Gell-Mann matrices
Encyclopedia
The Gell-Mann matrices, named for Murray Gell-Mann
, are one possible representation of the infinitesimal generators of the special unitary group
called SU(3).
This group (a real Lie algebra
in fact) has dimension eight and therefore it has some set with eight linearly independent
generators, which can be written as
, 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
, because any element of SU(3) can be written in the form
, where 
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
of the group. A particular choice of this representation is
and
. These matrices are traceless, Hermitian, and obey the extra relation 
. These properties were chosen by Gell-Mann because they then generalize the Pauli matrices
.
In this representation it is clear that the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices
and 
, which commute with each other. There are 3 independent SU(2) subgroups: 
, 
, and 
, where the x, y, z must consist of linear combinations of 
and 
.
These matrices form a useful representation for computations in the quark model
, and, to a lesser extent, in quantum chromodynamics
.
Murray Gell-Mann
Murray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
, are one possible representation of the infinitesimal generators of the special unitary group
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
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. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
in fact) has dimension eight and therefore it has some set with eight linearly independent
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
generators, which can be written as
, with i taking values from 1 to 8. They obey the commutationCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
relations
where a sum over the index k is implied. The structure constants
are completely antisymmetric in the three indices and have valuesAny set of Hermitian matrices which obey these relations are allowed. A particular choice of matrices is called a group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
, because any element of SU(3) can be written in the form
, where 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
 |
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|
 |
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and
. These matrices are traceless, Hermitian, and obey the extra relation . These properties were chosen by Gell-Mann because they then generalize the Pauli matricesPauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. 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 is the set of linear combinations (with real coefficients) of the two matrices
and , which commute with each other. There are 3 independent SU(2) subgroups: , , and , where the x, y, z must consist of linear combinations of and .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—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....
, and, to a lesser extent, in quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
.
See also
- Generalizations of Pauli matricesGeneralizations of Pauli matricesIn 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. In this article we give a few classes of such matrices....
- Unitary groupUnitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
s and group representationGroup representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s - Quark modelQuark modelIn physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....
, colour charge and quantum chromodynamicsQuantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
External links
- The quark model, by J. J. J. Kokkedee