SU(3)

Prerequisites: SU(2). Full path (5 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Imagine an object that can be in three basic states, for example a quark that can carry one of three colours, usually called red, green and blue. Quantum mechanics allows any complex superposition of the three. SU(3) contains all the ways of mixing these three states into each other such that total probability is kept and such that an overall common phase is left out. A three-by-three complex matrix has nine complex entries; these two conditions leave eight real parameters, which is where the eight "angles" come from.

At most two independent generators can be diagonal at the same time. States can therefore be labelled by two numbers at once, like isospin and hypercharge when SU(3) acts on the up, down and strange quark flavours, and the multiplets of SU(3) are drawn as patterns of points in a plane. Unlike for SU(2), the three-dimensional representation and its complex conjugate are genuinely different. This is the difference between quarks and antiquarks. A quark and an antiquark can combine into a colour neutral state, and so can three quarks, but two quarks cannot.

The Lie group $SU(3)$ describes abstract "rotations" in a space with three complex dimensions. Each "rotation" is characterized by eight abstract "angles".

Concrete

Representations

The diagram below shows the defining (3-dimensional) representation of $SU(3)$ in its upper branch and the 8-dimensional adjoint representations of the same group in its lower branch. The adjoint representation can be rewritten such that it acts on 8-dimensional vectors (as opposed to 3x3 matrices) by regular matrix-vector multiplication.

For more groups and their representations see Fun with Symmetry.

Abstract

$SU(3) = \{U\in\mathbb{C}^{3\times 3}: U^\dagger U = 1,\ \det U = 1\}$ is compact, connected and simply connected, of dimension $8$ and rank $2$ . Its centre is $\mathbb{Z}_3 = \{1,\omega,\omega^2\}\cdot 1$ with $\omega = \mathrm{e}^{2\pi i/3}$ . The Lie algebra is the compact real form of $A_2 = \mathfrak{sl}(3,\mathbb{C})$ .

Generators. With the Gell-Mann matrices $\lambda_a$ and $T_a = \frac12\lambda_a$ one has $[T_a, T_b] = i f_{abc}T_c, \qquad \{T_a,T_b\} = \frac13\delta_{ab} + d_{abc}T_c, \qquad \mathrm{tr}(T_aT_b) = \frac12\delta_{ab}.$ The totally antisymmetric structure constants are, up to permutations, $f_{123}=1$ , $f_{147}=f_{246}=f_{257}=f_{345}=\frac12$ , $f_{156}=f_{367}=-\frac12$ , $f_{458}=f_{678}=\frac{\sqrt3}{2}$ . The diagonal $T_3$ and $T_8$ span a Cartan subalgebra.

Representations. Irreducible representations are labelled by Dynkin labels $(p,q)$ with $p,q\in\mathbb{Z}_{\geq 0}$ , realised on tensors with $p$ upper and $q$ lower indices that are symmetric in each set and traceless. Their dimension is $\dim(p,q) = \frac12(p+1)(q+1)(p+q+2),$ so $\mathbf 3 = (1,0)$ , $\bar{\mathbf 3} = (0,1)$ , $\mathbf 8 = (1,1)$ , $\mathbf{10} = (3,0)$ . Complex conjugation exchanges $(p,q)\leftrightarrow(q,p)$ , so $\mathbf 3$ and $\bar{\mathbf 3}$ are inequivalent. Rank two means two independent Casimir operators, a quadratic and a cubic one. With the normalisation above the quadratic Casimir is $C_2(p,q) = \frac13\left(p^2 + q^2 + pq + 3p + 3q\right),$ which gives $C_2(\mathbf 3) = \frac43$ and $C_2(\mathbf 8) = 3$ . The centre acts on $(p,q)$ as $\omega^{p-q}$ ; the class of $p-q$ modulo $3$ is called triality. Some tensor products: $\mathbf 3\otimes\bar{\mathbf 3} = \mathbf 1\oplus\mathbf 8, \qquad \mathbf 3\otimes\mathbf 3 = \mathbf 6\oplus\bar{\mathbf 3}, \qquad \mathbf 3\otimes\mathbf 3\otimes\mathbf 3 = \mathbf 1\oplus\mathbf 8\oplus\mathbf 8\oplus\mathbf{10}.$ A singlet can only appear in a product of total triality zero. In QCD the quarks are in $\mathbf 3$ and the gluons in the adjoint $\mathbf 8$ , and the colour singlets in $\mathbf 3\otimes\bar{\mathbf 3}$ and $\mathbf 3\otimes\mathbf 3\otimes\mathbf 3$ are the mesons and baryons.

Why is it interesting?

$SU(3)$ is at the heart of the so-called "eightfold way", a scheme that organizes the large "zoo" of hadron particles into neat geometrical patterns (octets and decuplets).

$SU(3)$ is also the gauge group of the strong nuclear interaction. It describes how particles with "color charge" (quarks and gluons) interact.

Path to this page

Leads to: Quantum Chromodynamics.