# Fermions with mass, and null spinors

## Main Question or Discussion Point

Are spinor wave functions describing e.g. electrons, necessarily describing them as massless?

Spinors representing physical entities are often described as corresponding to null vectors in space-time, which suggests that they can only describe massless entities.

Nevertheless, the Dirac equation includes a mass term.

So if spinor wave functions for e.g. electrons do correspond to electrons having mass, then are the spinors non-null, or still somehow null? (Or is this a really stupid question?)

I may be confusing different phenomena, but neither the web nor textbooks seem to have anything much to say on this, apart from one (apparently non-peer-reviewed, but interesting) article found by Google: "Complex Four-vector Algebra" by Jonathan Scott, April 2005, which says:

"Spinor and twistor theory has been covered very thoroughly in a two-volume
work on “Spinors and Space-Time” [7] by Roger Penrose and Wolfgang Rindler. However, spinors and twistors are limited to describing massless objects associated with null four-vectors" (and, interestingly, he goes on to say: "...but when spinor results are converted to complex four-vector algebra, they often hold for non-null quantities as well.")

So Scott has read the literature - and I've scanned a copy of this book too.

The only alternative I can imagine is that the spinor is specifying only the direction of the spin axis in space, and that this is the only aspect that is encoded by the ratio of the two complex coefficients c1/c2 (forming a spinor of the coefficients: (c1 c2)Transpose) in...

Psi = c1 |Spin Up> + c2 |Spin Down> ?

... i.e. in such a way as to avoid the spin-component wave functions themselves having to correspond to null-vectors in spacetime?

I tried leaving a query about this on Mendel Sachs's site a couple of months ago, but the lack of an answer suggests that maybe the question is too stupid to justify a reply.

Any takers?