Gell-Mann matrices

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.

These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark model. Gell-Mann's generalization further extends to general SU(n). For their connection to the standard basis of Lie algebras, see the Weyl–Cartan basis.

In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the trace of the pairwise product results in the ortho-normalization condition

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of SU(2) are conventionally normalized. In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices ${\displaystyle \lambda _{3}}$ and ${\displaystyle \lambda _{8}}$, which commute with each other.

There are three significant SU(2) subalgebras:

where the x and y are linear combinations of ${\displaystyle \lambda _{3}}$ and ${\displaystyle \lambda _{8}}$. The SU(2) Casimirs of these subalgebras mutually commute.

However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations.