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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 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 center of SU(n) is isomorphic to the cyclic group /, and is composed of the diagonal matrices ζ I for ζ an n th root of unity and I the n × n identity matrix. Its outer automorphism group for n ≥ 3 is Z / 2 Z , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} while the outer automorphism group of SU(2) is the trivial group .
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 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.
The theorem was proved first by Gell-Mann and Low in 1951, making use of the Dyson series. [1] In 1969, Klaus Hepp provided an alternative derivation for the case where the original Hamiltonian describes free particles and the interaction is norm bounded. [2] In 1989, G. Nenciu and G. Rasche proved it using the adiabatic theorem. [3]
The upper and lower indices are frequently not distinguished, unless the algebra is endowed with some other structure that would require this (for example, a pseudo-Riemannian metric, on the algebra of the indefinite orthogonal group so(p,q)). That is, structure constants are often written with all-upper, or all-lower indexes.