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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. [2] Gell-Mann's generalization further extends to general SU. For their connection to the standard basis of Lie algebras, see the Weyl–Cartan basis.
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]
and predicted in 1962 that it would have a strangeness −3, electric charge −1 and a mass near 1680 MeV/c 2. In 1964, a particle closely matching these predictions was discovered [7] by a particle accelerator group at Brookhaven. Gell-Mann received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles.
In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case.
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 ...
An explicit representation of these generators is given by = /, wherein the (= …) are the Gell-Mann matrices. The symbol G μ ν a {\displaystyle G_{\mu \nu }^{a}\,} represents the gauge invariant gluon field strength tensor , analogous to the electromagnetic field strength tensor , F μν , in quantum electrodynamics .
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).
According to the figure, a bull week is followed by another bull week 90% of the time, a bear week 7.5% of the time, and a stagnant week the other 2.5% of the time. Labeling the state space {1 = bull, 2 = bear, 3 = stagnant} the transition matrix for this example is