Hermitian matrix model of Minkowski space

Core idea

Identify $v=(t,x,y,z)\in\mathbb R^{1,3}$ with the Hermitian matrix

X(v)= \begin{pmatrix} t+z&x-iy\\ x+iy&t-z \end{pmatrix}.

Then

\det X(v)=t^2-x^2-y^2-z^2.

For the $(-+++)$ Minkowski quadratic form fixed at Minkowski vector space, this says

\det X(v)=-q(v).

Thus the determinant is the opposite of the quadratic form used in this collection; it is not being identified with $q$ itself.

The SL(2,C) action

For $A\in SL(2,\mathbb C)$ , define

X\longmapsto AXA^\dagger,

where $A^\dagger=\overline A^{\mathsf T}$ . This map preserves Hermitian matrices and

\det(AXA^\dagger)=\det A\,\det X\,\overline{\det A}=\det X.

It therefore preserves $q$ and determines a real-linear Lorentz transformation. Composition of matrices makes this a homomorphism from $SL(2,\mathbb C)_{\mathbb R}$ into $SO^+(1,3)$ .

Null vectors and rank one

A nonzero future null vector corresponds to a nonzero positive-semidefinite rank-one Hermitian matrix. Such a matrix has the form $zz^\dagger$ for $z\in\mathbb C^2\setminus\{0\}$ , and rescaling $z$ by a nonzero complex scalar preserves its null ray. This gives the celestial sphere its $\mathbb{CP}^1$ description.

References

Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §§1.2–1.3. Publisher record.
Rolf Berndt, An Introduction to Symplectic Geometry, AMS, 2001, Appendix A. Publisher record.