introduction-weak-interaction-volume-one

'Heisenberg picture' may be used in the 'interaction picture' . From (2 .6 .5 )
we may write the differential form of the Heisenberg equation of motio n
j 0(tl =
[0(t), H i . (2 .6 .6 )
at
We shall now transform some of the results which we obtained in 2 . 2
concerning the free scalar field into the 'interaction picture' . The destruction
operator a(k) defined in (2 .2 .13) is now redefine d
a(k, t) = eJHo t a(k) ,-''Hot , (2 .6 .7 )
and the creation operator is similarly redefined . Thus the Fourier decompositio n
of the field become s
(x) __ lV~2~
k
(a(k, t)ei kx + al- (Is, t)e j1 ) . (2 .6 .8 )
We now wish to solve (2 .6 .7) . The only term in the scalar field Hamiltonian
(2 .2 .15) which does not commute with a(k) i s
w(k)a 1 (k)a(k) • (2.6 .9 )
Using the method outlined in Schwebor, Bethe, de Hoffman ;
Mesons and Fields (Vol .
Fields, in
I), Evanston 1955, section 15b, we obtai n
a(k, t) = a(k .)e-j w(k)t (2 .6 .10 )
an d
a t(k,t) = at (
k)e -Ica()t , (2 .6 .11 )
where
w(k) k
0
= J( m 2 ~ k 2 ) . (2 .6 .12)
At this point, we may substitute with (2 .6 .10) and (2 .6 .11) in (2 .6 .8) and we
obtain for the scalar interacting fiel d
Is ( x ) 5
(a(k)e,,dx+ at (k ) e jkx } (2 .6 .13 )
ko= o('r_)
summing over all allowed momenta k . Often the field is decomposed
1(x) _ Cr(x) . 0-(x) , (2.6 .14 )
where ()$ are the parts of the field containing only destruction or creatio n
operators respectively . The vacuum definition
a(k) 0> = 0, (2 .6 .15 )
which implies that it is not possible to remove a particle from the vacuum, na y
now be replaced by
I0> _ 0 , (2.6 .16 )