The operator 
is referred to as the chirality operator which
transforms as:
With the momentum 
along the z-axis, the helicity operator
becomes:
Since
we have:
It follows that the solutions to the Dirac equation for a
massless fermion are eigenstates of 
. This is not the case
for a massive fermion. However, the wavefunction can be written as a
linear combination of the right-handed and left-handed components:
where
Consider a 4-fermion interaction 
which is assumed
to be invariant under the chiral transformation defined above. The
interaction amplitude can be written as:
Chiral invariance leads to:
In order for the amplitude 
to be invariant under a Lorentz
transformation, the operator O can be shown to be a linear
combination of the following:
where
It follows that:
and
Of these five possible operators, only the vector and
axial-vector operators V and A anticommute with 
. Therefore,
From equation (
), one has a=-b since
and 
anticommutes with
. The operator O becomes:
where a is a constant. The amplitude M takes the form:
where the currents 
and 
are given below:
This is the modification of the current-current Fermi interaction --- at a point --- to accommodate the axial-vector contribution originally absent in Fermi's theory.
It should be pointed out that theoretically, one could arrive at a form of the weak charged current in which only S, T and P contribute. However, the experimental evidences are in favor of the (V,A) form.
The following conclusions follow from the V-A form
of the weakly charged
current [Che-79]:
leads to a neutrino with positive helicity which is not allowed.
(where 
describes the neutrino) leads
to an antineutrino still with negative helicity: this is not allowed.
is conserved. Since T reverses both spin and
momentum, the original state is unchanged.
Naturally, one would hope that all weak processes are described the
V-A interactions with a universal coupling constant.
