kinds of objects can be "red" or "green" or "blue"?

(2) What kinds of physical interactions can change the colors of such objects?

If you have a basic knowledge of how the strong interaction is modeled in the Standard Model, you should at least be able to make reasonable guesses at the answers to those two questions. Can you?

 

[PeroK]

@PeroK What does the $\lambda_1 \vec{red} = \vec{green}$ mean using the Gel-Mann matrices representation?

My point was that:

Last Sunday you were asking about QED:

https://www.physicsforums.com/threa...ity-of-an-electron-emitting-a-photon.1048457/

Monday was Dirac spinnors:

https://www.physicsforums.com/threads/please-help-with-derivation-for-dirac-spinors.1048518/

Thursday was the quark-neutrino mixing matrix:

https://www.physicsforums.com/threads/evaluating-the-quark-neutrino-mixing-matrix.1048582/

Yesterday was mesons:

https://www.physicsforums.com/threa...f-mesons-such-as-pion-a-new-particle.1048624/

And, today it's GCD and Gell-Mann matrices!

 

[James1238765]

@PeroK yes i admit, cmon it was fun though, isn't it? :) But seriously, I honestly believe that the proverbial blind man touching an elephant *will* in fact get a rather good idea of what the elephant *is* if he persistently pokes the elephant again and again for hundreds of times, and attempts to connect the dots.

 

[phinds]

@PeroK ... I honestly believe that the proverbial blind man touching an elephant *will* in fact get a rather good idea of what the elephant *is* if he persistently pokes the elephant again and again for hundreds of times, and attempts to connect the dots.

And there's your problem. Science is not an elephant and that approach won't really work.

 

[PeterDonis]

I honestly believe that the proverbial blind man touching an elephant *will* in fact get a rather good idea of what the elephant *is* if he persistently pokes the elephant again and again for hundreds of times, and attempts to connect the dots.

I think "hundreds of times" is way too optimistic. A better estimate might be "millions or billions of times, or even more--maybe never". There are much more efficient ways of learning.

If you want to take that horribly inefficient approach for yourself, that's your choice, although it will take many lifetimes for you to make significant progress, if you do at all.

But taking that approach here at PF will not serve you well. We are under no obligation to pander to horribly inefficient methods of learning.

cmon it was fun though, isn't it? :)

Your smiley face here betrays a huge blind spot: maybe it's been fun for you. But that doesn't mean it's fun for all the others who set out to try to help you, and find out that your definition of "help" is to wait for you to ponderously ask a million or a billion questions as you randomly poke elephants.

Long before you get to that number of questions, our patience will run out and you will end up having all your threads closed and eventually, if you don't take the hint, being banned from the forums because you are either unable or unwilling to be respectful of the time and effort of other posters. Please take heed.

 

[PeterDonis]

And with that said, this thread is closed.