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Wave Equations

Derivation of the Dirac Equation and the Weyl Spinor

In 1928 P.A.M. Dirac tries to resolve the problem with negative energy by writing the equation in linear form (in natural units) :
m = bE - ap --------- (23)
where a and b are to be determined by demanding that the square of Eq.(23) takes the same form of Eq.(15), i.e., m2= E2-p2. This scheme does not quite work since it is impossible to eliminate the cross terms between E and p. A more promising way is to break up the energy equation into the form such as Eq.(24) :
m2 = [bE - (axpx + aypy + azpz)][bE + (axpx + aypy + azpz)] = b2E2 - [(axpx)2 + (aypy)2 + (azpz)2)] ---------- (24)
Beside the trivial case of a=b=1, a new formulation appears if the ai's and b take the forms of the 2x2 Pauli matrices, and 2x2 identity matrix respectively (note that ai2=1, and the Pauli matrices are more often denoted as i) :
---------- (25)
By substituting the canonical quantization rules of Eqs.(3a), (3b) into Eq.(24), we obtain (in natural units and ignoring the y, z components for the time being) :
[i( + ax)][i( - ax)] v = m2 v ---------- (26)
where v is a two components single column matrix, e.g.,
v = ---------- (27)
Dirac Equation Eq.(26) can be split into two :
[i( - ax)] v = m u ---------- (28a)
[i( + ax)] 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)
where the gamma matrices i and the 4-components field (usually known as Dirac spinor) can be expressed in the forms :
---------- (29b)
and the repeated index in the expression indicates a sum. This is the Dirac equation cast in stone inside the Westminster Abbey (see Figure 02e). The problem with negative energy is still there, but the new form of equation turns out to be the one that governs all the particles in the material world, which consists of spin 1/2 particles, while particles of integral spin carry out interaction between the spin 1/2 particles. Massive fermions, such as the electrons, are described by the 4-component Dirac spinors; while the 2-component variety is known as Weyl spinor (defined in Eqs.(28a,b)), which is used in the Standard Model to designate the left-handed neutrino and electron. For massless fermion such as the neutrino, the v and u components can be identified to the left-handed and right-handed parts respectively, and the equations are decoupled into :

Helicity Reversal Neutrino, Parity Violation 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

Figure 02f Helicity
[view large image]

Figure 02g Neutrino, Parity Violation [view large image]

the 21st century when neutrino has 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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