PSL(2,C)–proper Lorentz isomorphism

Statement

The spin covering descends to an isomorphism

\boxed{\; PSL(2,\mathbb C)_{\mathbb R} \xrightarrow{\;\sim\;} SO^+(1,3) \;}

in the category of real Lie groups. Explicitly, a class $[A]$ acts on Hermitian $2\times2$ matrices by $X\mapsto AXA^\dagger$ .

Why the qualifiers matter

The map $SL(2,\mathbb C)\to SO^+(1,3)$ itself is not an isomorphism: it has kernel $\{\pm I\}$ . Quotienting by this kernel gives $PSL(2,\mathbb C)$ . The subscript $\mathbb R$ records that its complex manifold structure has been forgotten; $SO^+(1,3)$ is not a complex Lie group in this statement.

The target is also only the identity component of $O(1,3)$ . Spatial parity and time reversal lie in other components and are not represented by $PSL(2,\mathbb C)$ .

Compatible boundary action

Projectivizing the future null cone identifies the celestial sphere with $\mathbb{CP}^1$ . Under that identification, this Lorentz action becomes the Möbius action of $PSL(2,\mathbb C)$ .

References

Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §§1.2–1.3. Publisher record.
Gregory L. Naber, The Geometry of Minkowski Spacetime, 2nd ed., Springer, 2012, Chapter 2. Publisher record.