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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]
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. Here, we offer another useful Hermitian matrix using an abstract example.
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. . Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are kn
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.
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 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 ...
Using the cross product as a Lie bracket, the algebra of 3-dimensional real vectors is a Lie algebra isomorphic to the Lie algebras of SU(2) and SO(3). The structure constants are f a b c = ϵ a b c {\displaystyle f^{abc}=\epsilon ^{abc}} , where ϵ a b c {\displaystyle \epsilon ^{abc}} is the antisymmetric Levi-Civita symbol .