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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. [1] 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 ...
By convention, in the physics literature the generators are defined as the traceless Hermitian complex matrices with a / prefactor: for the () group, the generators are chosen as ,, where are the Pauli matrices, while for the case of () one defines = where are the Gell-Mann matrices. [6]
Gell-Mann referred to the scheme as the eightfold way, because of the octets of particles in the classification (the term is a reference to the Eightfold Path of Buddhism). [3] [15] Gell-Mann, along with Maurice Lévy, developed the sigma model of pions, which describes low-energy pion interactions. [49]
Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients, [ 6 ] [ 7 ] which results in skew-Hermitian matrices .
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension . [2] [3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by
Both the American physicist Murray Gell-Mann and the Israeli physicist Yuval Ne'eman independently and simultaneously proposed the idea in 1961. [1] [2] [a] The name comes from Gell-Mann's (1961) paper and is an allusion to the Noble Eightfold Path of Buddhism. [3]
The Gell-Mann matrices λ a are eight 3 × 3 matrices which form matrix representations of the SU(3) group. They are also generators of the SU(3) group, in the context of quantum mechanics and field theory; a generator can be viewed as an operator corresponding to a symmetry transformation (see symmetry in quantum mechanics).
The color of a gluon is similarly given by , which corresponds to the particular Gell-Mann matrix it is associated with. This matrix has indices i and j. These are the color labels on the gluon. At the interaction vertex one has q i → g ij + q j. The color-line representation tracks these indices. Color charge conservation means that the ends ...