2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationgroups, SO(3), SU(2)

What a group is — treating symmetries as "things that can be multiplied"

There are six ways to put an equilateral triangular plate back so that it fits exactly over its original place: three rotations by 0°, 120° and 240°, and three flips about an axis through a vertex. Doing two in succession gives one of the six again, and there is an operation that does nothing and operations that undo. A group is the word for exactly these properties and nothing more.

Label the vertices 0, 1, 2 anticlockwise, and write $e$ (do nothing), $r$ (rotate by 120°), $r^2$ (240°), and $s_k$ (flip about the axis through vertex k). The product $ab$ is read "first b, then a" (the direction of composition of maps $a\circ b$).

Swapping the order changes the result. That the order of multiplication matters is the protagonist all the way to the end of this book.

That each of the six elements appears exactly once in each row and each column reflects the fact that $ax=b$ has exactly one solution $x=a^{-1}b$, that is, that division is possible. Of the 36 ordered pairs, $ab\ne ba$ for 18 Computed.

Definition If there is a set $G$ and a rule (product) $(a,b)\mapsto ab$ that makes one element of $G$ from two elements, satisfying the following, $G$ is called a group.

1. Associative law: $(ab)c=a(bc)$.
2. Identity: there is $e$ with $ea=ae=a$ for all $a$.
3. Inverses: for each $a$ there is $a^{-1}$ with $aa^{-1}=a^{-1}a=e$.

If $ab=ba$ for every pair, the group is commutative (an abelian group); otherwise it is non-commutative. The "non-abelian" of "non-abelian gauge theory" has this meaning. The second half of the table below is groups of matrices, where $U^\dagger$ is the transpose with complex conjugation (1-01).

Groups of matrices are $n\ge2$. SO(2) is a commutative group of the same form as U(1).

For groups of matrices, compact means bounded and closed (the limit of a sequence of elements is again an element) in the space of entries Known. U(n) is bounded because each column has length 1, and $U^\dagger U=I$ is preserved in the limit. GL(n, ℝ) is not bounded. "Real parameters" is the number of real numbers that specify an element (the dimension of the group, 1-03); for SO(3) it is 2 for the direction of the axis and 1 for the angle.

A subgroup is a subset closed under products and inverses ($\{e,r,r^2\}\subset D_3$, SU(n) ⊂ U(n), and so on). A homomorphism is a map $\varphi:G\to H$ satisfying $\varphi(ab)=\varphi(a)\varphi(b)$, and the set of elements sent to the identity is called the kernel $\ker\varphi$.

The kernel of this homomorphism is $\mathrm{SU}(n)$. The kernel of $\theta\mapsto e^{i\theta}$ is $2\pi\mathbb{Z}$. The map sending the rotations of D₃ to +1 and the flips to −1 is also a homomorphism. A subgroup $N$ is normal if $gNg^{-1}=N$ for all $g$, and a kernel is always normal. Then the quotient group $G/N$, which identifies $g$ and $gn$ ($n\in N$), can be formed, and for a surjective homomorphism $\varphi$ the following holds (the homomorphism theorem) Known.

Example: $\mathbb{R}/2\pi\mathbb{Z}\cong\mathrm{U}(1)$ (an angle is a real number with differences of integer multiples of 2π forgotten). The centre $Z(G)$ (the elements that commute with every element) is also a normal subgroup, and the centre of SU(n) is the n elements $\omega I$ ($\omega^n=1$), of the same form as $\mathbb{Z}_n$ Known. It appears again in 7-02, which treats confinement.

A rotation in three dimensions is a linear map that preserves the inner product ($R^{\mathsf T}R=I$) and makes no mirror image ($\det R=1$).

Every rotation has an axis

Every element of SO(3) has a line that does not move (Euler's rotation theorem) Known. A real orthogonal matrix is unitary as a complex matrix (1-01), so its eigenvalues have absolute value 1, and their product is $\det R=1$. They are roots of a cubic with real coefficients, so complex eigenvalues come in pairs $\lambda,\bar\lambda$ (product $|\lambda|^2=1$), and the remaining real eigenvalue is 1. If all are real, they are three values ±1 with product 1, so again they include 1. The eigenvector of that eigenvalue is the axis. Odd dimension matters here: some rotations in four dimensions have no fixed point other than the origin. With axis $n$ and angle $\theta$ it has the following form (Rodrigues' formula) Known.

The trace is $\operatorname{tr}R(\theta,n)=1+2\cos\theta$, from which the angle can be read.

The mismatch from the order of rotations

Rotate by the same angle θ about the x axis and the y axis. "x first" is $A=R_y(\theta)R_x(\theta)$, "y first" is $B=R_x(\theta)R_y(\theta)$. The mismatch $D=AB^{-1}$ is also a rotation, so we measure the difference by its angle φ.

For small θ, $\varphi\approx\theta^2$, and the axis of D is almost in the −z direction ((−0.087, 0.087, −0.992) at θ = 10°) Computed. The mismatch is second order in θ, and to first order rotations look commutative. Extracting this gives the commutator $[X,Y]=XY-YX$ of 1-03. At θ = 180° both are diagonal matrices and commute.

Next is a group of complex 2×2 matrices.

The two columns are orthonormal. If the first column is $(a,b)$, then $|a|^2+|b|^2=1$, and the second column is a vector of length 1 orthogonal to it, $e^{i\chi}(-\bar b,\bar a)$, with $\det U=e^{i\chi}=1$.

The real and imaginary parts of a, b form a point on the unit sphere of $\mathbb{R}^4$, and as a space, SU(2) is the three-dimensional sphere $S^3$ Known. Using the Pauli matrices (1-04), every element can be written with a unit vector $n$ and a real number θ.

Writing $n\cdot\sigma=n_1\sigma_1+n_2\sigma_2+n_3\sigma_3$, it has the following form.

In components, $a=\cos\frac\theta2-in_3\sin\frac\theta2$, $b=(n_2-in_1)\sin\frac\theta2$. The equality on the right follows from $(n\cdot\sigma)^2=I$ (from $\sigma_j\sigma_k+\sigma_k\sigma_j=2\delta_{jk}I$ and $|n|=1$; how to check it is in 1-04) and the matrix exponential (1-01). At θ = 2π it is $U=-I$ whatever n is, and only at θ = 4π does it become $U=I$. This half angle is the protagonist of the next section.

Identify vectors of $\mathbb{R}^3$ with traceless Hermitian matrices.

$UXU^\dagger$ is also Hermitian and traceless, so it is the matrix of some vector $x'$. This fixes a linear map $\rho(U):x\mapsto x'$ Known.

1. It preserves length: $|x'|=|x|$ follows from $\det(UXU^\dagger)=\det X$. SU(2) is connected and ρ is continuous, so the determinant cannot move from +1, and $\rho(U)\in\mathrm{SO}(3)$.
2. It is a homomorphism: $\rho(UV)=\rho(U)\rho(V)$ follows from $(UV)X(UV)^\dagger=U(VXV^\dagger)U^\dagger$.
3. Its kernel is ±I: if $\rho(U)=I$, then U commutes with σ₃ (it is diagonal) and with σ₁ (its diagonal entries are equal), so $U=\lambda I$, $\det U=\lambda^2=1$.
4. It is surjective, and the angle is halved: $\rho(U(\theta,n))=R(\theta,n)$. Every rotation has the form $R(\theta,n)$ (the axis theorem of 04), so $U(\theta,n)$ is its preimage, and the map is surjective.

A simple case of 4: if n is the z axis, $U=\mathrm{diag}(e^{-i\theta/2},e^{i\theta/2})$ gives $U\sigma_1U^\dagger=\cos\theta\,\sigma_1+\sin\theta\,\sigma_2$. The two half angles multiply together to give the angle θ. From the homomorphism theorem,

Two elements $\pm U$ correspond to one rotation, so it is called a double cover. A 360° rotation goes in SU(2) only from $I$ to $-I$, and returning takes 720°.

Definition A (finite-dimensional) representation of a group G is a pair of a vector space V and a homomorphism $\pi:G\to GL(V)$ (continuous, for groups of matrices); that is, an assignment of an invertible matrix to each element such that $\pi(gh)=\pi(g)\pi(h)$ ($\dim V$ is the dimension). Examples: the trivial representation (everything to 1), writing D₃ by rotations and reflections of the plane, the action of SO(3) on $\mathbb{R}^3$ and of SU(2) on $\mathbb{C}^2$ (the defining representation), and the 3-dimensional representation ρ.

Representations of U(1) are labelled by integers

For $e^{i\theta}\mapsto e^{iq\theta}$ to be a representation of U(1), since θ and θ + 2π are the same element, $e^{2\pi iq}=1$, that is, q is an integer. These exhaust the continuous 1-dimensional unitary representations Known. For ℝ, $x\mapsto e^{iqx}$ is a representation for any real q, so what makes the labels integers is compactness. The physical argument that if the gauge group of electromagnetism (2-03) is U(1) then charges line up as integer multiples of a unit rests on this fact (the title of C. N. Yang 1970 states this correspondence) Physics.

Irreducible and completely reducible

A subspace W with $\pi(g)W\subset W$ for all g is called invariant, and a representation with no invariant subspaces other than 0 and V is called irreducible. The action of SO(3) on $\mathbb{R}^3$ is irreducible; representing D₃ by permutations of the coordinates of $\mathbb{R}^3$ leaves the line (1, 1, 1) invariant, so it is not irreducible. A finite-dimensional representation of a compact group splits into a direct sum of irreducible representations (complete reducibility) Known. If the inner product is averaged over the whole group, every $\pi(g)$ preserves that inner product, and the orthogonal complement of an invariant subspace is also invariant. For a finite group the average is the sum over all elements divided by the number of elements. For a compact group, the same can be done with an invariant integral on the group (its existence is known).

Representations of SU(2) and the sign at 360°

SU(2) has exactly one finite-dimensional complex irreducible representation in each dimension, up to equivalence (change of basis) Known. The one of dimension m + 1 is $\Pi_m$, acting by $(\Pi_m(U)f)(z)=f(U^{-1}z)$ on the space of homogeneous polynomials of degree m in 2 variables, and in physics $j=m/2$ is called the spin. $-I$ acts as $f(-z)=(-1)^mf(z)$, so the ones that can be representations of SO(3) are only those with m even (odd dimension).

The character χ is the trace of the representation matrix, measured at $U(\theta,e_3)$. At 720° every one returns to the value of the dimension Computed. For half-integer spin the sign flips at 360° — the $-I$ of Figure 3 appears, on the side of representations, as this sign.

The problem statement (0-03) begins with conditions on the group.

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R⁴ and has a mass gap Δ > 0.

Jaffe and Witten's description of the problem says "at the level of classical theory, replace the gauge group U(1) of electromagnetism by a compact gauge group G". In a word, the gauge group is the group of symmetries that can be applied independently at each point of spacetime. To be exact, a choice of one element g(x) of G at each point x is a gauge transformation, and G is the group in which it takes values (3-01).

Simple. As an abstract group, "simple" means that the only normal subgroups are {e} and the group itself, and in this sense SU(2) is not simple (its centre {±I} is a normal subgroup). In the terminology of Lie groups the meaning is different: a connected non-commutative group is called "simple" if its Lie algebra is simple (1-03). The description of the problem does not define simple, but in the sense of abstract groups every SU(N) (N ≥ 2) would be excluded (centre $\mathbb{Z}_N$), so this book reads it in the sense of Lie groups. SU(N) (N ≥ 2) is simple in this sense (the list is in 1-04) Known, and the commutative U(1) is not included.

Compact. The same description states that with a compact gauge group and a compactified spacetime, Wilson's lattice approximation reduces the functional integral to a finite-dimensional integral. An element of G is put on each edge of the lattice and integrated over the whole of G (5-01), so it matters that the size of G is finite.

Representations. The same description says that in the theory of the strong force (QCD) the gauge group is SU(3), and quarks transform in the fundamental representation (this chapter's defining representation). The gauge field takes values in the Lie algebra of G. Under transformations that apply the same element at each point it transforms in the adjoint representation (1-04), and under general gauge transformations an extra term is added (3-02). The field strength transforms in the adjoint representation. The pure Yang–Mills theory the problem treats has no quarks, only the gauge field (3-07). Lattice models with G a finite group are treated in 5-10.

Next is 1-03 Lie groups and Lie algebras: the "second-order mismatch" is extracted as the commutator of the infinitesimal part of the group.

What this chapter can say

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