Шаблон:Main article
The special unitary groupSU is the group of unitary matrices whose determinant is equal to 1.[1] This set is closed under matrix multiplication. All transformations characterized by the special unitary group leave norms unchanged. The Шаблон:Math 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 on the colour quantum numbers of the quarks that form the fundamental (triplet) representation of the group.
The group Шаблон:Math is a subgroup of group Шаблон:Math, 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 Шаблон:Math group is 9. Furthermore, multiplying a U by a phase, Шаблон:Math leaves the norm invariant. Thus Шаблон:Math can be decomposed into a direct product Шаблон:Math. Because of this additional constraint, Шаблон:Math has dimension 8.
Generators of the Lie algebra
Every unitary matrix Шаблон:Mvar can be written in the form
where <math>\lambda_k</math> are the 8 linearly independent matrices forming the basis of the Lie algebra of Шаблон:Math, in the triplet representation. The unit determinant condition requires the <math>\lambda_k</math> matrices to be traceless, since
<math>\det(e^A)=e^{\operatorname{tr} (A)}</math>.
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,
where <math>f_{jkl}</math> are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol <math>\epsilon_{jkl}</math> of Шаблон:Math.
In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric Шаблон:Mvars. Note <math>f_{ljk}=\frac{-i}{4}\mathrm{tr}([\lambda_l , \lambda_j] \lambda_k) </math>.
where <math>d_{jkl}</math> are the completely symmetric coefficient constants. They vanish if the number of indices from the set {2,5,7} is odd. In terms of the matrices,
A slightly differently normalized standard basis consists of the F-spin operators, which are defined as <math>\hat{F_i}=\frac{1}{2}\lambda_i</math> for the 3, and are utilized to apply to any representation of this algebra.
The Cartan–Weyl basis of the Lie algebra of Шаблон:Math is obtained by another change of basis, where one defines,[2]
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.[3]
Шаблон:Colend
All other commutation relations follow from hermitian conjugation of these operators.
These commutation relations can be used to construct the irreducible representations of the Шаблон:Math group.
The representations of the group lie in the 2-dimensional Шаблон:Math plane. Here, <math>\hat{I}_3</math> stands for the z-component of Isospin and <math>\hat{Y}</math> is the Hypercharge, and they comprise the (abelian) Cartan subalgebra of the full Lie algebra. The maximum number of mutually commuting generators of a Lie algebra is called its rank: Шаблон:Math has rank 2. The remaining 6 generators, the ± ladder operators, correspond to the 6 roots arranged on the 2-dimensional hexagonal lattice of the figure.
Casimir operators
Шаблон:Main article
The Casimir operator is an operator that commutes with all the generators of the Lie group. In the case of Шаблон:Math, the quadratic operator Шаблон:Math is the only independent such operator.
In the case of Шаблон:Math group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,[4]
These Casimir operators serve to label the irreducible representations of the Lie group algebra Шаблон:Math, 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.
The eigenvalue ("anomaly coefficient") of Шаблон:Tmath is[6]
<math>(p-q)(3+p+2q)(3+q+2p)/18 .</math>
It is an odd function under the interchange Шаблон:Math. Consequently, it vanishes for real representations Шаблон:Math, such as the adjoint, Шаблон:Math, i.e. both Шаблон:Tmath and anomalies vanish for it.
Representations of the SU(3) group
Шаблон:See also
The irreducible representations of SU(3) are analyzed in various places, including Hall's book.[7] Since the SU(3) group is simply connected,[8] the representations are in one-to-one correspondence with the representations of its Lie algebra[9] su(3), or the complexification[10] 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.[11] (See also the section of Young tableaux below: Шаблон:Mvar is the number of single-box columns, "quarks", and Шаблон:Mvar 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 <math>H_1</math> and <math>H_2</math> are linear combinations of the elements <math>\hat{I}_3</math> and <math>\hat{Y}</math>, but normalized so that the eigenvalues of <math>H_1</math> and <math>H_2</math> are integers.)
This is to be compared to the representation theory of SU(2), where the irreducible representations are labeled by the maximum eigenvalue of a single element, h.
and the corresponding Haar measure is[13]
<math>\mu(SU(3))=64 \sin\left(\frac \phi 2\right)^2
\sin\left(\frac 12 (\theta+\phi/2)\right)^2
\sin\left(\frac 12 (\theta-\phi/2)\right)^2</math>
such that <math>-2\pi\leq \phi\leq 2\pi</math> and <math>-3\pi\leq\theta\leq 3\pi</math>,
An Шаблон:Math 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 Шаблон:Math multiplets, namely the eigenvalues of its quadratic Casimir and of Шаблон:Mvar3.
Since <math>[\hat{I}_3,\hat{Y}]=0</math>, we can
label different states by the eigenvalues of <math>\hat{I}_3</math> and <math>\hat{Y}</math> operators, <math>|t,y\rangle</math>, for a given eigenvalue of the isospin Casimir. The action of operators on this states are,[14]
All the other states of the representation can be constructed by the successive application of the ladder operators <math>\hat{I}_\pm, \hat{U}_\pm</math> and <math>\hat{V}_\pm</math> 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).
The product representation of two irreducible representations <math>D(p_1,q_1)</math> and <math>D(p_2,q_2)</math> is generally reducible. Symbolically,
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 Шаблон:Tmath appears Шаблон:Tmath times in the reduction of this direct product of <math>D(p_1,q_1)</math> with <math>D(p_2,q_2)</math>.
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 the case of the irreducible representation Шаблон:Tmath is
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 Шаблон:Mvar, one must look outside the group. It is necessary to distinguish different Шаблон:Tmath for similar values of Шаблон:Mvar and Шаблон:Mvar.
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,
Шаблон:Equation box 1_{\nu_2}</math> are the basis states of <math>D(p_2,q_2)</math>. Also <math>{\phi^{\mu_1}}_{\nu_1},{\phi^{\mu_2}}_{\nu_2}</math> are the basis states of the product representation. Here <math>\nu_1, \nu_2</math> represents the combined eigenvalues <math>({i^2}_1, {i^z}_1, y_1)</math> and <math>({i^2}_2, {i ^z}_2, y_2)</math> respectively.
Thus the unitary transformations that connects the two bases are
<math>\psi
\begin{pmatrix}
\mu_1 & \mu_2 & \gamma\\
& & \nu
\end{pmatrix}
=\sum_{\nu_1,\nu_2}\begin{pmatrix}
\mu_1 & \mu_2 & \gamma\\
\nu_1 & \nu_2 & \nu
\end{pmatrix}
{\phi^{\mu_1}}_{\nu_1}{\phi^{\mu_2}}_{\nu_2}
</math>
This is a comparatively compact notation. Here,
<math>\begin{pmatrix}
\mu_1 & \mu_2 & \gamma\\
\nu_1 & \nu_2 & \nu
\end{pmatrix}
</math>
are the Clebsch–Gordan coefficients.
Orthogonality relations
The Clebsch–Gordan coefficients form a real orthogonal matrix. Therefore,
If an irreducible representation <math>{\scriptstyle{\mu}_{\gamma}}</math> appears in the Clebsch–Gordan series of <math>{\scriptstyle{\mu}_1\otimes{\mu}_2}</math>, then it must appear in the Clebsch–Gordan series of <math>{\scriptstyle{\mu}_2\otimes{\mu}_1}</math>. Which implies,
<math>\begin{pmatrix}
\mu_1 & \mu_2 & \gamma\\
\nu_1 & \nu_2 & \nu
\end{pmatrix}
=\xi_1\begin{pmatrix}
\mu_2 & \mu_1 & \gamma\\
\nu_2 & \nu_1 & \nu
\end{pmatrix}</math>
Where <math>\xi_1=\xi_1(\mu_1,\mu_2,\gamma)=\pm1</math>
Since the Clebsch–Gordan coefficients are all real, the following symmetry property can be deduced,
where the spring constant, the mass and Planck's constant have been absorbed into the definition of the variables, Шаблон:Math.
It is seen that this Hamiltonian is symmetric under coordinate transformations that preserve the value of <math>V=x^2+y^2+z^2</math>. Thus, any operators in the group Шаблон:Math keep this Hamiltonian invariant.
More significantly, since the Hamiltonian is Hermitian, it further remains invariant under operation by elements of the much larger Шаблон:Math group.
Apart from the tensorial trace of the operator<math>\hat{A}_{ij},</math>, which is the Hamiltonian, the remaining 5 operators can be rearranged into their spherical component form as
The eight operators (consisting of the 5 operators derived from the traceless symmetric tensor operator Шаблон:Mvar and the three independent components of the angular momentum vector) obey the same commutation relations as the infinitesimal generators of the Шаблон:Math group, detailed above.
As such, the symmetry group of Hamiltonian for a linear isotropic 3D Harmonic oscillator is isomorphic to Шаблон:Math group.
}}
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 <math>\hat{N_i}=\hat{a}_{i}^\dagger \hat{a}_{i}</math> amounts to
The Hamiltonian has 8-fold degeneracy. A successive application of Шаблон:Mvar and Шаблон:Mvar† on the left preserves the Hamiltonian invariant, since it increases Шаблон:Mvar by 1 and decrease Шаблон:Mvar by 1, thereby keeping the total
Since the operators belonging to the symmetry group of Hamiltonian do not always form an Abelian group, a common eigenbasis cannot be found that diagonalizes all of them simultaneously. Instead, we take the maximally commuting set of operators from the symmetry group of the Hamiltonian, and try to reduce the matrix representations of the group into irreducible representations.
Шаблон:See also
Hilbert space of two systems
The Hilbert space of two particles is the tensor product of the two Hilbert spaces of the two individual particles,
where <math>\mathbb{H}_1</math> and <math>\mathbb{H}_2</math> are the Hilbert space of the first and second particles, respectively.
The operators in each of the Hilbert spaces have their own commutation relations, and an operator of one Hilbert space commutes with an operator from the other Hilbert space. Thus the symmetry group of the two particle Hamiltonian operator is the superset of the symmetry groups of the Hamiltonian operators of individual particles. If the individual Hilbert spaces are Шаблон:Mvar dimensional, the combined Hilbert space is Шаблон:Math dimensional.
Clebsch–Gordan coefficient in this case
The symmetry group of the Hamiltonian is Шаблон:Math. As a result, the Clebsch–Gordan coefficients can be found by expanding the uncoupled basis vectors of the symmetry group of the Hamiltonian into its coupled basis. The Clebsch–Gordan series is obtained by block-diagonalizing the Hamiltonian through the unitary transformation constructed from the eigenstates which diagonalizes the maximal set of commuting operators.
Young tableaux
Шаблон:Main article
A Young tableau (plural tableaux) is a method for decomposing products of an SU(N) group representation into a sum of irreducible representations. It provides the dimension and symmetry types of the irreducible representations, which is known as the Clebsch–Gordan series. Each irreducible representation corresponds to a single-particle state and a product of more than one irreducible representation indicates a multiparticle state.
Since the particles are mostly indistinguishable in quantum mechanics, this approximately relates to several permutable particles. The permutations of Шаблон:Mvar identical particles constitute the symmetric groupSШаблон:Mvar. Every Шаблон:Mvar-particle state of SШаблон:Mvar that is made up of single-particle states of the fundamental Шаблон:Mvar-dimensional SU(N) multiplet belongs to an irreducible SU(N) representation. Thus, it can be used to determine the Clebsch–Gordan series for any unitary group.[16]
Constructing the states
Any two particle wavefunction <math>\psi_{1,2}</math>, where the indices 1,2 represents the state of particle 1 and 2, can be used to generate states of explicit symmetry using the symmetrizing and the anti-symmetrizing operators.[17]
Starting from a multiparticle state, we can apply <math>\mathbf{S_{12}}</math> and <math>\mathbf{A_{12}}</math> repeatedly to construct states that are:[17]
Symmetric with respect to all particles.
Antisymmetric with respect to all particles.
Mixed symmetries, i.e. symmetric or antisymmetric with respect to some particles.
Constructing the tableaux
Instead of using ψ, in Young tableaux, we use square boxes (□) to denote particles and i to denote the state of the particles.
Файл:Young tableaux 1.jpegA sample Young tableau. The number inside the boxes represents the state of the particles
The complete set of <math>n_p</math> particles are denoted by arrangements of <math>n_p</math> □s, each with its own quantum number label (i).
The tableaux is formed by stacking boxes side by side and up-down such that the states symmetrised with respect to all particles are given in a row and the states anti-symmetrised with respect to all particles lies in a single column. Following rules are followed while constructing the tableaux:[16]
A row must not be longer than the one before it.
The quantum labels (numbers in the □) should not decrease while going left to right in a row.
The quantum labels must strictly increase while going down in a column.
Case for N = 3
For N=3 that is in the case of SU(3), the following situation arises. In SU(3) there are three labels, they are generally designated by (u,d,s) corresponding to up, down and strange quarks which follows the SU(3) algebra. They can also be designated generically as (1,2,3). For a two-particle system, we have the following six symmetry states:
{
\begin{array}{|c|}
\hline
1 \\
\hline
2 \\
\hline
\end{array}
\atop
\frac 1 \sqrt 2 (ud-du)
}
\qquad
{
\begin{array}{|c|}
\hline
1 \\
\hline
3 \\
\hline
\end{array}
\atop
\frac 1 \sqrt 2 (us-su)
}
\qquad
{
\begin{array}{|c|}
\hline
2 \\
\hline
3 \\
\hline
\end{array}
\atop
\frac 1 \sqrt 2 (ds-sd)
}
</math>
The 1-column, 3-row tableau is the singlet, and so all tableaux of nontrivial irreps of SU(3) cannot have more than two rows. The representation Шаблон:Math has
Шаблон:Math boxes on the top row and Шаблон:Mvar boxes on the second row.
Clebsch–Gordan series from the tableaux
Clebsch–Gordan series is the expansion of the tensor product of two irreducible representation into direct sum of irreducible representations.
<math>D(p_1,q_1)\otimes D(p_2,q_2)=\sum_{P,Q}\oplus D(P,Q)</math>. This can be easily found out from the Young tableaux.
Procedure to obtain the Clebsch–Gordan series from Young tableaux:
The following steps are followed to construct the Clebsch–Gordan series from the Young tableaux:[18]
Write down the two Young diagrams for the two irreps under consideration, such as in the following example. In the second figure insert a series of the letter a in the first row, the letter b in the second row, the letter c in the third row, etc. in order to keep track of them once they are included in the various resultant diagrams:
Take the first box containing an a and appends it to the first Young diagram in all possible ways that follow the rules for creation of a Young diagram:
Then take the next box containing an a and do the same thing with it, except that we are not allowed to put two aШаблон:'s together in the same column.
In each rows while counting from right to left, if at any point the number of a particular alphabet encountered be more than the number of the previous alphabet, then the diagram must be deleted. Here the first and the third diagram should be deleted, resulting in:
↑ 13,013,1Шаблон:Harvnb Note: There is a typo in the final quoting of the result - in Equation 10.121 the first <math>\varphi</math> should instead be a <math>\psi</math>.
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