Chapter 10: Group Theory — From Galois to Gauge

Galois: A Life Measured in Theorems

Évariste Galois (1811–1832) crammed the equivalent of a mathematical lifetime into fewer than twenty years. Twice rejected by the École Polytechnique, twice imprisoned for political activity, he wrote out his most important discoveries the night before a duel from which he knew he would not return. He was shot at dawn on May 30, 1832, and died the following day. He was twenty years old.

His question was ancient: for a polynomial equation of degree \(n\), can we write its roots using radicals (square roots, cube roots, etc.)? Quadratic, cubic, and quartic equations all have such formulas. For five centuries, mathematicians had sought a general formula for degree five — the quintic. Galois proved no such formula exists.

To do it, he invented an entirely new kind of mathematical object: the group. He attached to each polynomial a group of permutations of its roots — now called the Galois group — and showed that the solvability of the equation depends on the structure of this group. A quintic is solvable by radicals if and only if its Galois group is solvable (a precise technical term he also invented). The general quintic is not.

The Abstract Group

Galois worked concretely with permutations. Over the following decades, mathematicians extracted the abstract skeleton. A group is a set \(G\) with a binary operation satisfying four axioms:

Notice what is absent: commutativity. Groups need not satisfy \(ab = ba\). Those that do are called abelian (after Niels Henrik Abel, who independently proved the quintic result). Non-abelian groups are precisely what make particle physics nontrivial.

Lie Groups: Continuous Symmetries

Galois groups are finite: they permute a finite set of roots. Sophus Lie (1842–1899) asked a deeper question: what about groups where the elements vary continuously, like rotations parameterized by an angle? In the 1870s he developed the theory of Lie groups — groups that are also smooth manifolds, where the group operation is differentiable.

The key examples are the rotation groups. In three dimensions, the group \(\text{SO}(3)\)consists of all \(3\times 3\) orthogonal matrices with determinant 1. It is three-dimensional as a manifold (parameterized by three Euler angles) and it is the symmetry group of the sphere. Its “double cover” \(\text{SU}(2)\) —\(2\times 2\) unitary matrices with determinant 1 — turns out to be the symmetry group of quantum spin.

Lie groups come equipped with a Lie algebra: the tangent space at the identity, which captures the infinitesimal generators of the symmetry. For \(\text{SU}(2)\), the three generators satisfy \([J_i, J_j] = i\epsilon_{ijk}J_k\) — the angular momentum commutation relations of quantum mechanics.

Klein's Erlangen Program (1872)

At age 23, Felix Klein delivered his inaugural lecture at the University of Erlangen with a radical thesis: geometry is group theory. Every geometry, Klein proposed, is defined by a group of transformations, and geometric properties are precisely those quantities that remain invariant under the group.

Klein's program unified all the seemingly unrelated geometries that had proliferated since the discovery of non-Euclidean geometry. It also pointed toward a future where physics itself would be organized around symmetry groups.

The Bridge to Physics

The 100-year gap between Galois (1832) and the first physical applications makes group theory's eventual triumph all the more striking. The applications came in waves: