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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 ...
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]
The Lie algebra (3) can be written as the set of 3×3 traceless Hermitian matrices. Physicists generally discuss the representation theory of the Lie algebra s u {\displaystyle {\mathfrak {su}}} (3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.
The Gell-Mann–Nishijima formula, developed by Murray Gell-Mann and Kazuhiko Nishijima, led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval Ne'eman, in 1961. The hadrons were organized into SU(3) representation multiplets, octets and decuplets, of roughly the same mass, due to the ...
Gell-Mann matrices — a generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3). Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear-quadratic regulator (LQR) systems.
The complexification of the Lie algebra () is (;), the space of all n × n complex matrices with trace zero. [15] A Cartan subalgebra then consists of the diagonal matrices with trace zero, [ 16 ] which we identify with vectors in C n {\displaystyle \mathbb {C} ^{n}} whose entries sum to zero.
The original current algebra, proposed in 1964 by Murray Gell-Mann, described weak and electromagnetic currents of the strongly interacting particles, hadrons, leading to the Adler–Weisberger formula and other important physical results.
t a = λ a /2 are the Gell-Mann matrices λ a divided by 2; a is a color index in the adjoint representation of SU(3) which take values 1, 2, ..., 8 for the eight generators of the group, namely the Gell-Mann matrices; μ is a spacetime index, 0 for timelike components and 1, 2, 3 for spacelike components;