How unique are the quantum numbers we commonly use?

Question

We use the eigenvalues of the Cartan generators (=diagonal generators) of a given gauge group as quantum numbers in physics. Are these numbers somehow fixed and if not, what transformations are allowed?

The easiest example is $SU(2)$ with just one Cartan generator $H_1$, which is commonly written in terms of the Pauli matrix $\sigma_3= \begin{pmatrix} 1&0\\0&-1 \end{pmatrix}$: $H_1 = \frac{1}{2} \sigma_3$ and therefore

$$H_1= \begin{pmatrix} \frac{1}{2} &0\\0&-\frac{1}{2} \end{pmatrix} $$

Would $H_1 = \begin{pmatrix} \frac{1}{7} &0\\0&-\frac{1}{7} \end{pmatrix} $ or $H_1= \begin{pmatrix} -\frac{1}{2} &0\\0&\frac{1}{2} \end{pmatrix}$ equally "work"?

A bit more involved example would be $SU(3)$, which has two Cartan generators $H_1=\frac{1}{2} \lambda_3$ and $H_2=\frac{1}{2} \lambda_8$, where $\lambda_3$ and $\lambda_3$ denote Gell-Mann matrices.

How unique are the diagonal entries of these matrices? In what ways are we allowed to transform the Cartan generators (and with them of course the corresponding quantum numbers)?

(One allowed transformation is certainly which one we call $H_1$ and which one $H_2$, i.e. permuations. $H_1 \leftrightarrow H_2$ )

Answer 1

Here we will for simplicity just consider an arbitrary finite-dimensional complex$^1$ semisimple Lie algebra $\mathfrak{g}$.

I) One may show that the CSAs are precisely the maximal toral Lie subalgebras of $\mathfrak{g}$. In particular CSAs are abelian.

Also the Killing form $\kappa:\mathfrak{g}\times \mathfrak{g}\to \mathbb{C}$ (which is non-degenerate) has a non-degenerate restriction to a CSA, so a CSA is canonically an inner product space, and canonically isomorphic to its dual vector space.

Moreover, all CSAs of $\mathfrak{g}$ have the same dimension (called the rank $r$), and are conjugated with each other, i.e. related via inner automorphisms of the Lie algebra $\mathfrak{g}$. So in this sense all choices of CSA are equivalent.

II) Consider from now on an arbitrary but fixed given choice of CSA $\mathfrak{h}\subset \mathfrak{g}$.

Obviously, one may pick an arbitrary basis $(H_1, \ldots, H_r)$ for $\mathfrak{h}$.

A root $\alpha\in \mathfrak{h}^{\ast}$ belongs to the dual vector space of $\mathfrak{h}$. Its defining property is $$\tag{1}\exists x\in \mathfrak{g}\backslash\{0\}~ \forall H\in\mathfrak{h} :~~[H,x]~=~\alpha(H)x.$$

From a physics perspective, the Lie algebra element $x\in \mathfrak{g}$ plays the role of a generalized raising/lowering creation/annihilation operator and the root $\alpha\in \mathfrak{h}^{\ast}$ plays the role of a generalized quantum number.

Note in particular that the definition (1) in principle does not depend on a choice of basis $(H_1, \ldots, H_r)$.

NB: Be aware that authors often use other associative/invariant bilinear forms than the canonical Killing form $\kappa$. This may induce a non-canonical isomorphism $\mathfrak{h}^{\ast}\cong \mathfrak{h}$ and non-canonical normalizations of roots.

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$^1$ Many results and properties for complex Lie algebras continues to hold for real Lie algebras, although sometimes in modified form.