2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationPauli, Gell-Mann, adjoint representation

SU(2) and SU(3) — seeing the matrices in which the gauge field takes values, at a size you can handle

The gauge field has, at each point of spacetime and for each direction, one element of the Lie algebra $\mathfrak{g}$ of the gauge group G (3-02). For G = U(1) of electromagnetism there is one component. For SU(N) the Lie algebra becomes a space of matrices.

The count: the diagonal entries are purely imaginary, N of them, and the entries above the diagonal are complex, $N(N-1)/2$ of them. From the real degrees of freedom $N^2$, the trace-zero condition removes one. For SU(3) this is 8, which is why there are 8 kinds of gluon (PDG).

In physics one writes $X = -i\theta^a T_a$ ($T_a$ Hermitian, repeated indices summed). The $U = \exp(-\tfrac{i\theta}{2}\,n\cdot\sigma)$ of 1-02 is this form with $T_a = \sigma_a/2$. This book normalises $\operatorname{tr}(T_aT_b) = \tfrac12\delta_{ab}$, and writes the structure constants of 1-03 like this.

$f_{abc}$ is real and totally antisymmetric. The PDG review of QCD uses the same convention Known.

The properties of the Pauli matrices (1-02 §05) are packed into the following identity. $\varepsilon_{abc}$ is the symbol with $\varepsilon_{123} = 1$ whose sign changes when two indices are swapped.

Taking the trace gives $\operatorname{tr}(\sigma_a\sigma_b) = 2\delta_{ab}$; swapping a and b and subtracting gives $[\sigma_a, \sigma_b] = 2i\varepsilon_{abc}\sigma_c$. With $T_a = \sigma_a/2$ this takes the form of formula (2), and $f_{abc} = \varepsilon_{abc}$. Example: $T_1T_2 = \tfrac{i}{4}\sigma_3$, $T_2T_1 = -\tfrac{i}{4}\sigma_3$, so $[T_1, T_2] = iT_3$.

Rotations have the same relation. $(L_a)_{bc} = -i\varepsilon_{abc}$ satisfies $[L_a, L_b] = i\varepsilon_{abc}L_c$, and $e^{-i\theta L_3}$ is the rotation by angle θ about the z axis Computed. su(2) and so(3) are the same Lie algebra. The Lie algebra sees only the neighbourhood of the identity, so the fact that SU(2) covers SO(3) twice (1-02 §06) does not appear here.

A representation of su(2) is a triple of Hermitian matrices $J_1, J_2, J_3$ satisfying formula (2) of $f = \varepsilon$. We build matrices that go up and down one step at a time.

If $J_3 v = mv$ then $J_3(J_\pm v) = (m \pm 1)J_\pm v$, so $J_\pm$ moves eigenvalues by 1. $J^2 = J_1^2 + J_2^2 + J_3^2 = J_-J_+ + J_3^2 + J_3$ commutes with every $J_a$. The dimension is finite, so there is a top $v$ ($J_+v = 0$); if its eigenvalue is j, then $J^2 v = j(j+1)v$. Going down gives $\lVert J_- v_m\rVert^2 = (j+m)(j-m+1)\lVert v_m\rVert^2$, and this coefficient cannot be negative, so the sequence must stop exactly at $m = -j$. Going from j to −j in integer steps, 2j is one of 0, 1, 2, … Known.

The dimension is $2j+1$, and there is one irreducible representation in each dimension (the $\Pi_m$ of 1-02 has m = 2j). Example: j = 1/2 is $J_a = \sigma_a/2$. j = 1, after a change of basis, is the $L_a$ of §02.

What plays the role of the Pauli matrices in su(3) is the 8 Gell-Mann matrices. All are Hermitian and traceless, with $\operatorname{tr}(\lambda_a\lambda_b) = 2\delta_{ab}$.

$\lambda_{4,5}$ does in entries 1 and 3, and $\lambda_{6,7}$ in entries 2 and 3, the same as the Pauli matrices, and $\lambda_8$ is a diagonal matrix orthogonal to $\lambda_3$. Setting $T_a = \lambda_a/2$ and extracting with $f_{abc} = -2i\operatorname{tr}([T_a, T_b]T_c)$ gives the following table.

Table 2: the non-zero ones with a < b < c (the others are fixed by antisymmetry or are 0). The 64 commutators are reproduced, and the deviation from the Jacobi identity is at most 3.3×10⁻¹⁶ Computed.

Inside there are three su(2). $\{T_1, T_2, T_3\}$, $\{T_4, T_5, \tfrac12 T_3 + \tfrac{\sqrt3}{2}T_8\}$ and $\{T_6, T_7, -\tfrac12 T_3 + \tfrac{\sqrt3}{2}T_8\}$ each satisfy formula (2) of $f = \varepsilon$ Computed. They are the SU(2) for each pair of entries (1,2), (1,3), (2,3).

$T_3$ and $T_8$ commute. The largest number of linearly independent mutually commuting elements one can take (the dimension of the space they span) is called the rank; su(3) has rank 2, and su(N) rank N − 1 Known. Indeed, the elements of su(3) commuting with both $T_3$ and $T_8$ are only the linear combinations of these two, and in su(N) too the elements commuting with the N − 1 diagonal ones were only their linear combinations (N = 2–6) Computed.

In a representation, $T_3$ and $T_8$ are commuting Hermitian matrices, so they can be diagonalised simultaneously, and the pair of eigenvalues $(\mu_3, \mu_8)$ of a common eigenvector is called a weight (the two-dimensional version of the m of su(2)). In the defining representation $\mathbf{3}$ one just reads off the diagonal entries.

The three points form an equilateral triangle, and $\bar{\mathbf{3}}$ (the complex conjugate) is −1 times it. The matrix units $E_{ij}$ (only the (i, j) entry is 1) satisfy $[H, E_{ij}] = (h_i - h_j)E_{ij}$ for diagonal H, and are ladders that move weights by $\mu(e_i) - \mu(e_j)$. These 6 differences are called roots, and form a regular hexagon of side 1. $E_{12} = T_1 + iT_2$ is the $J_+$ of §03. $E_{ij}$, like $J_\pm$, is a linear combination of elements of su(3) with complex coefficients, not an element of su(3) itself, but on representations it acts in the same way.

Irreducible representations are labelled by pairs of non-negative integers (p, q) (highest weight $p\,\mu(e_1) - q\,\mu(e_3)$), and the dimension is given by the following formula Known. (1, 0) is $\mathbf{3}$, (0, 1) is $\bar{\mathbf{3}}$, and (1, 1) is the adjoint representation.

The rule visible in the figure: the multiplicity increases by 1 from the outer shell inwards, and once the shells have become triangles the multiplicity stops increasing and stays constant to the inside. All 16 computed cases had this form Computed.

Detour: the SU(3) of flavour and the SU(3) of colour

A group acts on its own Lie algebra by conjugation. This is the adjoint representation, and its derivative is the commutator (1-03).

The matrices on the far right satisfy formula (2) again (equivalent to the Jacobi identity). For SU(2), $-i\varepsilon_{abc}$ is the $L_a$ of §02, so the map SU(2) → SO(3) of 1-02 is the adjoint representation itself, corresponding to j = 1.

Under a gauge transformation g that does not depend on space, the gauge field changes as $A \mapsto gAg^{-1}$ (the general form is in 3-02). For U(1), $gAg^{-1} = A$ and the photon carries no charge, but for SU(N) the gauge field itself is charged and interacts with itself. In the PDG too, gluons are in the adjoint representation and quarks in the fundamental representation Known.

$\sum_a T_aT_a$ commutes with every $T_b$ (the same as the $J^2$ of §03). By Schur's lemma (for an irreducible complex representation, a matrix that commutes with all representation matrices is a constant multiple of the identity), on an irreducible representation it is a constant multiple of the identity, and the constant is called the Casimir Known.

Table 4: computed with generalised Gell-Mann matrices Computed. The bottom row is the ratio of the Killing form $B(X, Y) = \operatorname{tr}(\operatorname{ad}_X\operatorname{ad}_Y)$ to $\operatorname{tr}(XY)$, which was 2N. For anti-Hermitian $X \ne 0$, $\operatorname{tr}(X^2) \lt 0$, so B is negative definite, and compactness shows in the sign.

That the Tr of the action is "an invariant quadratic form on the Lie algebra of G" was seen in 1-03 §09. On a compact simple Lie algebra (whose complexification is also simple), the invariant symmetric bilinear form is unique up to a constant multiple Known, so the difference between B and tr in Table 4 is also absorbed into g. $C_A$ enters the coefficient of asymptotic freedom. The PDG one-loop coefficient $b_0 = (11C_A - 4n_fT_R)/(12\pi)$ is $11N/(12\pi)$ for pure Yang–Mills without quarks Known. The N of "11N" in 7-01 and 4-06 is $C_A$.

The set of elements that commute with every element is called the centre. For SU(N) it is the scalar matrices of determinant 1 Known.

An element of the centre commutes with all representation matrices, so on an irreducible representation it acts as a scalar by Schur's lemma (§06). For SU(3), comparing diagonal entries gives $\omega I = \exp(4\pi i\,T_8/\sqrt3)$, and a vector of weight $(\mu_3, \mu_8)$ is multiplied by $e^{4\pi i\mu_8/\sqrt3}$. The highest weight of V(p, q) is $\mu_8 = (p+2q)/(2\sqrt3)$, so the whole representation is multiplied by $\omega^{p+2q} = \omega^{p-q}$. This value held at every weight point (216 points in all) of the 16 representations with p, q ≤ 3 Computed. $k = p - q \bmod 3$ is called the N-ality. In the column of Table 3, $\mathbf{3}$ is 1, $\bar{\mathbf{3}}$ is 2, and the adjoint is 0.

It is 0 for the adjoint because $zXz^{-1} = X$ for a scalar matrix z. The gauge field does not feel the centre. So gluons alone cannot screen charges with non-zero N-ality Physics, and the centre plays the leading role in arguments about confinement (7-02, 5-10). Holland et al. (2003) argue that for $G_2$, whose centre is trivial (§09), charges are screened by gluons and the string can break Physics.

The roots of §05 exist for every compact simple Lie algebra. For rank r, the roots line up in an r-dimensional space as the simultaneous eigenvalues of r mutually commuting elements. The inner product $(\alpha, \beta)$ is, as in §05, the inner product of the coordinates when the generators are made orthonormal for the invariant quadratic form (for su(3) it is the ordinary inner product on the $(\mu_3, \mu_8)$ plane, and the roots have length 1).

The heart of the classification is the "2j is an integer" of §03. For each root α, $E_\alpha$, $E_{-\alpha}$ and their commutator form an su(2), and seeing another root β from this su(2), the quantity corresponding to m is $(\alpha,\beta)/(\alpha,\alpha)$, which is an integer multiple of 1/2 (including integers) Known. In su(3), with α = (1, 0) ($E_{12}$; the su(2) is $\{T_1, T_2, T_3\}$) and β = $(\tfrac12, \tfrac{\sqrt3}{2})$ ($E_{13}$), m = 1/2. Multiplying the two directions together gives the following for the angle θ between them.

This is because the left-hand side is a product of integers, and is less than 4 if $\beta \ne \pm\alpha$. The ratio of the two factors is the square of the ratio of lengths, so the ratio of the lengths of two non-orthogonal roots is one of 1, √2, √3 (because the pairs of integers with product 1, 2 or 3 are limited).

There are many roots, so we choose representatives. Draw a line through the origin containing no root (a hyperplane, for rank 3 or more), and call the roots on one side the positive roots. The positive roots that cannot be written as a sum of two positive roots are the simple roots; their number equals the rank, and the angles between them are at least 90° Known. So the angles between simple roots take four values: 90°, 120°, 135°, 150°.

What recovers the whole from the simple roots is reflection. The reflection in a root α is the folding across the line (hyperplane) perpendicular to α.

Reflections take roots to roots, and repeatedly applying the reflections in the simple roots to the simple roots gives all the roots Known. The coefficients in (12) are the integers of (11), so every root is a sum of simple roots with integer coefficients. The coefficients are either all at least 0 or all at most 0 Known.

In rank 2 there are $A_1\times A_1$ (4 roots; it splits into two su(2) and is not simple), $A_2$ (6 roots; the regular hexagon of §05 itself), $B_2$ (8 roots) and $G_2$ (12 roots). The dimension is rank + number of roots (for su(3), the 2 diagonal directions and $E_{ij}$ for each of the 6 roots, giving 8). The coefficients in Figure 3 were also all at least 0 or all at most 0 Computed.

The problem statement (0-03) asks for "any compact simple gauge group G". Pushing the restrictions of §08 through for rank 3 and above, the candidate Lie algebras are exhausted by four series $A_n = \mathfrak{su}(n+1)$, $B_n = \mathfrak{so}(2n+1)$, $C_n = \mathfrak{sp}(n)$, $D_n = \mathfrak{so}(2n)$ and five exceptions $G_2, F_4, E_6, E_7, E_8$ Known. According to Baez's survey, Killing announced the classification in a letter of 1887 and published it in papers over the following three years (he counted 6 exceptions). Cartan's thesis of 1894 constructed them all and found that the two of dimension 52 are isomorphic, bringing the exceptions to 5.

Table 6: the series agree with the formulas for n ≤ 8 Computed. Spin(m) is the simply connected group that covers SO(m) twice, and Sp(n) is the group of unitary matrices with quaternion entries. The matrix of the integers $2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j)$ between simple roots (the factors of formula (11)) is called the Cartan matrix, and the order of the centre is its determinant Known. The form of the centre (ℤ₄ or ℤ₂×ℤ₂) was also computed from this matrix. $h^\vee$ (the dual Coxeter number) is only cited as a table value in this chapter.

For SU(N), $h^\vee = N$, agreeing with the $C_A = N$ of §06. If the squared length of the long roots is set to 1 (for SU(N) this corresponds to $\operatorname{tr}(T_aT_b) = \tfrac12\delta_{ab}$, the same as the roots having length 1 in §05), then $C_A = h^\vee$ for general G as well Known. In the normalisation common in mathematics, "squared length 2", the adjoint Casimir becomes $2h^\vee$. Comparing the two normalisations for SU(3) gave 3 and 6 Computed. The G of the problem statement may be any row of this table, and the group enters the coefficients of perturbation theory through numbers like these. Groups divided by the centre, such as SU(2)/ℤ₂ = SO(3), have the same Lie algebra, and the difference appears in the topology of bundles (3-05 §03).

Next is 1-05 Manifolds and tangent spaces: the language for putting this chapter's matrices at each point of spacetime.

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