A practical representation
In this section yet another representation of spinors will be constructed.
At this point, it may seem a little bit artifical, but later on it will become
extremely useful in calculating scattering amplitudes numerically.
THE MASSLESS CASE
Let us start by going back to the massless case. Denoting all spinors by u (independently
of them being particles or anti-particles) the sum over helicities from the completeness
relation can then be written as
Let us now make a little detour by defining two four vectors that will span a base of spinors,
namely a light-like vector k0 and a space-like one, k1, with the properties
Then basic spinors u±(k0) can be defined by
clearly a negative helicity spinor for a massless particle with momentum k0. Its
corresponding positive helicity spinor can be chosen to be
simple calculation shows that indeed
as anticipated. From these two basic spinors, spinors for arbitrary light-like momentum (for massless
particles) can be constructed through
which clearly is a good way as long as the scalar product in the denominator remains finite.
It is easy to convince oneself that these spinors satisfy the basic equation above, and that they
are indeed eigenstates of the helicity operator.
SPINOR PRODUCTS AND OTHER USEFUL IDENTITIES
Before proceeding with the construction of massive spinors let us first calculate some products
of these spinors. Basically, for massless spinors there are only two combinations which do
not vanish - this should be clear when keeping in mind that the helicities are used to label
orthogonal states. These combinations are
Obviously it is sufficient to calculate just one of them, the other one emerges through complex
conjugation. To do the calculation, at some point explicit Dirac indices will be exhibited,
they are labelled by α, β, etc..
This result by itself does not look particularly simple. However, at this point there still is
freedom in choosing the exact form of the two "gauge" vectors k. For instance choosing
results in
This clearly is a very suitable form for numerical applications - spinor products are not much
more costly than vector products in terms of CPU. It is tempting to understand this as a hint
that indeed spinor products are more fundamental than vector products. For instance, it is easy
to check that
INCLUDING MASSES
To continue, let us now construct suitable spinors for massive particles in this helicity
formalism. The starting point here is the following: As long as polarizations of massive
particles are unimportant, because they are not observed, the only thing that really
matters is that the completeness relations for the massive spinors are recovered. To this end,
let us from now on denote the chiral spinors forming the helicity base with w and the "real"
ones with u. Let us then try the following ansatz:
where the sign reflects whether the fermion is for a particle (+) or an antiparticle (-).
Then
as expected. This is good news: We now have a representation of massive spinors, which can be
used for numerical calculations. Of course, the same reasoning as above still holds true for
the calculation of spinor products, the only difference now is that there is an extended
prefactor now that includes the mass.