introduction-weak-interaction-volume-one

and from (2 .6 .13) and (2 .6 .14 )
(4 + )t Cp (2 .6 .17 )
Now let us attempt to calculate the value of the commutato r
[4(x) , 4(x')1 ,
(2 .6 .18 )
where x and x' are two position space-time four-vectors . Since two creation
or two destruction operators commute,
[( x ) , 0(xiF
V
o(k)
1
(27) 3 . 2
koz W(~
w(_' )
2
,1
o o
only cross-terms will contribute, and thu s
T
r a(s), a-1(k`)J ejkx-jk x
i (
L
[a t(), a(k')] e ')
d3k (ejk(x-x') _ ejk(x-x') )
,
ko
(2 .6 .19)
anticipating going to the limit of normalization and using the approximation
(2 .2 .10) . We defin e
_ 1
d :
A(x)
k sin itx
(2Tv)3 k W(k)
,
0 0
so tha t
E( x), (x' ;) = j b(x-x' )
From (2 .6 .20) we see that ,6.(x) is a real odd function of x, and is a Lorentz
invariant scalar . Thus
O(_, t = 0 ) -
1 cd 3'~
w(k) sin kX - 0
(2~ ) 3 J
so that the 'equal times' commutator
0 .
Since the Ls, function is a Lorentz invarian t
2s(x) 0 x2 - x2 - t2 > 0 ,
and thus
the 0 function vanishes outside the 'light-cone '
(2 .6 .20 )
(2 .6 .21 )
(2 .6 .22)
J x ` -
it( ,
and similarly the commutator in (2 .6 .21) vanishes for points with space-lik e
separation :
(x-x') 2 0 . (2 .6 .26 )
If the commutator (2 .6 .18) was non-vanishing for space-like separated points
x and x', then this would imply that the measurements of the field at two
separated points in space interfered with each other, necessitating a signal
travelling faster than the velocity of light .
upholds the special theory of relativity .
The vanishing commutator thu s