The Dirac equation for a massive fermion of momentum is:
However, the free particle must satisfy the energy-momentum requirement
which allows the determination of the coefficients and :
where are the Pauli matrices and I is the unit matrix. In covariant form, equation () can be written as:
equation () leads, after some manipulation, to the following:
In the case of a massless fermion, a neutrino for instance, the above relations are decoupled:
Each of these equations is based on the relativistic energy-momentum relation and therefore has one positive and one negative solutions. For ,
where is the helicity operator. describes a left-handed neutrino (negative helicity) whereas describes a right-handed (positive helicity) antineutrino. The other solution with satisfies
In this case, and describe a right-handed antineutrino and a left-handed neutrino, respectively.
Consider the case and the operator
The action of the operator on the the spinor u gives:
So, the operator projects out the left-handed spinor whereas
projects out the right-handed spinor .