introduction-weak-interaction-volume-one

1g-t = z (1 + '( 5 )y , (3 .6 .8b )
which are eigenatates of y- 5 :
Y5 4 = ± if± . (3 .6 .9 )
We usually define i1+ to have positive chirality and r- negative
chirality . Writing (45) 1
= 1( 1 + Y4)y (3 .6 .10a )
x = 4 (1 - Y4 ) -v , (3 .6 .lob )
we have
r49 - X
I+ _ '-z (1 t Y 5 ) 1V (3 .6 .11a)
2 L-( ep - x)
1 + x
(1 - Y5 ) 2 +
(3 .6 .11b )
where
= (3.6 .11c)
We note that, if we project with positive chirality, we obtain th e
two-component spinor - X) and if with negative chirality (0 + X) ,
and thus two of the components of the original four-component spinor ar e
now redundant .
We observe that, in the special case of a massless Dirac particle, th e
Dirac equation is invariant under the chirality transformations, sinc e
(4 .6 .8a) reduces to
r ( a / axr) = 0 . (3 .6 .12 )
Evidently the wave functions of free particles with finite rest mass
chirality noninvariant . Nevertheless, narshak and Szdarshen (46) suggeste d
that the complete four-fermion interaction might be invariant under
1 5
applied to each field separatel y 2 . This implies that the Dirac operator s
appearing in the weak Hamiltonian obey the relations
[o. , Y 5] + = 0 , (3 .6 .13a )
0i = 0i Y5 . (3 .6 .13b )
1. These are ;mown as 'non-chiral-projected' spinors .
2. This is obviously only plausible if the weak Hamiltonian is parity non -
invariant .