Master Dissertation

Axiom 0 (Well-definedness) Let g = g1, . . . , gn ∈ S(R n ), then
〈Tn, (g1 ⊗ · · · ⊗ gn)〉 : D0 → D0,
is a well-defined operator-valued distribution with
〈Tn, (g1 ⊗ · · · ⊗ gn)〉 = dx1 · · · dxnTn(x1, . . . , xn)g(x1) · · · g(xn).
Note that the S-matrix and Tn are operator-valued distributions, in the
sense that if φ ∈ D0, then
∞
1
S(g)φ = 1 +
n!
n=1
∞
1
= φ +
n!
n=1
d 4 x1 . . . d 4
xnTn(x1, . . . , xn)g(x1) . . . g(xn) φ
d 4 x1 · · · d 4 xnTn(x1, . . . , xn)(φ(g(x1)) · · · φ(g(xn))).
We can express the inverse of S(g) by a similar perturbation series as
follows
S(g) −1 ∞
1
= 1 + d
n!
n=1
4 x1 . . . d 4 xn ˜ Tn(x1, . . . , xn)g(x1) . . . g(xn)
= (1 + T ) −1 ∞
= 1 + (−T ) r ,
using Theorem 1.7. From
=
∞
n=1
∞
r=1
we see that
n 1
λ
n!
−
∞
n=1
r=1
d 4 x1 . . . d 4 xn ˜ Tn(x1, . . . , xn)g(x1) . . . g(xn) =
1
n! λn
˜Tn(X) =
∞
(−T ) r
r=1
d 4 x1 . . . d 4 r xnTn(x1, . . . , xn)g(x1) · · · g(xn) ,
n
(−1)
r=1
r
where Pr is the set of all partitions of X
Pr
Tn1 (X1) . . . Tnr(Xr), (5.1)
X = X1 ∪ . . . ∪ Xr , |X| = n , Xj = ∅ , |Xj| = nj.
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