The Gell-Mann matrices, developed by
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
, are a set of eight
linearly independent 3×3
traceless Hermitian matrices used in the study of the
strong interaction
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.
They span the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
group in the defining representation.
Matrices
:
Properties
These matrices are
traceless,
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
, and obey the extra trace orthonormality relation (so they can generate
unitary matrix group elements of
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
through
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
). These properties were chosen by Gell-Mann because they then naturally generalize the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
for
SU(2) to SU(3), which formed the basis for Gell-Mann's
quark model
In particle 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. The quark model underlies "flavor SU(3)", or the Ei ...
.
Gell-Mann's generalization further
extends to general SU(''n''). For their connection to the
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
of Lie algebras, see the
Weyl–Cartan basis.
Trace orthonormality
In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the
trace of the pairwise product results in the ortho-normalization condition
:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
.
This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of ''SU''(2) are conventionally normalized. In this three-dimensional matrix representation, the
Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices
and
, which commute with each other.
There are three independent
SU(2) subalgebras:
*
*
and
*
where the and are linear combinations of
and
. The SU(2) Casimirs of these subalgebras mutually commute.
However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations.
Commutation relations
The 8 generators of SU(3) satisfy the
commutation and anti-commutation relations
:
with the
structure constant
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pro ...
s
:
The
structure constant
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pro ...
s
are completely antisymmetric in the three indices, generalizing the antisymmetry of the
Levi-Civita symbol of . For the present order of Gell-Mann matrices they take the values
:
In general, they evaluate to zero, unless they contain an odd count of indices from the set , corresponding to the antisymmetric (imaginary) s.
Using these commutation relations, the product of Gell-Mann matrices can be written as
:
where is the identity matrix.
Fierz completeness relations
Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz ''completeness relations'', (Li & Cheng, 4.134), analogous to that
satisfied by the Pauli matrices. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices, the following identities hold,
:
and
:
One may prefer the recast version, resulting from a linear combination of the above,
:
Representation theory
A particular choice of matrices is called a
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
, because any element of SU(3) can be written in the form
, where the eight
are real numbers and a sum over the index is implied. Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity transformation, since that leaves the commutator unchanged.
The matrices can be realized as a representation of the
infinitesimal generators of the
special unitary group called
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
. The
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of 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.
Casimir operators and invariants
The squared sum of the Gell-Mann matrices gives the quadratic
Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
, a group invariant,
:
where
is 3×3 identity matrix. There is another, independent,
cubic Casimir operator, as well.
Application to quantum chromodynamics
These matrices serve to study the internal (color) rotations of the
gluon fields associated with the coloured quarks of
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(cf.
colours of the gluon). A gauge colour rotation is a spacetime-dependent SU(3) group element
, where summation over the eight indices is implied.
See also
*
Casimir element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operat ...
*
Clebsch–Gordan coefficients for SU(3)
*
Generalizations of Pauli matrices
*
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s
*
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
*
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
*
Qutrit
A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states.
The qutrit is analogous to the classical radix-3 trit, just as ...
*
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...
References
*
*
*
*
*
{{Matrix classes
Matrices
Quantum chromodynamics
Mathematical physics
Theoretical physics
Lie algebras
Representation theory of Lie algebras