Linked Questions

16
votes
4answers
10k views

Trace and adjoint representation of $SU(N)$

In the adjoint representation of $SU(N)$, the generators $t^a_G$ are chosen as $$ (t^a_G)_{bc}=-if^{abc} $$ The following identity can be found in Taizo Muta's book "Foundations of Quantum ...
11
votes
1answer
3k views

The conjugate representation in $\mathfrak{su}(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $\mathfrak{su}(2)$ and that for the diagonal operator $T_3$, \begin{align} ...
2
votes
3answers
1k views

Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
2
votes
2answers
1k views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
1
vote
1answer
597 views

Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
1
vote
1answer
123 views

Product of generators in fundamental representation of $SU(N)$

I'm trying to prove equation 25.20 in Schwartz: $$T^a T^b=\frac{1}{2N}\delta ^{ab}+\frac{1}{2}d^{abc}T^c + \frac{1}{2}if^{abc}T^c,\tag{25.20}$$ where $T^a$ are the fundamental representation ...
2
votes
1answer
114 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 ${\...
2
votes
1answer
64 views

Why do Gell-Mann matrices have this normalization?

This might be a stupid question, but why is the normalization of the Gell-Mann matrices (basis of the $\mathrm{su}(3)$ Lie algebra) chosen to be $$\mathrm{trace}(\lambda_i\lambda_j)=2\delta_{ij}$$ ...
1
vote
0answers
32 views

Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting ...