Master Dissertation

Chapter 7
Splitting of Numerical
Distributions
7.1 The Singular Order of a Distribution
We expect the operator-valued distributions to be expanded in terms of
free fields as follows1 Dn(x1, . . . , xn) =
:
ψ(xj)d k n(x1, . . . , xn)
ψ(xl) ::
A(xm) :,
k
j
where the numerical distributions d k n ∈ S ′ (R 4n ) have the causal support
property (5.20). We have to assume them to be tempered in order to use
the Fourier transformation. Our aim is to split these distributions
where
d k n(x) = rn(x) − an(x),
supp rn ⊂ Γ + n−1 (xn) and supp an ⊂ Γ − n−1 (xn).
We have assumed translational invariance thus we may assume xn = 0 and
thus only consider
d(x) := d(x1, . . . xn−1, 0) ∈ S ′ (R m ), m = 4(n − 1).
In the Hilbert space setting of our earlier example we would do the
splitting as
rn(x) = χn(x)d k n(x), where χn(x) =
n−1
j=1
l
Θ(x 0 j − x 0 n) =
m
n−1
1 Note that the double dots denote normal ordering. See Definition 4.4
2 We write it like this to show what it would be if xn = 0.
49
j=1
Θ(x 0 j) 2 .