Equation (2-101) requires P = 0, thus kO = E = in Visual Studio .NET

Generation PDF417 in Visual Studio .NET Equation (2-101) requires P = 0, thus kO = E =

Equation (2-101) requires P = 0, thus kO = E =
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Ikl, and Eq. (2-102) implies
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(2-103)
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L . kl/J =
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y5l/J
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Therefore the chirality equals the helicity (it is opposite for negative energy solutions). Let us label the independent solutions of (2-101) by their chirality: e- ik . x u (k) 2 l/J(x) = { eik . x v (k) with k = 0, kO = Ikl > 0
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Y5u (k) = u (k) Y5v (k) = v (k)
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In the usual representation Y5 =
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QUANTUM FIELD THEORY
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where
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 two-component 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 two-dimensional 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 (2-20) 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
(2-104)
whereas for y5 = - 1, l/J =
(~) and
+ P . a)x =
(2-105)
In both cases, we have a two-component theory, and the Dirac equation' is equivalent to the pair of Weyl equations. The so-called 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 (2-97) reads
C = ( - i<J2 . ) 1<J2
Therefore, the CP operation acts according to
IjIcP(t, x)
= I]CIjI*(t,
- x)
+- il]<J21j1*(t, -
(2-106)
for chirality y5 = 1 respectively. We observe that the Lorentz invariant normalizations (2-43a, 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.
2-5 DIRAC PROPAGATOR 2-5-1 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 first-order 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
(2-107)
The introduction of yO will be justified soon. Any solution IjI is a linear superposition of plane wave solutions
Owing to the relations (2-43), we may write
d 3 x u(a)(k) e-ik-xyoljl(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) e-ik'(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
(2-108)
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. (2-40) and (2-41) to recast K(xz, Xl) in a more compact form
K(xz X1)=e(t z - t 1 , 3 )J 2E(2n)3 [(~+m)e-ik'(X2-XI)+(~_m)eik'(X2-XI)] dk
(2-109)
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)
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