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m = bE -

where

m

Beside the trivial case of a=b=1, a new formulation appears if the a

---------- (25) |

[i(

where v is a two components single column matrix, e.g.,

v = | ---------- (27) |

Eq.(26) can be split into two : [i( _{} - a_{x}_{})] v = m u ---------- (28a)[i( _{} + a_{x}_{})] u = m v ---------- (28b)where u is another two components column matrix similar to v. It is a mathematical property of the differential equations that more components will be created when transforms a second-order equation to two first-order equations, the components would be mixed between the first-order equations. Eqs.(28a) and (28b) are equivalent to the original equation written down by Dirac. The original form of the 4-components Dirac equation is obtained by stacking these 2-components equations on top of one another : | |

## Figure 02e Dirac Equation [view large image] |

---------- (29a) |

---------- (29b) |

according to the right-hand rule.

Introduction of this Weyl spinor in 1929 had been criticized as unphysical since it violates the conservation of parity considered to be universal at that time. For particles with mass, the sign of helicity varies according to the frame of reference as shown in Figure 02f. But it becomes an intrinsic property of massless particle as it travels with the speed of light and cannot be overtaken (Figure 02f). It is also equivalent to parity in this case. Figure 02g shows that the helicity changes handedness upon a reflection, hence the parity is violated. This fact was vindicated by a 1956 experiment, which demonstrated the absence of right-handed neutrino. Other experiment also shows that only right-handed anti-neutrino exists in this world (Figure 02g). The perception has changed again in the 21^{st} century when neutrino has
| ||

## Figure 02f Helicity |
## Figure 02g Neutrino, Parity Violation [view large image] |
been found to possess tiny mass. It makes physics more complicated (or more interesting) and the re-work is still in progress (see "Neutrino"). |

See "Majorana Neutrino" for a possible replacement of the Weyl neutrino.

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