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Massless Fermion

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 (gif) can be written as:







equation (gif) 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 .

Bernward Krause
Mon Jan 15 14:57:06 MET 1996