All Questions
19
questions
1
vote
1
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55
views
How can I calculate action of $\mathfrak{su}(3)$ or other simple algebra ladder operators on "states" from the algebra commutators?
I wanted a way to "derive" Gell-Mann matrices for $\mathfrak{su}(3)$ and generalise this to other semi-simple algebras $\mathfrak{g}$. The way I wanted to approach this is start from the ...
- 785
2
votes
1
answer
450
views
Why, in QCD, are quarks in the fundamental representation of $SU(3)$?
QCD is built from the notion that Dirac's Lagrangian should be invariant under gauge colour transformations.
Here, quarks are elements of $\psi_{\alpha,f,c}(x)$, where $\alpha$, $f$ and $c$ stand for ...
- 343
1
vote
0
answers
1k
views
How to decompose tensor products of $SU(3)$ representations? [duplicate]
Formally, one can arrange the quark flavors in a $SU(n)$ fundamental representation. One can then do tensor products for flavor and spin to construct other representations like baryons and mesons. An ...
- 11
3
votes
2
answers
313
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Normalisation of QCD Lagrangian
In QCD, and more generally in representations of $\mathfrak{su}(N)$, there is a freedom to choose the normalisation of the generators,
$$
\mathrm{Tr} \, \left[R(T^a) R(T^b)\right] = T_R \delta^{ab}.\...
- 264
2
votes
1
answer
171
views
Physical significance of the reality of an ${\bf N}$ representation: how the nature of interactions is affected?
Background The fundamental representation of ${\rm SU(N)}$ is denoted by ${\bf N}$ and the conjugate of the fundamental is denoted by ${\bar{\bf N}}$. If the representation ${\bf N}$ is related to ${\...
- 26.8k
0
votes
0
answers
221
views
What is the application of dimension $6$ representation of $SU(3)$ in particle physics?
As we know, the $uds$ transforms in fundamental representations of $SU(3)$. It has the antifundamental partner. According to representation theory,
$$
\mathbf{3} \otimes \mathbf{\bar{3}}= \mathbf{8} \...
8
votes
1
answer
5k
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$\mathfrak{su}(3)$ structure constants
The $\mathfrak{su}(3)$ structure constants $f^{abc}$ are defined by $$[T^a,T^b] = i f^{abc} T^c,$$ with $T^a$ being the generators of the group $\mathrm{SU}(3)$. They are usually written out in a very ...
- 109
1
vote
1
answer
77
views
What prohibits fundamental fermions transforming like the $6$ and $6^*$ IRR's of $SU(3)$?
The lowest IRRs of SU(3) are 3,3* (the fundamental reps), 6,6*, and 8 (the adjoint rep). The quark fields are chosen to transform as 3, 3*, and the gluons as 8 under SU(3), but there is no ...
- 417
3
votes
0
answers
134
views
What is the physical meaning of Lie congruence classes?
Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1)
$$
\lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
1
vote
2
answers
1k
views
Breaking of $SU(3)$ symmetry by bi-fundamental representation
Are there any general theorems which fix the possible symmetry breaking patterns of Lie groups (such as $SU(3)$) by vacuum expectation values of fields in specific representations (such as the quark ...
- 1,783
12
votes
2
answers
2k
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$SU(3)$ vs $SO(3)$ color gauge
I have kind of a dumb question: what would happen if the color gauge group is $SO(3)$ instead of $SU(3)$, assuming there are still colors and physical states are still color singlets? Will we e.g. get ...
- 175
5
votes
2
answers
664
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Quark model extension to all six flavors
Gell-Mann's $SU(3)$ quark model is extremely successful at describing the bound states of the three light quarks $u,d,s$. The bound states fall neatly into the irreducible representations of $\...
1
vote
0
answers
195
views
Why does the $U(2n)$ flavor symmetry break down to a $U(1)$ group and an $SU(2n)$ group?
I am studying quantum field theory using Srednicki's textbook. Problem 83.1 is:
Suppose that the color group is $G_C=SO(3)$ rather than $SU(3)$, and that each quark flavor is represented by a Dirac ...
- 1,653
1
vote
0
answers
683
views
SU(3) adjoint representation's invariant tensors
Considering a complex scalar field $\varphi^a$ that transform in the adjoint representation (8) of SU(3).
A quartic interaction term SU(3) invariant is
$$\lambda C^{abcd}\varphi^{\dagger a} \varphi^...
- 21
0
votes
0
answers
456
views
Non-singlet and singlet flavor combination [duplicate]
In the perturbative QCD, specific partonic distribution functions (pdf) linear combination is called "flavor singlet" or "flavor nonsinglet" combination.
What is the combination, and why are they ...
- 857