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Equation (2101) requires P = 0, thus kO = E = in Visual Studio .NET
Equation (2101) requires P = 0, thus kO = E = Decoding PDF417 In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Generate PDF417 2d Barcode In VS .NET Using Barcode printer for .NET framework Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. Ikl, and Eq. (2102) implies
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e and > are the polar angles of k. Similarly, v+(k) = Cii~(k) = v_(k) = Ciir(k) fi = fi
e+(k) = e(k) = i(J;:~\k) =  a+(k)) =  u+(k) i(J~a:~~~)=  a_(k)) =  u_(k) There exists only two independent solutions, for a given k. Experimental observation shows that only negative chirality neutrinos exist. Neutrinos have the helicity  1, antineutrinos the helicity + 1. This will be better understood in terms of twocomponent spinors. Indeed the reason for using fourcomponent spinors no longer holds for the massless Dirac equation where the algebra {ai, aA = 2(5ij may be realized by the three twodimensional Pauli matrices. The identification ai + (Ji leads to positive helicity, positive energy particles, whereas ai +  (Ji gives negative helicity. Such spinors, initially introduced by H. Weyl, were rejected because they were incompatible with parity conservation (which reverses the sign of helicity). This is not a serious objection any more since neutrinos are involved in weak interactions which do not conserve parity. We have already introduced in (220) the corresponding chiral representations of a matrices: For positive chirality, y5 =
5=(1 0) + 1, l/J = (~) and y. pl/J = 0 reduces to
(  pO
+ p. 0) > = 0
(2104) whereas for y5 =  1, l/J =
(~) and
+ P . a)x =
(2105) In both cases, we have a twocomponent theory, and the Dirac equation' is equivalent to the pair of Weyl equations. The socalled charge conjugation C (the neutrinos have no charge!) connects the two chiralities and changes the sign of the energy. There is no C in variance if nature uses only neutrinos of a definite chirality. Actually, since the parity operation P also connects the two types of solutions THE DIRAC EQUATION
is antidiagonal), the combined operation CP leaves the Weyl equations invariant. In the new representation, the matrix C of (297) reads C = (  i<J2 . ) 1<J2
Therefore, the CP operation acts according to
IjIcP(t, x) = I]CIjI*(t,  x) + il]<J21j1*(t,  (2106) for chirality y5 = 1 respectively. We observe that the Lorentz invariant normalizations (243a, b) of the massive solutions have to be modified in the massless case. Then we shall write u(a)(k)yOu(P)(k) = 2Ebap v(a>(k)y V<P>(k) = 2Eb ap
and leave it to the reader to construct the appropriate plane wave solu"tions.
25 DIRAC PROPAGATOR 251 Free Propagator
In Chap. 1 we developed the concept of Green functions of a classical scalar field. We will extend it here to spin i particles. We consider first free propagation. Let us try to determine the solution of the Dirac equation at time t2 as a function of its value at an earlier time t 1. This is possible since we deal with a firstorder equation. We thus look for a kernel K(X2' Xl) such that ljI(t2' X2) = d 3X l K(t2' X2; th
Xl)yOIjl(th
(2107) The introduction of yO will be justified soon. Any solution IjI is a linear superposition of plane wave solutions Owing to the relations (243), we may write
d 3 x u(a)(k) eikxyoljl(O, x) = a<a>(k) d3 x v(a>(k) eik xyOIjl(O, x) = Ha)*(k) QUANTUM FIELD THEORY
Therefore, t/J(tz, Xz) = J(~:~3 ~ ~ J
d3X1 [u(a)(k) u(a)(k) eik'(x,xl) + v(a)(k) v(a)(k) eik . (x,x1)Jy t/J(t 10 Xl) Interchanging the order of integration, we find the desired kernel
X . Z, t x) (2n E
~"[u(a)(k) 'x' u(a)(k) e ik ~ \61 (X'XI) + da)(k) v(a>(k) eik . (x,x 1 )] for tz > t1
(2108) Notice that K depends only on (xz  Xl), which is a reflection of the translation in variance of the free equation. We may also use the projectors A (k) of Eqs. (240) and (241) to recast K(xz, Xl) in a more compact form K(xz X1)=e(t z  t 1 , 3 )J 2E(2n)3 [(~+m)eik'(X2XI)+(~_m)eik'(X2XI)] dk
(2109) We denote this retarded kernel by K ret . Let us show directly that it is a Green function of the Dirac equation. Acting on Kret(xz, Xl) with W)Z  m) yields (iilz  m)Kret(Xz,Xl)

