introduction-weak-interaction-volume-one

g. Similarly we see that
b (r) (g) (2 .4 .4 )
is the annihilation operator for antiparticles with polarization r and
momentum 2, and this will occur in place of b t in the adjoint field operator .
bt (r) (s)
creates an antiparticle and
(2 .4 .5 )
at (r) (g) (2 .4 .6 )
creates a particle . For an electron, which has
a spin of -4-, we can have two
possible spin orientations, parallel to the momentum 2, or antiparallel to it ,
corresponding to the polarizations r =1 and r = 2 respectively .
are known as left- and right-handed states of the electron .
These state s
The operator s
(2 .4.3), (2 .4.4), (2 .4.5) and (2 .4 .6) obey the anticommutation relation s
[a() , at(r1) (r. ' )J +
[b(r) (a) , bl-(r' ) (a' )] = 1 (2 .4 .7 )
+ rr qq'
and all other anticomautators vanish .
From (2 .4 .1), we may calculate that
H = > - E (a t(r) (s) a(l) (s) + b() b(r)()) (2 .4 .8 )
a, r
Since we are now concerned with half-spin particles (fermions) which obey the
Pauli exclusion principle3 (10), the value of the occupation operator (2 .2 .18)
can only be either one or zero . From (1 .8 .3) we may write the current density o f
a spinor fiel d
Sr ( x ) — (je / 2 ) C "~,~( x) , Y(x)] . (2 .4 .9 )
This current density is adjusted so that the vacuum expectation values of
all S vanish :
r
= 0 . (2 .4 .10 )
Furthermore, (2 .4 .9) implies
Q = -j S4 (x) d3x
e (at(r)(g)a(r)(g) -
(2 .4 .1 1 )
bt(r)(3)b(r)(a) )
Si,
2 .5 Transformational Properties of the Spinor Field .
Under space inversion the spinor field behaves
P ("X( P ( x)) ) _ G p Y4 V( x )
(2 .5 .1 )
where
E -- ±1, P
— j
(2 .5 .2)