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Compactness and normalized invariant form for SU(2)

Statement

Assume the axiom of countable choice. Let $G = S U (2) = {U \in M_{2} (C) : U^{*} U = I_{2}, \det U = 1}$ with its real Lie-group structure. Define the Pauli matrices $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ , and $σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ , and put $T_{a} = - \frac{i}{2} σ_{a}$ for $a \in {1, 2, 3}$ . Then $G$ is compact and noncommutative, its real Lie algebra has basis $T_{1}, T_{2}, T_{3}$ with $[T_{a}, T_{b}] = \sum_{c = 1}^{3} ϵ_{a b c} T_{c}$ , and the real bilinear form $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ on $s u (2)$ is positive definite, symmetric, and invariant under ${Ad}_{g} X = g X g^{- 1}$ , with $κ (T_{a}, T_{b}) = δ_{a b}$ .

Statement references

David Tong, Gauge Theory, Chapter 2, Yang-Mills Theory

Matthijs Vákár, Principal Bundles and Gauge Theories

Facts & Assumptions

Given: The matrix space $M_{2} (C)$ with its underlying eight-dimensional real Euclidean topology and the displayed Pauli matrices.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every countable family of nonempty sets has a choice function; it is inherited by the supplied Lie-group tangent-bracket construction.

[F1]

Unitary and special unitary Lie groups: $S U (2)$ is a finite-dimensional real Lie group and its Lie algebra is the real vector space of traceless skew-Hermitian $2 \times 2$ matrices.

[F2]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: a subset of Euclidean $R^{8}$ is compact exactly when it is closed and bounded.

Proof

1.1A1F1F2algebra

The entries of $U^{*} U - I_{2}$ and $\det U - 1$ are continuous real polynomial functions of the eight real coordinates of $U$ . Their common zero set $S U (2)$ is therefore closed. Since $U^{*} U = I_{2}$ , each column of $U$ has Euclidean norm one, so every matrix entry has modulus at most one; hence $S U (2)$ is bounded in $R^{8}$ . By [F2], $S U (2)$ is compact. Its real Lie-group structure is supplied by [F1].

1.2A1algebra

The matrices $g = diag (i, - i)$ and $h = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ satisfy $g^{*} g = h^{*} h = I_{2}$ and $\det g = \det h = 1$ , so both lie in $S U (2)$ . Direct multiplication gives $g h = (\begin{array}{cc} 0 & i \\ i & 0 \end{array})$ and $h g = (\begin{array}{cc} 0 & - i \\ - i & 0 \end{array})$ ; therefore $g h \neq h g$ and the group is noncommutative.

1.3F1algebra

By [F1], every element of $s u (2)$ has the form $(\begin{array}{cc} i α & z \\ - \overline{z} & - i α \end{array})$ with $α \in R$ and $z \in C$ . Thus $T_{1}, T_{2}, T_{3}$ form a real basis. Multiplying the displayed matrices gives $[T_{1}, T_{2}] = T_{3}$ , $[T_{2}, T_{3}] = T_{1}$ , and $[T_{3}, T_{1}] = T_{2}$ ; antisymmetry and bilinearity give the stated $ϵ_{a b c}$ formula.

2.1step 1.3algebra∎

The same finite matrix products give $- 2 {Tr}_{2} (T_{a} T_{b}) = δ_{a b}$ . Since each $X \in s u (2)$ has real coordinates in this basis, $κ (X, X)$ is the sum of their squares, and $κ$ is positive definite; trace cyclicity gives symmetry. For every $g \in S U (2)$ , $κ ({Ad}_{g} X, {Ad}_{g} Y) = - 2 {Tr}_{2} (g X Y g^{- 1}) = - 2 {Tr}_{2} (X Y) = κ (X, Y)$ , so the form is $Ad$ -invariant.

Compactness and normalized invariant form for SU(2)

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources