In
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
,
Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In m ...
are the expansion coefficients of
total angular momentum
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's ...
eigenstate
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s in an uncoupled
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt ...
(1833–1872) and
Paul Gordan
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous ...
(1837–1912), who encountered an equivalent problem in
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...
.
Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing
hadronic decays, where a
flavor-SU(3) symmetry exists (the ''
eightfold way'') that connects the three light
quarks
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
:
up,
down, and
strange
Strange may refer to:
Fiction
* Strange (comic book), a comic book limited series by Marvel Comics
* Strange (Marvel Comics), one of a pair of Marvel Comics characters known as The Strangers
* Adam Strange, a DC Comics superhero
* The title c ...
.
SU(3) group
The
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
SU is the group of
unitary matrices
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if
U^* U = UU^* = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate ...
whose determinant is equal to 1. This set is closed under
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. All transformations characterized by the special unitary group leave norms unchanged. The symmetry appears in the light quark flavour symmetry (among ''up'', ''down'', and ''strange'' quarks) dubbed the
Eightfold Way (physics). The same group acts in
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the study 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 of ...
on the colour quantum numbers of the quarks that form the fundamental (triplet) representation of the group.
The group is a subgroup of group , the group of all 3×3 unitary matrices. The unitarity condition imposes nine constraint relations on the total 18 degrees of freedom of a 3×3 complex matrix. Thus, the dimension of the group is 9. Furthermore, multiplying a ''U'' by a phase, leaves the norm invariant. Thus can be decomposed into a direct product . Because of this additional constraint, has dimension 8.
Generators of the Lie algebra
Every unitary matrix can be written in the form
:
where ''H'' is
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 me ...
. The elements of can be expressed as
:
where
are the 8 linearly independent matrices forming the basis of 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 ident ...
of , in the triplet representation. The unit determinant condition requires the
matrices to be traceless, since
:
.
An explicit basis in the fundamental, 3, representation can be constructed in analogy to the Pauli matrix algebra of the spin operators. It consists of the
Gell-Mann matrices
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the fundame ...
,
:
These are the generators of the group in the triplet representation, and they are normalized as
:
The Lie algebra structure constants of the group are given by the commutators of
:
where
are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol
of .
In general, they vanish, unless they contain an odd number of indices from the set , corresponding to the antisymmetric s. Note
.
Moreover,
:
where
are the completely symmetric coefficient constants. They vanish if the number of indices from the set is odd. In terms of the matrices,
:
Standard basis

A slightly differently normalized standard basis consists of the ''F-spin'' operators, which are defined as
for the 3, and are utilized to apply to ''any representation of this algebra''.
The Cartan–Weyl basis of the Lie algebra of is obtained by another
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are conside ...
, where one defines,
:
:
:
:
:
Because of the factors of ''i'' in these formulas, this is technically a basis for the complexification of the su(3) Lie algebra, namely sl(3,C). The preceding basis is then essentially the same one used in Hall's book.
Commutation algebra of the generators
The standard form of generators of the group satisfies the
commutation relation
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
The commutator of two elements, ...
s given below,
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
All other commutation relations follow from hermitian conjugation of these operators.
These commutation relations can be used to construct the irreducible representations of the group.
The representations of the group lie in the 2-dimensional plane. Here,
stands for the z-component of
Isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle.
Isospin is also known as isobaric spin or isotopic spin.
Isospin symmetry is a subset of the flavour symmetr ...
and
is the
Hypercharge
In particle physics, the hypercharge (a portmanteau of hyperonic and charge (physics), charge) ''Y'' of a subatomic particle, particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charg ...
, and they comprise the (abelian)
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 ...
of the full Lie algebra. The maximum number of mutually commuting generators of a Lie algebra is called its ''rank'': has rank 2. The remaining 6 generators, the ± ladder operators, correspond to the 6
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
arranged on the 2-dimensional hexagonal lattice of the figure.
Casimir operators
The Casimir operator is an operator that commutes with all the generators of the Lie group. In the case of , the quadratic operator is the only independent such operator.
In the case of group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,
:
These Casimir operators serve to label the irreducible representations of the
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
algebra , because all states in a given representation assume the same value for each Casimir operator, which serves as the identity in a space with the dimension of that representation. This is because states in a given representation are connected by the action of the generators of the Lie algebra, and all generators commute with the Casimir operators.
For example, for the triplet representation, , the eigenvalue of is 4/3, and of , 10/9.
More generally, from
Freudenthal's formula, for generic , the eigenvalue of is
:
.
The eigenvalue ("anomaly coefficient") of is
:
It is an ''odd function'' under the interchange . Consequently, it vanishes for real representations , such as the adjoint, , i.e. both and anomalies vanish for it.
Representations of the SU(3) group
The irreducible representations of SU(3) are analyzed in various places, including Hall's book. Since the SU(3) group is simply connected, the representations are in one-to-one correspondence with the representations of its Lie algebra su(3), or the complexification of its Lie algebra, sl(3,C).
We label the representations as ''D''(''p'',''q''), with ''p'' and ''q'' being non-negative integers, where in physical terms, ''p'' is the number of quarks and ''q'' is the number of antiquarks. Mathematically, the representation ''D''(''p'',''q'') may be constructed by tensoring together ''p'' copies of the standard 3-dimensional representation and ''q'' copies of the dual of the standard representation, and then extracting an irreducible
invariant subspace
In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''. More generally, an invariant subspace for a collection of ...
. (See also the section of Young tableaux below: is the number of single-box columns, "quarks", and the number of double-box columns, "antiquarks").
Still another way to think about the parameters ''p'' and ''q'' is as the maximum eigenvalues of the diagonal matrices
:
.
(The elements
and
are linear combinations of the elements
and
, but normalized so that the eigenvalues of
and
are integers.)
This is to be compared to the
representation theory of SU(2)
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abe ...
, where the irreducible representations are labeled by the maximum eigenvalue of a single element, ''h''.
The representations have dimension

:
their
irreducible characters are given by
[ Note: There is a typo in the final quoting of the result - in Equation 10.121 the first should instead be a .]
:
and the corresponding Haar measure is
such that
and
,
:
An multiplet may be completely specified by five labels, two of which, the eigenvalues of the two Casimirs, are common to all members of the multiplet. This generalizes the mere two labels for multiplets, namely the eigenvalues of its quadratic Casimir and of
3.
Since
, we can
label different states by the eigenvalues of
and
operators,
, for a given eigenvalue of the isospin Casimir. The action of operators on this states are,
Senner & Schulten
/ref>
:
:
:
:
:
:
:
Here,
:
and
:
All the other states of the representation can be constructed by the successive application of the ladder operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
s and and by identifying the base states which are annihilated by the action of the lowering operators. These operators can be pictured as arrows whose endpoints form the vertices of a hexagon (picture for generators above).
Clebsch–Gordan coefficient for SU(3)
The product representation of two irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s and is generally reducible. Symbolically,
:
where is an integer.
For example, two octets (adjoints) compose to
:
that is, their product reduces to an icosaseptet (27), decuplet, two octets, an antidecuplet, and a singlet, 64 states in all.
The right-hand series is called the Clebsch–Gordan series. It implies that the representation appears times in the reduction of this direct product of with .
Now a complete set of operators is needed to specify uniquely the states of each irreducible representation inside the one just reduced.
The complete set of commuting operators
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of ...
in the case of the irreducible representation is
:
where
: .
The states of the above direct product representation are thus completely represented by the set of operators
:
where the number in the parentheses designates the representation on which the operator acts.
An alternate set of commuting operators can be found for the direct product representation, if one considers the following set of operators,
:
Thus, the set of commuting operators includes
:
This is a set of nine operators only. But the set must contain ten operators to define all the states of the direct product representation uniquely. To find the last operator , one must look outside the group. It is necessary to distinguish different for similar values of and .
:
Thus, any state in the direct product representation can be represented by the ket,
:
also using the second complete set of commuting operator, we can define the states in the direct product representation as
:
We can drop the from the state and label the states as
:
using the operators from the first set, and,
:
using the operators from the second set.
Both these states span the direct product representation and any states in the representation can be labeled by suitable choice of the eigenvalues.
Using the completeness relation,
Here, the coefficients
are the Clebsch–Gordan coefficients.
A different notation
To avoid confusion, the eigenvalues can be simultaneously denoted by and the eigenvalues are simultaneously denoted by . Then the eigenstate of the direct product representation can be denoted by[ ]
:
where is the eigenvalues of and is the eigenvalues of denoted simultaneously. Here, the quantity expressed by the parenthesis is the Wigner 3-j symbol
In quantum mechanics, the Wigner's 3-j symbols, also called 3''-jm'' symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address exactly the same physical problem, the 3-' ...
.
Furthermore, are considered to be the basis states of and are the basis states of . Also are the basis states of the product representation. Here represents the combined eigenvalues and respectively.
Thus the unitary transformations that connects the two bases are
:
This is a comparatively compact notation. Here,
:
are the Clebsch–Gordan coefficients.
Orthogonality relations
The Clebsch–Gordan coefficients form a real orthogonal matrix. Therefore,
:
Also, they follow the following orthogonality relations,
:
:
Symmetry properties
If an irreducible representation appears in the Clebsch–Gordan series of , then it must appear in the Clebsch–Gordan series of . Which implies,
:
Where
Since the Clebsch–Gordan coefficients are all real, the following symmetry property can be deduced,
:
Where .
Symmetry group of the 3D oscillator Hamiltonian operator
A three-dimensional harmonic oscillator is described by the Hamiltonian
:
where the spring constant, the mass and the Planck constant have been absorbed into the definition of the variables, .
It is seen that this Hamiltonian is symmetric under coordinate transformations that preserve the value of . Thus, any operators in the group keep this Hamiltonian invariant.
More significantly, since the Hamiltonian is Hermitian, it further remains invariant under operation by elements of the much larger group.
More systematically, operators such as the Ladder operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
s
: and
can be constructed which raise and lower the eigenvalue of the Hamiltonian operator by 1.
The operators are not hermitian, but hermitian operators can be constructed from combinations of ,
: and .
There are nine such operators for .
The nine hermitian operators formed by the bilinear forms are controlled by the fundamental commutators
:
:
and seen to ''not'' commute among themselves. As a result, this complete set of operators don't share their eigenvectors in common, and they cannot be diagonalized simultaneously. The group is thus non-Abelian and degeneracies may be present in the Hamiltonian, as indicated.
The Hamiltonian of the 3D isotropic harmonic oscillator, when written in terms of the operator amounts to
: