introduction-weak-interaction-volume-one

interaction, and, in fact, we find that it is not . The third component o f
isospin, 1 3 , is nevertheless also conserved in the electromagnetic interaction ,
since it is charge-dependent . From (5 .1 .16) we see that isospin is given by
I = (id - 1) / 2 , (5 .1 .18 )
where M is the multiplicity of a particular particle, i .e . the number of
particles in its multiplet . Thus, for the nucleons ,
I = . (5 .1 .19 )
We now consider field quantization using isospin . We may introduce
single eight-component field for the nucleon, to replace our original tw o
spinor ones . We distinguish between different components of this field by th e
two indices z and T, which denote the spin and isospin, respectively, of eac h
component . z may assume values between 1 and 4 inclusive, and T either + . or -1 .
Hence our field become s
z,T (x) _ p > r =l (e jPx uz(+)(r)(p) a„(r) ( )
+ e Jpx uz(-)(r)(-2) bT (r)(PI/, (5 .1 .20 )
where uz(1)(r)(p) are, as usual, plane wave solutions to the Dirac equation ,
of the form (1 .8 .5), with positive and negative energies respectively, polarizatio n
state r, and momentum p . The operator aT (r) is the destruction operator fo r
both protons and neutrons with polarization r, and similarly, bt (r) is th e
i
creation operator for antiprotons and antineutrons with polarization r. Fo r
the pions, we must introduce three separate fields : a Hermitian one for the T ` 0 ,
and complex ones for the "TN ''
described by scalar fields of the form (2 .2 .3) :
4)0(x)
=
Since the pions have zero spin, they may b e
( 1 / 2 ) ( e kx a0(k) t e Jkx a0 (k) ) , (5 .1 .21 )
(x) _ / ( l / 2W ) (eJkx a+(k ) - e jkx a t ( k ) ) , (5 .1 .22 )
q) t (x) ( 1A5W)(- ei a_( k ) + e jfix at(K) ) , (5 . 1 . .23 )
in obvious notation . The minus sign in front of the creation and destruction
operators of the - v 11 in (5 .1 .22) and (5 .1 .23) originates from the phase convention 2 .
The charge of the pions is given by
= e
k ( a*(Y) a+( 'k) - a t (k) a _Q=2) ) , (5 .1 .24 )
which is equivalent to (2 .2 .23) . The operator (5 .1 .24) is, in fact, identica l
to T 3 except for the factor of e . We might continue to consider interacting
a