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Gell-Mann matrices
 
 
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The Gell-Mann matrices, named for Murray Gell-Mann
Murray Gell-Mann

Murray Gell-Mann is an United States physicist who received the 1969 Nobel Prize in physics for his work on the theory of particle physicss.Among his many accomplishments, he formulated the quark model of hadronic resonances, and identified the SU flavor symmetry of the light quarks, extending isospin to include strange quark, which he als...
, are one possible representation of the infinitesimal generator
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
s of the special unitary group
Special unitary group

In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
 called SU(3).

This group (a real Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 in fact) has dimension eight and therefore it has some set with eight linearly independent
Linear independence

In linear algebra, a family of vector spaces is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection....
 generators, which can be written as gi, with i taking values from 1 to 8. They obey the commutation relations where a sum over the index k is implied.






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The Gell-Mann matrices, named for Murray Gell-Mann
Murray Gell-Mann

Murray Gell-Mann is an United States physicist who received the 1969 Nobel Prize in physics for his work on the theory of particle physicss.Among his many accomplishments, he formulated the quark model of hadronic resonances, and identified the SU flavor symmetry of the light quarks, extending isospin to include strange quark, which he als...
, are one possible representation of the infinitesimal generator
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
s of the special unitary group
Special unitary group

In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
 called SU(3).

This group (a real Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 in fact) has dimension eight and therefore it has some set with eight linearly independent
Linear independence

In linear algebra, a family of vector spaces is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection....
 generators, which can be written as gi, with i taking values from 1 to 8. They obey the commutation relations where a sum over the index k is implied. The structure constants are completely antisymmetric in the three indices and have values
Any set of Hermitian matrices which obey these relations are allowed. A particular choice of matrices is called a group representation
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
, because any element of SU(3) can be written in the form exp(i ?j gj), where ?j are real numbers and a sum over the index j is implied. Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

An important representation involves 3×3 matrices, because the group elements then act on complex vectors with 3 entries, i.e., on the fundamental representation
Fundamental representation

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
 of the group. A particular choice of this representation is
and gi = ?i /2. These matrices are traceless, Hermitian, and obey the extra relation Tr
Trace

Trace may refer to:Mathematics, computing and electronics:* Trace of a square matrix or a linear transformation* Trace of a surgery on a manifold...
(?i?j) = 2dij. These properties were chosen by Gell-Mann because they then generalize the Pauli matrices
Pauli matrices

The Pauli matrices are a set of 2 × 2 complex number Hermitian matrix and Unitary matrix matrix Usually indicated by the Greek letter 'sigma' , they are occasionally denoted with a 'tau' when used in connection with isospin symmetries....
.

In this representation it is clear that the Cartan subalgebra
Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent Lie algebra subalgebra of a Lie algebra that is self-normalising ....
 is the set of linear combinations (with real coefficients) of the two matrices ?3 and ?8, which commute with each other. There are 3 independent SU(2) subgroups: , , and , where the x, y, z must consist of linear combinations of ?3 and ?8.

These matrices form a useful representation for computations in the quark model
Quark model

In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, i.e., the quarks which give rise to the quantum numbers of the hadrons....
, and, to a lesser extent, in quantum chromodynamics
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
.

See also

  • Generalizations of Pauli matrices
    Generalizations of Pauli matrices

    In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the properties of the Pauli matrices....
  • Unitary group
    Unitary group

    In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrix, with the group operation that of matrix multiplication....
    s and group representation
    Group representation

    In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
    s
  • Quark model
    Quark model

    In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks, i.e., the quarks which give rise to the quantum numbers of the hadrons....
    , colour charge and quantum chromodynamics
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....