introduction-weak-interaction-volume-one

matrix is one such that
Wt = u tU = I, (2 .3 .11 )
I being the identity element . An antiunitary operator transform s
AAt = A A - I* (2 .3 .12 )
One interesting feature of time reversal is that it transforms all outgoing
states into incoming ones and vice-versa . Using the interacting field (Heisenberg
definition) (9), we obtai n
an (k,
T
'co) >
%tut 'co)
,
where a t acting on the vacuum creates an incoming particle, and a
out
an
outgoing one .
The charge conjugation operator, C, acts on the scalar field :
C ((P (x) ) = e C t( x ) , (2 .3 . 1 4 )
which implies the transformatio n
C1 , k> = C-~ jp, k> , —
(2 .3 .15 )
where 4
represents a particle in the field and
its corresponding antiparticle
. If the transformations C and T commute, which is probable, since they
are physically unconnected, we see that
E T
C- C .
If 4)(x) is a Hermitean field, then
E — c
(2 .3 .13 )
(2 .3 .17 )
± 1 .
(2 .3 .16 )
2 .4 The FreeSpinorField .
We recall the Dirac equation (1 .7 .10) and Fourier decompose the stat e
vector 'y(x) which we now reinterpret as a field operator (10) . Thu s
9. 9
(ejgx u,, (-') (r) (2.) a(r) (2.)
.~ jqx uo,(-)(r) (-g) bb(r) (4 (2 .4 .1 )
where
u ac (r ) (g) (2 .4 .2 )
is a Dirac spinor with polarization (spin) state r and momentum g,
runs from 1 to 4 according to the gamma matrices .
a (r) (s) (2 .4 .3 )
is the annihilation operator for a particle with polarization r and momentum
and