User:Marozols/Sandbox

Generalized Gell-Mann matrices[edit]

Construction[edit]

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[edit]

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.