quantum field theory - How is the invariant speed of light encoded in $SL(2, mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to describe spin $\frac12$ particles.

Therefore, I was wondering how the invariant speed of light is encoded in $SL(2, \mathbb C)$?

This curious fact of nature, is encoded in $SO(3,1)$, because this is exactly the group that leaves the Minkowski metric invariant. In contrast, $SL(2, \mathbb C)$ is just the group of complex $(2 \times 2)$ matrices with unit determinant.

Answer

In the spinorial representation of a Lorentz transformation, we represent an event in spacetime as an element of the 4D vector space of Hermitian $2\times2$ matrices:

$$X = t\,\mathrm{id} + x\,\sigma_x+y\,\sigma_y+z\,\sigma_z\tag{1}$$

where the $\sigma_j$ are, as always, the Pauli matrices and $(t,\,x,\,y,\,z)$ the four spacetime co-ordinates.

A member $\zeta\in SL(2,\,\mathbb{C})$ of the double cover of $SO(1,\,3)$ acts on such an event by the so-called spinor map:

$$X\mapsto \zeta^\dagger\,X\,\zeta\tag{2}$$

so we now seek to encode the line interval invariance into (2); in terms of the spacetime event $X$.

Exercise: Check that the Minkowskian length of $X$ is, indeed encoded as $\det X$.

Thus, from (2), the invariance of the spacetime interval length under the action of $\zeta$ is

$$\det(\zeta^\dagger\,X\,\zeta) = |\det\zeta|^2 \det X = \det X\tag{3}$$

and the fulfilling of (3) for all Hermitian $X$ is a necessary and sufficient condition for $\zeta$ to leave the spacetime interval invariant. In particular, unimodularity of $\zeta$ is sufficient (but not necessary) for this conservation.

This answers your question, but take heed that the whole set of matrices that conserve the interval is precisely $U(1)\times SL(2\,\mathbb{C})$, i.e. the set of matrices of the form $e^{i\,\phi}\,\zeta;\,\phi\in\mathbb{R}$, where $\zeta$ is unimodular. But note that the phase factor makes no difference to the spinor map (2); therefore, the group of spacetime interval conserving matrices $U(1)\times SL(2\,\mathbb{C})$ breaks up into equivalence classes of spinor maps, with precisely one member of $PSL(2,\,\mathbb{C})\cong SO^+(1,\,3)$ for each spinor map. The double cover $SL(2,\,\mathbb{C})$ comprises two members for each distinct spinor map, i.e. pairs of the form $\pm\zeta$.