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 :

and

where are the Pauli matrices and
**I** is the unit matrix. In covariant form,
equation () can be written as:

where

With

where

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 ,

and

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

and

In this case, and describe a right-handed antineutrino and a left-handed neutrino, respectively.

Consider the case and the operator

where

The action of the operator on the the spinor **u** gives:

Similarly,

So, the operator projects out the left-handed spinor whereas
projects out the right-handed spinor .

Mon Jan 15 14:57:06 MET 1996