introduction-weak-interaction-volume-one

antiparticle's is always aligned parallel to it . We define an operato r
( p ) (i • 2)/)a\ , (2 .5 .26 )
called helicity, which is negative for the solution u(p) and positive for v(p) .
These states are said to be left- and right-handed respectively . In general ,
helicity, H, is define d
H = o< v/c (2 .5 .27 )
and hence it is only constant for massless particles . The relation {2 .5 .27 )
may be obtained from the Dirac equation and this is done in Huirhead :
ElementaryParticlePhysics, P ergamon 1972, pp . 41-46 .
2 .6 Interacting Fields .
In the time-dependent Hamiltonian form of the
Schrddinger equation
(2 .1.6), we assume that the total Hamiltonian may be written as the sum of a
free and an interacting Hamiltonian, which we label H0 and HI respectively .
We now define the state vector (x, t) :
1) (x, t) = ejHet t) (2 .6 .1 )
and the operator 0(x, t) :
0 (x, t) = ejH ° t' e- ;H°t . (2 .6 .2 )
At
t 0 (2 .6 .3 )
the total Hemiltonian, H, is identical to the free Hamiltonian, H 0 .
(2 .6.1), (2.6.2), and (2 .1.6), we thus obtain
From
J Cl) ( xt t) . 'C() (x, t ) ,
~ gI (2.6.4 )
and hence we see that the state vectors in the 'interaction picture' (14) hav e
the same dependence on the interacting Hamiltonian as those in the 'Schrddinger
picture' have on the total Hamiltonian . Let us rewrite (2 .6.2) a s
0 (x, t) = e jHot 0 (x, 0) e 0 .
We now interpret 0(t) defined in (2 .6 .5) as being an operator in the 'Heisenberg
picture', ignoring the position vector ( ::) . In the 'Heisenberg picture' the
free Hamiltonian in the 'interaction picture' acts as the total Hamiltonian ,
and hence the operator in the 'interaction picture'
(2 .6 .5)
is identical to the one
in the 'Heisenberg picture' for the case of a non-interacting system . Thu s
both equations of motion and commutation relations for operators from th e