Master Dissertation

Microlocal Spectrum Condition.
For the numerical distribution tn ∈ D ′ (M n ), n ≥ 2 of any time-ordered
product,
WF (tn) ⊂ Γn,
where
Γn =
(x1, k1; . . . ; xn, kn) ∈ T ∗ M n \ {0}|∃g ∈ Gn and
an immersion (x, γ, k) of G, in which ke is
future directed whenever x s(e) /∈ J − (x r(e)) and
such that ki =
m:s(m)=i
km(xi) −
n:r(n)=i
kn(xi) ,
where t and s runs over all curves terminating and starting at xi,
respectively.
where
J − (x) := {y ∈ M|y < x and there exists γ causal, connecting x and y}
is the set of all points in M in the past of x that can be connected with x
by a causal curve.
The microlocal spectrum condition may be seen as a replacement of the
spectrum condition (Axiom 4iii of the Wightman Axioms). Further it
ensures that the wave front set has the following property.
Figure 9.1: Illustration of Lemma 9.7.
Lemma 9.7. Let (x, k) ∈ Γn then there exists a pair (xm, km) such that
km /∈ V + (0).
75