Master Dissertation

in the sense that if h ∈ D(M), then
〈: φ(x1) · · · φ(xn) :, Ψ〉(h n
) = : φ(x1) · · · φ(xn) : Ψ h(x1) · · · h(xn) (9.1)
= i −n
∂nΨ ∂f(x1) · · · ∂f(xn) h(x1) · · · h(xn)
We want to generalize this.
Definition 9.2. The microlocal domain of smoothness D is defined by
where
D = {Ψ ∈ H|Φ(f) is infinitely often differentiable at f = 0
∂nΨ and for all n ∈ N, WF
∂f n
(T ∗ M n )± def
={x, k) ∈ T ∗ M|ki ∈ V − (0)}.
⊂ (T ∗ M n )−},
Now we may use the idea of equation (9.1) to define an operator-valued
distribution on D.
Definition 9.3. We call the operator-valued distribution : φ(x1) · · · φ(xn) :
on D defined by (9.1) a Wick monomial. 2
Now we would like to generalize the Wick monomial to a polynomial. That
is, for arbitrary indices l = (l1, . . . , ln) we would like to make sense to
To this end we need the following definition.
: φ l1 (x1) · · · φ ln (xn) : . (9.2)
Definition 9.4. A partial diagonal ∆l1,...,lj , where l1 + · · · + lj = n is a
subset of M n on the form
∆l1,...,lj (M) = {(x1, . . . , x1),
. . . , (xj, . . . , xj)|xi
∈ M, i = 1 . . . , j} M
l1 times
lj times
j .
Now we want to define the polynomial (9.2) as a restriction of ∂ n Ψ
∂f1···∂fn to
arbitrary partial diagonals. This is done in practice by a pullback. 3
Define the partial diagonal map
by
δn,l : M n → ∆l1,...,ln (M)
δn,l : (x1, . . . , xn) ↦→ (x1, . . . , x1,
. . . , xn, . . . , xn)
l1 times
ln times
2 Note that there are different definitions of a Wick monomial and polynomial in the
literature. The definitions used here are not the most used ones.
3 Remember that we considered the pullback of a distribution by a function in Sec-
tion 4.2 of [4].
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