# 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 Dynkin diagram/Cartan matrix and find out the roots of the algebra and weights of the fundamental representation. This gives me in turn the algebra/structure constants (from Chavalley-Serre relations for example). So, I have the algebra from the Dynkin diagram. Now, I can construct the $$r=\text{rank}(\mathfrak{g})$$ number of commuting Cartan subalgebra generators directly from the weights of the representation (and ofcourse choosing the standard Cartesian/Euclidean orthogonal basis for writing down the weight vectors explicitly). The problem is I am not able to find the non-diagonal non-commuting generators usually denoted in weight theory as $$E_{\alpha}$$ in the Cartan-Weyl basis in which the Cartan generators take the form I just derived. If I had this, I could construct explicit matrix representations for $$E_{\alpha}$$ and doing a trivial basis change, the Gell-Mann matrices. But for this, one possible way is to know the analogue of

$$J_{\pm} | j,m \rangle =\sqrt{j(j+1)-m(m \pm 1)} |j, m \pm 1 \rangle$$

for $$\mathfrak{su}(3)$$. This leads to the titular question: How to find the action of the ladder operators/non-Cartan generators on states/weight vectors? I have found no book which does this, which makes me wonder if all of this is too trivial. Thomson's particle physics textbook on pg. 226 states the results which is intuitively obvious:

$$T_-u=d, T_+d=u, U_+ s=d , U_-d =s, V_+s =u, V_-u=s$$ and conjugates of these where $$u,d$$ and $$s$$ are weight vectors representing up, down and strange quarks and $$T_{\pm},U_{\pm}$$ and $$V_{\pm}$$ are the $$\mathfrak{su}(3)$$ ladder operators sometimes called isospin, U spin and V spin generators respectively and they are given as particular linear combinations of the Gell-Mann matrices with coefficients $$\pm \frac{1}{2}$$.

If I try to retrace whatever is done for $$\mathfrak{su}(2)$$ I would need to express the $$E_{\alpha}E_{-\alpha}$$ combination in terms of Casimir operators and Cartan generators (e.g. $$J^-J^+=J^2-J_z^2-J_z$$ in the case of $$\mathfrak{su}(2)$$) and there are plenty of them for large algebras, so I am not sure how to do this in a systematic way. Plus, I think there might be an alternative without using the Casimirs because they don't technically belong to the algebra but the universal enveloping algebra, but I might be wrong in thinking this.

Ultimately I need a way to derive matrix representations of the $$\mathfrak{su}(3)$$ and more preferably any simple algebra from the commutation relations in the Cartan Weyl basis alone and ofcourse along with known diagonal form of the Cartan generators and knowledge of the weight/root system. But for that I think I first need to know the answer to the titual question

• A less descriptive form of the above question in MSE website by me can be found here. Commented Mar 28 at 7:26
• The body of the question is much more restrictive than what the title suggests. Are you just interested in the adjoint irrep or irreps in general? Commented Mar 28 at 15:07
• WP not detailed enough? Many books, such as Greiner-Mueller, Quantum Mechanics, Symmetries, beat this to a pulp... Commented Mar 28 at 16:29
• @CosmasZachos Thanks, I will look into Greiner-Mueller, now but no: WP does not mention a way to derive the matrix representation or give the action on states with coefficients worked out. Additionally I discussed this recently with members in the hbar chat and the conclusion was that this is surprisingly not trivial! Commented Mar 28 at 22:11
• @ZeroTheHero I am looking for fundamental irreps that produce the Gell-Mann matrices. I guess, the titular question was long enough so that I couldn't put that in. Commented Mar 28 at 22:13

This is actually non-trivial to do in practice. One practical reason is there is no uniform phase convention for various irreps of $$su(3)$$ (or $$su(n)$$). In other words, there could be many equivalent actions, which basically differ only by scattered relative signs. Technically, this is the same as stating there is no unique way of fixing the relative phases of the various $$su(2)$$ strings of weights in your irrep.

First note there is a general method for arbitrary irreps due to Gelfan'd and Zeitlin. However, this is prescriptive rather than descriptive, but you can check out the general expression for matrix elements in Chapter 10 of

Raczka, R., & Barut, A. O. (1986). Theory of group representations and applications. World Scientific Publishing Company.

One way to proceed is to use the oscillator representation. The set of harmonic oscillator kets $$\vert n_1,n_2,n_3\rangle$$ with $$n_1+n_2+n_3=\lambda$$ form a basis for the carrier space of the irrep $$(\lambda,0)$$ of $$su(3)$$. Matrix elements are easy to get using the realization $$C_{ij}\mapsto a_i^\dagger a_j$$ and in particular you can get anything you need for the irrep $$(1,0)$$ this way.

You can then construct $$(1,1)$$ (which is the adjoint of $$su(3)$$ and thus recover from this the Gell-Mann matrices) by constructing the irrep $$(0,1)$$ and then coupling $$(1,0)\otimes(0,1)$$ and recovering $$(1,1)$$ from there. The problem is in constructing $$(0,1)$$. You can do this by tensoring $$(1,0)\otimes (0,1)$$ and fetching the antisymmetric states in there, which span $$(0,1)$$. The phase issue comes up because the overall sign of any antisymmetric combination is not clearly defined. For instance, the state $$\vert\psi_{12}\rangle=\frac{1}{\sqrt{2}} \left(\vert 100\rangle \vert 010\rangle - \vert 010\vert 100\rangle\right)$$ is antisymmetric and lives in $$(0,1)$$, but so does $$\vert\tilde{\psi}_{12}\rangle=-\vert\psi_{12}\rangle$$. You will get different sets of Gell-Mann matrices depending on your use of $$\vert{\psi}_{12}\rangle$$ or $$\vert\tilde{\psi}_{12}\rangle$$: they will differ by signs somewhere.

There are other ways. Bég and Ruegg in

Bég MA, Ruegg H. A set of harmonic functions for the group SU (3). Journal of Mathematical Physics. 1965 May 1;6(5):677-82

actually construct a set of harmonic functions for $$su(3)$$ which generalize the spherical harmonics of $$su(2)$$. These harmonic functions span irreps of the type $$(\lambda,\lambda)$$ in $$su(3)$$ so you could specialize their results. Their method works for $$su(n)$$. Once you have that you can figure out the action and recover the Gell-Mann matrices.