User:Marozols/Sandbox

Generalized Gell-Mann matrices

Construction

Let $\;E_{jk}$ be the matrix with 1 in the $jk$-th entry and 0 elsewhere. Consider the space of $d\times d$ complex matrices, $\mathbb {C} ^{d\times d}$, for a fixed d. Define the following matrices

• For $k<j$, $f_{k,j}^{d}=E_{kj}+E_{jk}$.

• For $k>j$, $f_{k,j}^{d}=-i(E_{jk}-E_{kj})$.

• Let $h_{1}^{d}=I_{d}$, the identity matrix.

• For $1<k<d$, $h_{k}^{d}=h_{k}^{d-1}\oplus 0$.

• For $k=d$, $h_{d}^{d}={\sqrt {\frac {2}{d(d-1)}}}(h_{1}^{d-1}\oplus (1-d))$.

The collection of matrices defined above are called the generalized Gell-Mann matrices, in dimension d.

Properties

The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert-Schmidt inner product on $\mathbb {C} ^{d\times d}$. By the dimension count, we see that they span the vector space of $d\times d$ complex matrices.

In dimensions 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.