Master Dissertation

Now by Theorem 4.5 from [4] we may define a Wick polynomial as an
operator-valued distribution on D defined as the pullback of a Wick
monomial by the a diagonal map δn,l. 4
First we calculate the set of normals of the partial diagonal map.
The transpose of the Jacobi matrix of the partial diagonal map is
δ ′ n,l (x1, . . . , xn) t =
⎛
∂x1
⎜
⎝
(δn,l)1
⎞t
. . . ∂xn(δn,l)1
.
. ..
⎟
. ⎠
∂x1 (δn,l)n
⎛
1 . . . 1
⎜ 0 . . . 0
⎜
= ⎜ .
⎝ 0 . . .
. . . ∂xn(δn,l)n
0 . . .
1 . . . 1 0 . . .
0 1 . . . 1 0
⎞
0
0 ⎟
. ⎟ ,
⎟
0 ⎠
0 . . . 0 1 . . . 1
where row j contains lj ones. Hence
δ ′ n,l (x1, . . . , xn) t η =
⎛
⎜
⎝
η1 + . . . + ηl1
ηl1+1 + . . . + ηl1+l2
.
ηl1+...+ln−1+1 + . . . + ηl1+...+ln
⎞
⎟
⎠ .
Note that the matrix product is independent on x. The set of normals is
Nδn,l = {(δn,l(x), η) ∈ ∆l(M) × R n |δ ′ n,l (x)t η = 0}
= {(y, η) ∈ ∆l(M) × R n li+1
| ηli+j = 0 for all i = 0, . . . , n − 1},
Where we have set l0 := 0.
j=1
Definition 9.5. By a Wick polynomial : φ l1 (x1) · · · φ ln (xn) : we mean the
operator-valued distribution
when it exists, that is, if
.
: φ l1 (x1) · · · φ ln (xn) : def
= δn,l ∗ : φ(x1) · · · φ(xn) :
Nδn,l ∩ WF (: φ(x1) · · · φ(xn) :) = ∅.
4 Note that we proved the theorem exactly for the case of a diagonal map.
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