Master Dissertation

5.3 Splitting of Distributions
In our previous example we saw that the splitting of Dn into advanced and
retarded parts could be done by use of the Heaviside function. This was
because we were dealing with operators in Hilbert space. But in our
setting we are considering distributions which cannot simply be multiplied
with discontinuous step-functions. In this section we consider the
properties of the advanced and retarded distributions.
First a technicality.
Theorem 5.3. Let Y = P ∪ Q, P = ∅, P ∩ Q = ∅, |Y | = n1 ≤ n − 1 and
x /∈ Y .
If {Q, x} > P ,|Q| = n2, then we have
If {Q, x} < P , then we have
R ′ n1+1(Y, x) = −Tn2+1(Q, x)Tn1−n2 (P ). (5.16)
A ′ n1+1(Y, x) = −Tn1−n2 (P )Tn2+1(Q, x). (5.17)
Proof. Equation (5.16) is proved in [6]. Therefore we only prove equation
(5.17).
A ′ n1+1(Y, x) =
˜Tn3 (X)Tn−n3 (Y ′ , x),
where P2 is the set of all partitions Y = X ∪ Y ′ such that X = ∅.
Now let
and
Now since
causality implies
P2
Y ′ = Y1 ∪ Y2, where Y1 = Y ′ ∩ P and Y2 = Y ′ ∩ Q
X = X1 ∪ X2, where X1 = X ′ ∩ P and X2 = X ∩ Q
Q ∪ {x} < P, Y2 < Y1, X2 < X1 and Y1 > Y2 ∪ {x},
A ′ n1+1(Y, x) =
˜Tn3 (X)Tn−n3 (Y ′ , x)
P2
=
˜T (X2) ˜ T (X1)T (Y1)T (Y2, x), (5.18)
P 0 4
subscripts are omitted for simplicity. P 0 4
form
is the set of all partitions of the
P 0 4 : P = X1 ∪ Y1, Q = X2 ∪ Y2 and X1 ∪ X2 = ∅.
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