Gell-Mann matrices
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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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
3×3
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
used in the study of the strong interaction 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 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 In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
,
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 In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
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 r ...
). 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) 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 special ...
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 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 Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
of the pairwise product results in the ortho-normalization condition :\operatorname(\lambda_i \lambda_j) = 2\delta_, where \delta_ 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 In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
is the set of linear combinations (with real coefficients) of the two matrices \lambda_3 and \lambda_8, which commute with each other. There are three independent
SU(2) 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 special ...
subalgebras: *\ *\, and *\, where the and are linear combinations of \lambda_3 and \lambda_8. 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 : \begin \left \lambda_a, \lambda_b \right&= 2 i \sum_c f^ \lambda_c, \\ \ &= \frac \delta_ I + 2 \sum_c d^ \lambda_c, \end with the structure constants : \begin f^ &= -\frac i \operatorname(\lambda_a \lambda_b, \lambda_c , \\ d^ &= \frac \operatorname(\lambda_a \). \end The structure constants f^ are completely antisymmetric in the three indices, generalizing the antisymmetry of the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
\epsilon_ of . For the present order of Gell-Mann matrices they take the values :f^ = 1 \ , \quad f^ = f^ = f^ = f^ = f^ = f^ = \frac \ , \quad f^ = f^ = \frac \ . 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 : \lambda_a \lambda_b = \frac ( lambda_a,\lambda_b+ \) = \frac \delta_ I + \sum_c \left(d^ + i f^\right) \lambda_c , 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, :\delta^\alpha _\beta \delta^\gamma _\delta = \frac \delta^\alpha_\delta \delta^\gamma _\beta +\frac \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta and :\lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = \frac \delta^\alpha_\delta \delta^\gamma _\beta -\frac \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta ~. One may prefer the recast version, resulting from a linear combination of the above, :\lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = 2 \delta^\alpha_\delta \delta^\gamma _\beta -\frac \delta^\alpha_\beta \delta^\gamma _\delta ~.


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 to ...
, because any element of SU(3) can be written in the form \mathrm(i \theta^j g_j), where the eight \theta^j 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 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 special ...
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 of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
generators, which can be written as g_i, 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, : C = \sum_^8 \lambda_i \lambda_i = \frac 3 I where I\, 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 field In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in ...
s associated with the coloured quarks of quantum chromodynamics (cf. colours of the gluon). A gauge colour rotation is a spacetime-dependent SU(3) group element U=\exp (i \theta^k (,t) \lambda_k/2), where summation over the eight indices is implied.


See also

* Casimir element *
Clebsch–Gordan coefficients for SU(3) 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 irreduc ...
* 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 to ...
s * Killing form *
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