Connectedness, Covers, and the Fundamental Group

The global topology of a Lie group — how many connected pieces it has, and whether its loops contract — is invisible to the Lie algebra but decisive for physics. It is why $SU(2)$ rather than $SO(3)$, and $SL(2,\mathbb{C})$ rather than the Lorentz group, are the symmetry groups that act in quantum mechanics. This page develops connectedness, the universal cover, and the link between ${\pi }_1(G)$ and projective representations. It builds on lie-groups.md and feeds QFT/preliminaries.md.

Connectedness and discrete components

A Lie group can have several connected components. The component containing the identity is the identity component $G^0$ — a normal subgroup, and itself a Lie group of the same dimension. The set of components is the component group $${\pi }_0(G)=G/G^0,$$ a discrete group. The Lie algebra only sees $G^0$ (it is the tangent space at the identity), so groups differing only in their components share an algebra.

Example — the Lorentz group $O(1,3)$ has four components, sorted by the signs of $\det \Lambda =\pm 1$ (proper/improper) and ${\Lambda }^0{}_0\gtrless 0$ (orthochronous/non-orthochronous). The identity component is the proper orthochronous subgroup $S{O}^{+}(1,3)={\mathcal{L}}_{+}^{\uparrow }$; the other three are reached from it by parity $P$, time-reversal $T$, and $PT$. See QFT/preliminaries.md § The Lorentz group.

Simple connectedness and ${\pi }_1$

A connected space is simply connected if every loop contracts to a point — equivalently its fundamental group ${\pi }_1(G)$ (loops up to continuous deformation, under concatenation) is trivial. For a connected Lie group ${\pi }_1(G)$ is always abelian and discrete, and sits at the centre of the universal cover (below).

Examples:

• $SU(2)\cong S^3$ is simply connected: ${\pi }_1=\{e\}$.
• $SO(3)\cong {\mathbb{RP}}^3$ has ${\pi }_1(SO(3))={\mathbb{Z}}_2$ — there are two classes of loops, contractible and not. Physically: a $2\pi$ rotation is a non-contractible loop, a $4\pi$ rotation contracts (the "belt trick").
• $U(1)\cong S^1$ has ${\pi }_1=\mathbb{Z}$ (winding number).
• $SO(2)\cong S^1$: likewise ${\pi }_1=\mathbb{Z}$.

The universal cover

Every connected Lie group $G$ has a universal cover $G$: the unique simply connected Lie group with the same Lie algebra, together with a surjective covering homomorphism $$\pi :G\to G,\ker \pi \cong {\pi }_1(G),$$ whose kernel is a discrete central subgroup isomorphic to ${\pi }_1(G)$. Thus all connected groups with a given algebra $\mathfrak{g}$ are quotients $\tilde{G}/Z$ of the one simply connected group by discrete central subgroups $Z$ — the exact statement of Lie's third theorem (see lie-groups.md § The Lie correspondence).

Projective representations and why covers matter

Quantum mechanics represents symmetries only up to phase — states are rays — so the relevant notion is a projective representation $\rho (g_1)\rho (g_2)=\omega (g_1,g_2)\rho (g_1{g}_2)$ (see reps-basics.md § Projective representations). The central theorem:

Projective reps of $G$ = ordinary reps of its universal cover $\tilde{G}$ (for connected $G$, and modulo genuine algebra-level central extensions).

So the true symmetry group acting on a quantum Hilbert space is $\tilde{G}$, not $G$. Consequences:

• Rotational symmetry $SO(3)$ is realised through $SU(2)$, whose extra irreps are the half-integer spins — a $2\pi$ rotation multiplies a spin-$\frac{1}{2}$ state by $-1$, tracking the non-contractible loop in ${\pi }_1(SO(3))={\mathbb{Z}}_2$.
• Lorentz symmetry $S{O}^{+}(1,3)$ is realised through $SL(2,\mathbb{C})$, whose spinor reps are the building blocks of relativistic fermions. See physics-bridge.md.
• For the Poincaré group one likewise passes to the universal cover to obtain the physically relevant projective reps; the little-group analysis in QFT/preliminaries.md § Little groups uses exactly this.

The phases $\omega$ are classified by the group cohomology $H^2(G;U(1))$; the same extension mechanism appears for finite groups in products.md § The extension problem and physically as anomalies.

References

• Hall, Lie Groups, Lie Algebras, and Representations, Ch. 1, 13.
• Weinberg, The Quantum Theory of Fields, Vol. 1, §2.7 (projective reps and covers).
• Nakahara, Geometry, Topology and Physics, Ch. 4 (${\pi }_1$, covering spaces).