introduction-weak-interaction-volume-one

In order to discover more about the degrees of freedom and nature of the field ,
it is useful to perform a Fourier decomposition
into a series of harmonic components .
1
V
Thus we obtain
(5) on it, or to resolve i t
1V
ejkz a(k) + e jkz at
0(x)
,17 c3
(
(k) ) . (2 .2 .3 )
`
The normalization constant (1/j 2o,;) is so chosen so that the operators a(k) and
al" (k) fulfill the commutation relations (6 )
al-(LO]
Ca (k) , a (k' )1 _ [a t (k) ,
0 (2 .2 .4 )
[a (k) , at (103 _ E kk, (2 .2 .5 )
where
[X , Y] = X.Y - Y .X (2 .2 .6 )
V represents a large normalization volume, which is usually considered to b e
a rectangular box with sides L x , Ly and L z . k_ is an energy-momentum fourvector3
, which, because of (2 .2 .1), obeys the relatio n
k2 4- m 2 = 0 ,
and, by normalization ,
ki Li = ni 2i ( i = x, y, z ) . (2.2 .8 )
Thus the total number of state vectors _ , An (7), in the interval d3k i s
An = (1/(217) 3 ) V d3k .
(2 .2 .9 )
When V is large,
it is often convenient to replace the summation over allowe d
values of the momentum by the formul a
f(k) =
k
f(k) . being a slowly-varying function giving all allowed momenta . The equating
of the two sides of (2 .2.10) involves a slight error, but this tends to zer o
as the normalization volume V becomes very large .
We now consider the operator a(k) and its Hermitean adjoint at (k) .
From the commutation relations (2 .2 .4) and (2 .2 .5) we may guess that a(k )
and a1 (k) are the destruction (or annihilation) and the creation operator s
respectively. We begin with a normalized state
(2.2 .7 )
1
(2 77 f (k) d3k (2 .2 .10 )
(2 .2 .11 )
1 TO
containing n bosons, each with a momentum k . Here we employ the Dirac notation
(8) : X> is a ket, and represents the wave function or state vector of th e
particle or system of particles X .