/sci/ - Science & Math

[Anonymous No.3122649]

amateur QCD student here: looking for some help with my SU(3) structure constants. Trying plug this junk into my TI-89t, only some of it is working...

> http://en.wikipedia.org/wiki/Special_unitary_group#SU.283.29
Define my commuter brackets as [a,b] = ab - ba
Constants equal to 1 and 1/2 work
> Constants equal to <span class="math">\fraq{\sqrt{3}}{2}[/spoiler] don't

> http://www.wolframalpha.com/input/?i={{0%2C0%2C1}%2C{0%2C0%2C0}%2C{1%2C0%2C0}}*{{0%2C0%2C-i}%2C{0%2C
0%2C0}%2C{i%2C0%2C0}}
seems either way <span class="math">\lambda4[/spoiler] and <span class="math">\lambda5[/spoiler] will always have a 0 in the center, so no scalar can make their combination equal to <span class="math">\lambda8[/spoiler]

of course I see there are multiple sources confirming these values for the structure constants so I figure I'm missing something. Can anyone point it out?

[Anonymous No.3122670]

<span class="math">f_{458} = f_{678} = \frac{\sqrt{3}}{2} [/spoiler] is the trouble :(

[Anonymous No.3122698]

sorry brah, can't help ya. i have enough trouble w/ the pauli matricies :(

[Anonymous No.3122748]

>>3122698
:( I think I got the pauli matrices... but I need more dimensions

[Anonymous No.3124890]

[Anonymous No.3124905]

>>3122649
this is nothing but matrix multiplication and then identifying components in the basis given by your Gell-Mann matrices. If you are studying QCD and can't even do that, well are you sure you are suited for physics?

[Josef !!nUf2NflSAyw No.3124939]

>>3122649
(Forgive me for not typesetting vectors in bold or with funny arrows.)
Let <span class="math">T_a=\frac12\mathbf\lambda_a[/spoiler]. The Lie algebra of the <span class="math">\mathbf T_a[/spoiler] is then<div class="math">[T_a,T_b]=if^c_{\;ab}T_c</div>with the antisymmetric structure constants
<span class="math">f^1_{\;23}=1[/spoiler]
<span class="math">f^1_{\;47}=f^2_{\;46}=f^2_{\;57}=f^3_{\;45}=f^5_{\;16}=f^6_{\;37}=\frac12[/spoiler]
<span class="math">f^4_{\;58}=f^6_{\;78}=\frac{\sqrt3}2[/spoiler]
<span class="math">\mathrm{rest}=0[/spoiler].

[Anonymous No.3125163]

>>3124939
>>3124905
I have a computer science background, but no courses in quantum physics or group theory. My error is probably fairly simple misunderstanding, or calculator quirk.

Josef, I see you have written out the definition similar to other pages on the web. I can't balance the equations for all the structure constants.

This I get: <span class="math">f^{1}_{23} = 1<span class="math">
\left[ {\begin{array}{ccc}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array} \right)\] \cdot \[ \left( \begin{array}{ccc}0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\] = \[ \left( \begin{array}{ccc} i & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & 0 \end{array} \right)\]
Fairly easy to see the similarity to \lambda_{3} even though commutator brackets aren't defined that way.
>http://www.wolframalpha.com/input/?i={{0%2C1%2C0}%2C{1%2C0%2C0}%2C{0%2C0%2C0}}*{{0%2C-i%2C0}%2C{
i%2C0%2C0}%2C{0%2C0%2C0}}

This I don't: f^{4}_{58}
\left[ {\begin{array}{ccc}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0\end{array} \right)\] \cdot \[ \left( \begin{array}{ccc}0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right)\] = \[ \left( \begin{array}{ccc} i & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -i \end{array} \right)\]
> http://www.wolframalpha.com/input/?i={{0%2C0%2C1}%2C{0%2C0%2C0}%2C{1%2C0%2C0}}*{{0%2C0%2C-i}%2C{0%2C
0%2C0}%2C{i%2C0%2C0}}
which doesn't at all resemble \lambda_{8}[/spoiler][/spoiler]

[Josef !!nUf2NflSAyw No.3125177]

>>3125163
Sci doesn't support matrices (evil trolling feature or whatever).
a) Paste screenshot of typeset document
b) Send me the source to latex-compile it myself
c) Come up with a groundbreaking idea

[Anonymous No.3125188]

>>3125163

Take <span class="math">f^{1}_{23} = 1[/spoiler]. So I make a combination of <span class="math">\lambda_{1} and \lambda_{2}[/spoiler]
>http://www.wolframalpha.com/input/?i={{0%2C1%2C0}%2C{1%2C0%2C0}%2C{0%2C0%2C0}}*{{0%2C-i%2C0}%2C{
i%2C0%2C0}%2C{0%2C0%2C0}}
Fairly easy to see structure constant scaling <span class="math">\lambda_{3}[/spoiler].

But take <span class="math">f^{4}_{58} = \frac{\sqrt{3}}{2}[/spoiler], in combination:
> http://www.wolframalpha.com/input/?i={{0%2C0%2C1}%2C{0%2C0%2C0}%2C{1%2C0%2C0}}*{{0%2C0%2C-i}%2C{0%2C
0%2C0}%2C{i%2C0%2C0}}
That doesn't at all resemble <span class="math">\lambda_{8}<span class="math">.

I defined some custom functions like Josef mentioned, and indeed they balance for structure constants 1 and 1/2, but not \frac{\sqrt{3}}{2}.[/spoiler][/spoiler]

[Josef !!nUf2NflSAyw No.3125215]

>>3125188
There are two structure constants involving 4 and 5.
And by the way, your notation is misleading. The lower indices are the ones from the commutator, the upper one is the index for summation. (Took me a bit to find out where your error was because of that.)

[Anonymous No.3125220]

>>3125188
Trying to clean these links up

<span class="math">\lambda_{1} \cdot \lambda_{2}[/spoiler]
http://www.wolframalpha.com/input/?i={{0%2C1%2C0}%2C{1%2C0%2C0}%2C{0%2C0%2C0}}*{{0%2C-i%2C0}%2C{i%2C
0%2C0}%2C{0%2C0%2C0}}

<span class="math">\lambda_{4} \cdot \lambda_{5}[/spoiler]
http://www.wolframalpha.com/input/?i={{0%2C0%2C1}%2C{0%2C0%2C0}%2C{1%2C0%2C0}}*{{0%2C0%2C-i}%2C{0%2C
0%2C0}%2C{i%2C0%2C0}}

commutator brackets are defined with a dot product correct?

[Josef !!nUf2NflSAyw No.3125223]

>>3125220
Matrix product is "." in Mathematica/on Wolfram Alpha, yes.

[Anonymous No.3125244]

>>3125215
okay, yeah, thanks. I was plugging in the first two numbers into the brackets, not the last two. This "index of summation," I'm not sure I'm clear on that. Are matrices being added somewhere?

<span class="math">\lambda_{5} \cdot \lambda_{8}[/spoiler]
http://www.wolframalpha.com/input/?i={{0%2C0%2C-i}%2C{0%2C0%2C0}%2C{i%2C0%2C0}}*{{1%2Fsqrt%283%29%2C
0%2C0}%2C{0%2C1%2Fsqrt%283%29%2C0}%2C{0%2C0%2C-2%2Fsqrt%283%29}}
>Still has a pesky zero in the center...

[Anonymous No.3125256]

>>3125244
>pesky zero...
oh wait, I guess it should for <span class="math">\lambda_{4}[/spoiler]...

[Anonymous No.3125261]

The su(3) Lie Algebra gives me a hardon.

[Josef !!nUf2NflSAyw No.3125262]

>>3125244
Summation convention:<div class="math">[T_a,T_b]=if^c_{\;ab}T_c=\sum_{c=1}^8if^c_{\;ab}T_c</div>

[Anonymous No.3125286]

>>3125215
yeah, thanks Josef. I balanced it for <span class="math">f^{4}_{58}[/spoiler] using the last two indices w/ matrix halving :)

[Anonymous No.3125345]

and got without the matrix halving now too...

I'm guessing physics just has some convention for halving the group generators in the structure constants definition? meh, whatever...

[Josef !!nUf2NflSAyw No.3125359]

The <span class="math">\lambda[/spoiler] structure constants are simply twice as large as the one I've posted above.