Dirac spinor as null vectors

In this paper, on page 9, the authors show that a spinor is equivalent to a null vector with a bit of extra structure (just one real parameter I think?):

https://arxiv.org/abs/1312.3824

They then go on to explain that a Dirac spinor contains two spinors and therefore two null vectors, and we can combine these into the 4-velocity and 4-spin of the particle by taking their sum and difference.

But what if we stick with the interpretation of two null vectors, ie. two rays of light? Because of the variation of the wavefunction over space, the null vectors would not tend to form straight lines, whereas in special relativity light always travels in straight lines. And of course the Dirac equation was made for special relativity. But what if we suppose that spacetime is actually curved, even around a free electron, and the wavefunction, through these null vectors, is giving us the clue to that curvature? Can we distort the manifold to the point where these apparently curved tracks of light become geodesics?

I don't know how that would then lend itself to a probabilistic interpretation but I'm just curious about this piece of it.

I found this a really helpful paper but honestly much of it is too advanced for me so I'd appreciate if you could answer in more intuitive language and less abstract mathematical! I haven't taken a graduate level quantum course. But anything is appreciated.

• Hi Adam, for future reference, please link to the abstract page, rather than the PDF page (consider, for instance, mobile users). – Kyle Kanos Jun 13 at 0:23